Gérard Mourou

Marie Antier

Jérémy Le Dortz

Jérôme Bourderionnet

Arnaud Brignon

Christian Larat

Eric Lallier

Christophe Simon-Boisson

Within the more than three years which I spent on the XCAN project at the Ecole Polytechnique, I saw, heard and learned a lot, not only about physics, but also for life. I am highly grateful to everybody who took part in this experience, and in particular I would like to thank

• Jean-Christophe Chanteloup, my thesis' advisor, who rst o all accepted me as his PhD student. I highly appreciated his constant availability and the help he provided on any kind of question, in particular when it came to dealing with French administration. Also, he gave me the opportunity to travel to Japan and the USA, two journeys which were highly enjoyable not only from a professional, but also from a personal point of view.

• Louis Daniault, my co-advisor, who gave me countless private lessons on everything related to femtosecond lasers, ber ampliers and coherent beam combining. He never got tired

Introduction

Ultrafast lasers are nowadays used in a wide range of disciplines such as material processing, medical treatments, microscopic imaging and fundamental science [1,[START_REF] Sibbett | The development and application of femtosecond laser systems[END_REF]. Most of these applications make use of the high peak powers and intensities which can be obtained when a femtosecond laser pulse is strongly focused.

For instance, in material processing applications such as laser drilling, the energy of the laser pulse can be used to vaporize material so that a hole is created. The advantage of using femtosecond pulses for this process is that due to the short pulse duration the energy transfer to the surrounding material is largely suppressed. Hence, very clean and neat holes can be obtained [3].

The same principle applies to laser-based surgery, where the energy of the focused laser is used to create very neat and accurate cuts. Thanks to the short pulse duration, the surrounding tissue remains largely undamaged, which has a positive impact on the subsequent healing processes.

Moreover, as some organic tissues such as the human eye are transparent to certain wavelengths, the laser can be focused to deeper layers of the tissue. Thus, these layers can be treated while the rest of the tissue remains intact [START_REF] Callou | Advances in femtosecond laser technology[END_REF].

Several nonlinear microscopy techniques make use of the high intensities obtained when a femtosecond laser pulse is strongly focused in the interior of a sample. In this case, the material is not destroyed as in the previous examples, but a nonlinear process such as two-photon uorescence is excited. By scanning the position of the focal spot in all three spatial dimensions, a three-dimensional picture of the sample can be obtained [START_REF] Rudolph | Microscopy with femtosecond laser pulses: applications in engineering, physics and biomedicine[END_REF].

Last but not least, femtosecond lasers are used in dierent areas of fundamental science, for instance in ultrafast spectroscopy techniques [START_REF] Dantus | Ultrafast Spectroscopy[END_REF] or for the generation of frequency combs for high precision metrology experiments [START_REF] Holzwarth | Optical Frequency Synthesizer for Precision Spectroscopy[END_REF]. Another point of interest is the use of secondary sources, that is, the generation of radiation based on the interaction of a light pulse with matter. Examples include the generation of coherent light at wavelengths not accessible with classical lasers. For instance, high harmonic generation makes use of the very high peak power of femtosecond pulses to generate extremely short laser pulse bursts in the extreme ultraviolet (EUV) or soft-X-ray wavelength region [8]. In contrast, coherent light in the terahertz regime can be obtained i.e. by optical rectication of the femtosecond pulses in a second order nonlinear crystal [START_REF] Tonouchi | Cutting-edge terahertz technology[END_REF]. Such coherent light sources can be used in a variety of research elds ranging from the fundamental understanding of atom or molecule dynamics to exploring spectroscopic imaging methods operating in the range of the newly generated wavelengths.

It is also possible to generate high energy electron beams via laser wakeeld acceleration, where a plasma wave is created in the wake of the laser pulse. The strong electric elds associated with this wave can then be used to accelerate previously injected electron beams to very high energies.

Besides, protons and ions can be accelerated by focusing laser pulses on a foil target, so that charged particles are expelled from this foil and accelerated by the electric elds created when the laser impinges on the target [START_REF] Norreys | Laser-driven particle acceleration[END_REF].

The peak powers required by some of these applications are typically obtained at the expense of the average power of the laser system in use. However, the dierent applications would also benet from an increase in pulse repetition rate at a given peak power. For example, this could decrease the measurement times by increasing the signal-to-noise ratio of microscopic imaging applications which rely on ultrashort pulses provided by a secondary source.

Furthermore, there are other potential applications which are not accessible with the performances of current state-of-the-art ultrafast laser systems. A eld of particular interest is particle acceleration, where the advantages of laser-based techniques with respect to conventional accelerators are a better eciency and a reduced size and, potentially, cost, which could increase the accelerator's availability. However, even the peak powers and pulse repetition rates of state-of-theart large-scale ultrafast laser facilities are not sucient to produce particle beams which exhibit an energy and intensity high enough to be used for applications such as cancer therapy or nuclear waste transmutation [START_REF] Linz | Laser-driven ion accelerators for tumor therapy revisited[END_REF].

Even more ambitious studies investigate the tracking and remediation of space debris with ultrafast lasers. Indeed, the ablation caused by a laser incident on a piece of debris could change its orbit such that it re-enters earth's atmosphere and burns up. However, this would require a laser system operating in the kilohertz regime and emitting pulse energies of at least some joule [START_REF] Soulard | ICAN: A novel laser architecture for space debris removal[END_REF].

In consequence, there is a large interest in the development of ultrafast laser sources which could deliver both high peak and average powers. Current systems are limited in one of these two parameters due to the architecture in use: typical high peak power laser systems rely on Ti:Sapphire crystals as active laser medium, and are limited in average power and therefore repetition rate due to the heating of the crystal. In contrast, ber lasers benet from a very good heat removal thanks to the high surface-to-volume ratio of the ber and can therefore operate at very high repetition rates. However, as the light is conned in the small core of the ber, the beam reaches very high intensities. Due to the long interaction lengths within the ber, this favors the occurrence of nonlinear eects. These can in the worst case damage the ber itself and set thus a limit to the accessible peak powers.

There are several techniques to mitigate these problems by reducing the peak power of the beam in the ber. The most famous one is chirped pulse amplication (CPA, [START_REF] Strickland | Compression of amplied chirped optical pulses[END_REF]), which consists in the temporal stretching of the pulses prior to amplication, and the subsequent recompression of the amplied pulses. This technique is nowadays routinely used in ultrafast laser systems and has allowed to increase the available peak powers by several orders of magnitude. Another option is the use of so-called large mode area bers, whose core diameters are increased with respect to the ones of standard bers.

However, current state-of-the-art ultrafast ber laser systems already use the maximum peak power scaling capabilities of both of these techniques, so that new approaches need to be explored in order to allow for a further increase of peak power.

A method which has been proven to be well suited for such a power scaling is coherent beam combining (CBC, [14]). In this technique, the laser beam is spatially split into N sub-beams, each of which gets amplied independently. Afterwards, the sub-beams are coherently superposed in order to form one very powerful combined beam. If this combination shall be ecient, the spatial and spectral amplitudes and phases of all sub-beams need to be exactly matched. In this way, both the peak and the average power of the combined beam can theoretically be scaled by a factor of N with respect to what would have been possible if only a single amplier was used. This approach is particularly well suited for the implementation on ber lasers, since their waveguiding properties guarantee a repeatable and very high spatial beam quality. Indeed, already the coherent combination of a moderate number of bers has allowed to increase both the peak and average power of ultrafast ber laser systems to new records of 35 GW and 1.83 kW, respectively [START_REF] Kienel | 12 mJ kW class ultrafast ber laser system using multidimensional coherent pulse addition[END_REF][START_REF] Müller | 1.8-kW 16-channel ultrafast ber laser system[END_REF]. Furthermore, such systems have been shown to be suitable for high harmonic generation and subsequent fundamental science experiments and EUV imaging techniques [START_REF] Rothhardt | High-repetition-rate and high-photon-ux 70 eV high-harmonic source for coincidence ion imaging of gas-phase molecules[END_REF][START_REF] Tadesse | High resolution XUV Fourier transform holography on a table top[END_REF]. However, in order to access applications such as particle acceleration, the coherent combination of up to 10 000 bers has to be considered [START_REF] Mourou | The future is bre accelerators[END_REF]. Hence, the scalability of CBC system becomes a question of interest. Within this framework, recent calculations have shown that the phase matching of up to 10 000 parallel channels is feasible with o-the-shelf technical equipment [START_REF] Antier | kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining[END_REF].

In order to implement a rst proof-of-principle demonstration of a highly scalable CBC setup, the XCAN project, a collaboration between the Ecole Polytechnique, Thales TRT and Thales LAS France SAS, has been initiated. It consists of three main points: rst, the implementation of a 19-ber low power setup in order to investigate the applicability of the aforementioned phase matching technique in the femtosecond regime [START_REF] Le Dortz | Highly scalable femtosecond coherent beam combining demonstrated with 19 bers[END_REF]. Second, the setup of a seven ber amplier test system which is needed to analyze the impact of the amplication and the onset of nonlinear eects on both the phase matching algorithm and the characteristics of the combined beam.

And nally, the CBC of 61 ber ampliers in order to conrm the scalability of the previously demonstrated concepts to a considerably increased number of channels. This nal demonstrator is meant to deliver an average power of 750 W, at a repetition rate of 200 kHz and a pulse duration of ∼ 220 fs, corresponding to a peak power of 15 GW and a pulse energy of 3.5 mJ.

In this thesis, the work performed on the second of these three points, that is, the seven ber amplier setup, will be presented. In a rst chapter, the physical and technological fundamentals which are the basis of the laser system will be discussed. Afterwards, the principle of coherent beam combining will be explained, thus motivating our choice for a specic system architecture. In the third chapter, numerical simulations will be presented which allowed to estimate the maximum tolerable mismatches between the spatial and spectral properties of the dierent channels of our setup, and which played therefore an important role in the dimensioning of our laser system. In the last chapter, a complete description of the implementation of the seven ber setup will be given, and it will be characterized both under linear and nonlinear operating conditions. 1 Fundamentals This chapter provides an overview of the physical principles and technological developments which are the basis of this thesis. In a rst section, the spatial and spectral characteristics of femtosecond laser pulses are presented. Afterwards, a short overview of optical ber technology is provided, thus introducing the dierent types of bers used in our experimental setup. Finally, the amplication of femtosecond pulses with Yb-doped ber ampliers is discussed.

Femtosecond Laser Beams

An intuitive picture of a train of (femtosecond) laser pulses is shown in g. 1.1. Ultrashort pulses are emitted at a repetition rate ν = 1/T and each pulse contains a certain energy E. The pulse duration ∆t is given by the intensity full width half maximum (FWHM) of the temporal pulse prole. These parameters allow to dene two characteristic powers of a pulsed laser, which are the peak power P peak = E ∆t and the average power P av = E

T . In the following, a more rigorous mathematical description of femtosecond laser pulses will be given. At rst, the laser beam is considered in free space without dispersion, so that its spatial and temporal characteristics can be treated independently. Afterwards, the propagation in an optical ber will be described. In this case, the spatial beam prole is imposed by the waveguide structure of the ber, whereas the propagation in the ber medium can induce changes to the temporal and spectral shape of the pulse.

E

Femtosecond Pulses in Free Space

Spatial Characteristics

The spatial part of the laser beam can in principal have any shape and characteristics whichever.

However, in our setup, most of the beams are given as the fundamental mode of an optical ber, and are therefore well approximated by a Gaussian beam [START_REF] Saleh | Fundamentals of Photonics[END_REF], schematically shown in g. 1.2 and mathematically described by E Gaussian (x, y, z) = E 0 w 0 w(z) exp -x 2 + y 2 w(z) 2 exp -i kz + k

x 2 + y 2 2R(z) -ψ(z) .

(1.1)

In this expression, k = 2π λ 0 is the wavenumber, λ 0 the central wavelength and w denotes the radius of the beam, dened as the radial distance between the optical axis and the point where the intensity has dropped to 1/e 2 of its on-axis value. w 0 is the minimum radius or waist of the beam.

Using this waist, it is possible to dene the so-called Rayleigh range z R of the beam as z R = πw 2 0 λ 0 .

It indicates the distance at which the beam radius has increased to √ 2 times its initial value. In a general way, the beam width evolves with z according to w(z) = w 0 1 + z z R

2

. For z z R this simplies to w(z) = w 0 z/z R , so that the angle of divergence θ is given by θ = w(z) z = λ 0 πw 0 = w 0 z R .

Finally, R(z) is the radius of curvature of the wavefronts and ψ(z) is the Gouy phase. Both are included here for completeness and will not be of further interest within the framework of this thesis. Beam Quality Factor M 2

As indicated above, the Gaussian beam is only an approximation of any real beam. Moreover, a laser beam can take other shapes, which are much less well described by a Gaussian beam.

For instance, in our setup the intensity prole of the spatially ltered combined beam is given by an Airy pattern clipped at the rst zero, and with a dierent ltering conguration even a donut-shaped beam could be created (see section 2.2.1 and section 4.5.4 for more detail). It is therefore necessary to introduce a criterion applicable to arbitrary beam shapes which evaluates the spatial quality of the beam in question. This is usually done via the beam quality factor M 2 , dened as the ratio of the products of waist w 0 and divergence θ of the experimental beam and an ideal Gaussian one. Hence, for a Gaussian beam, M 2 = 1. In practice, the closer the measured M 2 gets to one, the more the experimental beam resembles a Gaussian one.

The beam width can be evaluated in dierent ways. One of the simplest methods is certainly the 1/e 2 -denition given above. However, this approach is inaccurate in a sense that it does not take into account the exact shape of the beam, so that the width of a Gaussian beam could be the same as that of a top-hat one, if they have the same radius at 1/e 2 , as illustrated in g. 1.3.

Therefore, the width of arbitrarily shaped beams is better evaluated by using the 4σor second moment method, which weights the beam intensity with respect to its distance from the beam center. The beam diameter in x-direction d x is then dened as d x = 4 I(x, y)(x -x) 2 dx dy I(x, y) dx dy ,

where I(x, y) is the intensity distribution of the eld and x is its centroid along the x-direction.

The diameter in y-direction can be found analogously. The beam diameter is thus dened as four times the standard deviation of the intensity prole. For a Gaussian beam, this diameter and the

1/e 2 one are equivalent.

x I(x)

1 e 2

Temporal Characteristics

Since the time and the frequency domain are related via a Fourier transform, short pulses necessarily exhibit a broad spectrum. If this spectrum is given by E(ω), then the temporal shape of the pulse can be found by

E(t) = F[E(ω)] = 1 √ 2π E (
ω)e -iωt dω .

(1.3)

Hence, the pulse prole in the time domain can also be understood as an innite sum of plane waves oscillating each at a frequency ω and having a complex spectral amplitude E(ω). Provided that all these plane waves are in phase at a certain time t 0 , their constructive interference leads to the creation of ultrashort pulses.

In a general way, E(ω) can be expressed via a real-valued envelope centered around a central frequency ω 0 and a spectral phase, so that E(ω) = A(ω-ω 0 )e iϕ(ω) . In the simplest case, ϕ(ω) = 0, which means that the temporal pulse shape is simply given by a Fourier transform of the spectral envelope, E(t) = F[A(ω -ω 0 )]. The shift of the amplitude by ω 0 in the frequency domain leads to an additional phase in the time domain, as can be shown by using the substitution ∆ω = ω -ω 0 , E(t) = 1 √ 2π e -iω 0 t A(∆ω)e -i∆ωt d∆ω = e -iω 0 t F[A(∆ω)].

(1.4)

In the time domain, the shape of the pulse is thus given by the Fourier transform of its spectral envelope, which is modulated by a carrier plane wave of frequency ω 0 . It should be noted that the maximum of the temporal envelope coincides with a maximum of the carrier wave, as illustrated in g. 1.4a.

1 Fundamentals

Re(E(t))

t t = 0

Spectral Phase

In a next step, the spectral phase is added to the spectral envelope, so that E(ω) = A(∆ω)e iϕ (ω) . Since the width of the spectrum is typically small compared to its central frequency, the phase can be developed in a Taylor series, yielding ϕ(ω

) = ϕ 0 + ϕ 1 (ω -ω 0 ) + ϕ 2 2 (ω -ω 0 ) 2 + ϕ 3 6 (ω -ω 0 ) 3 + O(ω -ω 0 ) 4 .
Using this approach, it is possible to study the inuence of each part of the spectral phase independently from the others [START_REF] Träger | Springer Handbook of Lasers and Optics[END_REF].

Adding the constant part ϕ 0 , the eld in the frequency domain becomes E(ω) = A(∆ω)e iϕ 0 .

Since ϕ 0 does not depend on ω, it translates directly into the time domain without being modied by the Fourier transform. Hence, the temporal eld reads E(t) = e iϕ 0 e -iω 0 t F[A(∆ω)].

(1.5)

The carrier plane wave gets thus shifted by the phase oset ϕ 0 with respect to the envelope, so that the maximum of the envelope does no longer coincide with one of the maxima of the carrier wave, as shown in 1.4b. This phase mismatch is called carrier-envelope phase.

In the context of coherent beam combining, the phase osets ϕ 0 of all the emitters need to be matched. In the presence of carrier-envelope phases, such a matching can give rise to slight mismatches in the positions of the pulse envelopes. However, this is not critical as long as the pulse duration is much larger than a few cycles of the carrier wave. Since this is the case within the framework of this thesis, carrier-envelope phases will not have any further impact.

A phase ϕ 1 which varies linearly with the frequency will result in a delay of the pulse, as can be shown by a simple calculation,

E(t) = 1 √ 2π
e -iω 0 t A(∆ω)e iϕ 1 ∆ω e -i∆ωt d∆ω = 1 √ 2π e -iω 0 t A(∆ω)e -i(t-ϕ 1 )∆ω d∆ω = e -iω 0 t F[A(∆ω)][t -ϕ 1 ].

(1.6)

A spectral phase of rst order results thus in a shift of the pulses by ϕ 1 in the time domain.

However, only the temporal envelope gets shifted, while the carrier wave remains at its initial position with a maximum at t = 0. As this or any other maximum of the carrier wave does not necessarily coincide with the new position of the envelope maximum, a rst order spectral phase generally introduces an additional carrier-envelope phase, as illustrated in g. 1.4c.

Mathematically, this can be described by substituting t = t -ϕ 1 , so that the time axis is centered around the maximum of the spectral envelope. This yields E(t ) = e -iω 0 t e -iω 0 ϕ 1 F[A(∆ω)][t ]

(1. [START_REF] Holzwarth | Optical Frequency Synthesizer for Precision Spectroscopy[END_REF] and shows that the carrier wave gains an additional phase oset ϕ 0 = -ω 0 ϕ 1 . Hence, if a constant phase ϕ 0 = ω 0 ϕ 1 is applied at the same time as ϕ 1 , both the carrier wave and the spectral envelope are subject to the same delay, and no carrier-envelope phase is acquired. This is referred to as pure delay and requires that a spectral phase ϕ = ϕ 1 ∆ω + ϕ 1 ω 0 = ϕ 1 ω is added to the envelope.

As will be seen below, such a phase is automatically applied when the pulse propagates in a non-dispersive medium.

A spectral phase of second order induces a frequency chirp and results in a temporal broadening of the pulse. This can be understood in terms of the group delay τ g = dϕ dω , which describes the delay accumulated by a frequency ω after propagation of a distance z. Since a second order spectral phase leads to a linear group delay τ g = ϕ 2 (ω -ω 0 ), the frequencies will arrive one after the other, so that the pulse undergoes a linear spectral chirp and a temporal broadening. For large ϕ 2 , this behavior results in temporal phase and pulse proles which are identical to the spectral ones.

The analytical expression of the temporal pulse prole depends in this case on its exact spectral shape. In order to give an instructive example, we will focus here on a pulse with a Gaussian spectral amplitude of width δω centered around ω 0 , A(∆ω) = A 0 exp -(∆ω) 2 2δω 2 . For comparison, the temporal shape of a Gaussian spectrum without a spectral phase is given by

E(t) = F[A(∆ω)] = 1 √ 2π A 0 exp (-iω 0 t) exp - ∆ω 2 2δω 2 exp (-i∆ωt) d∆ω = 1 √ 2π δωA 0 exp (-iω 0 t) exp - t 2 2δt 2 , (1.8)
where δt = 1/δω. Adding the second order spectral phase ϕ(ω) = ϕ 2 2 ∆ω 2 yields then

E(t) = 1 √ 2π A 0 exp (-iω 0 t) exp - ∆ω 2 2δω 2 exp i ϕ 2 2 ∆ω 2 exp (-i∆ωt) d∆ω = 1 √ 2π δω √ C A 0 exp (-iω 0 t) exp - t 2 2(Cδt) 2 exp -iϕ 2 t 2 2C 2 δt 4 , (1.9)
with the chirp parameter C 2 = 1 + ϕ 2 2 δt 4 ≈ ϕ 2 2 δt 4 , where the approximation holds for ϕ 2 δt 2 . The Gaussian envelope gets thus broadened by a factor C. In the same time, its amplitude is reduced by 1/ √ C, so that the energy conservation is still observed.

The frequency chirp postulated earlier can be conrmed by considering the instantaneous frequency ω(t), given by

ω(t) = - dϕ(t) dt = - d dt -ϕ 2 t 2 2C 2 δt 4 -ω 0 t = ϕ 2 t C 2 δt 4 + ω 0 ≈ t ϕ 2 + ω 0 , (1.10)
where again the approximation holds for ϕ 2 δt 2 . The instantaneous frequency increases thus linearly with time, leading to a frequency chirp as depicted in g. 1.4d. Moreover, at the point in time where the intensity of the chirped pulse has dropped to 1/e of its maximum value, t = Cδt = ϕ 2 δt , the instantaneous frequency is given by ω(t ) = 1 δt + ω 0 = δω + ω 0 , corresponding exactly to the point where the intensity of the frequency spectrum has dropped to 1/e. Since the frequency chirp is linear in time and centered around ω(0) = ω 0 , this direct correspondence is maintained for any other pairs of t and ω. This conrms the homothetic behavior between the time and frequency domain described above.

A spectral phase of third order (TOD, third order dispersion) gives rise to a quadratic group delay, so that the central frequency arrives rst (or last, depending on the sign of ϕ 3 ), while the other frequencies arrive later (earlier) in pairs of two. This leads to an oscillatory behavior in the time domain, so that the main pulse is followed (preceded) by several weaker post-(pre-) pulses.

An analytical treatment of this case is possible for Gaussian pulses [START_REF] Mcmullen | Chirped-pulse compression in strongly dispersive media *[END_REF], but beyond the scope of this thesis. Moreover, it will be seen in section 3.4.2 that the impact of TOD on the coherent combining eciency is very small.

Time-bandwidth Product

A common means to characterize the temporal and spectral quality of an ultrashort pulse is the time-bandwidth product, given as the product of the FWHM of the intensity spectrum ∆ν and the temporal pulse prole ∆t. In the ideal case, the pulse is Fourier transform limited or time-bandwidth limited, which means that the time-bandwidth product is minimized. This minimum value is calculated based on the measured spectrum, which is then Fourier transformed in order to access the theoretical temporal prole and its FWHM. However, due to spectral phases the actual experimental pulse width might be larger than the calculated one. Hence, the timebandwidth product based on the measured pulse width is in general larger than the one based on the calculation, and indicates how close the pulse is to the ideal transform limited case.

Pulse Propagation in Free Space

On the optical axis and by neglecting the Gouy phase, the spatial phase of a Gaussian beam is given by ϕ = kz. Hence, after propagating a distance z, the beam has accumulated a phase ϕ = kz. In free space and if dispersion can be neglected, k = n 0 ω/c, where n 0 is the refractive index of air. Thus, after propagation of a distance z, the temporal prole of the pulse becomes

E(t) = 1 √ 2π E(ω)e -iωt e iω n 0 c z dω = 1 √ 2π e -iω 0 t e iω 0 n 0 c z A(∆ω)e -i∆ω(t-n 0 c z) d∆ω .
(1.11)

The pulse envelope gets thus shifted to a time t ϕ 1 = t -n 0 c z. At the same time, the carrier wave acquires an additional phase ω 0 n 0 c z, so that it reaches its maximum at t ϕ 0 = t -n 0 c z. Given that t ϕ 1 = t ϕ 0 = t , both the carrier wave and the envelope propagate at the same velocity v ϕ = c/n 0 , and no carrier-envelope phase is generated. Besides, by identifying ϕ 1 = n 0 c z one sees that the overall phase applied to the pulse becomes ϕ = ω 0 ϕ 1 + ϕ 1 ∆ω = ϕ 1 ω, which corresponds to the pure delay introduced above.

By traveling a certain distance in a dispersion-free medium, the pulse get thus shifted to a time t , but remains otherwise unchanged, as expected. As will be seen in the next section, this behavior changes dramatically when it comes to propagation in a dispersive medium, such as an optical ber.

Femtosecond Pulses in Optical Fibers

Spatial Considerations

In an optical ber, the light is guided due to total internal reection at the boundary of the ber core and cladding (for more details on optical bers, see section 1.2). In the ber, the superposition of the reected waves forms transversal standing waves, the so-called linearly polarized modes LP of the ber. As every single wave is superposed with its own reection, the radial components of their wavevectors cancel out, and the resulting modes propagate in z-direction according to the propagation constant β LP , given by the z-component of the respective wavevectors. Hence, the phase accumulated by a given mode LP after propagation of a distance z in the ber is simply given by ϕ = β LP z, and is constant along the transverse beam prole. In a single mode ber, this prole can be approximated by a Gaussian beam of width w, where w depends on the ber parameters [START_REF] Saleh | Fundamentals of Photonics[END_REF]. Furthermore, as only the lowest order mode LP 01 can propagate in such a ber, β LP can be replaced by β.

Dispersion

Due to the dispersive character of the ber medium, its refractive index n depends on the wavelength, so that n = n(ω) and therefore also β = β(ω). In order to take into account the frequency dependence of the propagation constant, β is commonly approximated by a Taylor series, given by β(∆ω

) = β 0 + β 1 ∆ω + β 2 2 ∆ω 2 + β 3 6 ∆ω 3 + O(∆ω 4
). The phase is then recovered by multiplication of β(∆ω) with z, so that it is possible to identify each term of βz with one term of the spectral phase discussed in section 1.1.1. This means that the propagation in a dispersive medium gives rise to spectral phases of all orders, leading to the same eects on the temporal pulse shape as described above.

Fundamentals

In consequence, a beam which covered a distance z in an optical ber has gained a phase

oset β 0 z = β(ω 0 )z = ω 0 n 0 c z = ω 0
vϕ z, where v ϕ = c/n 0 is the phase velocity and n 0 = n(ω 0 ) the refractive index of the central wavelength in the ber medium. In the same time, the envelope has accumulated a delay of β 1 z = z ∂β ∂ω | ω=ω 0 = z vg , with v g = ∂ω/∂β being the group velocity.

Since phase and group velocity are generally not the same in a dispersive medium, this will give rise to a carrier-envelope phase.

Furthermore, β 2 z can be identied as the group delay dispersion, giving rise to a linear frequency chirp and a temporal pulse broadening as discussed in section 1.1.1 for the spectral phase of second order. β 2 is a material constant given by β 2 200 fs 2 /cm for fused silica at a wavelength of λ = 1030 nm. Finally, β 3 z corresponds to a third order spectral phase. Since the material constant β 3 410 fs 3 /cm for fused silica at λ = 1030 nm is positive, this leads to the creation of post-pulses.

In order to estimate the impact of the dispersion on the nal pulse shape, it is convenient to refer to the characteristic length scales associated with the dierent terms of the wavevector.

They are dened as L D,2 = ∆t 2 /β 2 for the group delay dispersion and L D,3 = ∆t 3 /β 3 for the third order dispersion, and are an estimation of the distance which a pulse can propagate in the ber before the eects of the respective phase become important [START_REF]Nonlinear Fiber Optics[END_REF]. In our case, ∆t ≈ 225 fs, so that L D,2 ≈ 2.5 m and L D,3 ≈ 280 m. Considering that the front end of our laser system (cf. section 4.1) consists of about 90 m of ber per channel, the contribution of the third order dispersion is negligible. In contrast, a second order phase of ϕ 2 = β 2 z ≈ 1.8 × 10 6 fs 2 will be accumulated due to the group delay dispersion. However, at the beginning of the chain, the pulses are temporally stretched by a factor of about 2000, which corresponds to a spectral phase of ϕ 2 ≈ Cδt 2 = 36 × 10 6 fs 2 . The additional second order phase induced by the ber GVD will thus have no signicant impact on the pulse duration. Nevertheless, if not taken into account and properly canceled, the newly accumulated phase can be detrimental to subsequent temporal pulse compression. Moreover, GVD can have a negative impact on the coherent combining eciency, as will be shown in section 3.4.2.

Nonlinear Eects

For high intensity laser pulses, nonlinear eects may occur during pulse propagation. In the femtosecond regime, the most important ones are those related to the optical Kerr eect, which describes a change of the refractive index of a medium induced by the laser eld itself, and proportional to its intensity. Hence, the refractive index of the medium becomes n = n 0 + n 2 I, which introduces an intensity-dependent nonlinear phase dϕ NL = β dz = 2π λ n 2 I(x, y, z, t) dz. As this phase depends on the exact shape of the intensity prole, and since this shape changes constantly during propagation due to the newly accumulated nonlinear phase, the exact impact of the nonlinear phase on the pulse shape can only be simulated numerically by calculating the intensity prole for every increment dz. However, a qualitative understanding of the behavior of the beam when subject to a nonlinear phase can be obtained by considering the temporal and spatial domain independently one from another.

In the spatial domain, the optical Kerr eect generates a refractive index which is higher at the center of the beam than at the edges. The resulting nonlinear phase will lead to a curvature of the wavefronts such that the beam gets focused. In the same time, its diraction gets stronger, so that it tends to re-diverge. If the peak power of the beam does not exceed the critical power P crit = π(0.61) 2 λ 0 8n 0 n 2 [START_REF] Boyd | Nonlinear Optics[END_REF], the diraction is stronger than the focusing eect, and the beam is properly guided in the ber. In the contrary, if P crit is exceeded, the self-focusing eect enhances itself due to continuously increasing peak intensities, so that the beam nally collapses on itself, thereby destroying the material it propagates through. In an optical ber with refractive indices n 0 = 1.45

and n 2 = 2.5 × 10 -20 m 2 /W operating at a wavelength λ 0 = 1032 nm, the critical power is given by P crit = 4.3 MW. This value is rarely reached in standard bers, since the peak power has to be limited well below P crit in order to avoid distortions of the temporal and spectral pulse shape due to self-phase modulation (SPM) or damage of the ber output surface.

Self-phase modulation is the result of a nonlinear phase in the time domain. Assuming a Gaussian temporal pulse prole approximated by a parabola, ϕ NL ∝ I(t) = -t 2 2δt 2 gives rise to a temporal phase of second order and thus to a frequency chirp which is inversely proportional to the pulse duration (squared). For an initially transform-limited pulse, the second order nonlinear phase leads to a spectral broadening, just as a spectral phase of second order gives rise to a temporal broadening. Furthermore, due to the increased spectral width the impact of dispersion on the temporal pulse shape gets more important, resulting in an increased temporal broadening of the pulse.

On the contrary, an initially stretched pulse exhibits already a strong frequency chirp. Moreover, the impact of the nonlinear temporal phase on the frequency chirp and the spectral broadening is reduced by a factor of C. Hence, the overall eect of the SPM on the pulse prole is quite low, so that the pulse essentially maintains its temporal and spectral shape. However, since for highly stretched pulses the temporal and spectral phase are homothetic, the temporal phase induced by SPM translates directly into a spectral one. It has thus to be taken into account when it comes to pulse compression. Furthermore, any real pulse shape is unlikely to be perfectly parabolic, so that the nonlinear phase contains higher order terms which in any case lead to arbitrary deformations of the spectral phase.

Finally, also pre-or post-pulses which are often present due to various reasons such as imperfect ber coupling or parasitic reections from a mirror are aected by nonlinear eects. Indeed, for largely stretched pulses SPM leads to an energy transfer from the main pulse to these pre-or post-pulses as well as to the creation of additional pulses at the harmonics of the initial pulse positions. The presence of a secondary pulse in the time domain leads to an oscillating spectral phase and oscillations in the frequency spectrum. Because of the homothety of the two domains, these oscillations are recovered in the temporal pulse shape, where they give rise to an oscillating nonlinear phase generated by SPM. This phase is then translated back into the frequency domain,

where it leads to a nonlinear spectral phase whose oscillations add to those already present on the initial spectral phase. Hence, the amplitude of these oscillations is increased, which results in a relative power increase in the pre-or post-pulses [START_REF] Schimpf | The impact of spectral modulations on the contrast of pulses of nonlinear chirped-pulse amplication systems[END_REF]. Thus, even if the main pulse could be recompressed perfectly, at high levels of nonlinearity the transfer of energy to the secondary 1 Fundamentals pulses due to SPM would still have a detrimental eect on the energy contained in the main pulse.

In order to quantify the general impact of the nonlinear phase on the prole of the femtosecond pulses, a commonly used parameter is the B-integral, which is a measure of the overall nonlinear phase accumulated at the center of the beam of peak power P peak during propagation along a distance z. It is dened as

B = dϕ NL = 2π λ 0 n 2 I c (z) dz = γ NL P peak (z) dz, (1.12)
where I c is the peak intensity at the center of the beam and γ NL = 2π

λ 0 n 2 A e
is the so-called nonlinear parameter. A e is the eective mode area of the beam in the ber. It was introduced in order to take into account that the intensity of the fundamental mode reaches its maximum at the beam center, which means that the nonlinear phase accumulated elsewhere is inferior to the estimation based on I c . However, as the spatial intensity prole in the ber is the result of a superposition of reected beams, where each individual beam changes position constantly, and is therefore subject to varying amounts of nonlinear phase, the overall nonlinear phase is averaged over the entire spatial beam prole. It is thus less important than the one expected by considering only the intensity at the center of the ber. In order to take this eect into account, I c is replaced by the peak power P peak of the beam and the eective mode area A e = πw 2 , which is by a factor of two larger than the "normal" area of a Gaussian beam. Hence, for a given peak power the intensity used for calculating the B-integral is decreased with respect to I c , which also reduces the resulting nonlinear phase.

Equation (1.12) implies that there are only two ways to limit the B-integral and thus the amount of nonlinear eects: by reducing either the intensity of the beam or the distance z covered in the medium. Reducing z is not the best option due to the characteristics of beam amplication in optical bers (see section 1.3.2 for more detail). However, there are several well-established techniques which allow to decrease the intensity, either by reducing the peak power of the beam, or by increasing its area. The former one can be achieved by chirped pulse amplication, the latter one by the use of large mode area bers, as explained in sections 1.2.2 and 1.3.3, respectively.

Optical Fibers

In principal, optical bers are very long and thin cylindrical waveguides. They are most commonly used in telecommunication systems because of their high signal transmission rate, their low losses and their robustness. However, there are numerous other applications, one of them being high power ber lasers and ampliers. Since the whole front end of the laser system used in this thesis, from the master oscillator to the power ampliers, relies on ber technology, this section will give a short introduction into the dierent types of optical bers used in our setup.

Working Principle

An optical ber is typically made of silica, where dierent dopants lead to dierent refractive indices in the small ber core and the cladding around it. In the most common case, the refractive indices are constant within the ber core and cladding, resulting in a so-called step-index ber.

A light signal can be guided inside of a ber core due to total internal reection at the boundary between the core and the cladding (cf. g. 1.5). In order to keep the reection angle inside of the ber larger than the critical angle of the total internal reection, the light must enter the ber at an angle inferior to the maximum tolerable input angle θ max = arcsin

1 n n 2 core -n 2
clad , where n is the index of refraction of the medium surrounding the ber, n core is the one of its core, and n clad the one of the cladding. Typically, the dierence between these two indices is very small, and the ber is called weakly guiding.

In order to characterize a ber, one often refers to its numerical aperture NA, dened as

NA = n sin θ max = n 2 core -n 2
clad . This is another way of describing the acceptance angle of the ber, that is, the cone of light which is guided when coupled into the ber. Light from outside of this cone is not guided, since it is transmitted in the cladding rather than reected back in the ber core, and therefore lost. A complete description of the spatial shape of the beam inside the ber, and therefore also at its output, can be obtained by solving Maxwell's equations with the boundary conditions imposed by the ber. The solutions are a set of guided modes, and involve a combination of Bessel functions and a cosine along the azimuthal angle [START_REF] Saleh | Fundamentals of Photonics[END_REF]. However, the intensity distribution of the fundamental mode can be well approximated by the one of a Gaussian beam. The emergence of higher order modes is characterized by the so-called V -parameter of the ber, given by V = akNA, where a is the radius of the ber core and k is the wavenumber. For V < 2.405, only the fundamental mode 1 Fundamentals can propagate in the ber, and the ber is called truly single-mode. For higher V -parameters, eective single mode operation can be obtained if the propagation or guidance of higher order modes is suppressed, for instance by bending the ber and thereby reducing its waveguiding ability.

Typical truly single mode bers in the infrared wavelength region have a core diameter of about 5.5 µm. However, as the guided mode extends beyond the core-cladding interface, one often uses the mode eld diameter (MFD) in order to characterize the ber. It is dened as

MFD = 2 √

A e /π, and for the core diameter given above, one nds MFD = 6.6 µm [START_REF]Polarization Maintaining 980 nm Telecommunication Fibers[END_REF].

Furthermore, every mode of the ber can take two dierent polarization states, and is therefore degenerate. In a perfect ber, both polarizations would propagate in the same way, thus remaining independent one from another and conserving the initial state of polarization. However, any real ber exhibits imperfections such as a slightly elliptical core. This introduces some random birefringence in the initially isotropic ber. Moreover, local distortions of the ber, such as bending or twisting, enhance this birefringence. Hence, the two polarization states of the mode propagate with dierent velocities, thus breaking the mode degeneracy. Furthermore, since the birefringence is induced locally in the ber, the polarization eigenmodes change locally as well.

The projection of one local eigenmode on the subsequent one leads to polarization mode coupling [START_REF] Thévanaz | Advanced Fiber Optics[END_REF]. In consequence, the polarization in an optical ber changes arbitrarily, and the output polarization state is random.

An optical ber can be used as a laser amplier if the ber core is doped with laser-active ions such as erbium (Er 3+ ) or ytterbium (Yb 3+ ). The pumping necessary for the amplication is then performed optically, which means that an additional pump laser needs to be coupled into the ber. This can be achieved in two ways: rst, the pump beam can be coupled directly into the ber core. This method is interesting because of its simplicity, however, in the case of single mode bers it requires a pump laser of high beam quality. As those are somewhat limited in output power, the output power of a core-pumped single mode ber amplier stays limited as well. The alternative is the use of so-called double-clad bers. In this type of ber, an outer cladding is added around the rst inner one, so that the signal stays conned in the ber core, whereas the pump beam is guided by total reection at the surface between inner and outer cladding, as illustrated in g. 1.6. As in this case the pump beam is not required to be single-mode, both the numerical aperture and the diameter of the inner cladding can be increased, so that even poor-1.2 Optical Fibers quality pump beams can be eciently coupled into the ber. Those are often readily available in the form of low cost laser diodes, and can provide considerable pump powers. Hence, double-clad bers are much better suited for high power amplication.

Special Types of Fibers

Polarization-Maintaining Fibers

Polarization-maintaining (PM) bers are used if the polarization state of the input beam needs to be conserved. For this purpose, a strong birefringence is introduced in the ber, so that the two polarization eigenmodes are well dened all along the ber. Random mode coupling is still possible due to imperfections in the fabrication, bending or stress, but is largely suppressed, so that the initial polarization state is maintained within the ber. The birefringence is induced by breaking the ber symmetry, either by making the cladding itself asymmetric or by introducing stress bars parallel to the ber core which are made of a dierent material than the cladding, as can be seen in g. 1.7a.

Large Mode Area Fibers

As the name suggests, large mode area (LMA) bers are optical bers where the ber core and therefore also the guided mode have a larger area than in standard bers. The advantage is that the larger mode area reduces the intensity of the light in the ber, so that the impact of nonlinear eects gets reduced. However, if such a ber shall be truly single-mode, the larger mode area and therefore larger core diameter implies that the numerical aperture has to be reduced. This in turn requires a smaller dierence between the indices of refraction of the core and cladding of the step-index ber, which is dicult to manufacture. In consequence, the ber is more sensitive to changes in the refractive index induced for example by stress or bending, and the waveguiding ability of the ber gets reduced. Therefore, the core diameters of truly single-mode step-index LMA bers are limited to about 15 µm. In contrast, dierent higher order mode suppression techniques allow to achieve an eectively single-mode operation of bers with core diameters of up to 30 µm [START_REF] Limpert | Yb-doped large-pitch bres: eective single-mode operation based on higher-order mode delocalisation[END_REF].

Photonic Crystal Fibers

A photonic crystal ber (PCF) is microstuctured, that is, its cross section is interspersed with air holes. This leads to a variety of dierent ber designs, each having its particular characteristics and applications [33]. In the PCF used within the framework of this thesis, the air holes are inserted in the ber cladding, and the light is guided in the solid ber core. The exact guiding mechanisms of such a PCF are complicated due to the silica-air hole structure of the ber cladding, and the characteristics of the guided modes can only be determined numerically. A simplied model supposes an eective refractive index of the microstructured cladding which is inferior to the one of the solid ber core. The light guidance can then be explained by a modied form of total internal reection, comparable to the one in a step-index ber [START_REF] Birks | Endlessly single-mode photonic crystal ber[END_REF]. However, a PCF has the advantage 1 Fundamentals that the eective refractive index of the cladding can be tuned precisely by adjusting the size and spacing of the air holes. Thus, for large core diameters PCFs are more readily manufactured than step-index ones, and MFDs of 35 µm have been demonstrated [START_REF] Limpert | Low-nonlinearity single-transverse-mode ytterbiumdoped photonic crystal ber amplier[END_REF].

Since a main limitation to even larger mode areas is bending of the ber, unbendable rod-type

PCFs have been developed. This allows to design so-called large pitch bers with MFDs exceeding 100 µm [START_REF] Limpert | Yb-doped large-pitch bres: eective single-mode operation based on higher-order mode delocalisation[END_REF], whose output powers are largely increased with respect to standard ber ampliers.

Furthermore, a PCF can be made double-clad by inserting a second layer of air holes in the ber cross section. It is also possible to obtain polarization-maintaining PCFs by inserting stress bars in the microstructured cladding. In the extreme case, the dierence in refractive index of the two polarization eigenmodes is that high that only one of the modes is guided. In this case, the ber is polarizing (PZ).

Due to the air holes in the ber cross section, a PCF needs to be collapsed after cleaving. This means that the air holes are sealed o in order to prevent water vapor or other contaminants to enter. Since this necessarily removes the microstructure near the ber end, the light is not guided anymore and diverges in the collapsed part of the ber. However, this can be an advantage since the larger area of the diverging beam reduces the risk of damage of the ber output surface.

Furthermore, it is possible to add an endcap to the collapsed ber end. This can be used to further increase the beam area, since the endcap can have a larger diameter than the ber itself.

In our setup, rod-type bers cannot be used due to the geometric constraints applied to the ber endings, as will be explained in section 2.2.1. Thus, we use a state-of-the-art exible double-clad polarizing ytterbium-doped large mode area ber with a core diameter of 40 µm and an inner cladding diameter of 200 µm (a DC-200/40-PZ-Yb LMA ber) [START_REF]Modal properties of the DC-200/40-PZ-Yb LMA ber. Whitepaper[END_REF]. A microscopic photo of its cross section is shown in g. 1.7c, where the air holes of the cladding and the stress bars are clearly visible. In this ber design, the guidance of other modes than the fundamental one is suppressed when the ber is coiled to a certain radius. Therefore, the outer surface of the outer cladding is polished such that the ber coils automatically in the required way. Furthermore, since both polished parts (only one of which is visible in g. 1.7c) are parallel to the plane of the stress bars, the polarization of the output beam is automatically aligned with respect to the plane of the ber bending. Moreover, the doping and therefore the gain of the amplier contributes to the eectively single-mode behavior of the ber, since only the parts of the modes which overlap with the doped ber core are amplied. As this overlap is small for higher order modes, they are amplied much less than the fundamental one, which becomes therefore predominant. Hence, the output beam of the DC-200/40-PZ-Yb is eectively single mode and spatially and spectrally stable only if the ber is coiled correctly and operated at a minimum pump power.

Finally, a 1 mm diameter endcap was added to the ber ending in order to enlarge the output beam and avoid surface damage.

Multicore Fibers

The bers used within the framework of this thesis all have only one single core, as it is most common for optical bers. However, bers with multiple cores do exist, and although they follow 1.2 Optical Fibers the same physical principles as single core bers, they have some peculiarities typical to the multicore design, such as mode coupling between the dierent cores [START_REF] Saitoh | Multicore Fiber Technology[END_REF]. They are of interest for CBC laser systems due to their compactness and the low phase dierences between the beams conned in the dierent cores [START_REF] Ramirez | Coherent beam combining with an ultrafast multicore Yb-doped ber amplier[END_REF]. The cross section of such a ber used in a CBC experiment is shown in g. 1.7b. However, multicore ber ampliers are still a relatively new technology, with no commercially available products and a limited number of cores [39]. Hence, at the current state of the art, they are not well suited for the CBC of very high channel numbers. (b) a 7-core PCF, with the doped cores highlighted in blue, taken from [START_REF] Ramirez | Coherent beam combining with an ultrafast multicore Yb-doped ber amplier[END_REF], and (c) the DC-200/40-PZ-Yb LMA ber used as power amplier in our laser setup.

Ytterbium-doped Femtosecond Fiber Ampliers

A ber amplier is an optical ber whose core is doped with rare-earth ions such as erbium (Er 3+ ) or ytterbium (Yb 3+ ). If a pump beam is coupled in the ber, the ions are transferred in an excited state, so that the laser signal can be amplied by stimulated emission of photons. Due to the long interaction lengths in optical bers and the ecient heat removal resulting from their high surface-to-volume ratio, these ampliers are very well suited for the amplication of high repetition rate signals to high average powers.

The characteristics of a specic amplier, such as its bandwidth and gain, are determined by the doping ions and the hosting material, as will be explained in more detail in the rst part of this section. The following two parts include some considerations on the pump beam, and a discussion of chirped pulse amplication.

Signal Amplication and Gain

The wavelength range which can be amplied by the ber core depends on its spectroscopic properties, which are for their part determined by the dopants and the host itself. In the ber ampliers used in our setup, the doping ion is Ytterbium, which can in principle be described as a two-level system. However, as the ytterbium ions are integrated in the amorphous structure of the silica ber, the electric elds of the surrounding atoms give rise to the Stark eect which causes a splitting and shifting of the initial Yb 3+ energy levels. Moreover, since every ion sees slightly dierent electric elds, the individual energy shifts vary from one ion to another. Averaging over all ions thus yields a smooth energy distribution for both levels. Furthermore, it can be assumed that each state is in thermal equilibrium, which means that the ions are most likely in a lower sublevel of the respective manifold. As a consequence, absorption is more likely to take place at lower wavelengths, where ions from a lower sublevel of the ground state are excited, as schematically shown in g. 1.8a. Conversely, emission is favored at higher wavelengths, corresponding to the emission of a photon from a lower sublevel of the excited state. For intermediate wavelengths, both processes have similar probabilities. The resulting exact absorption and emission cross sections of an Yb-doped silica ber are shown in g. 1.8b.

The amplication process is characterized by the frequency-dependent gain coecient γ(ν), which describes the change in photon number per unit length. It is determined by the dierence between the number of absorbed and emitted photons, so that γ(ν) = N 2 σ e (ν) -N 1 σ a (ν), (1.13) where N 1,2 is the number density of Yb-ions in the ground and excited state, respectively, so that N 1 + N 2 = N , where N is the total number of Yb-ions. σ a,e (ν) are the frequency-dependent absorption and emission cross sections of Yb:glass, as shown in g. 1.8b. The gain coecient depends thus both on the wavelength and the relative population of the two states.

It is worth noting that even if there is no population inversion, the gain coecient can be positive in the long wavelength region where σ e (ν) > σ a (ν). This is the case when only a small pump power is injected in the amplier, and results in a low, but broadband gain at wavelengths well above 1000 nm, as shown in g. 1.9a. In this case, amplication at higher wavelengths is favored.

On the other hand, for higher pump powers and therefore larger excited state populations, the overall gain increases, and gets peaked around 1032 nm. Hence, wavelengths around this value get amplied most.

Due to this large gain prole around 1032 nm, Yb-doped bers are well suited for the amplication of the broadband spectra inherent to femtosecond pulses. However, as the gain coecient depends on the wavelength, some regions of the pulse spectrum are amplied stronger than other parts. The spectral envelope of the pulses changes thus with amplication. If this envelope was initially centered around the central frequency of the gain spectrum, its width gets reduced, a (a) Gain proles as a function of pump power (from low to high) for pumping at 910 nm.

(b) Gain proles as a function of pump wavelength for high pump powers.

Figure 1.9: Gain proles of an Yb-doped ber ampliers in dependence of the properties of the pump laser. Pictures taken from [START_REF] Paschotta | Ytterbium-doped ber ampliers[END_REF].

phenomenon which is referred to as gain narrowing. Furthermore, if the gain spectrum is shifted with respect to the pulse spectrum, the amplication process additionally shifts the maximum of the pulse spectral envelope towards the maximum of the gain curve.

The overall gain of the amplier, given by the ratio between its in-and output power, depends not only on the gain coecient γ, but also on the power of the input signal. For increasing signal power, the upper state population and therefore the gain decreases. The saturation power P s indicates the power at which the gain has dropped to half of its maximum value, and is given by P s = hν τ (σa+σe) A e [START_REF] Desurvire | Erbium-doped ber ampliers. Principles and applications[END_REF], where h is Planck's constant and τ is the upper state lifetime of Yb:glass, given by τ ∼ 1 ms. For a standard Yb-doped ber, P s ∼ 5 mW, whereas it is as high as P s ∼ 180 mW in the PCF used as power amplier in our setup.

Depending on the signal power, the amplier works in two very dierent regimes. The so-called small signal approximation holds for P P s . The behavior in this case can be understood by assuming that every photon of the signal induces the creation of a second photon by stimulated emission. Hence, the number of photons and therefore the signal power increases exponentially, so that P (z) = P (0)e γ(ν)z . On the other hand, in the heavily saturated case where P P s , the number of photons is much higher than the number of atoms in the excited state. Therefore, the number of newly created photons is proportional to the number of excited atoms, and the signal grows linearly according to P (z) = P (0) + γ(ν)P s z [START_REF] Saleh | Fundamentals of Photonics[END_REF].

As ber ampliers can be very long and can therefore provide very high gains, they typically operate in both regimes. For instance, the input power of 100 mW of the power amplier used in our setup is somewhat below its saturation power, whereas the output signal of 25 W is largely saturated.

For pulsed lasers, the impact of the repetition rate of the pulse train on the gain needs to be considered as well. When a CW pump beam is used, the excited state population builds up in between two pulses, since there is no possibility for stimulated emission, and spontaneous emission is suppressed as long as the time between two pulses is well inferior to the upper state lifetime τ .

If this is not the case, there is a large risk that spontaneously emitted photons are amplied (ASE, amplied spontaneous emission), which reduces the upper state population and thereby the gain, and mixes unwanted frequencies in the output signal. Since the upper state lifetime of Yb 3+ is τ ∼ 1 ms, the pulse repetition rate in our setup should be kept well above 1 kHz in order to avoid ASE.

Besides, due to the build-up of a large upper state population in between the pulses, the leading edge of each pulse is subject to a high gain. For high pulse energies, it partially depletes the excited state, so that the gain is reduced for the trailing edge of the pulse. Hence, the leading part of the pulse is amplied stronger than the trailing one. When the pulses are chirped, this implies that the leading frequencies (the lower ones for a positive chirp) are amplied stronger than the trailing ones, which results in a re-shaping of the pulse spectrum. For low-energy pulses, this behavior is often negligible since the gain depletion per pulse is weak, so that the gain remains approximately constant over the whole pulse. However, for high-energy pulses, serious spectral distortions can be observed, and the gain decreases considerably. The pulse energy necessary to reduce the gain to 1/e of its initial value is called saturation energy, and given by E s = P s τ . In the power amplier used in our setup, E s ∼ 180 µJ, so that the pulse energies should be kept well below this value in order to avoid spectral deformations due to gain depletion. In the experiments, the output average power was set to 25 W per amplier, which means that even at the lowest pulse repetition rate of 750 kHz, the pulse energy does not exceed E ∼ 33 µJ and is thus well inferior to E s .

Optical Pumping

The spectroscopic properties of the gain medium also determine the wavelength of the optical pumping. Considering the absorption spectrum in g. 1.8b, there are two obvious candidates for pumping of Yb:glass: 910 nm and 975 nm. Since the emission cross section at 910 nm is almost zero, this wavelength allows to transfer a very high amount of ions to the excited state. This implies that the gain curve is very similar to the spectral distribution of the emission cross section, with a strong peak around 975 nm, as shown in g. 1.9b. If the signal to be amplied is located within the broad peak around 1032 nm, there will thus be a large risk for ASE at 975 nm. The simplest method to overcome this problem is to use this wavelength for the pump laser. Since at 975 nm the absorption and emission cross sections are almost equal, the upper state population is limited to about 50 %. In consequence, the gain in the longer wavelength region is reduced by about a factor of two, and is largely suppressed at the pump wavelength. Hence, spontaneous emission at this wavelength is reabsorbed rather than amplied, and has thus no negative impact on the amplication process. Therefore, and because of the large absorption cross section at 975 nm, this wavelength is typically used for the optical pumping. A convenient side eect of choosing a pump wavelength which is closer to the signal one is that also the quantum defect is reduced, that is, the amount of energy which is lost in the conversion of a pump photon to a signal one.

During operation, care has to be taken that there is always an input signal of constant average power coupled into the ber amplier. This ensures that during turn-on of the pump laser, the amplied signal power increases continuously and the gain remains moderate. After a certain time, the system reaches a steady state operation, where both the gain and the amplied signal power remain constant and at a moderate level. However, if for some reason the input signal is interrupted while the pump laser is running, very high gains are built up. Since furthermore an optical ber can act as laser cavity, these gains can amplify spontaneously emitted photons to power levels largely exceeding those obtainable in steady state operation. This process is known as Q-switching and can in the worst case lead to an optical damaging of the ber amplier.

When it comes to coupling the pump laser into the ber, there are two main options. On the one hand, it is possible to inject both the signal and the pump beam at the same end of the ber amplier, which is referred to as forward pumping. On the other hand, the pump can be injected at the opposite end of the amplier, which is called backward pumping. For both cases, the signal and pump power are qualitatively shown in g. 1.10 as a function of position in the ber amplier.

For both pumping schemes, the increase in signal power changes from exponentially to linear as the signal power itself changes from small to saturated. However, in the case of forward pumping, at some point all the pump power has been absorbed, so that the amplication ceases and the signal power starts to decrease slowly due to losses in the ber amplier. This behavior does not occur for backward pumping since in this case the pump is strongest at the ber end. In contrast, in this conguration care has to be taken that there is enough pump power near the beginning of the ber, in order to avoid a diminishment of signal power due to ber losses. The forward pumping scheme is slightly less ecient than the backward pumping one, since it exhibits a more pronounced ASE near the ber beginning [START_REF] Desurvire | Analysis of gain dierence between forward-and backward-pumped erbiumdoped ber ampliers in the saturation regime[END_REF]. Moreover, when nonlinear eects need to be considered, forward pumping is less advantageous since the signal power increases slower near the ber ending. This implies that quite some ber length is needed in order to amplify an already strong signal to its maximum power. Since the B-integral scales with the signal intensity, this small endmost increase in signal power comes along with a large increase of nonlinear eects. In practice, the length of the ber amplier has to be chosen as a compromise between amplied signal power and B-integral. In a backward pumping scheme, this problem does not occur since the signal power increases steadily till the ber ending. However, in our setup backward pumping is not possible due to the spatial constraints which are imposed on the ber outputs. Hence, a forward pumping scheme was chosen and the ber length of 1.6 m was calculated carefully in order to nd the best working point.

Chirped Pulse Amplication

In order to minimize distortions of the spatial and spectral pulse shape induced during amplication of high intensity pulses by self-focusing or self-phase modulation, one commonly uses chirped pulse amplication (CPA, [START_REF] Strickland | Compression of amplied chirped optical pulses[END_REF]). In this technique, the laser pulses are temporally stretched prior to amplication. This is achieved by passing the pulses through a suitable dispersive optical element, such as a pair of diraction gratings or a chirped ber Bragg grating (CFBG). These induce a spectral phase of several orders, which leads to a frequency chirp and a temporal stretching of the pulses, so that their peak power is reduced, as depicted in g. 1.11. After amplication, the spectral phase is removed by passing the pulses through another optical element with the inverse dispersion of the one used as stretcher. Due to the high peak powers attained in this process, the optical element in question usually relies on pulse propagation in free space. For largely stretched pulses, it most often consists of a diraction grating pair.

The description given above somehow neglects the presence of additional dispersive elements between the stretcher and the compressor, such as the laser amplier itself. The stretched pulse accumulates thus an additional spectral phase during propagation through these elements. In setups which rely essentially on free-space optics, these phases are often very small and can therefore be neglected. In contrast, in the all-ber front end of our laser system, much larger spectral phases are accumulated. Hence, in order to recover the initial pulse shape after compression, the dispersion has to be taken into account in the design of the compressor, so that the spectral phase induced by the compressor should cancel the overall accumulated spectral phase rather than only the one induced by the stretcher. Unfortunately, the spectral phase components accumulated in the bers due to GVD and TOD are both positive, while a grating compressor induces a second and third order spectral phase which are proportional to each other, but of opposite sign. This implies that in such a conguration a total compensation of the spectral phase is only possible if the spectral phase components of the stretcher can be chosen arbitrarily, as it is the case for a CFBG.

While the dispersion is generally well known and can therefore be included in the design of the CPA system, things are much more complicated when it comes to self-phase modulation. As in this case the accumulated phase is highly dependent on the initial conditions and on the exact behavior of the amplier, the overall accumulated nonlinear phase can not be predicted precisely.

Moreover, if developed in a Taylor series, the dierent orders of the nonlinear phase may have dierent signs or relative weights than those of the grating compressor, which means that if one order is well compensated, the relative contribution of another order may be increased. It is thus not possible to cancel the complete spectral phase of the pulse, so that the pulse does not recover its initial shape after compression. Indeed, the remaining spectral phase can give rise to a pedestal in the temporal pulse shape, or a broadened pulse prole. The exact temporal shape of the compressed pulse is often a compromise between these two, and depends on the specic alignment of the compressor.

1 Fundamentals

Conclusion

In this chapter, the physical and technological bases of this thesis have been presented. In a rst section, the physics of femtosecond laser pulses and their propagation in optical bers have been discussed. It has been concluded that care has to be taken to properly manage the dispersion inherent to the all-ber front end of our laser system, and to avoid excessive nonlinear eects which are detrimental to the temporal and spectral pulse quality.

In a next section, dierent types of optical bers used in our setup have been introduced. In particular, the design and functioning of the photonic crystal ber used as power amplier have been presented.

Finally, phenomena related to the amplication of femtosecond pulses to high average powers have been discussed. It has been seen that Yb-doped ber ampliers are particularly well suited for this task, but that care has to be taken in order to avoid deformations of the pulse spectrum due to inappropriate amplication conditions. Finally, also the CPA technique has been presented.

The combination of all these principles and technologies is at the basis of the next chapter, where dierent methods of coherent beam combining of femtosecond ber ampliers will be discussed.

Coherent Beam Combining

In the previous chapter, the limitations in power scaling of femtosecond ber ampliers due to nonlinear eects have been discussed. The two main approaches used to overcome these limits have been presented, namely the use of LMA bers and the CPA technique. However, both methods are hitting their limits. A further scaling of the mode area of ber ampliers is dicult due to increasing mode instabilities [START_REF] Eidam | Experimental observations of the threshold-like onset of mode instabilities in high power ber ampliers[END_REF]. In order to reach higher peak powers with CPA, the pulses would need to be stretched more, which would require huge, complex and cost-intensive systems for their recompression. This is why new approaches for the power scaling of femtosecond ber ampliers are being investigated. The current records in terms of peak and average output powers of ber laser systems have been obtained using coherent beam combining (CBC) [14], which will be introduced in this chapter. In a rst part, general aspects and dierent architectures will be discussed and evaluated with respect to their scalability to very large channel numbers exceeding 1 000 bers.

This will motivate our choice for one specic setup, which will be presented in more detail in the second part of this chapter.

It should be noted that the techniques presented below are limited to those suitable for the power scaling of femtosecond ber ampliers in master oscillator power amplier (MOPA) conguration. Other combining techniques such as coherent pulse synthesis [START_REF] Manzoni | Coherent pulse synthesis: towards sub-cycle optical waveforms[END_REF], CBC of other laser architectures [START_REF] Leshchenko | Coherent combining of relativisticintensity femtosecond laser pulses[END_REF] or passive combining techniques using a joint cavity [START_REF] Brignon | Coherent Laser Beam Combining[END_REF] have been excluded from this description of the state of the art, in order to keep it strictly focused on the topic of this thesis.

General Aspects of Coherent Beam Combining

A schematic of a general MOPA CBC setup is shown in g. 2.1. The beam emitted by the master oscillator is spatially split into N sub-beams, each of which is amplied independently of the others. The channels are then recombined coherently, so that one single, very powerful beam is obtained. This procedure can be interpreted as an extension of the already existing concepts of mode area increase and peak power reduction, since the use of multiple ber ampliers increases the overall available eective area. In the same time, the peak power in a single amplier is largely reduced compared to the one obtained after the coherent combination.

In short pulse regimes, this concept can be expanded in the temporal domain, so that a pulse burst is amplied rather than single pulses. After amplication, the burst is recombined temporally so that a single pulse output with increased peak power is obtained.

Furthermore, temporal and spatial beam combining can be implemented simultaneously by using a spatial CBC architecture as amplication stage within the temporal combining setup.

In order to ensure an ecient beam combination, all the characteristics of the individual pulses, that is, their spatial and spectral envelopes and phases as well as their polarizations, need to be perfectly matched [START_REF] Goodno | Perturbative analysis of coherent combining eciency with mismatched lasers[END_REF][START_REF] Leshchenko | Coherent combining eciency in tiled and lled aperture approaches[END_REF]. The combining eciency η is then dened as the ratio between the power in the combined beam P comb and the total output power of the ber ampliers P tot , so that η = P comb P tot .

(

It should be noted that CBC does not only increase the peak and average power of the laser system, but has also a positive eect on the quality of the output beam. This is due to the fact that only features common to all pulses combine coherently. Aberrations of a single beam with respect to the average are still present in the combined beam, but reduced in intensity by a factor of N . Hence, the combined beam can exhibit spatial and spectral properties which are superior to those of the individual beams.

There are several points which need to be considered when setting up a CBC system: the combination architecture itself, the implementation of temporal beam combining, and the phase matching. In the following, dierent approaches are presented for these points, and they are evaluated in terms of scalability to a very large number of channels.

Spatial Combining Architectures

The spatial combining architectures can be divided in the two main classes of lled and tiled apertures, whose advantages and disadvantages are discussed below.

Filled Aperture

In a lled aperture geometry, the beams are combined in the near eld. This can be done either by combining the beams two by two using (polarizing) beam splitters, as shown in g. 2.2a, or in a collective approach which makes use of a diractive optical element (DOE, g. 2.2b). In both cases, the decrease in combining eciency due to spatial and spectral disparities converges towards (a) Filled aperture conguration using beam splitters.

(b) Filled aperture conguration using a DOE.

(c) Tiled aperture conguration. a constant value for an increasing number of channels [START_REF] Goodno | Perturbative analysis of coherent combining eciency with mismatched lasers[END_REF][START_REF] Klenke | Performance scaling of laser ampliers via coherent combination of ultrashort pulses[END_REF], so that the overall combining eciency is only limited by the losses intrinsic to the combining elements.

Using beam splitters as combiners, very good eciencies of over 90 % have been demonstrated

experimentally for the CBC of a moderate number of beams [START_REF] Daniault | Coherent beam combining of two femtosecond ber chirped-pulse ampliers[END_REF][START_REF] Müller | 1 kW 1 mJ eight-channel ultrafast ber laser[END_REF]. Moreover, the current record in average output power of a ber laser system of P av = 1.83 kW has been obtained by combining 16 rod-type ampliers using thin lm polarizers (TFP) as beam combiners, where the combining eciency of η = 0.82 is limited by thermal lensing in some optical components [START_REF] Müller | 1.8-kW 16-channel ultrafast ber laser system[END_REF].

However, CBC with beam splitters is somewhat limited when it comes to the scalability to very large channel numbers, since N -1 beam splitters are necessary to combine N beams. Hence, the component count and therefore the complexity and cost of the system increases linearly with the number of channels. Moreover, if one considers the losses α inherent to the beam splitters, the combined power is proportional to N (1 -α) N -1 [48]. This function rst increases steeply, reaches then a maximum at N max = -1/(ln(1 -α)) ≈ 1/α and decreases afterwards. Hence, once the maximum output power is reached it cannot be increased further by adding more beams.

Assuming high-quality AR-coated beam splitters with α = 0.001, the number of beams is limited to N max ≈ 1000, with a corresponding maximum combining eciency of η ≈ 0.37.

The problem of an increasing number of combining elements can be overcome by using DOEs, which act as N × 1 beam splitters. However, the losses intrinsic to these elements are largely superior to those of 50/50 beam splitters or TFPs. In a recent demonstration where eight beams have been combined using a pair of DOEs, the intrinsic losses lead to a decrease in combining eciency of about 10 % [START_REF] Zhou | Two-dimensional combination of eight ultrashort pulsed beams using a diractive optic pair[END_REF]. Although this value could be reduced by improving the manufacturing process of the DOEs, a remaining problem is the angular dispersion induced by these elements.

Indeed, even when a pair of DOEs is used for the beam combination, the decrease in eciency due to uncorrected dispersion is estimated to be about 5 % when combining about 200 beams of a spectral width of ∼ 12 nm [START_REF] Zhou | Coherent combination of ultrashort pulse beams using two diractive optics[END_REF].

Tiled Aperture

In a tiled aperture geometry, the beams are arranged in an array and collimated in order to obtain a near eld. The combined beam is then found in the far eld, as depicted in g. 2.2c. This has the advantage that no combining elements are needed, and only one lens has to be put in the beam path in order to perform the optical Fourier transform necessary to obtain the far eld. However, as the near eld is composed of the Gaussian output beams of the ber ampliers, its intensity is periodically modulated. This gives rise to secondary lobes in the far eld which surround the main central lobe. Since this central lobe is typically considered as the combined beam, the side lobes put an upper limit to the combining eciency which can therefore never reach 100 %. However, this upper limit is independent of the number of beams to be combined. Moreover, a beam array can be readily extended to include a large number of channels, which explains why the current record numbers of combined bers have been obtained in tiled aperture congurations, involving 64 beams in CW regime [START_REF] Bourderionnet | Collective coherent phase combining of 64 bers[END_REF] and 19 beams in femtosecond regime [START_REF] Le Dortz | Highly scalable femtosecond coherent beam combining demonstrated with 19 bers[END_REF]. Therefore, this geometry has been chosen for our laser system.

Temporal Beam Combining

In temporal beam combining techniques, a pulse burst is prepared prior to amplication, and temporally combined afterwards. Hence, the amplitudes and phases of all the individual pulses in the burst need to be matched at the temporal combining stage.

Unfortunately, this is not fully possible for setups operating at high pulse energies, since the rst pulse of the train partially depletes the gain of the amplier. In consequence, the following pulses are amplied less strongly and exhibit therefore lower amplitudes and nonlinear phases. In order to overcome this problem, an amplitude shaping of the pulse train is typically performed prior to amplication, so that the gains experienced by the individual pulses are more alike. This allows to match either the amplitudes or the nonlinear phases of the pulses in the train, but not both at the same time. Hence, certain mismatches in at least one of the two properties remain, leading to a decrease in combining eciency and the creation of pre-or post-pulses which deteriorate the contrast of the combined pulse [START_REF] Kienel | Energy scaling of femtosecond ampliers using actively controlled divided-pulse amplication[END_REF]. Furthermore, it should be noted that at a given pump power temporal combining can only increase the peak power of the output pulses, while the average power remains constant and limited by the aforementioned pump power.

There are two dierent temporal combining schemes which are currently being investigated. In the divided pulse amplication (DPA, [START_REF] Zhou | Divided-pulse amplication of ultrashort pulses[END_REF]) technique, the pulse burst is generated by separating the pulses in two dierent polarization states. This can be achieved in two ways. On the one hand, it is possible to use birefringent crystals, as has been done in the rst experimental demonstration of DPA [START_REF] Zhou | Divided-pulse amplication of ultrashort pulses[END_REF]. However, as the delay that can be induced by a crystal of reasonable size is limited, this approach is only applicable to unchirped femtosecond pulses, and therefore not suitable for the amplication to high peak powers. On the other hand, polarizing beam splitters and suitable delay lines can be used, so that larger delays can be obtained [START_REF] Zaouter | Femtosecond ber chirped-and divided-pulse amplication system[END_REF]. Hence, this method is compatible with temporally stretched pulses. Moreover, when it comes to the amplication to high energies, the amplitude shaping described above has to be taken into account. This implies that the amplitudes and phases of the amplied pulse burst vary considerably from those of the input pulses. Hence, independent delay lines are needed for the temporal pulse separation and recombination, and an active phase stabilization has to be implemented. This has the further advantage that any chirped-pulse multi-channel spatial CBC setup can be inserted between the temporal splitting and combining stages. Indeed, the current record in output peak power of a femtosecond ber amplier system has been obtained by coherently combining four temporal and eight spatial pulse replicas. It was thus possible to demonstrate a pulse energy of 12 mJ at a repetition rate of 56 kHz and an overall combining eciency of 78 % [START_REF] Kienel | 12 mJ kW class ultrafast ber laser system using multidimensional coherent pulse addition[END_REF]. However, creating more than four replicas is dicult due to the strongly increasing system complexity and the intrinsic mismatches between the amplitudes and nonlinear phases of the amplied pulses in the burst, which lead to a decrease in combining eciency. Furthermore, the use of DPA is limited to systems using exclusively bers whose amplication characteristics are independent of the polarization state of the beam.

In contrast, in the coherent pulse stacking technique [START_REF] Zhou | Coherent pulse stacking amplication using low-nesse Gires-Tournois interferometers[END_REF], the pulse burst is carved directly out of the master oscillator signal. The recombination is performed in a modied form of a Fabry-Pérot interferometer, called Gires-Tournois interferometer, which is aligned such that one round trip time corresponds exactly to the repetition rate of the pulses in the burst. The pulse burst is then prepared such that all but the last pulse interfere destructively at the resonator output port, so that their power remains in the interferometer and adds up. The last pulse is then used to extract the stored energy via constructive interference at the resonator output. Using this method and a pulse burst comprising 81 individual pulses, pulse energies of ∼ 10 mJ have been achieved at a pulse duration of 540 fs [START_REF] Pei | Near-complete stored energy extraction from ber ampliers in ultrashort >10mJ energy pulses using coherent pulse stacking amplication (Conference Presentation)[END_REF]. A combination of this technique with spatial beam combining seems possible, but has not been demonstrated yet.

Phase Matching Techniques

As pointed out above, phase dierences among the beams have to be balanced in order to obtain en ecient CBC. For femtosecond setups, this means that all orders of the spectral phase need to be matched. Orders higher than zero are typically constant in time, so that a static adjustment is sucient [START_REF] Hanna | Coherent combination of ultrafast ber ampliers[END_REF]. This means that the group delay of the dierent channels need to be matched at all orders, so that delay and dispersion dierences among the pulses are suppressed. In contrast, the zeroth order phase uctuates due to acoustic and thermal noises. It is therefore essential to implement a real time phase oset correction. The necessary bandwidth of this correction depends strongly on the employed material and the experimental conditions. For instance, it has been concluded from measurements of phase uctuations in an Yb-doped ber amplier that a bandwidth of at least several kilohertz would be necessary to obtain a phase correction on the order of λ/10 peak-to-valley. This value is considered as the maximum tolerable residual phase error for an ecient CBC [START_REF] Augst | Coherent beam combining and phase noise measurements of ytterbium ber ampliers[END_REF]. However, various implementations demonstrate residual phase errors below λ/40 RMS with considerably lower bandwidths in the range of 500 Hz to 1 kHz [START_REF] Antier | kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining[END_REF][START_REF] Daniault | Coherent beam combining of two femtosecond ber chirped-pulse ampliers[END_REF]. An extreme case was reported for the coherent combining of the dierent cores of a multicore ber, where a bandwidth of 2 Hz was sucient due to the low phase dierences among the cores [START_REF] Ramirez | Coherent beam combining with an ultrafast multicore Yb-doped ber amplier[END_REF].

Similarly, various devices can be used for the phase correction, such as piezomechanical actuators used for translating a mirror [START_REF] Müller | 1 kW 1 mJ eight-channel ultrafast ber laser[END_REF] or stretching a ber [START_REF] Antier | kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining[END_REF], electro-or acousto-optical modulators [START_REF] Daniault | Coherent beam combining of two femtosecond ber chirped-pulse ampliers[END_REF][START_REF] Augst | Coherent beam combining and phase noise measurements of ytterbium ber ampliers[END_REF], and in the case where a high bandwidth is not required, spatial light modulators [START_REF] Ramirez | Coherent beam combining with an ultrafast multicore Yb-doped ber amplier[END_REF].

In the following, dierent phase matching methods are presented, and each of them is evaluated with respect to its suitability for a highly scalable CBC system in tiled aperture conguration.

Passive Phase Matching

In passive phase matching approaches, the laser system is set up such that the phases of all emitters match automatically. In a MOPA conguration, this can be achieved by using a so-called Sagnac interferometer, where all pulses traverse the same optical components and ampliers. They accumulate thus automatically the same spectral phase, so that no external phase correction is necessary. This allows to achieve very high combining eciencies. For instance, η = 0.96 has been shown for the coherent combination of two ampliers [START_REF] Daniault | Passive coherent beam combining of two femtosecond ber chirped-pulse ampliers[END_REF]. However, combining more than two channels using this architecture has not been demonstrated yet, and seems dicult due to increasing system complexity and the large amount of B-integral accumulated over the long distances which the beams cover in the ampliers [START_REF] Brignon | Coherent Laser Beam Combining[END_REF]. Moreover, as such a setup necessarily implies a lled aperture conguration, it is not well suited for CBC of a very large number of beams.

Hänsch-Couillaud Interferometry

Hänsch-Couillaud interferometry relies on the use of polarizing beam splitters as combining elements. The beams need thus to be combined two by two in a lled aperture conguration.

Supposing that both beams have the same power and are orthogonally polarized, the polarization of the combined beam should be linear and turned by 45 • with respect to the one of the input beams. Deviations from this ideal polarization are attributed to phase uctuations between the two beams, which allows for the generation of an error signal for the active phase correction. This method has been proven to be well applicable to CBC of a moderate number of beams yielding very good combining eciencies of more than 90 % [START_REF] Müller | 1 kW 1 mJ eight-channel ultrafast ber laser[END_REF], but is not suitable for a use in tiled aperture conguration.

Frequency Tagging

Another commonly used method for phase matching is the so-called locking of optical coherence via single-detector electronic-frequency tagging (LOCSET). It consists in modulating the phase of each channel with a unique, small amplitude frequency dither. This allows the identication of every channel in the combined beam by demodulating the recorded power signal at the respective frequencies, and the subsequent deduction of each individual phase with respect to the phases of all other channels. The advantage of this technique is that the detection is performed with one single device, and combining eciencies higher than 90 % with residual phase errors as low as λ/40 RMS have been achieved by combining two femtosecond ber ampliers using LOCSET [START_REF] Daniault | Coherent beam combining of two femtosecond ber chirped-pulse ampliers[END_REF]. However, the number of beams which can be phase locked is limited by the number of available tagging frequencies, and is estimated at N ≈ 200. Hence, the scalability of systems using LOCSET remains limited, even if the algorithm itself seems well suited for the application in a tiled aperture conguration.

Techniques to overcome this scaling limitation have been proposed [START_REF] Ahn | Cascaded multi-dithering theory for coherent beam combining of multiplexed beam elements[END_REF][START_REF] Klenke | Sequential phase locking scheme for a lled aperture intensity coherent combination of beam arrays[END_REF], but require several combining stages and one detector per stage. Hence, they make void one of the main qualities of LOCSET and are best suited for lled aperture congurations.

Hill-Climbing Algorithm

The stochastic parallel gradient descent (SPGD) algorithm is a phase matching method which does not require the measurement of individual phases. Instead, small phase variations are randomly applied to all the channels, and the corresponding power of the combined beam is measured. Based on this result, the algorithm varies the phase osets such that the combined power is maximized.

This approach has the advantage of a relatively simple implementation which requires neither multiple detectors nor RF electronics, and was successfully implemented to phase lock moderate numbers of channels [START_REF] Ramirez | Coherent beam combining with an ultrafast multicore Yb-doped ber amplier[END_REF][START_REF] Zhou | Two-dimensional combination of eight ultrashort pulsed beams using a diractive optic pair[END_REF]. However, the convergence time of the algorithm increases linearly

with N [START_REF] Brignon | Coherent Laser Beam Combining[END_REF], so that it is not suitable for the real time phase matching of a very large number of beams. Nevertheless, SPGD algorithms can still be used to nd optimal congurations for static or slowly varying parameters, and will be applied in this context in our setup (cf. section 2.2.2).

Phase-Intensity Mapping

A recently proposed phase matching method for CBC in tiled aperture conguration relies on the so-called phase-intensity mapping (PIM), which converts relative phase dierences within the ber array into intensity dierences. Behind the PIM device, the measured intensity of each beam in the array is compared to the one which would be obtained if all the beams of the input array were perfectly in phase. From these results, an iterative algorithm calculates the phases that need to be applied to the input beam array in order to obtain the best possible match between the measured and the calculated intensities, which corresponds to a phasing of the beam array. Simulations have shown that phase locking of almost 500 beams should be possible with this technique [START_REF] Kabeya | Active coherent combining of laser beam arrays by means of phase-intensity mapping in an optimization loop[END_REF],

and an experimental implementation using 37 CW laser beams yielded a good residual phase error of λ/25 [START_REF] Kabeya | Ecient phase-locking of 37 ber ampliers by phase-intensity mapping in an optimization loop[END_REF].

Although this method seems to be suitable for the phase locking of femtosecond lasers, it could not be considered as phase matching technique for the XCAN laser system since it was developed only after the project had started.

Collective Interferometric Measurements

In collective interferometric phase measurements, each beam interferes with a common reference beam, which allows to determine the relative phase between each emitter and the reference.

This can be achieved either in the temporal or in the spatial domain. In the rst case, each beam is superposed with the same frequency shifted reference beam, and the resulting beat signals are recorded with photodiodes. The phase of each beam can then be deduced by demodulating the beat signals at the shifting frequency. This approach is somehow similar to the LOCSET technique, with the advantage that only one modulation frequency is needed, but with the drawback that it requires N photodiodes and that the phase is only measured with respect to the reference beam.

In the spatial domain, interference can be obtained by superposing each beam with one and the same tilted reference beam. The resulting fringes are then detected on a fast CCD camera, and the relative phase between each beam and the reference can be calculated based on the exact position of the fringes on the camera screen. This approach is thus only limited by the number of beams which can be imaged on the camera, and by the time needed for the acquisition and analysis of the camera images. Besides, it has been shown that by using exclusively readily available equipment, the phase matching of an array of about 10 000 beams at a bandwidth of 1 kHz is theoretically possible [START_REF] Antier | kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining[END_REF]. Moreover, the highest number of coherently combined channels in the femtosecond regime has been obtained using this method on passive bers [START_REF] Le Dortz | Highly scalable femtosecond coherent beam combining demonstrated with 19 bers[END_REF]. This shows that this technique is well suited for femtosecond CBC of very large ber arrays, and has therefore been chosen for the XCAN setup. More details on the exact implementation are given in section 2.2.2.

However, this approach has also some disadvantages. First of all, the use of a reference beam is cumbersome and somehow complicates the setup. Furthermore, since the phases of all beams are locked with respect to the reference beam, there is no guarantee that they are locked in phase with each other. Hence, an individual phase oset has to be applied to each beam in order to ensure constructive interference in the combined beam.

Self-Referenced Interferometric Measurement

In order to overcome the limitations of collective interferometric measurements, self-referenced methods relying on the spatial interference between adjacent beams have been proposed. The quadriwave lateral shearing interferometry technique uses a diraction grating to spatially split each beam into several replicas, which overlap with those of the adjacent beams after having propagated a certain distance. The phase of a single beam is then calculated based on all the interference patterns originating from this beam [START_REF] Bellanger | Collective phase measurement of an array of ber lasers by quadriwave lateral shearing interferometry for coherent beam combining[END_REF]. Hence, several fringe patterns need to be evaluated in order to obtain the phase of one beam, which makes this approach more timeconsuming than the one presented above, and therefore less well scalable to very large channel numbers. Nevertheless, this technique has been used in the CW regime in order to coherently combine a record number of 64 bers [START_REF] Bourderionnet | Collective coherent phase combining of 64 bers[END_REF]. A related approach suggests to omit the grating and use the interference fringes emerging in the free space propagation of the near eld beam array instead [START_REF] Prevost | Coherent beam combining of a narrow-linewidth long-pulse Er3+-doped multicore ber amplier[END_REF]. However, CBC using this technique has not been demonstrated yet.

In conclusion, self-referenced techniques are very promising, but not yet mature enough to assure an easy and reliable phase matching for very large ber arrays. Therefore, we decided to rely on the collective measurement presented before which uses a common reference beam.

Implementation in the XCAN Setup

In the previous section it has been concluded that a tiled aperture arrangement in combination with an interferometric phase measurement technique are best suited for the coherent combination of a very large number of channels. These congurations have thus been implemented in the XCAN setup, and will be presented in more detail in this section.

Tiled Aperture Conguration

In a tiled aperture conguration, highest combining eciencies can be obtained for highest ll factors of the near eld aperture. In consequence, the beam width has to be close to the array pitch, which can be achieved by collimating the beams after having them diverge over a certain distance. This has the side eect that each beam is clipped by its respective collimating lens, so that a certain part of the beam is not correctly collimated and therefore lost for the subsequent combining process, as illustrated in g. 2.3a.

Such a conguration results in a side-by-side arrangement of circular beams, so that a hexagonal ber array leads to the highest aperture lling [START_REF] Brignon | Coherent Laser Beam Combining[END_REF]. Such an array is shown in g. (a) A part of the beam is not collimated by the designated lens, and therefore lost for the subsequent combining process. Mathematically, the resulting near eld can be described using four variables: a Dirac comb X p describes the position of the beam centers on a hexagonal grid with pitch p. In order to take into account the nite size of the beam array, it is multiplied by a function Π A describing the geometrical shape of the global aperture. Each individual beam can be expressed via a Gaussian envelope of width w, E w (x, y) = exp(-x 2 +y 2 w 2 ), which is multiplied with the aperture of the collimating lens Π L . The near eld E NF as depicted in g. 2.4a can then be expressed by the For simplicity, the lens aperture ΠL has not been included in the scheme. convolution of these two products, so that

E NF = (X p × Π A ) ⊗ (E w × Π L ).
(2.

2)

The far eld E FF is given by the Fourier transform of E NF . According to the convolution theorem, it can thus be expressed via the Fourier transforms of the individual components of the near eld. Both the Dirac comb X and the Gaussian envelope E transform into themselves, but exhibit a modied period p = λf /p and width w = λf /(πw), respectively, where f is the focal length of the lens used to perform the Fourier transform experimentally. The Fourier transform of the global aperture Π A has to be calculated numerically. As the hexagonal array is composed of circular beams, the aperture has the shape of a hexagon with rounded corners. For a seven ber beam array, it can therefore be well approximated by a circular aperture (cf. the green and blue lines in g. 2.3b), whose Fourier transform is given by the Airy function Ai. For larger beam numbers, the hexagonal features of the aperture become more pronounced, and F(Π A ) converges towards the Fourier transform of a hexagon, which is shown in g. 2.4c. It should be noted that the Fourier transforms of both apertures exhibit signicant side lobes around the main lobe.

Furthermore, the width of this main lobe is inversely proportional to the size of the aperture itself.

Finally, the Fourier transform of the lens aperture Π L depends on the geometrical shape of the collimating lens. In our setup, we use a hexagonal array of hexagonal microlenses, so that F(Π L )

is given by the Fourier transform of a hexagon.

The far eld can then be written as

E FF = (X p ⊗ F(Π A )) × (E w ⊗ F(Π L )).
(2.

3)

The convolution of the new Dirac comb with F(Π A ) yields a hexagonal distribution of peaks with pitch p , whose width depends on the size of the beam array. Their side lobes stretch out in between the peaks. This basic pattern is modulated by the convolution of F(Π L ) with the Gaussian envelope E w . If the aperture of the collimating lens is suciently large compared to the diameter of the beams, the clipping is negligible and the envelope reduces to E w . In contrast, if the beams are noticeably clipped, the convolution has to be considered, and due to the side lobes of F(Π L ) the resulting envelope exhibits side lobes as well.

The multiplication of the basic pattern with this envelope leads to a far eld with one strong central lobe, whereas the other peaks are reduced in intensity, as shown in g. 2.4b. Since for a given Π L the width of the envelope scales as w ∝ 1/w, more side lobes will appear if the beam width in the near eld is reduced, resulting in a decrease in combining eciency. Conversely, larger beams in the near eld result in a better suppression of the side lobes in the far eld and therefore a higher combining eciency. Hence, larger w in the near eld should be preferred. However, larger beams also result in an enhanced clipping at the collimating lenses, which leads not only to power losses directly at the lenses, but also to the appearance of side lobes of the envelope at the edges of the far eld, so that the combining eciency gets reduced. In consequence, the optimum beam size is a compromise between aperture lling and microlens clipping. Simulations used to calculate the best beam size will be presented in the next chapter.

Moreover, the combining eciency is very sensitive to misalignments in the hexagonal beam array. Previous studies have investigated this eect for a rectangular beam array in terms of decrease of the Strehl ratio, which compares the on-axis power of the simulated system with an ideal one [START_REF] Bellanger | Design of a ber-collimated array for beam combining[END_REF]. In the following chapter, these simulations will be adapted to the hexagonal array used in our setup, in order to nd the maximum error tolerances acceptable on our ber array.

Finally, the tiled aperture geometry has an impact on the choice of the specic ber amplier to be used. Since high combining eciencies can only be obtained with a large aperture lling, empty space between the beams in the near eld should be avoided. Hence, beam collimation with individual lenses and corresponding mounts is suboptimal, and the use of microlens arrays is preferred. Their typical pitches in the range of 100 µm to a few millimeters have to be applied to the ber array as well. The use of unbendable rod type bers becomes therefore impractical, and we use the DC-200/40-PZ-Yb LMA ber presented in section 1.2.2 instead. This guarantees state of the art performances in terms of average and peak power of a single emitter, while also keeping the exibility necessary to insert the bers in an array with a pitch on the order of a few millimeters. However, as this ber is polarizing, our spatial CBC setup cannot be combined with the polarization-dependent DPA technique. Hence, no temporal beam combining scheme was implemented.

Interferometric Phase Measurement

As explained in section 2.1.3, a collective interferometric phase measurement consists in the superposition of the beam array with a reference beam. In our setup, we work with a superposition in the spatial domain, so that a single fast camera is used as common detector for all beams. A small power fraction of the near eld is then imaged on the camera screen, and superposed with a plane wave reference beam which is aligned with a small angle α to the camera, as depicted in g. 2.5a. This results in N fringe patterns on the camera, as shown in g. 2.5b, and allows to correct both the relative delay ϕ 1 and the phase osets ϕ 0 , as will be explained in the following.

(a) (b)

Delay Adjustment

Intuitively it is easy to understand that the pulses of the dierent ampliers must arrive at the same time if they shall recombine coherently. Hence, in a rst step, the relative delay between the pulses needs to be adjusted. This can be done by optimizing the contrast of the interference fringes on the camera, as illustrated in g. 2.6 for chirped pulses. If the delay between the pulses is superior to their coherence time, no interference is possible and the camera images the Gaussian beams of the near eld, as shown in g. 2.6a. When the delay is inferior to the coherence time, but not fully compensated, interference is possible only for the parts of the pulses which overlap both in the time and frequency domain, as shown in g. 2.6b. This results in an interference pattern with a bad contrast, since the non-interfering parts of the pulses add an incoherent background.

Finally, when the delay between the pulses is well compensated, the whole temporal prole of the pulse contributes to the interference, and the contrast is maximized, as illustrated in g. 2.6c.

This approach is appealing due to its simplicity, but it entails several issues.

First, this method is not very precise. It has been seen experimentally that relative delays inferior to ∼ 60 fs, which is more than 25 % of the pulse duration, do not visibly deteriorate the fringe contrast, but have a negative impact on the combining eciency. Hence, in the experiments with a seven beam array, the delays were rst adjusted roughly based on the contrast of the interference fringes. Afterwards, the ne tuning was performed by measuring the combining eciency in the far eld and manually adjusting the delays accordingly. However, this is only possible for small beam arrays, where the adjustment of the parameters of one beam has a noticeable impact on the overall combining eciency. This is not the case for N = 61 beams, where a single beam represents only 1.6 % of the overall beam array. Hence, the changes in combining eciency due to the relative delay of a single beam are not necessarily distinguishable from those due to noise, which makes a one-by-one alignment impossible.

Second, for very large beam arrays, the tilt angle of the reference beam can become problematic.

As the wavefronts are tilted with respect to the camera, the arrival time τ of the reference pulse at the screen varies linearly with its horizontal position. Since the delays of the beam array are adjusted with respect to this arrival time, they will also vary linearly with position, as illustrated in g. 2.7. For instance, if one considers an interference pattern with ve fringes per beam, the phase dierence between the extreme edges of two outward beams is 15 λ for a seven ber array.

This translates into a relative delay of 50 fs, which is on the order of magnitude of the precision in the adjustment of the delays based on the fringe contrast. Hence, in the seven ber array the tilt of the reference beam will have no further impact on the delay alignment, and therefore on the combining eciency.

In contrast, if one considers a 61 beam array under the same conditions, the delay between the two outmost beams becomes 150 fs, which is about two thirds of their coherence time. As this is detrimental to the combining eciency, a way to compensate this linear delay dependence has to be found.

Third, this delay adjustment method works well only if all the channels have the same second order spectral phase as the reference beam. If this is not the case, for instance because the reference beam is not power amplied and therefore not subject to nonlinear phases, interference is only possible between a certain part of the temporal envelope of the stronger chirped signal pulse and the temporally unchanged reference beam, as shown in g. 2.6d. Hence, there is always an incoherent background mixed to the interference pattern, and the contrast remains moderate for a wide range of delays. In such a case it is therefore not possible to compensate the relative delays among the pulses solely by considering their contrast on the camera screen.

In order to overcome these three inconveniences, an SPGD algorithm is intended to be implemented in the 61 ber setup for the precise optimization of the relative delays.

Phase Oset Correction

Once the delays are adjusted such that for all beams there are interference fringes visible on the camera, these can be used to measure the relative phase osets between the signal beams and the reference one. This is done using an algorithm as described in [START_REF] Antier | kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining[END_REF]. It calculates the relative phases based on the position of the interference fringes on the camera. If a beam is perfectly in phase with the reference beam, the rst maximum of the interference pattern will be located exactly at the center of the Gaussian envelope. Deviations from this position are due to phase dierences, and can therefore be used to determine these latter ones. This could readily be done using Fourier transforms. However, as these calculations are relatively time-consuming and the feedback algorithm needs to run in the kilohertz regime, an approach using correlation calculations is implemented instead.

This method is based on the calculation of a library of reference interference patterns R i (ϕ i ), each corresponding to a xed phase oset ϕ i . Subsequently, the correlation coecient between the measured fringe pattern S and every R i is calculated in real time. This is depicted in g. 2.8

for three dierent R i . The measured phase corresponds then to the phase used to calculate the library pattern R i which yields the highest correlation coecient. The precision of this phase measurement is given by the number N P of reference patterns in the library, since the phase is probed in intervals of 2π/N P .

As mentioned in section 2.1.3, this method locks the phases with respect to the reference beam, which means that the phase locking does not automatically guarantee constructive interference in the far eld. Hence, it is necessary to dene a set point for each beam to which its phase oset shall be locked. Although an optimization of these osets with respect to the combining eciency would have been possible by hand in the seven ber setup, better results have been obtained using a dedicated SPGD algorithm.

Conclusion

In the rst part of this chapter, dierent CBC architectures and phase matching techniques have been presented. It has been concluded that for a CBC setup which should be scalable to more than 1 000 channels, a tiled aperture conguration in combination with an interferometric phase measurement is the best choice.

In the second part, these two approaches have been presented in more detail. In tiled aperture conguration, highest combining eciencies are obtained with a hexagonal ber array. Nevertheless, misalignments in this ber array can lead to decreases of the combining eciency. Besides, the interferometric phase measurement allows for an easy matching of both phases and delays of the beams, but especially the latter suers from imprecisions in the adjustment.

In order to estimate the maximum imprecisions tolerable in our setup, the losses in combining eciency due to various alignment errors both in the spatial and spectral domain have been calculated numerically. The results of these simulations are presented in the next chapter.

As mentioned in section 2.2.1, an accurate positioning of the bers in a hexagonal array is essential for a high eciency CBC in tiled aperture conguration. Moreover, also the spectral properties of the beams need to be very well matched. In this section, the maximum error tolerances for our specic setup will be determined based on numerical simulations.

General Setup

We suppose a hexagonal array of Gaussian beams, each of which gets both collimated and truncated by a hexagonal microlens. The far eld is found by an optical Fourier transform of this composite near eld, and the combined beam is given by the central lobe of the far eld. Spatial ltering with a circular aperture allows to select the central part of the far eld, which is then again Fourier transformed in order to access the recovered near eld. A schematic of this setup is shown in g. The output of each individual ber is supposed to be a Gaussian beam of the form

E i (x, y) = E 0,i exp - (x -x f,i ) 2 + (y -y f,i ) 2 w 2 0,i , (3.1)
where E 0 is the amplitude of the beam, w 0 its waist at 1/e 2 , and (x f , y f ) are the coordinates of the beam center. Hence, the total output power is given by P tot = |E i | 2 dx dy.

Each individual beam gets both collimated and truncated by a hexagonal microlens centered around (x f,i , y f,i ) and described by the function

Π L,i (x, y) =    1, (x, y) inside the hexagon 0, else, (3.2)
so that the truncated beam is given by E t,i = E i × Π L,i (x, y). It should be noted that due to this truncation, there is no overlap between the individual E t,i . The sum of all the truncated beams then yields the near eld Due to the microlens clipping, the overall power in this near eld P NF = |E NF | 2 dx dy is smaller than the total power P tot . In order to quantify this power loss it is convenient to dene the microlens transmission t µ-lens = P NF P tot .

E NF = E t,i = E 0,i exp - (x -x f,i ) 2 + (y -y f,i ) 2 w 2 0,i × Π L,i (x, y),
(3.4)

Although there is no direct dependence in the formula, it is intuitively clear that the microlens transmission depends on the diameter 2w 0 of the beams at the microlens array with respect to the short diagonal d of the microlens. In the following, this ratio will be called microlens ll factor τ r = 2w 0 /d.

In a next step, the near eld gets Fourier transformed. This generates the far eld E FF = F(E NF ), which consists of a very powerful main lobe, six secondary lobes arising due to the threefold sym-metry of the near eld, and several other, fainter lobes, as explained in section 2.2.1 and shown in g. 3.2b. In these and all following 2D images of a 61 ber far eld, the peak intensity has been normalized to one and the maximum of the color scale has been set to 0.25, so that the side lobes become clearly visible. Figure 3.2c displays a vertical lineout of the far eld, showing the relative amplitude of some of those fainter lobes close to the main lobe. The radius has been normalized

with respect to the position of the secondary lobes, so that the rst minimum next to the main lobe is located at r ml = 0.13.

Then, the far eld gets spatially ltered by a circular aperture. This can be described by multiplication of E FF with a truncating function, so that one nds for the remaining truncated eld

E tr (r) = E FF × disc r (x, y) with disc r (x, y) =    1, x 2 + y 2 ≤ r 0, x 2 + y 2 > r, (3.5)
where r is the radius of the aperture. This radius determines how much power is transmitted into the truncated eld, which can be described by the truncation eciency η tr (r) = P tr (r) P FF ,

where P tr is the power in the truncated eld, given by P tr (r) = |E tr (r)| 2 dx dy, and However, there is also the shape of the recovered near eld E out which needs to be considered.

P FF = |E FF | 2 dx
Given by the far eld of the truncated eld E tr , it can be found by a second Fourier transform, so that E out = F(E tr ). Since the spatial ltering removes the higher spatial frequencies of E FF , the recovered near eld becomes more homogeneous and Gaussian-like for smaller aperture radiuses r.

Indeed, if the ltering is performed at the rst extinction of the far eld (r = r ml ), a very clean and smooth recovered near eld can be obtained (cf. left column of g. 3.3b). For a ltering at the second extinction (r = 0.25 for N = 61 beams), the recovered near eld has a donut-like shape, as can be seen in the middle column of g. 3.3b. For larger ltering radii, E out becomes more structured. Finally, for a ltering which only removes the secondary lobes, it recovers the hexagonal contour of the initial ber array (r = 0.9, right column of g. 3.3b). Considering this behavior, the truncating aperture radius has to be chosen as a function of transmitted power and desired output beam shape.

Typically, the combined beam is considered to be the main lobe of the far eld. In order to calculate the combining eciency η as dened in eq. (2.1) (section 2.1), it is thus necessary to know P comb = P ml , which can be obtained by truncating the far eld at the position of its rst minimum r ml . Hence, E ml = E FF × disc r ml , and consequently P ml = |E ml | 2 dx dy. Thus, η = P comb P tot = P ml P tot .

(3.7)

It should be noted that η = η tr (r ml ), since η takes into account the microlens transmission, whereas η tr (r ml ) does not. Hence, η tr (r ml ) corresponds to the eciency measured in the far eld,

η FF = η tr (r ml ) = P ml P FF , (3.8)
and using eq. (3.4) and the identity P FF = P NF it follows that η = η FF × t µ-lens .

(3.9)

The next sections will describe the impact of spatial misalignments and spectral disparities on the combining eciency η. The maximum tolerable error on the alignment or the spectral properties was dened as the value corresponding to an eciency loss of one percentage point.

As far as not otherwise mentioned, the following parameters have been used in the simulations:

• central wavelength λ 0 = 1032 nm

• ber pitch and microlens short diagonal d = 3.2 mm

• microlens pitch p = 3.2 mm

• microlens ll factor τ r = 0.9

• microlens focal length f = 50.

mm

It should be noted that the microlens focal length has not been chosen arbitrarily, but has been calculated based on the array pitch p, the microlens ll factor τ r and the beam divergence θ at the ber output, so that f = τ r p/(2θ). The beam divergence is specic to the ber amplier in use and is given as a function of the waist and M 2 of the beam. In our case, we have w 0 = 15 µm and M 2 = 1.3, so that it follows θ = M 2 /w 0 = 0.0285 rad and consequently f = 50.5 mm. Furthermore, the beam divergence can be used to calculate the Rayleigh range of the uncollimated beam, which equals z R = w 0 /θ = 0.68 mm.

Microlens Fill Factor

In a rst step, the combination eciency η has been calculated as a function of the microlens ll factor τ r . As explained in section 2.2.1, for small τ r the microlens clipping is nearly zero, but the small beam diameter in the near eld gives rise to numerous and powerful side lobes in the far eld (see g. 3.4a (top) for an example with N = 7 beams). This decreases the combining eciency η. In contrast, for large τ r , the side lobes in the far eld are less prominent and there is thus a higher amount of power in the main lobe, as shown in g. 3.4a (bottom). However, there is also more power lost due to microlens clipping. In order to nd the best compromise between microlens clipping and power contained in side lobes, η has been calculated as a function

Spatial Alignment Errors

In this section, the inuence of errors in the alignment of the ber and microlens array on the coherent combination eciency is investigated. For every misalignment, a schematic of the simulated situation and a curve showing the drop in eciency for increasing misalignments will be presented. If applicable, the maximum tolerable alignment errors will be considered relative to the absolute value of the corresponding parameter of the ber array.

Misalignments within the Fiber Array

In this part, the impact of misalignments in the ber array is simulated. It is supposed that the alignment of each ber is carried out as precise as possible, so that errors are randomly distributed around zero, with higher probabilities for errors closer to zero. Hence for every misalignment m the errors were simulated by drawing N random variables from a Gaussian distribution with standard deviation m and applying them as errors to the N beams. This leads to irregular intensity distributions in the far eld, as exemplarily shown in g. 3.5a for a far eld calculated with 61 beams and errors on the lateral alignment of ∆x = 5 µm.

As every set of errors leads to slightly dierent far elds and corresponding combining eciencies, 50 runs were executed for each m. The combining eciency is then given as the mean of the 50 individual results, whereas the error bars represent their standard deviation. They indicate thus not the standard error of the mean, but rather the uctuations of the simulated eciencies.

Since the XCAN project aims at the coherent combining of 61 bers, but the experiments presented in this thesis have been conducted with only seven channels, simulations have been run using both of these numbers. seven random errors is shown, together with the Gaussian probability distribution they are drawn from (red line). Another way to describe this eect is to consider that in the seven ber array, one outlying variable will impact one ber out of seven, that is, 15 % of the total ber number.

In the case of 61 bers, this reduces to 1.6 %. From another point of view one could say that in the 61 ber array, there is a sort of averaging among the errors which will smooth out the impact of individual values, and which is much less important in the case of seven bers.

It should be emphasized that the errors indicated on the x-axis of the eciency curves are the standard deviations of the distribution from which the N errors have been drawn. This means on the one hand that about 32 % of the errors will be higher than the indicated value. For instance, in the example given in g. 3.5c, 23 of the 61 values are larger than the indicated error of 1 µm, with three of them having more than 2.5 times its value. On the other hand, when it comes to an estimation of eciency losses based on errors measured on the experimental setup, care has to be taken not to consider the highest measured peak-to-valley error, but rather the standard deviation of all the experimental values.

Another option consists of course in simulating the far eld and thereby the combining eciency by directly applying the measured errors to the respective beams in the model. This is interesting especially in the seven ber setup, since in this case the combining eciency can vary considerably for dierent sets of errors, as has been explained above and is also reected in the large error bars in g. 3.5b. Hence, calculating the combining eciency directly with the measured errors leads to a more precise result than an estimation based on their standard deviation. In contrast, such an estimation might be adequate for a 61 ber setup, since the simulated combining eciencies depend much less on the actual distribution of the errors among the channels.

Especially for large errors, the central lobe of the far eld may become shifted, or the minima next to it may blur out. It was therefore decided that the combining eciency would always be calculated by considering the amount of power which is in the area of the main lobe of an ideal far eld. As the far eld with errors becomes more arbitrary for larger errors, the amount of power in the main lobe area varies more strongly, giving rise to larger uctuations in the calculated eciencies. This explains the larger standard deviations found for larger errors (cf. g. 3.5b), even if the high number of runs per error guarantees a reliable result for the mean eciency η.

It should be noted that for all spatial errors, the combination of the beams remains coherent.

This means that the combining eciency can even drop to zero, if the alignment errors are such that they result in a destructive interference in the area of the main lobe.

In the following, dierent misalignments within the ber array will be presented independently one from another.

Fiber Tilt

If a ber is tilted by an angle θ x with respect to the optical axis, the collimated beam will still be parallel to the optical axis, but shifted by a distance ∆x, as shown in g. 3.6a. The nal eld reads then

E = E 0 exp - (x -∆x) 2 + (y -∆y) 2 w 2 0 , (3.10)
where ∆x is given by ∆x = f tan θ x . This shifting of the beam impacts the combining eciency in two ways: on the one hand, the microlens clipping is increased, and on the other hand the far eld eciency is reduced due to the irregularities in the hexagonal beam array.

The corresponding eciency curves are shown in g. 3.6b. It can be seen that the overall losses in combining eciency are dominated by the reduced microlens transmission. In order to limit these overall eciency losses to one percentage point, the tilts on the bers should not exceed 2.5 mrad. This corresponds to 8.9 % of the ber numerical aperture or the divergence of the emitted beam.

Fiber Displacement

If a ber is placed at a distance ∆x from the optical axis, the beam collimated by the microlens will be tilted by an angle θ x (see g. 3.7a). This corresponds to a tilt of the wavefronts and can be described in the near eld by

E = E 0 exp - (x -∆x) 2 + (y -∆y) 2 w 2 0 exp (-ik(xθ x + yθ y )) , (3.11)
where the approximation θ x ≈ tan θ x = ∆x f has been used. This latter one is justied since the ∆x are typically in the order of a few micrometers, whereas f equals 50.5 mm. Given the resulting curve shown in g. 3.7b it is apparent that errors on ber displacement should be limited to about one micrometer. This corresponds to only 0.03 % of the ber pitch and is therefore one of the f f ∆x

θ x θ x (

Focal Errors

If the ber is placed on the optical axis at a distance ∆f from the focal point of the microlens, the beam will not be collimated anymore, but rather focused at a distance s from the microlens, as shown in g. 3.8a and 3.8b for positive and negative ∆f , respectively. These distances are related by the thin lens equation * , given by [START_REF] Saleh | Fundamentals of Photonics[END_REF]. In our case, zR ≈ 0.7 mm and f = 50.5 mm, meaning that zR/f can be neglected. Writing furthermore s -f = ∆f , the previous equation simplies to s /f -1 = f /∆f , which is equivalent to the expression given in eq. (3.12).

1 f = 1 s + 1 s = 1 f + ∆f + 1 s ⇔ s = f (f + ∆f ) f + ∆f -f = f 2 ∆f + f.
(s/f -1) 2 +(z R /f ) 2
Such an error on the focal distance results in spherical wavefronts with radius r converging towards the point z = s . In the plane of the microlenses, the resulting curved wavefront can be described by

r 2 = x 2 + z 2 ⇔ z(x, ∆f ) =    r + √ r 2 -x 2 , ∆f < 0 r - √ r 2 -x 2 , ∆f > 0, (3.13)
where r = s = f 2 ∆f + f . This curved wavefront translates into a phase term, so that the nal expression for the eld of the beams reads

E = E 0 exp - x 2 + y 2 w 2 0 exp (-ikz(x, y, ∆f )) , (3.14)
and the resulting drop in eciency is shown in g. 3.8c. As can be deduced from the curve, errors in the focal distance should not exceed 0.08 mm, which is 0.15 % of the microlens focal length and 12 % of the Rayleigh range of the beam.

One might argue that an error on the focal distance changes the beam size on the microlens array, and thereby the microlens transmission. This is certainly true, however, it can be shown * that the relative change in beam radius caused by the maximum focal errors applied in g. 3.8c

does not exceed 1 %, which translates directly to a relative change of the microlens ll factor of 1 %. This implies that the losses due to an altered microlens ll factor are negligible, especially with respect to the eciency drop already caused by the spatial phase mismatches.

Output Power

There might be dierences in the output power of the dierent ber ampliers, resulting in power dierences in the near eld as illustrated in g. 3.9a.

(a) Example of a near eld with ∆P0 = 0.5.

(b) Eciency losses due to disparities in output power. In order to model these disparities, the power amplitude P 0 was normalized to one, and errors ∆P 0 have been added. However, especially for large ∆P 0 there is a certain risk that a negative total amplitude P 0 + ∆P 0 < 0 is found. Since such an amplitude has no physical meaning, it was set to zero whenever this case occurred. Finally, the amplitude of the electric eld E 0 is related to the power amplitude by E 0 = √ P 0 + ∆P 0 , so that one nds for the Gaussian beam

E =    √ P 0 + ∆P 0 exp -x 2 +y 2 w 2 0 , P 0 + ∆P 0 > 0 0, else. (3.15)
It should be noted that after applying these errors the total power P tot might be higher than the one of the ideal case with E 0 = 1. The overall power in the main lobe of the far eld P ml might therefore also be higher than in the ideal case. However, since the combining eciency is given as the ratio of those two, the extra power will have no impact on η.

The resulting eciency curve is shown in g. 3.9b and indicates that power dierences of up to 25 % from the nominal value are acceptable.

Polarization

The combined beam is polarized along p, which is the mean of the polarizations of all individual beams. Hence, if a a beam is polarized along a vector p that is turned by an angle θ with respect to the mean polarization direction, only the projection of p on p can be considered for the coherent combination, as illustrated in g. main dierence between these two cases is that for mismatched output powers, the total power emitted by the ber ampliers is transmitted to the far eld, and losses in combining eciency are solely due to power disparities. In contrast, in the case of polarization mismatches, a part of the emitted power is lost in the orthogonal polarization direction. Hence, in this case the decrease in combining eciency is due to both an overall power loss and power disparities in the remaining near eld.

The corresponding eciency curve is shown in g. 3.10b and suggests that polarization disparities should be limited to ∆θ = 7 • .

Beam Ellipticity

Typically, the output mode of the PCF ber used in the experiments is not exactly circular.

Therefore it was studied at which point a beam ellipticity might impact the combining eciency.

An elliptic beam can be modeled by decomposing the beam waist w 0 into its components in xand y-direction and by multiplying them with an ellipticity constant , so that one nds w x = w 0 and w y = w 0 / . In this way the area of the beam A = πw x w y = πw 2 0 and thus its relative power stay constant, and the expression for the Gaussian beam reads

E = E 0 exp - x 2 w 2 x - y 2 w 2 y . (3.17)
In a rst step it was supposed that all beams exhibit the same ellipticity . parallel to the long diagonal of the hexagon.

Furthermore, due to the elliptic beam shape in the near eld, the envelope of the far eld becomes asymmetric as well, and somehow squeezed in the direction of the long diagonal of the ellipse. This results in a suppression of the side lobes of the far eld in this direction, as can be seen in g. 3.11a (bottom left and right). Due to this asymmetric behavior, the related losses in combining eciency are less important in the case where the long diagonal of the beam ellipse is oriented parallel to the long diagonal of the hexagonal aperture of the microlens, which corresponds to > 0. Hence, in the experimental setup, the bers should be aligned accordingly.

In a next step, variations in the ellipticity were applied, so that w x = ( + ∆ )w 0 . In order to stay close to the experimental situation, the measured mean ellipticity = 1.07 was added to each beam. The corresponding eciency curve is shown in g. 3.11c and indicates that dierences in ellipticity of up to ∆ = 0.12 won't strongly impact the combination eciency.

Microlens Array Misalignments

In this section, the impact of a misalignment of the microlens array on the combining eciency will be studied. Since there is only one microlens array, each error needs to be applied only once, so that there is no need for a statistic approach as presented in the previous section. This is also why there are no error bars on the eciency curves.

For each error one or two example far elds will be presented. However, those are for illustrative purposes only, as the errors applied in each case largely exceed the acceptable range. They were chosen in order to show the general eect of such an error on the far eld. As explained in section 3.1 (page 44), the peak intensity of the far elds has been normalized to one and the color axis adjusted such that the side lobes become clearly visible.

Besides, it should be noted that any misalignment of the microlens array results in a change in the position and/or the shape of the truncating mask Π L (x, y) which gives rise to an enhanced microlens clipping. However, it has be seen in the section before that in the case of a displacement between each ber and its respective microlens, already random errors of some micrometers cause a considerable drop in combining eciency. In this section, the errors will be common to all microlenses. Nevertheless, it will be seen that the tolerable errors on the microlens array positioning remain on the order of several micrometers. Hence, eciency losses caused by an enhanced microlens clipping due to a microlens array displacement can be neglected. In contrast, if the clipping is due to a tilt of the microlens array, its inuence on the combining eciency will have to be discussed.

Microlens Displacement

A displacement of the microlens array in xor y-direction is described by the same geometry as a displacement of a ber, but with all of the bers being subject to one and the same displacement (see g. 3.12a). Thus, the mathematical description does not change with respect to section 3. displacement should not exceed 1.4 µm in either direction, which is 0.04 % of the array pitch.

As consequence of such a displacement, the envelope of the far eld gets shifted in xor ydirection, as shown in g. 3.12c and 3.12d for a displacement of 4 µm in xand y-direction, respectively.

Microlens Rotation

A rotation of the microlens array corresponds to a displacement of the bers in xand y-direction, where x f if the position of the ber. Thus, the expression for the Gaussian beam reads

E = E 0 exp - (x -∆x rot ) 2 + (y -∆y rot ) 2 w 2 0 exp (-ik(xθ xrot + yθ yrot )) (3.20)
where θ xrot = ∆xrot f . This leads to a sort of smearing of the side lobes, as if the far eld was spun around the central lobe, as shown in g. 3.13c and 3.13d for a rotation of 1 mrad and 1.7 mrad, respectively.

As can be seen in g. 3.13a, the displacement between the ber and the corresponding microlens gets larger with increasing distance from the center of the array. It is thus not surprising that the impact of a rotation of the microlens array is much more violent for a 61 ber array than for one containing only seven bers. The two corresponding curves are shown in g. 3.13b in black and gray, respectively. It follows that for 61 bers the rotation has to be as precise as 0.16 mrad, whereas for the downscaled version of seven bers, 0.5 mrad are still acceptable.

Microlens Pitch

Just as for the microlens rotation, a dierence between the pitch p of the microlens array and the pitch d of the ber array results in a displacement of each ber which respect to the center of the corresponding microlens. If the dierence in pitch is given by ∆p = p -d, then the displacement can be described by ∆p tot = x l p ∆p, where x l is the center of the microlens. This results in a wavefront tilted by an angle θ x l = ∆ptot f = x l p ∆p f , as shown by the dotted lines in g. 3.14a. If the far eld is modeled with these tilted wavefronts, the impact of a pitch dierence on the combining eciency is very high, as shown in g. 3.14b (top), yielding a far eld as shown in g. 3

.14c (top).

However, if the phase osets between the individual beams are well adjusted, one obtains an overall segmented wavefront as shown by the dashed lines in g. 3.14a, which can be approximated by a parabola (solid line). Such a wavefront can be easily compensated in an experiment by adjusting the distances of the subsequent imaging lenses.

A mathematical description can be derived by considering that the slope θ x l of the wavefronts increases linearly with x l . Hence, the parabolic wavefront z can be calculated by generalization of θ x l to all x and subsequent integration,

z = θ x dx = ∆p pf x dx = 1 2 ∆p pf x 2 .
(3.21)

The individual wavefront segments z l are then given by the tangents on this parabola in the points x l , so that

z l = ∆p pf x l (x - 1 2
x l ). They are by more than a factor of ten less sensitive than in the case without compensation, and independent of the number of bers. This latter one is due to the fact that after the compensation of the overall second order phase, the phase of each individual beam does not depend anymore on the distance of the beam from the microlens center. Therefore, there is no reason that beams closer to the edges of the hexagon are impacted stronger by the dierence in pitch. Nevertheless, this dierence should be kept below 9.5 µm in order to limit the eciency losses to one percentage point. Although this implies that the outmost bers are shifted by almost 40 µm with respect to their microlenses, the eciency losses due to an enhanced microlens clipping are still largely inferior to those induced by the parabolic wavefronts of the beams.

The maximum tolerable pitch dierence of 9.5 µm corresponds to 0.3 % of the microlens array pitch and highlights again the high precision which is necessary on both the ber and the microlens array, so that a perfect match between the two can be achieved.

Microlens Focal Distance

An error in the focal distance of the microlens array is described by the same equations as an error on a single ber, only that all bers are subject to the same error (see g.

Microlens Tilt

A schematic of a tilted microlens array is shown in g. 3.16a. It can be described as a combination of a displacement between the bers and their respective lenses, an error on the focal distances, and a tilt of each single microlens. While this latter one does not impact the phase of the beam, it has to be taken into account that the beams are not clipped by the original hexagonal shape of the lenses Π L (x), but rather by its projection on the x-axis, given by Π Lx (x) = Π L (x) cos ϑ x , where ϑ x is the tilt angle. Furthermore, each lens gets shifted by a distance ∆x = x f (1 -cos ϑ x )

with respect to the ber position, which also enhances the microlens clipping. However, even for the largest tilt angles ϑ x used in the simualations, the calculated microlens transmissions remain almost constant, which implies that the losses in combining eciency are largely dominated by the displacements and the altered focal distances between the bers and their respective microlenses.

Those focal distances ∆f depend on the position of the ber and can be described by ∆f = x f tan ϑ x .

(3.24)

Inserting this term into eq. (3.13) yields the parabolic phase z(x, y, ∆f ) which has to be added to the Gaussian beam. Furthermore, the wavefront tilt angle θ x induced by the relative displacement between bers and microlenses can be deduced from geometrical considerations and is given by

θ x = ∆x f + ∆f = x f (1 -cos ϑ x ) f + ∆f , (3.25)
with the ∆f derived above. Finally, the expression for a beam including all phase terms reads

E = E 0 exp - x 2 + y 2 w 2 0 exp (-ik(xθ x + yθ y + z(x, y, ∆f ))) . (3.26)
A tilt on the microlenses in one direction will somehow squeeze the far eld in this direction, as is shown in g. 3.16c and 3.16d for a tilt of 3 • in xand y-direction, respectively. Just as for the microlens displacement, there are nearly no dierences between the eciency losses related to a tilt along the one or the other axis.

However, as for the rotation of the microlens array, the tilt also becomes more important with the distance from the array center, and is thus more critical for larger ber arrays. This can also be seen in the curve given in g. 3.16b which implies that for a seven ber array, tilts of up to 2 • are acceptable, whereas they should be limited to 1 • for the nal setup of 61 bers.

Gap Between Lenses

Due to constraints in the manufacturing of the microlens array, there is a small gap g between the individual lenses where no light is collected. This results in an enhanced beam clipping at the microlens array, so that the microlens transmission gets reduced and the beam diameter in the near eld decreases. This leads to an increase of power both in the side lobes of the far eld and at its borders. An example near eld with a gap of g = 800 µm between the lenses is shown in g. 3.17a, and leads to a far eld with strongly increased side lobes as depicted in g. 3.17b. (c) Eciency losses due to a gap between the microlenses.

Figure 3.17: Gap between the microlenses.

From this eciency curve it can be concluded that the gap should not exceed 30 µm, which corresponds to 0.9 % of the microlens pitch. However, according to the datasheet of the microlens array used in the experiments, a gap between 100 µm and 200 µm has to be expected, which would lead to eciency losses between 3.5 % and 7.5 %.

Summary

In this section, the inuence of various errors in the spatial alignment of the ber and the microlens array on the coherent combining eciency have been modeled. An overview of the maximum error tolerances is given in table 3.1. As far as comparable, these results are in good agreement with previously published error tolerances for coherent beam combining in lled aperture conguration [START_REF] Goodno | Perturbative analysis of coherent combining eciency with mismatched lasers[END_REF].

Furthermore, a statement about the criticality of the misalignments is given, that is, how dicult it will be to meet the stated precisions in the experimental setup. The ber and microlens mounts which have been implemented in the nal laser system in order to comply with these spatial requirements will be presented in detail in section 4.3.

The most critical error is a relative displacement between the bers and their corresponding microlenses, which has to be controlled on a micrometer scale. Mismatches on the tilts, and focal distances and polarizations of the individual beams also have an impact, but are less critical.

Finally, the acceptable errors on beam ellipticity and power dierences are so high that they largely exceed any error likely to be present in an actual experimental setup. Therefore, they need to be considered only marginally when setting up the experiment.

However, it should be kept in mind that the given error tolerances have been calculated by applying only one error at a time. In reality, the errors will occur all at the same time, thus inuencing one another. For instance, if a focal error ∆f is added to a ber displacement ∆x, the wavefront will be tilted by an angle θ x = ∆x/(f + ∆f ) rather than by the angle θ x = ∆x/f used in the simulation of the ber displacement. Hence, the eciencies will change.

Moreover, all the simulations presented here were performed using a microlens ll factor of τ r = 0.9. For dierent ll factors, the eciencies vary slightly. For instance, the losses found for a tilted ber are mainly due to an enhanced microlens clipping. Hence, for a lower τ r and thus smaller beams, the tilts can be higher before the microlens clipping becomes important. This is of particular interest since in the experimental setup, a ll factor of only τ r = 0.7 has been measured. An eciency curve using this value has been calculated, but remains quite close to the one based on τ r = 0.9, and has therefore not been shown. In the same way, the impact of a gap between the microlenses decreases for lower ll factors. Indeed, for the experimental ll factor of τ r = 0.7, a gap of up to 60 µm can be tolerated.

Spectral Disparities

Since the experiments are conducted with femtosecond pulses, the spectral properties of the beams may inuence the combining eciency. In order to estimate the losses in eciency due to disparities in the spectral amplitude or phase of the pulses, eq. (3.1) was expanded in the spectral dimension, so that the electric eld of each beam becomes E(x, y, ω) = E(x, y)E(ω),

where E(ω) = A(ω -ω 0 )e iϕ(ω) as in section 1.1.1.

Since there is no interference between the dierent wavelengths of the spectrum, the observed far eld can be considered as the incoherent superposition of the far elds of every incremental wavelength. Thus, the power P of an electric eld E is given by P = |E(x, y, ω)| 2 dω dx dy, and the combining eciency can be calculated using the same formula as in eq. (3.7).

Spectral Envelope

The spectral envelope A(ω -ω 0 ) might not be exactly the same for all of the beams. However, as each frequency increment is considered independently from the others, mismatches in spectral envelope translate into power discrepancies among the beams. Since it has been seen in section 3.3.1 that those do not have a strong impact on the combining eciency, spectral envelope mismatches have not been studied in more detail.

Spectral Phase

Considering additionally the spectral phase, the electric eld reads E = E(x, y)A(ω -ω 0 ) × e iϕ(ω) .

(3.28)

As detailed in section 1.1.1, the spectral phase can be developed in a Taylor series, where the dierent coecients correspond to a phase oset (ϕ 0 ), a delay (ϕ 1 ), a group delay dispersion and therefore a frequency chirp (ϕ 2 ), and a third order dispersion (ϕ 3 ). This allows to separate E into four independent electric elds, each describing one separate term of the spectral phase,

E ϕ 0 = E(x, y)A(∆ω) × exp (iϕ 0 ) (3.29) E ϕ 1 = E(x, y)A(∆ω) × exp (iϕ 1 ∆ω) (3.30) E ϕ 2 = E(x, y)A(∆ω) × exp i ϕ 2 2 ∆ω 2 (3.31) E ϕ 3 = E(x, y)A(∆ω) × exp i ϕ 3 6 ∆ω 3 . (3.32)
Furthermore, the nonlinear phase ϕ NL can simply be expressed as

E ϕ NL = E(x, y)A(t) × exp (iϕ NL (t)) , (3.33)
where A(t) = F(A(∆ω)) is the amplitude of the pulse in the time domain.

If all beams had the same spectral phase, this would have no impact on the combining eciency.

However, there may be dierences in the spectral phases of the individual beams, which give rise to a transfer of power from the main lobe to a speckle pattern around it. Since this speckle is removed at the spatial ltering pinhole, the power in the combined beam is reduced with respect to the overall output power, and the combining eciency decreases.

The impact of the dierent spectral phase terms on the combining eciency η was studied in the same way as it has been done for the spatial errors. Hence, for each phase coecient 50 runs have been executed, each using a set of random errors drawn from a Gaussian distribution with standard deviation ∆ϕ n . The eciency curves show the mean value and the standard deviation of the 50 eciency values that were calculated, in black for the average power and in gray for the peak power, as explained below.

General Considerations

As mentioned before, the overall power in the far eld can be considered as the sum of the powers contained in the far elds of each incremental frequency. However, except for the zeroth order, the spectral phase grows with ∆ω, so that frequencies close to the center of the spectrum are less heavily impacted by the spectral phase than those at the edges. In consequence, for these latter frequencies the phase dierences between the individual beams are higher, so that the far elds become more arbitrary, with less power concentrated in the main lobe and more in a speckle pattern around it. This behavior is shown in g. 3.18 for three dierent wavelengths for beams with a Gaussian spectral envelope, a pulse duration of ∆t ≈ 225 fs and a rst order phase of ∆ϕ 1 = 250 fs. This phase has been chosen much higher than any value that could be expected in real experiments, in order to clearly show the general behavior of the far eld under the inuence of spectral phases. It should also be noted that the rst wavelength corresponds to the central one and exhibits therefore an ideal far eld.

In the following, dierent consequences resulting from this frequency dependence will be presented. As above, the phases used in all the examples are highly exaggerated in order to give an instructive illustration of the eect in question.

Inhomogeneous Spectral Content of the Far Field

For small spectral phase coecients, the phase dierences between the central and the edge frequencies may remain below 2π. In this case, there is a clear relation between the phase of each incremental frequency, so that the corresponding far elds resemble each other, only that the speckle pattern becomes more pronounced for the larger spectral phases belonging to the frequencies at the edge of the spectrum. In contrast, for large phase coecients, the phase dierence between two frequency components largely exceed 2π, so that each frequency exhibits a far eld which is not related to those of the other frequencies, and has its own individual speckle pattern. This implies that dierent frequencies have dierent intensities at dierent locations in the far eld, so that the spectral content of the far eld is spatially inhomogeneous. wavelengths around 1038 nm can be found at this shifted position, but are very faint in the main lobe. In contrast, at a dierent wavelength of λ = 1040 nm, the intensity maximum is located to the right of the main lobe, as can be seen in g. 3.19c. This means that this wavelength appears mostly at another position in the far eld than the wavelength considered before, which conrms the spatial variations of the spectral content of the far eld.

In conclusion, the tiled aperture conguration gives rise to spatio-temporal inhomogeneities in the far eld.

Impact of the Spectral Envelope

Since the combining eciency is dened as the amount of power in the area of the main lobe of an ideal far eld, the frequency-dependent increase of the speckle pattern in the far eld implies that frequencies at the edges of the spectrum give rise to a stronger decrease in combining eciency than those at the center. In consequence, the spectral shape of the pulses has an impact on this eciency. For instance, by comparing a Gaussian spectrum |A G (∆ω , as shown in g. 3.20a, one can observe that for the super-Gaussian spectrum the relative amplitude of frequencies close to the center of the spectrum is increased, whereas it is strongly reduced at the wings. Hence, for the super-Gaussian spectrum the impact of the frequencies at the edges of the spectrum on the combining eciency is reduced as well, resulting in higher overall eciencies. This is illustrated in g. 3.20b which shows curves for losses in average power eciency due to disparities in the second order spectral phase, calculated with both the Gaussian and the super-Gaussian spectra dened above. However, it should be noted that the combining eciency obtained with the super-Gaussian spectrum highly depends on its width. In the case presented here, the Gaussian and the super-Gaussian spectrum have the same intensity FWHM. As can be seen in g. 3.20a, this means that the reduction of intensity at the edges of the super-Gaussian spectrum is somewhat stronger than the increase near the center. This could be easily changed by normalizing the spectra to another parameter, for instance their energy. The corresponding super-Gaussian spectrum would then be broader than the one presented here, resulting in a reduced combining eciency.

)| 2 = exp -(∆ω) 2
The impact of the spectral shape on the combining eciency is increased in the case where a nonlinear phase is considered in combination with a CPA conguration, so that the spectral phase is directly proportional to the power spectrum of the pulses. In this case, a super-Gaussian spectrum implies a nonlinear phase which is approximately constant over the central frequency region, and can therefore be corrected by the feedback algorithm. Hence, the losses in combining eciency are strongly reduced.

Peak Power Losses

Due to the increasing speckle pattern in the far eld (cf. g. This implies that there are not only losses in the average power of the combined pulse, but additionally that its peak power gets reduced due to the temporal broadening. This decrease can be calculated by comparing the peak power of the combined and compressed pulse P peak = P spec (t = 0) * with the ideal peak power P peak, ideal , so that the peak power eciency η peak is given by

η peak = P peak P peak, ideal × η ideal , (3.34)
with η ideal as dened in section 3.2.

Asymptotic Behavior for Very Large Phase Mismatches

It should be noted that disparities among the spectral phases destroy the coherence between the beams. For example, in the extreme case where the relative delay between the pulses becomes larger than the pulse duration, the pulses arrive one after the other and can thus not interfere.

Hence, the resulting far eld is just the incoherent superposition of all the beams. However, the combining eciency is still dened as the amount of power conned in the area of the main lobe of an ideal far eld, so that even an incoherently combined beam exhibits a certain non-zero combining eciency η inc . Since spectral phases cause incoherence, but can not lead to destructive interference, combining eciencies below η inc are not possible. However, η inc depends on the size of the near eld and therefore the number of beams, since for larger beam numbers the size of the main lobe of the far eld decreases with respect to the size of the incoherently combined beam, as shown in g. 3.22. In consequence, for N = 7 beams, the eciency will not drop below η inc = 38 %, whereas it can decrease to η inc = 4 % in the case of N = 61 bers.

This can be generalized by considering that η inc is roughly proportional to the area of the main lobe of the far eld A ml ∝ 1 (N D /2) 2 , where N D is the normalized diameter of the near eld as dened in section 2.2.1. By substituting N 2 D = (4N -1)/3 one obtains A ml ∝ 12/(4N -1) → 3/N (N → ∞), which means that η inc decreases rst steeply and tends then slowly towards zero. This implies that also the requirements on the matching of the spectral phase increase considerably when the ber number is increased from a very low to a higher value, as is the case in the examples above.

However, for already very large N , the changes atten out. Furthermore it should be kept in mind that these incoherent eciencies represent an asymptotic behavior found in extreme cases for very large spectral disparities. In any real CBC setup, the combining eciency still depends on N , but the spectral mismatches need to be limited to values which can be considered as small perturbations of an ideal system.

Modeled Eciency Curves

In the following, the decrease in combining eciency due to disparities among the dierent components of the spectral phase will be modeled both for the average and the peak power of the combined beam. As for the spatial misalignments, the maximum tolerable error is set to a loss of one percentage point in average power combining eciency.

The beams are supposed to have a Gaussian spectral envelope

A G (∆ω) = exp - (∆ω) 2 2δω 2 , (3.35)
whose width δω is related to the intensity FWHM wavelength λ FWHM and the central wavelength

λ 0 by δω = πc √ ln 2 λ FWHM λ 2 0 . (3.36)
We assume λ FWHM = 7 nm. The corresponding pulse length is given by ∆t = 2 √ ln 2/δω = 223 fs, which corresponds well to the experimental pulse width of 216 fs retrieved from autocorrelation measurements on the combined beam (see section 4.5.5 for more detail).

Phase Oset ϕ 0

The results for a dierence in phase oset are shown in g. In an experiment, phase uctuations will be introduced by any kind of acoustic, mechanical or, on a longer timescale, thermal noise. The feedback algorithm used for the coherent combination has thus to be able to correct for all these noises in order to reduce the phase uctuations on a single ber to values below λ/48 rms.

Delay ϕ 1

Intuitively it is easy to understand that the pulses of all the channels need to arrive at the same time if they shall recombine coherently. However, small delays between the pulses are acceptable, as can be seen in the eciency curves in g. 3.24. This implies that the delays should be limited to about 35 fs for a seven ber setup and 22 fs for a 61 ber setup, which corresponds to an optical path dierence of 10.5 µm or 6.6 µm, respectively. This result may be generalized by considering the delay of 35 fs or 22 fs with respect to the pulse duration of about 223 fs, so that the errors on the delays should not be signicantly higher than 15 % of the pulse duration in the seven ber setup, and 10 % in the 61 ber setup.

Group Delay Dispersion ϕ 2

The second order phase is induced by the dispersion of the ber medium. It can therefore be expressed as a dierence in ber length, as indicated in the top axis of the eciency curves shown in g. 3.25. It can be seen that the dierences in ϕ 2 should not exceed 7500 fs 2 for seven bers and 5000 fs 2 for 61 bers, which corresponds to a relative error of ∆ϕ 2 /∆t 2 = 0.15 or 0.1, respectively, and a dierence in ber length of 37.5 cm or 25 cm.

Third Order Dispersion ϕ 3

As the group delay dispersion, the third order dispersion (TOD) is related to beam propagation in a medium. The resulting eciency curves are shown in g. 3.26, and imply that the acceptable dierence in ber length is on the order of 32 m for seven bers and 20 m for 61 bers. Since this is much larger than the ∼ 30 cm required for the matching of the GDD, no special care has to be taken for the TOD adjustment.

Higher Order Dispersions ϕ 4 , ...

Since already the third order dispersion is not critical for the combining eciency, no higher order dispersions have been modeled. Since global phase osets have no impact on the combining eciency, the overall B-integral of the beams has been neglected in the simulations, and only variations ∆B have been considered * .

However, it should be kept in mind that negative B-integrals have no physical meaning. Hence, if a relative dierence of ∆B = 1 is applied to the beams, this implies that the absolute B-integral is high enough to tolerate such dierences. From a more application-oriented point of view, one could also say that a relative dierence of ∆B = 1 is not very probable for an absolute B-integral of, for instance, B = 1.5 rad, since this would imply that one beam has a B-integral of 0.5 rad, whereas for another beam one may nd B = 2.5 rad. Such values are unlikely to be obtained under normal experimental conditions, especially when an eort is made to equalize the B-integrals of the dierent beams.

Furthermore, it should be noted that in the experiments ∆B is expected to increase with growing B. This is due to the fact that for an increasing amount of nonlinearities the behavior of each amplier becomes more sensitive to its exact working conditions, so that dierences among the channels become more pronounced. Hence, even if the absolute value of B has no impact on the combining eciency, for increasing B-integral a decrease in combining eciency has to be expected due to increasing ∆B.

The spectrum of the beams is supposed to be Gaussian as dened in eq. (3.35). Nevertheless, assuming that phases of order zero and one are corrected by the feedback algorithm and the delay adjustment procedure, their contributions have to be removed from the nonlinear phase. Hence, in eq.

Summary

In this section it was modeled at which point the combining eciency is aected by disparities in the spectral properties of the beams. A summary of the results obtained for the average power is shown in table 3.2.

In the literature, similar simulations for CBC in tiled aperture conguration are restricted to losses in peak power eciency, but show results comparable to the ones obtained here [START_REF] Leshchenko | Coherent combining eciency in tiled and lled aperture approaches[END_REF][START_REF] Cui | Spectral phase eects and control requirements of coherent beam combining for ultrashort ultrahigh intensity laser systems[END_REF].

Furthermore, the tolerances stated in [START_REF] Goodno | Perturbative analysis of coherent combining eciency with mismatched lasers[END_REF] for the phase and delay mismatches in lled aperture congurations are in good agreement with the average power eciencies given in the table below.

Especially a phase oset and a delay between the pulses prove to be very critical and will have to be controlled precisely in the experiments. This will be done using the procedures detailed in In contrast, dispersions of an order of two or higher and dierences in the B-integrals will probably not pose major problems.

Conclusion

In this chapter, a numerical model was developed which gives a complete description of the hexagonal ber and microlens array used for the coherent beam combining.

This allowed rst of all to estimate the impact of misalignments in these two arrays and of spectral disparities among the beams on the combining eciency. It was calculated which errors on the alignment are acceptable if the eciency should not drop by more than one percentage point. These maximum tolerances will have to be taken into account when it comes to the design of the experimental setup. The solutions which have been found in order to meet these requirements are presented in the next chapter.

Furthermore, this model also allows to simulate the experimental beam array by taking into account the measured misalignments. It is thus a valuable tool for evaluating the impact of each of these misalignments on the combining eciency, and to estimate the maximum eciency which can be expected in the experiments.

Seven Fiber Laser System

In this chapter, the setup and characterization of a tiled aperture CBC system consisting of seven high power ber ampliers is presented. It is the predecessor of the 61 ber setup aimed for in the XCAN project, and has been implemented in order to identify and overcome a variety of technical and scientic challenges which might occur in the upscaled version of 61 bers. Hence, the setup presented here is designed such that it is suitable to accommodate all 61 ampliers, and is therefore partly overdimensioned for only seven channels.

Front End Laser Head Back End Figure 4.1: Schematic of the three main parts of the seven ber laser system.

The laser system can be divided into three main parts (cf. g. 4.1), which will be described in this chapter. In the front end, the initial laser pulses are prepared for the coherent combining and the power amplication, and are then power amplied. The following section describes measures which were taken in order to characterize and reduce the passive phase noise in this front end. A third section introduces the next part of the seven ber laser system, the laser head, which is the mechanical structure that allows to obtain a hexagonal array of collimated beams. Next, the laser back end is presented, where the coherent beam combining, the spatial ltering and the temporal recompression take place. Finally, the performances of the system are characterized both in linear and nonlinear working regimes.

Laser Front End

A schematic of our laser front end is shown in g. 4.2, and a photo of it is displayed in g. 4.3.

The front end has to meet several requirements:

First, it needs to prepare the initial laser pulse sequence for power amplication. This includes temporal and spectral broadening of the pulses, reduction of the pulse repetition rate by pulse picking, and shaping of the spectral phase in order to guarantee best recompressibility. Furthermore, the whole front end has to be designed such that its B-integral is minimized. In g. 4.2, the parts belonging to the amplication are colored in dierent shades of blue, whereas the pulse pickers are yellow and the phase shaping components exhibit a red color gradient.

Second, the front end is used to prepare the pulses for the coherent combination. This means that the initial laser beam has to be split into eight sub-beams (seven for the CBC and one as reference beam for the phase measurement), and the optical path lengths of these sub-beams have to me matched. Furthermore, the real time phase correction takes also place in the front end.

Elements belonging to this domain are colored in green in g. Third, the front end should be fail-safe, robust, easy to maintain, and convenient to use. Therefore, we designed an all-PM-single-mode-ber front end using o-the-shelf parts wherever possible.

Moreover, in order to ensure an easy path length adjustment, the ber lengths of all parallel components were meant to be matched to ±1 cm.

Finally, our front end is intended to be as similar as possible to the one which will be used in the 61 ber setup. This means that any elements which might be critical in the 61 ber setup have been included in the seven ber one, even if they were overdimensioned for this case, or if their use complicated the setup. This concerns in particular the pulse picking system and the number of pre-ampliers which were included in the front end, as will be explained below.

In the following, the individual parts of the front end will be presented in more detail.

Pulse Generation and Temporal Stretching

The master oscillator is a Mikan from Amplitude Systèmes delivering 225 fs pulses at a repetition rate of 55 MHz. In a rst step, these pulses are passed through a 3 m long ber, so that the pulse spectrum gets broadened due to self-phase modulation. This allows to pre-compensate the spectral gain narrowing induced by subsequent amplication stages.

Afterwards, a rst pulse picker can be used to reduce the pulse repetition rate, as will be described in more detail below. Subsequently, a chirped ber Bragg grating (CFBG) stretcher from Teraxion is used to increase the pulse duration to 500 ps. This stretcher was designed such that it pre-compensates the overall spectral phase induced by both the GVD in the all-ber front end and the xed geometry of the grating compressor.

The pulse duration in the seven ber setup is by a factor of ten smaller than the 5 ns aimed for in the 61 ber setup. It was chosen as a preliminary set point, since the 5 ns compressor was not available in time, whereas a much more compact 500 ps compressor could be readily provided. In consequence, in order to keep the overall B-integral at the level of about 5 rad aimed for in the 61 ber setup, the pulse repetition rate had to be increased by a factor of ten from the initial 200 kHz to 2 MHz, so that the pulse energy is decreased by this same factor of ten and the peak

Pulse Shaping

The stretcher is followed by a pulse shaper. Based on a spatial light modulator, this device allows to reshape both the spectral amplitude and phase of ultrashort pulses. This can be done either by applying a pre-dened mask, or by measuring and compensating the actual phase of the pulses using an iterative correction algorithm. In our setup, the pulse shaper is used to compensate for spectral phase mismatches between the stretcher and the compressor. These can arise due to unforeseen changes in the front end setup, such as an increase in overall ber length, which introduce additional spectral phases and thus degrade the compressibility of the combined beam. Moreover, if the system is operated in nonlinear regime, the accumulated nonlinear phase causes a further downgrading of the compressed pulse quality. To a certain extend, this can be compensated for by the pulse shaper. Besides, due to the limited bandwidth of the pulse shaper of about 18 nm, the spectrum is cut sharp between 1025 nm and 1043 nm.

Phase and Delay Adjustment

The beam is separated into eight sub-beams by a 1x8 spatial splitter, and each sub-beam is passed through a phase and delay control element (∆ϕ in the gures) provided by IDIL bres optiques.

These elements consist of two stages: First, a piezomechanical ber stretcher manufactured by

Optiphase is used for the real time phase adjustment. This device consists of 15 m of PM ber which is wound around a cylindrical piezo stack (cf. g. 4.4). If a voltage is applied to the piezo, ber stretcher VODL it changes in diameter, thus slightly stretching the ber and thereby changing its length. This changes also the spectral phase accumulated by the beam when traveling through the ber, and the zeroth order component of this phase is adjusted such that all beams remain in phase with the reference beam. Due to the small available driving voltage of ±10 V, the optical path length correction is limited to ±19 µm, corresponding to phase dierences of ±18.4 λ. Hence, in order to achieve a stable phase locking, phase uctuations must not exceed ±18.4 λ. This means that the laser system has to exhibit a good stability when operated in open loop and needs to be well isolated from any sources of noise. The measures taken on our experimental setup in order to achieve such conditions are discussed in section 4.2.

Second, a free space variable optical delay line (VODL) allows to adjust the optical path lengths of the channels. Figure 4.4 shows a photo of a phase and delay control element, where the VODL can be seen to the left. It consists of a motorized translation stage which can be used to move a ber port with respect to the xed position of its counterpart. This translation has a precision of 0.1 µm, which corresponds to 0.3 fs. This is by far high enough to avoid eciency losses due to mismatches in the optical path lengths. Furthermore, the motorization ensures both a precise and repeatable alignment and the possibility to implement an SPGD algorithm for the delay ne adjustment.

However, although all elements of the front end were supposed to feature well matched optical path lengths, dierences in ber length of up to ±9 cm between two channels have been measured. This is way too large to be compensated for by the VODLs which oer a free-space delay adjustment of ±3 cm or ±90 ps, corresponding to ±2 cm of ber length. Hence, additional pieces of ber of suitable length had to be added to each channel. Afterwards, the remaining path length dierences were matched by adjusting the VODLs.

Pulse Picking

A three-stage pulse picking system manufactured for us by AA Opto-electronic is included in the front end, which allows to reduce the pulse repetition rate. At a given average power, this increases the peak power and energy of the pulses and allows thus to study the behavior of the laser system in presence of nonlinear eects. The pulse picking is performed with acousto-optic modulators (AOM). A rst AOM with a short response time of 6 ns is placed at the very beginning of the front end in order to ensure a good extinction ratio at the initial pulse repetition rate of 55 MHz. A second pulse picker would be needed for a further reduction of the repetition rate in the 61 ber setup, but was not used in the seven ber system. The third stage consists of eight parallel pulse pickers with identical optical path lengths, characteristics and driving electronics.

In this way it is ensured that all parallel pulse pickers apply the same frequency shifts at the same time to the sub-beams.

The easiest way to obtain the intended repetition rate of 2 MHz would have been to use only the rst two pulse pickers. However, it was decided to work with the rst and the third stage in order to test the operation of the parallel pulse pickers and to reduce the repetition rate as late as possible. In this way, high peak powers and the corresponding nonlinear eects occur only near the end of the chain, which reduces their impact.

However, the pulse picking system is congured such that the pulse picking ratios of the three stages depend on one another, and need to be at least a factor of three per pulse picker. Hence, even if the second stage is bypassed, it is not possible to achieve pulse repetition rates between 55 MHz and 55 MHz/3 3 ≈ 2 MHz. In the experiments, the minimum ratio of three has been applied to the rst pulse picker, thus reducing the repetition rate to 18 MHz. The ratios applied to the other pulse picking stages were chosen as a function of the desired nal repetition rates between the nominal working point at 2 MHz and a much lower value of only 750 kHz. This latter one was used to study the impact of increased peak powers and thereby nonlinear eects on the combining eciency.

Pre-amplication

Several core pumped Yb-doped single-mode 6/125 * ber ampliers tailored for us by Keopsys, a Lumibird Group company, allow to raise the average power to ∼ 100 mW. They have been inserted wherever in the 61-channel front end the average power would have dropped below 1 mW, which is the typical input power of such ampliers. The endmost pre-amplier uses 10/125 LMA bers in order to minimize nonlinear eects which occur if the pulse repetition rate is strongly reduced.

Due to the gain narrowing inherent to these ampliers, the spectrum of the pulses changes from the at-top form it obtained due to the hard cut of the pulse shaper to the more triangular shape which can be seen in gs. 4.5a and 4.5b.

In the seven ber setup, the average power losses are much less important than in the 61 ber setup, since the pulse repetition rate stays higher and only one 1x8 spatial splitting stage needs to be included. Hence, the pre-ampliers operate with very high input signals, which implies that their gain is quite small. This in turn results in a shift of the central frequency of the spectrum towards higher wavelengths. In order to reduce this eect and to work with a spectrum which resembles more the one estimated for the 61 ber setup, attenuators have been included in the front end. Due to the limited number of attenuators available, this was only possible before the spatial splitting, so that the LMA pre-ampliers stayed largely saturated. The resulting output spectrum has a maximum around 1035 nm, as shown in the dark blue curve in g. 4.5a. For comparison, the output spectrum of the 61 ber setup was emulated by putting attenuators in front of all pre-ampliers of one channel, leading to the spectrum shown in the light blue curve in g. 4.5a, with a maximum at 1032 nm. The shift of the spectral bandwidth of this spectrum is due to a dierent alignment of the pulse shaper.

A behavior similar to the one emulated for the 61 ber setup can be observed for a reduced pulse repetition rate. Since the pulse picking also entails a reduced average power at the pre-amplier inputs, their gain is increased and the peak of the output spectrum is centered around 1032 nm, as shown in g. 4.5b.

Preparation for Power Amplication

After the last, large mode area pre-amplier, one of the sub-beams is separated in order to serve as reference plane wave for the interferometric phase measurement. The remaining seven subbeams are designated for power amplication. Prior to coupling into the power ampliers, a small fraction of about 2 % of their overall power is separated by a tap coupler and directed into a security box manufactured by Koheron. The purpose of this element is to rapidly turn o the pump lasers of the power ampliers in case of a sudden signal breakdown. If Q-switches due to ASE shall be avoided, this turn-o time needs to be well below to the build-up time of the upper state population, which depends on both the upper state lifetime of the Ytterbium ions and the available pump power. For our experimental conditions, this build-up time is estimated at 250 µs.

Therefore, the security box is congured such that it opens the hardware interlock of the pump lasers whenever the signal power of one channel stays below a certain threshold during more than 10 µs.

The remaining power of each of the seven sub-beams, typically between 80 mW and 100 mW, is combined with the pump beams and coupled into the power ampliers. This could have been done in free space. However, as our power ampliers have a small signal numerical aperture of only 0.03, free space coupling is complex and would require maintenance on a very regular basis. Since this would hinder a smooth and simultaneous operation of 61 ber ampliers, we decided to work with integrated components. This approach also allows to maintain the advantages of the all-ber front end. In a rst step, the signal ber is adapted from the initial 10/125 to a 25/250 LMA ber, which is then combined with the 105/125 multimode (MM) pump ber into a single 25/250 double-clad (DC) LMA ber using a pump + signal combiner. Subsequently, the mode area of this ber is matched to the 40/200 of the power ampliers using a mode eld adapter (MFA). These two components shown on the photo in g. 4.6 where spliced to the bers by Optical Engines.

Power Amplication

We use CW pump diodes provided by BWT operating at a wavelength of λ pump = 976 nm and providing a power of up to 60 W. In order to avoid variations in the operation conditions of the power ampliers, the pump wavelength is stabilized to ∆λ < 1 nm.

The power ampliers, made of DC-200/40-PZ-Yb LMA photonic crystal bers from NKT Photonics (see section 1.2.2 for more detail on this ber), deliver up to 35 W average signal power.

Based on simulations regarding the amplied signal power and the amount of nonlinear eects accumulated in the process, their length has been xed to 1.6 m.

This ber has been chosen since it is the commercially available PCF amplier with the largest front end power ampliers laser head MFD, thus providing highest peak powers. However, the specic properties of the ber, such as the exact position of its core, may vary among dierent batches and depend on the actual working conditions.

One of the characteristics of this ber is that it needs to be coiled in order to guarantee a stable monomode operation. As this entails a quite cumbersome ber arrangement which is inconvenient for our purposes, we rst operated the bers without coiling. However, this lead to spatially modulated spectra, as shown in g. 4.5c. This could be resolved by introducing a 90 • curvature near the ber end as can be seen in g. 4.7. Nevertheless, the mode remains spatially sensitive to external perturbations, resulting in pointing instabilities on the individual beams, and therefore also on the combined beam. Since we observed that this eect is largely reduced if the ber is coiled to at least one full circle, such a conguration is currently being implemented in the 61 ber setup (cf g. 5.1 on page 117). On the seven ber setup, we decided to stay with the mostly uncoiled bers in order to fully explore the advantages and limitations of this conguration.

Phase Noise Reduction

As explained in the previous section, the relative phase deviations of each channel must stay within a range of ±18.4 λ so that they can be compensated for by the ber stretchers. If this requirement shall be fullled, a good passive noise reduction has to be ensured. In order to determine the main sources of noise in our laser system, numerous phase noise measurements have been performed under varying conditions on all critical elements of the front end. The setup used for these measurements, the main sources of noise which have been identied, and the measures taken to reduce their impact will be presented below.

A schematic of the setup used for the phase noise measurement is shown in g. 4.8. The outputs of two channels are superposed using a slightly tilted beam splitter, thus creating interference patterns in the two resulting beams. In each path an aperture is positioned such that it selects one interference fringe whose power is recorded on a photodiode. The apertures are aligned such that one photodiode measures the quadrature of the signal of the other diode. The recorded signals are thus given by I 1 (t) ∝ cos ϕ(t) and I 2 (t) ∝ cos(ϕ(t) + π/2) = -sin ϕ(t), respectively, where ϕ(t)

is the phase dierence between the two beams. Hence, this phase dierence can be determined by ϕ(t) = arctan(I 2 (t)/I 1 (t)), and the phase noise is derived from the Fourier transform of this time-dependent phase. The measurements presented below were recorded during 100 s, with a sample rate of 10kS/s.

BS

Although this allows to record frequency components of up to 5 kHz, the dynamic range of the photodiode reaches its limits already at ∼ 1 kHz, resulting in a large noise oor at higher frequencies which overwrites the signal information.

These measurements allowed to identify the frequencies where the phase noise is most prominent. By changing the experimental conditions, these could be traced back to certain elements or components in the setup. Three typical curves of such measurements taken at the LMA pre-amplier output are shown in g. 4.9. While g. 4.9a shows a measurement under normal conditions, in g. 4.9b the air conditioning system of the laboratory was turned o, which leads to an overall decrease in phase noise, and to a disappearance of the broad peak at 5 Hz. Moreover, the bump at 100 Hz decreases considerably in intensity.

Given these observations, one can conclude that measures have to be taken to reduce the impact of the climate conditioning on the phase noise. The most obvious solution would consist in turning o the air conditioner during measurements. Under such conditions, a relatively stable operation could be achieved during the short time required to measure the phase noise of the free-running front end. However, coherent beam combining of the power ampliers was not possible due to the excessive heating of the lab and the resulting ongoing phase changes in the ampliers. Hence, it was impossible to run measurements with the air conditioning turned o. Therefore, several measures have been taken in order to reduce the impact of the air conditioning on the phase deviations of the ampliers.

As a rst action, enclosures have been installed over the optical tables, which shield the ber ampliers from air ows. However, they could not be closed completely since several bers, cables and hoses had to be passed on the table. Furthermore, as can be seen in the photo of the front end (g. 4.3), the pre-ampliers are arranged underneath the optical table, so that they can not be shielded by the enclosure. In order to still provide a minimum of shielding, the pigtails were covered under a plastic tarpaulin. Furthermore, the bers lying on the optical table were taped on the table in order to reduce their sensitivity to residual air ows.

A similar approach has been considered for the power ampliers. However, in this case the situation is more complicated since the bers dangle between 10 cm and 15 cm above the optical table and their arrangement changes from adjacent at their inputs to hexagonal at the laser head.

Hence, a suitable support would have to make up for the height dierence and the changing arrangement of the ampliers. Taping the bers to a slightly tilted breadboard was tested, but proved to be dicult due to spatial constraints and the relative stiness and fragility of the power ampliers. Since on top of that the results were not convincing, this approach was not pursued any further.

A second component which has a considerable impact on the phase noise is the air cooling of the LMA pre-ampliers. This can be seen by comparing g. 4.9b and g. 4.9c, which show phase noise spectra with these fans turned on and o, respectively. In the second case, several peaks, for instance the one at 31 Hz or those around 300 Hz, reduce in intensity or vanish completely.

Since turning o the air cooling is not possible due to the heating of the pre-ampliers, several fan models have been tested in order to nd a good compromise between cooling capacity and phase noise increase. The curves shown in g. 4.9 have been taken with the nal fan version.

Similar observations have been made on the VODLs, which initially featured an air cooling system in order to prevent any heating of the motorized delay stages. However, the fans of this cooling system gave rise to considerable amounts of phase noise, whereas no heating could be detected even with the fans turned o. Therefore, and since the supplier conrmed that the fans are not necessarily required, but have rather been installed to be on the safe side, it was decided to simply turn them o for all future measurements.

Finally, also the water cooling of the pump lasers of the power ampliers has an impact on the phase noise. As the chiller is dimensioned for the 61 ber setup, its cooling capacity is much stronger than what is needed for operating the pump diodes of only seven power ampliers.

Hence, instead of constantly adjusting the temperature of the cooling water, the chiller waits until the temperature exceeds a certain critical value before reducing it. In consequence, the water temperature constantly uctuates around a certain set point value. This in turn has an impact on the wavelength of the pump diodes, which depends on the water temperature and therefore uctuates slightly within the stabilization range of ∆λ < 1 nm. These uctuations also aect the gain of the power ampliers, which depends on the pump wavelength. Since the gain is related to the refractive index of the silica ber via the Kramers-Kronig relations, a change in gain translates directly into a refractive index change. Hence, also the phase of the light traveling through the ber changes, resulting in a phase shift of the power ampliers which depends on the cooling characteristics of their pump lasers.

Since this behavior occurs on relatively long time scales of several seconds, it was observed as oscillations in the correction voltage of the ber stretchers. In order to overcome these uctuations, a set point water temperature was determined experimentally at which the wavelength of the pump diodes remains mostly stable, and the water ow of the chiller was largely reduced so that it needs to be cooled constantly.

All the measures described above contribute to the passive reduction of phase noise. However, occasionally the remaining phase dierences still exceeded the ±18.4 λ allowed by the ber stretchers, resulting in uncompensated relative delays between the channels. This is mainly due to the uncoiled and dangling power ampliers, whose movements give rise to small local bendings of the bers. This leads not only to a direct change of the local refractive indices, but also to power losses due to local changes in the waveguiding properties, which impact the gain in the amplier and therefore also the refractive index. These local refractive index variations translate directly into phase uctuations. This behavior will be remedied in the 61 ber setup by taping the bers to breadboards.

Furthermore, in this upscaled setup the phase uctuations caused by the pump diodes will be reduced, since the cooling system will be operating at its optimal working point, thus ensuring a better temperature stability. As these measures could not yet be taken in the seven ber setup, the exceeding group delay was corrected manually by readjusting the VODLs whenever necessary.

The laser head is the structure where the individual ber ampliers are assembled in a hexagonal array. It has thus to meet the precisions stated in section 3.3, while also allowing for a safe and uncomplicated operation and maintenance.

In this section, the setup of our laser head and the associated ber array alignment process will be introduced. Subsequently, an estimation of the imprecisions within the nal ber array will be presented.

Fiber Array Setup

We use a simple ber stacking approach to assemble our hexagonal array. The principle is illustrated in g. 4.10a. The basic part is a high precision 60 • V-shape holder, in which high precision ceramic sleeves (green in the drawing) of an outer diameter of 3.2 mm are stacked so that they form a hexagonal array. They are hold in place by a top lid which is tightly screwed to the main part. A small foam layer (dark red) is integrated into the lid, which softens and equally spreads the mechanical pressure exerted on the sleeves. In order to insert the power ampliers in this structure, a ferrule of suitable size is put near the end of the bers, so that they can be easily slid into the sleeves. Finally, a second plate is slid onto the main part in order to dene the ferrule longitudinal position. This plate has boreholes whose diameters are slightly larger than the endcaps of the power ampliers. Thus, a part of the residual pump power which diverges largely in the endcaps exits through their lateral surfaces and is ultimately absorbed by this plate.

Nevertheless, there is still a non-negligible amount of pump light which propagates towards the microlenses. Hence, its light disperses into all directions, giving rise to a very large pattern of partly collimated unabsorbed pump beams just centimeters after the microlens array. In order to block this stray light, a metal blocking plate with a hole of the size of the composite seven beam array has been inserted in the beam path. However, some of the parasitic beams arisen in the rst ring of the hexagon still pass through this plate, so that a second aperture has been included in order to block whatever stray light remains.

Figure 4.11a shows a photo of the laser head, the microlens array and the blocking plate. A false-color image of the same assembly taken with an infrared camera while operating the laser system at full power is shown in g. 4.11b. While the microlens array and the blocking plate remain at a reasonable temperature, the ceramic sleeve array heats considerably. This is due to a heating of both the bers themselves and the second plate which absorbs an important part of the residual pump power and whose heat radiates into the sleeve array. Furthermore, it has been noticed that the resulting temperature increase leads to a small change of the beam pointing, observable as a displacement of the far eld. Since the heating of the bers could not be remediated without changing the whole setup, it has been decided to implement a water cooling on the second plate. This allowed to reduce the temperature in the laser head from almost 80 • C to less than 50 • C, as can be seen in g. 4.11c. However, the circulation of the water in the second plate gave rise to vibrations which for their part aected the pointing stability of the far eld.

Hence, the second plate was slightly pulled away from the main part. In this way, the mechanical motions of the two plates are decoupled and the beam pointing remains stable even at high power operation with active water cooling. However, as the second plate also serves as reference plane for the longitudinal alignment of the ber array, it has to be joined to the rst one for the insertion of the bers. The separation of the two plates has to take place once the ber array is properly aligned, so that the ferrules do not outreach the main part and are well decoupled from the water cooled shifted second plate.

Besides, the ber ending including the ferrule and the endcap was carefully designed in order to guarantee a repeatable, break-safe ber insertion into the ceramic sleeves and a safe removal in case of maintenance or damage. When adding the ferrule around the ber, special care was taken that the ber output mode was well centered with respect to the ferrule outer diameter. A schematic of the nal design and its insertion in the laser head is shown in g. A former version of the ber ending did feature neither the metallic sleeve nor the plastic hose shown in g. 4.12. However, it turned out that inserting the bers by hand in a closely packed array with a pitch of only 3.2 mm was precarious, especially since the available space is very limited. Moreover, it was almost inevitable to touch the ferrules themselves during the insertion process, which might pollute them and thus reduce the precision of the alignment. Therefore, metal sleeves have been added around the ferrules. Touching these rather than the ferrule itself guarantees a safe and proper insertion. Moreover, as soon as two adjacent bers have been inserted

successfully, a third one can simply be put onto the metallic sleeves of the other two and is then slid into its respective ceramic sleeve. Although slight corrections of the ber position and angle might be necessary, this mechanism largely facilitates the insertion process.

Furthermore, a plastic hose of a length of 10 cm surrounds the ber at its ending in order to stabilize the ber and make it less sensitive to mechanical vibrations. Moreover, it also provides a means to manipulate the ampliers without touching the bers themselves. This is necessary since there is only very little spacing between the metallic sleeves in the hexagonal array, so that grabbing an individual sleeve for removing or realigning the ber would be almost impossible.

It should be noted, however, that this design is the result of an iterative process, which was completed in parallel with operation of the seven ber setup. Therefore, the bers used in this setup do not yet feature the metallic sleeves and the plastic hoses.

Fiber Array Assembly and Characterization

The assembly of the hexagonal ber array starts with the stacking of the ceramic sleeves in the main part of the laser head. The bers are then inserted simply by sliding their endings into these sleeves, so that only very minor adjustments can be performed on their positioning, and the precision of the ber tilts and displacements is largely determined by the precision of the sleeve array. As these sleeves are rigid high precision components, imprecisions in their alignment arise most probably due to dust which gets between the sleeves during their assembly. Another source of imprecisions might be dust which gets into the sleeves once the array is assembled, and which hinders the correct insertion of the bers into the sleeves.

The polarization axis of the ber ampliers aligns automatically parallel to the coiling plane.

Therefore, all the beams should be polarized horizontally. However, as the polarizing characteristic of the bers is not perfect, there is a vertical component in each individual polarization direction. Due to the birefringence of the ber, this component has a dierent group delay than the horizontal one, so that the two cannot combine coherently and the vertical component needs to be removed. Moreover, the polarization self-alignment procedure is not perfect, so that small individual deviations of the horizontal polarization may occur. These have to be ltered as well.

For this purpose, the composite beam is sent through a half-wave plate and on a thin lm polarizer (TFP, cf. g. 4.17). The half wave plate is used to turn the polarization of the composite beam by 90 • so that it is then vertical and reected by the TFP. Moreover, as the exact polarization directions vary slightly among the individual beams, the wave plate allows to adjust the average polarization such that the sum of the power losses of each channel is minimized. Using this alignment procedure it was also possible to check that the dierences in individual beam polarization are below the 7 • stated in section 3.3.1, so that eciency losses due to polarization mismatches are avoided.

As described in section 2.2.1, the output beams of the ber ampliers are collimated by a microlens array. In order to meet the precisions on the alignment of this array stated in section 3.3.2, it was inserted in a mount featuring precision mechanics on all six spatial degrees of freedom.

The alignment of the microlens array was performed both in the near eld and the far eld.

In a rst step, the lateral position of the microlens array was adjusted such that the beams were well centered within their respective microlenses. This was observed on a camera, and let to a near eld as shown in g. 4.13b. In a second step, the spot size of the incoherent far eld was minimized by adjusting rst the rotation of the microlens array, and second its focal distance.

This alignment procedure gave a rst insight into the properties of the microlens array, the relative precision between itself and the ber array, and misalignments in the ber array itself.

In the far eld, the minimum spot size of each individual beam was not always obtained for the same longitudinal microlens position. Hence, the focal distances of some beams were not well matched, and a manual re-adjustment was performed. In order to do so, the microlens array position was xed and the positions of the bers in question were carefully corrected with respect to their focal spot sizes. The standard deviation of the remaining errors is as low as ∆f = 0.04 mm, and has therefore a negligible impact on the combining eciency.

Furthermore, it was observed that the far elds of some beams got somewhat distorted while scanning the longitudinal position of the microlens array (cf. g. 4.14). This is an indication of an astigmatism which could arise due to either errors on the lateral ber positioning, or production imprecisions on the microlens array. In the rst case, correcting this aberration is dicult due to the very limited possibilities for realignments in the ber array, whereas in the second case only a replacement of the microlens array could solve the problem. In consequence, these beam distortions needed to be tolerated in the seven ber setup. Finally, it was not possible to achieve a perfect overlap of the beams in the far eld. This can be attributed to a slight pitch mismatch between the ber and the microlens array. However, quantifying this mismatch proved to be dicult due to the very high precision required in the measurements.

The analysis of the near eld image allowed rst of all the measurement of the microlens ll factor. For this purpose, the relative diameter of the beams in the near eld was evaluated, yielding a ll factor of τ r = 0.7. This result is signicantly smaller than the ideal value of 0.93, which means that the microlens focal length has been chosen too small. This focal length was determined based on the theoretical M 2 and mode eld diameter of the output beam of the power ampliers. However, the divergence of the real beam is inferior to the upper limit stated in the (c) Far eld including measured alignment errors. Besides, the low microlens ll factor leads to an increased microlens transmission of t µ-lens = 0.98.

In consequence, microlens clipping due to a displacement of the beams on their respective microlenses is reduced. Such displacements correspond to misalignments in the ber tilts, which can therefore tolerate larger errors than those calculated in section 3.3.1.

Measuring these ber tilt errors in the near eld image based on the displacement of the beams with respect to their microlenses has been attempted, but proved to be dicult due to the high precision which is required by this approach. Therefore, the uncertainties on the measurements were as large as the measured values themselves. Hence, this method gives only the order of magnitude of the tilt errors, which is about ∆θ = 0.7 mrad. Given the low microlens ll factor stated above, this corresponds to eciency losses of 0.5 %.

Furthermore, also the positioning errors in the ber array have been estimated using the near eld. For this purpose, the interference fringes between the beams and the reference plane wave were considered (cf. g. 4.13c). Since a positioning error between a ber and its microlens leads to a tilted wavefront, the corresponding fringe pattern has either a fringe period varying from the average one or is tilted as well. Errors in the individual ber positioning can thus be inferred from deviations between the individual and the overall period or tilt angle of the interference fringes.

Unfortunately, this method is not more precise than the one used for the ber tilt estimation. The measured errors on the ber positioning are on the order of ∆x = 2.6 µm and lead to eciency losses of 4 %.

Finally, it was attempted to measure the gap between the microlenses based on a near eld image of the lenses. However, this is dicult since the imaging itself highly depends on the exact position of the camera on the optical axis, so that quite dierent gaps can be obtained for slightly varying camera positions. The best recorded picture is shown in g. 4.16a, but has to be mistrusted since there are diraction patterns clearly visible at the edges of the lenses. Hence, the imaging is probably not perfect, leading to a falsied estimate of the microlens gap. Also the measured value of ∼ 330 µm supports this hypothesis, since it largely exceeds the upper limit of 200 µm indicated by the manufacturer.

Given that this measurement was not satisfactory, a high precision photo of one microlens was taken (see g. 4.16b) in order to directly measure the width of the contact zone between the lenses. However, this approach was not very precise either, as the results vary between 66 µm and 88 µm. Moreover, one can not be sure that the gap is really restricted to the bright lines visible on the photo, since it is highly possible that also the surrounding areas do not properly transmit the light.

Since neither of the two measurements presented above is in accordance with the manufacturer's specications of 100 µm ≤ g ≤ 200 µm, an intermediate value of 150 µm has been assumed for the microlens gap. This would lead to a decrease of combining eciency of 3 %.

An overview of the eciency losses due to the dierent misalignments is given in table 4.1. It can be seen that the most important losses are due to the reduced microlens ll factor, and a new microlens array with a longer focal length has been purchased in order to reduce these losses in the 61 ber setup. Furthermore, also the gap between the microlenses has a noticeable impact on the combining eciency. Since this gap is an artifact of the manufacturing process of the array, there is no immediate way to reduce its size and therefore the related eciency losses.

Among the errors on the ber array, only the displacement exceeds the one percentage point several times, this may be due to dust which got into the ceramic sleeves. A thorough cleaning and subsequent reassembly of the sleeve array might therefore be helpful to decrease the alignment errors, but was not performed during characterization of the seven ber laser system since the current performance of the laser head was considered satisfactory.

The reduced microlens ll factor and the alignment errors on the ber array were inserted into the simulations presented in chapter 3 in order to obtain an estimate of the far eld which has to be expected in the experiments. The result is shown in g. 4.15c and exhibits a combining eciency of η err = 0.56. The microlens gap has not been included in these simulations because of the high uncertainty of the estimated values. For the gap of g = 150 µm listed in table 4.1, the far eld eciency would reduce by 2.5 % due to an increase of the side lobes, whereas the decrease in microlens transmission corresponds to 0.5 % and is therefore negligible.

Laser Back End

The laser back end starts right after the assembly of laser head and microlens array and includes all optical components which are necessary for the coherent beam combining itself, the spatial ltering of the far eld, and the temporal compression of the combined beam. A schematic of this back end is shown in g. 4.17. It should be noted that the scheme is not to scale, where in particular the focal distances of the lenses are arbitrary. Moreover, several apertures and imaging lenses have been omitted in order to keep the schematic as simple as possible. The back end starts behind the assembly of laser head, microlens array and blocking plate which was described in the previous section. Since the emitted composite beam still consists of both pump (green in g. 4.17) and signal (blue) light, the former one is removed with a dichroic mirror and blocked by a beam dump.

In a next step, it is assured that all subbeams exhibit one and the same polarization and orthogonal components are ltered by a half-wave plate and a rst TFP, as described in section 4.3.2.

However, the ltering of a single TFP is not strong enough to remove all the orthogonally polarized parts, so that a second TFP has been added later on in the back end.

Subsequently, three mirrors are used to separate small fractions of the beam, which are then used for various steps in the coherent beam combining process. Although the resulting beams are very faint, attenuators (not shown in g. 4.17) had to be used in all three cases in order to avoid saturating the measurement devices. These attenuators consist of two sets of a half-wave 95 plate and a polarizing beam splitter (PBS). The rst set is needed to remove remaining horizontal polarization components, so that the second set can be used to attenuate the beam without having to worry about the admixing of orthogonally polarized light.

Behind the rst mirror, the far eld of the composite beam is observed on a camera in the focal plane of a lens. This serves as a visual tool in the alignment process and gives a rst idea of the quality of the coherent combining.

The second mirror is used to create the measurement path needed for the phase measurement of the phase correction algorithm. The composite near eld is imaged on a 1 kHz fast camera and a 50/50 beam splitter is used to superpose it with a reference plane wave in order to create the interference fringes necessary for the phase measurement. Since the reference beam is not power amplied, a passive ber of suitable length has been added into its path in order to compensate for the missing length of active ber. At its output, the reference beam is collimated by a lens of a focal length of a few tens of centimeters and a diameter of two inches. Hence, the size of the resulting beam is much larger than the one of the camera screen, so that its intensity is fairly constant within the useful zone of the camera. Furthermore, as this camera is placed near the image focal plane of the lens, the wavefront of the collimated beam is approximately plane. The reference beam can thus be considered as a plane wave.

Behind the third mirror, another lens is used to create another far eld, whose side lobes are removed with a pinhole (not shown in g. 4.17). The power of the remaining central lobe is detected by a photodiode and fed to an SPGD algorithm which optimizes the static phase osets of the individual beams (cf. section 2.2.2). In this way, the combining eciency measured in the phase locked system is maximized.

Afterwards, the beam is passed through another set of a half-wave plate and a TFP. This allows not only to further lter the orthogonal polarization, but also provides a means of regulating the power in the subsequent part of the laser back end. For alignment purposes, it is often desirable to work with a beam of low average power. However, during the transition from low to high pump powers the relative phases and delays between the channels change due to the heating of the ber ampliers. Hence, they have to be adjusted in the high power regime once the whole setup is well thermalized. It is thus necessary to work with high output powers, but the polarization of the beam can be rotated such that most of the power traverses the TFP and is absorbed by a large, water cooled beam dump. The alignment of the subsequent components can thus be realized using only a part of the overall power, and the whole power is transmitted only in the nal measurements.

This stage could have been included at an earlier point in the back end, so that the additional polarization ltering in the beam diagnostic paths described above could have been omitted.

However, in this case a power adjustment in the main beam path would have implied a change of power in these secondary paths. Hence, a frequent readjustment of the power incident on the beam diagnostics would have been necessary. In order to avoid such complications and to keep the power in the secondary paths constant, it was preferred to place the large beam dump after these paths.

The next step in the laser back end is the spatial ltering of the far eld, which is obtained by passing the composite beam through a thin lens and a truncating aperture. This aperture is inserted in a water-cooled precision mount as shown in g. 4.18a. In this way, its position can be exactly matched to the main lobe of the far eld, and the water cooling avoids excessive heating of the mount.

Using a selection of lenses and pinholes with dierent diameters, the relative size of the ltering aperture in the far eld can be adjusted arbitrarily. It is thus possible to obtain dierent truncated beam shapes, as was shown in the simulations in section 3.1 and will be demonstrated experimentally in section 4.5.4. However, in order to obtain the combined beam as dened in section 2.1.1 and to measure the combining eciency, the far eld has to be truncated at the rst minimum next to the main lobe.

Besides, as the ltering pinholes are designed as the top of a cone carved in a small metal block, their edges are quite thin and therefore easily changed in shape by optical micro-machining. This is illustrated in g. 4.18b and g. 4.18c where an unused pinhole is compared to one which served as ltering aperture in measurements conducted at high power without water cooling. The impact of the side lobes of the far eld is visible in the discolored area near the hole itself. In order to avoid such or similar damages of the pinhole, it is preferable to choose the position of the aperture such that there is not too much power incident on the edges of the pinhole. This implies that the aperture should be placed at a power minimum of the far eld, for instance at the rst extinction next to the main lobe (as shown in g. 4.24a), or just before the secondary lobes (g. 4.27a (top)).

Behind the spatial ltering, the transmission of another mirror is used to image the ltered far eld on another camera. By adding a further lens, also the recovered near eld can be imaged.

This setup was also used to measure the beam quality factor M 2 of the combined beam.

The last element in the laser back end is the temporal compression stage, which relies on a 1760 l/mm multilayer dielectric reection grating pair (see photo in g. 4.19). A half-wave plate has to be added in front of the compressor in order to rotate the beam polarization back to horizontal, so that the light gets properly reected by the gratings. The output of the compressor represents in the same time the output of the whole laser system.

Experimental Results in Linear Regime

In this section, ve dierent measurements which were taken in order to fully characterize the performance of the laser system operated at full power in linear regime at 55 MHz will be presented.

In this context, full power means that the output power of the power ampliers is limited by the available pump power. Due to varying coupling and conversion eciencies among the power ampliers and slightly dierent polarization directions, similar signal output powers may be reached for quite dierent pump powers. Since one of the bers was limited to an average power of 25.4 W at the position of the spatial ltering pinhole, the other ampliers were adjusted accordingly, yielding an overall power of P FF = 178 W in the far eld.

Residual Phase Noise

The residual phase noise was measured between dierent sets of adjacent bers. The coherent superposition of only two beams obtained at the focal plane of a lens leads to interference fringes in the far eld, which allowed to measure the residual phase using a setup similar to the one described in the previous section and schematically shown in g. These measurements allow to access the relative phase between two bers, ϕ(t) = ϕ 1 (t) -ϕ 2 (t),

where ϕ 1,2 (t) are the phases of the two individual bers. From these results, the corresponding phase noise density has been be calculated. Additionally, the measurements in closed loop are characterized by the residual phase error between the two bers, ϕ res = σ[ϕ(t)], where σ denotes the standard deviation.

However, as the maximum tolerable phase deviation of λ/38 RMS obtained in the simulations (cf. section 3.4.2.2) corresponds to the phase error of a single ber, this parameter is much more relevant. Since all bers are phase locked to a common reference beam, they all share the same rapidly changing phase oset. Given that a phase common to all beams has no impact 99 on the combining eciency, it will be neglected in the following discussion. Besides, the phase of each single ber deviates slightly from this global phase. This can be described by small individual phase changes ϕ i (t) and the corresponding residual phase error of an individual ber of ϕ res,i = σ[ϕ i (t)].

Since the residual phases of the individual bers are not correlated with each other, the residual phase error between two bers can be written as ϕ res = ϕ 2 res,1 + ϕ 2 res,2 . Assuming that both bers are subject to the same noise and exhibit therefore the same individual residual phase error ϕ res,1 = ϕ res,2 = ϕ res,i , the relative error between two bers simplies to ϕ res = √ 2ϕ res,i . It is thus possible to calculate the residual phase error of a single ber based on the value measured between two bers. Although the other curves resemble the one displayed above, the residual RMS phase deviation can vary for dierent ber pairs. For instance, the measurement shown in g. 4.20b yields a residual phase error of ϕ res = λ/34 RMS between the two bers, which corresponds to an error of ϕ i = λ/48 RMS on each ber. Considering that in the seven ber setup the phase of each channel needs to be stabilized to λ/38 RMS in order to avoid losses in combining eciency, this is a very satisfying result. However, under comparable experimental conditions, the two bottommost bers of the hexagon exhibited a residual phase error of only ϕ res = λ/55 RMS. The reason for this unequal behavior can be found in the dierent positions and mountings of the bers: the upper bers are more exposed to the air ows of the lab's air conditioning system, whereas a part of one of the bottom bers was taped to a breadboard, which stabilizes its position and phase by somehow reducing the impact of air ows.

Nevertheless, the results measured on the dierent ber pairs remain comparable in a sense that they have no impact on the combining eciency.

All in all, the demonstrated phase stabilization is more than adequate for the seven ber setup.

In contrast, for the upscaled version of 61 bers, a residual phase error of λ/48 RMS on a single ber is required in order to avoid eciency losses due to phase mismatches. Since this is exactly the value measured for the two topmost bers in the seven ber setup, the phase stabilization is sucient even for the more severe requirements of the larger setup. Nevertheless, as the exact value of the residual phase noise highly depends on the day-to-day experimental conditions, a further reduction of the phase noise would be benecial in order to ensure a stable operation even under varying and possibly less favorable conditions. Such a reduction will be achieved in the 61 ber setup by fully closing the enclosure around the optical table, and by coiling each power amplier to a full circle and taping it to a breadboard.

Combining Eciency

Operating the laser system in closed loop, a far eld as shown in g. 4.21a has been obtained. The combining eciency has been determined by truncating it at its rst extinction and measuring the power contained in the remaining central lobe. It is given by P ml = 87 W, yielding a far eld eciency of η FF = P ml /P FF = 0.49. In order to obtain the overall combining eciency η, this value has to be multiplied with the microlens transmission t µ-lens . In the simulations, t µ-lens = 0.98 is obtained for our ll factor τ r = 0.7. In the experiments, only the order of magnitude of this value could be conrmed. This lack in precision is due to the measurement range of our power meter which is inadequate for this task. We thus use the theoretical microlens transmission in order to calculate the overall combining eciency η = η FF × t µ-lens = 0.48. This corresponds to 71 % of the simulated ideal eciency of η ideal = 0.67, and to 79 % of the maximum possible eciency for the experimental microlens ll factor, η(τ r = 0.7) = 0.61.

This simulated value drops to η err = 0.56 when also the alignment errors on the ber array are considered (see end of section 4.3.2), so that the measured combining eciency corresponds to 86 % of the theoretical maximum.

The remaining dierences between the simulated and the measured eciencies might be due to errors on the microlens array which have not been considered in the simulations. In particular, the enhanced side lobes of the measured far eld (cf. lineouts in g. 4.22) might be an indication of both a gap between the individual microlenses and a mismatch between the microlens and the ber array pitch. The supposed gap of g = 150 µm would cause a decrease of 2.5 percentage points.

If the remaining dierence between the measured eciency of η = 0.48 and the reduced simulated value η err = 0.56-0.025 = 0.535 was solely due to a pitch dierence between the microlens and the ber array, this discrepancy would equal ∆p = 23 µm. Since such a large mismatch between the high precision ber assembly and the likewise high precision microlens array seems not realistic, other factors impacting the combining eciency need to be considered.

In particular, the estimation of the spatial alignment errors within the ber array is not very accurate, so that the real misalignments might vary from those used in the simulations. This implies that the actual maximum possible eciency might be inferior to the simulated one, which could partly explain the dierence between the simulated and the measured value.

Finally, one needs to consider the impact of the spectral phase on the combining eciency. As it has been stated in the previous subsection, the residual phase error of ϕ i = λ/48 RMS on a single ber is low enough to avoid eciency losses due to phase oset dierences. The relative delays between the channels have been optimized with respect to the power in the main lobe of the far eld. In this process, no changes in transmitted power could be detected over a range of path length dierences of about ±10 µm. As this within the necessary precision of ±10.5 µm RMS stated in section 3.4.2.2, delay mismatches should not give rise to losses in combining eciency superior to one percentage point. Finally, mismatches of spectral phase of order two or higher are precluded by the matching of the ber lengths of all the channels to ±3 cm, as described in section 4.1. In conclusion, it is unlikely that spectral phase mismatches give rise to losses of combining eciency of more than one percentage point.

From the recorded far eld, the truncation eciency curve shown in g. 4.21c was calculated. As indicated in g. 4.21c, the relative radius of the main lobe for a seven ber system is r ml = 0.42. Hence, the truncation eciency η tr (r ml ) = 0.45 corresponding to this radius should also correspond to the measured η FF = 0.48. For a number of reasons, this is only approximately the case. First and most importantly, the eciency measurement based on the camera image is highly sensitive to the background noise of the camera and is therefore less accurate than the eciency calculated from the measured power of the truncated beam. Furthermore, the normalization of the radius of the measured truncation eciency curve is based on the position of the side lobes in the far eld. Since the measured far eld is slightly asymmetric, there are small dierences among the radial distances between each side lobe and the center of the main lobe.

Hence, the normalization should be performed with respect to the average position of the six side lobes. However, since the precision of the truncation eciency curve is intrinsically limited by the noise of the camera, it was decided that it would be sucient to perform the normalization with respect to the average position of the two side lobes of the vertical lineout. Nevertheless, there might be a small discrepancy between this two-side-lobe average radial position and the overall average of the six side lobes, which would give rise to a slight shifting of the truncation eciency curve. Moreover, dierent pixel sizes in the experimental and theoretical far eld images can lead to slightly dierent truncation eciencies for same aperture radii. Finally, it is not sure that the experimental truncation has been performed exactly at the rst minimum of the far eld. A slight dierence between the ideal and the actual size of the main lobe of the far eld with respect to the truncating aperture would also result in dierences between the measured eciency and the one deduced from the truncation eciency curve.

Given all these uncertainties, the experimental truncation eciency curve shown in g. 4.21c

should not be considered as a precise measurement, but rather as an approximative characterization of the general shape of the far eld.

Power Stability

The power stability of the combined beam has been measured by focusing it on a photodiode and recording the power signal over 100 s. The power variations are as low as 1.7 % RMS, and the power noise spectrum obtained from the signal is shown in g. 4.23. It should be noted that the power uctuations are not only due to residual phase uctuations, but are also caused by pointing instabilities of the far eld. Such instabilities result in small changes of the position of the far eld with respect to the spatial ltering aperture, which leads to small uctuations in the transmitted power and reduces thus the overall power stability of the combined beam.

As such pointing instabilities can also be observed on a single beam, they are not due to the CBC system, but are rather caused by vibrations in the spatial mode of each individual beam which are transmitted in the combined one. These vibrations can be due to either instabilities in the mode emitted by the ber ampliers, or to vibrations in the optomechanics used in the laser back end. Indeed, due to the long overall free space beam path of about 3 m, already minor pointing instabilities can become large enough to impact the power transmitted through the spatial ltering aperture which has a diameter of only 630 µm.

In order to reduce such instabilities, more adequate optomechanics will be used in the 61 ber setup. Furthermore, the stability of the modes emitted by the ber ampliers will be increased by properly coiling them and taping them on breadboards.

Spatial Beam Shape

As described in section 3.1, the shape of the truncated beam highly depends on the relative size of the spatial ltering aperture. This behavior has been observed experimentally for two dierent aperture sizes, the rst corresponding to a truncation at the rst extinction at a relative radius of r = r ml = 0.42, and the second to one at the second extinction at r ≈ 0.76 * . For our experimental conditions, this corresponds to aperture diameters of 630 µm and 1.2 mm, respectively. In the following, the spatial characteristics of the truncated beam are described for these two cases. As explained near the end of section 4.4, the recovered near eld has been observed in the image focal plane of a lens. The beam conserves its good spatial properties (see g. 4.24c), but recovers truncation eciency η tr = 0.58 is in good agreement with this value. Furthermore, it corresponds to 89 % of the eciency expected from the simulations including alignment errors, and to 71 % of the ideal case.

By comparing these results to the ones obtained by measuring the combining eciency (in tight ltering conguration), one notices that both measured eciencies correspond to 71 % of their respective simulated ideal values. This conrms the correct predictions of the simulations. At the same time, η tr measured in loose ltering conguration corresponds to 89 % of the simulated value including alignment errors, whereas the tightly ltered η amounts to only 86 % of the respective simulation. This might indicate that the estimation of the misalignments in the ber array is somehow imprecise, so that the simulated eld does not exactly correspond to the measured one.

As discussed in section 3.1, the larger ltering aperture lets pass higher spatial frequencies, so that the beam partially recovers its initial hexagonal shape in the near eld. A simulation of this behavior based on an ideal beam array and the experimental microlens ll factor of τ r = 0. The lower line of this gure shows the corresponding measured elds. The overall agreement between the simulations and the measurements is quite good, although misalignments in the initial near eld lead to distortions in the recovered elds, which are particularly well visible during beam propagation.

These measurements conrm that it is indeed possible to create donut-shaped beams by truncating the far eld at its second extinction. Once scaled to higher powers, such beams could be of interest for certain applications in particle acceleration [START_REF] Brabetz | Laser-driven ion acceleration with hollow laser beams[END_REF][START_REF] Vieira | des eets non-linéaires qui dégradent les qualités spectrales des impulsions et peuvent dans le pire des cas endommager la bre[END_REF].

Temporal Compression

The tightly ltered combined beam has been temporally compressed and send to an autocorrelator for temporal pulse shape characterization. The resulting autocorrelation trace is shown in g. 4.28a, where the blue curve shows the measurement and the red curve is a simulated autocorrelation. This latter one has been obtained by Fourier transforming the measured spectrum (cf.

g. 4.28b), and calculating the autocorrelation trace which corresponds to the resulting temporal pulse shape. In this way, also the factor of deconvolution between the autocorrelation and the temporal pulse shape could be determined. Based on the autocorrelation FWHM of 293 fs and the minimum pulse duration of 212 fs derived from the measured spectrum, this factor is given by 1.38. Since the simulated and the measured curve are highly similar, this very factor has been used to determine the pulse duration of the combined beam based on its autocorrelation, yielding a value of ∆t = 216 fs. Considering the spectrum with a FWHM of ∆λ = 8.76 nm, this corresponds to an experimental time-bandwidth product of 0.53. Given that in the simulation one nds a value of 0.52, this result conrms that the compression is almost Fourier transform limited.

The compressed beam has an average power of 71 W, corresponding to a compression eciency of 82 %.

Summary

In this section, experimental results obtained with the seven ber setup operated in linear regime have been presented. In a rst step, the quality of the phase locking was checked by measuring the residual phase noise. The result of ϕ i = λ/48 RMS on a single ber is low enough to ensure a stable operation not only in the seven ber setup, but even in the upscaled version with 61 channels. In a next step, the combining eciency η was determined. The measured value of 107 ϕ res,NL = λ/32 RMS deduced from the curve in g. 4.29 is almost the same as the ϕ res = λ/34 RMS found in linear regime. Similar results are obtained for other ber pairs and at a pulse repetition rate of 1 MHz. It has thus been concluded that the B-integral has no impact on the performance of the phase locking algorithm.

Combining Eciency

As for the measurements in linear regime, the system was operated in closed loop and the com- At the working point of 2 MHz, the combining eciency η = 0.45 is only three percentage points below the one obtained in linear regime. From the simulations presented in section 3.4.2.2 it can be deduced that this eciency loss corresponds to B-integral mismatches of ∆B = 0.8 rad, which is consistent with the overall B-integral of 5 rad, the specic amplication conditions and the fact that the average output power of the ampliers has been matched rather than their B-integrals.

However, it should be noted that in this nonlinear regime the quality of the matching of the delays between the channels decreases due to the dierent chirps that the pulses obtain in their respective amplication conditions. Hence, it is highly possible that a certain part of the overall eciency loss is due to a delay mismatch, and that the dierences in B-integral are inferior to the value stated above. More pronounced drops in combining eciency occur at very high B-integrals of 10 rad or 13 rad.

Indeed, for the latter value, the measured eciency drops below the theoretical limit deduced in section 3.4.2.1, although the far eld (bottom line in g. 4.31a) still shows the basic structure of one central lobe and six side lobes. This discrepancy is probably due to the fact that the overlap of the individual beams in the far eld is not perfect, so that the resulting experimental incoherent eciency is reduced.

Power Stability

The power stability of the combined beam depends both on its pointing stability and on residual phase uctuations. The former one is subject to instabilities due to the environmental conditions, as explained in section 4.5.3. It is thus independent of the spectral phase, and remains the same as in linear regime. Furthermore, the phase noise power spectra of the linear and nonlinear regime are very similar both in open and closed loop. Hence, the phase matching algorithm operates under the same conditions, leading to the same quality of phase matching.

In consequence, considering that the two main factors which have an impact on the power stability of the combined beam remain unaected by the nonlinear eects, it is safe to assume that also the power stability itself does not change with respect to the one observed in linear regime. However, the corresponding measurement could not be performed due to constraints in the schedule of the experiments.

Spatial Beam Shape

As pointed out in section 4.5.4, the spatial shape of the truncated beam depends on the spatial ltering aperture radius. For a tight ltering, only the main lobe of the far eld is conserved, so that only very minor parts of the speckle pattern close to it might be transmitted. These could have a slight impact on the spatial beam shape, but are unlikely to lead to major distortions, so that the clean shape of the combined beam is essentially conserved.

In contrast, if a loose ltering was applied, some of the speckle would be conserved. This speckle is due to both the nonlinear phase dierences and the spatial alignment errors in the ber array.

These latter ones have already been present in linear regime, and have let to a slight distortion of the experimentally recovered near eld (see bottom line of g. 4.27). In nonlinear regime, the contribution of the nonlinear phase would add up to these distortions, making the near eld more arbitrary and therefore less usable for any experiments. Furthermore, as described in section 3.4.2 the spectral content of the truncated beam could become spatially modulated.

At the same time, due to the larger ltering radius the spectral narrowing would be reduced, so that a larger spectrum could be conserved. However, this would not necessarily increase the spectral quality of the truncated beam, since also the larger spectral phases related to the frequencies closer to the edges of the spectrum would be transmitted.

In conclusion, it is probably unadvisable to apply a loose ltering to a beam which exhibits non-negligible amounts of nonlinear phase.

Temporal Compression

As in linear regime, the tightly ltered combined beam has been temporally compressed and characterized with an autocorrelator. It should be emphasized here that the phase of the spatially ltered combined beam corresponds to the mean of the phases of the individual beams. Hence, even if disparities among the nonlinear phases of the individual beams are removed in a CBC setup, the average B-integral is conserved. Hence, for increasing levels of nonlinearity, a degradation in the temporal pulse quality has to be expected. This is indeed the case, as can be seen in the autocorrelation traces of the combined beams at the ve dierent pulse repetition rates shown in g. 4.31b. Figure 4.31c represents the corresponding spectra, which exhibit oscillating features of increasing intensity, as had to be expected from the increasing secondary pulses in the autocorrelation traces.

At the working point of 2 MHz, the intensity of the secondary pulses remains at a reasonable level, and an autocorrelation FWHM of 350 fs has been measured. The corresponding pulse duration of 253 fs is somewhat larger than in linear regime, where the broadening is due to the nonlinear eects. However, the result remains acceptable.

Summary

In this section, the performances of our laser system have been characterized in nonlinear regime.

It was pointed out that neither the spatial quality of the combined beam, nor its phase or power stability change with respect to the linear regime. Furthermore, at the intended working point of 2 MHz or B ∼ 5 rad, both the combining eciency and the temporal shape of the compressed beam are somewhat degraded with respect to the linear regime, but remain at reasonable levels.

If the repetition rate is further decreased, the degradation increases.

However, the temporal distortion of the combined beam seems to be more detrimental to the overall system performance than the reduced combining eciency. Hence, the limiting factor in our setup is the temporal compression, just as for any conventional short pulse laser system.

Moreover, as the combining eciencies remain at reasonable levels even for increased amounts of nonlinearities, it could be considered to work at higher B-integrals in order to obtain higher peak powers, at the expense of the temporal pulse quality. Obviously, this is only possible if the presence of pre-or post-pulses can be tolerated.

Conclusion

In this chapter, the setup and characterization of a tiled aperture CBC system consisting of seven high power ber ampliers have been presented. The aim of building this setup was most importantly to conrm the correct operation of all components, and to identify and solve possible problems before scaling the system to 61 bers. Indeed, it has been observed that it is absolutely indispensable to coil the power ampliers to a full circle and to implement a good shielding of the whole setup from air ows in order to avoid phase and pointing instabilities in the beams.

At the same time, the assembly of a hexagonal ber array in the laser head by using a simple ber stacking approach worked well, and provided precisions close to the ones required in order to avoid losses in combining eciency. The careful design of a suitable ber ending allowed to make the ber insertion process easy and fail-safe.

Once assembled, the laser system was operated both in linear and nonlinear regime. The obtained results are highly satisfying, with combining eciencies of η = 0.48 and η = 0.45, respectively, and very good spatial and spectral beam properties.

The insights gained in these experiments provide valuable information for the design and assembly of the 61 ber system. Together with the very promising results obtained on the seven ber setup, they pave the way for a straightforward and fail-safe setup and operation of the upscaled CBC laser system.

5 Conclusion and Outlook

In this thesis, a seven high power ber amplier coherent beam combining system in tiled aperture conguration has been presented.

This work has been performed within the scope of the XCAN project, which aims at demonstrating the scalability of CBC setups by implementing an upscaled version of this seven ber system consisting of 61 ampliers. Such highly scalable architectures are of particular interest since potential applications such as laser-based particle acceleration would require the coherent combination of up to 10 000 bers.

It was therefore necessary to identify a system design which could theoretically support such a high number of bers and still guarantee a good combining eciency. A review of dierent architectures and phase matching techniques has shown that a tiled aperture conguration and an interferometric phase measurement method are best suited for this purpose. In such a setup, the beams are arranged in a hexagonal array and collimated with a microlens array to form the initial near eld distribution, and the combined beam is given as the main lobe of the corresponding far eld. The quality of the beam combining can then be evaluated via the combining eciency, given as the ratio between the average power contained in the combined beam and the overall average output power of the ber ampliers.

It has then been simulated at which point the combining eciency is sensitive to misalignments in the ber or the microlens array. It has been shown that a precision on the order of a micrometer is necessary on the lateral alignment of the bers in order to limit losses in combining eciency to less than one percentage point below the optimum value. In a similar way, misalignments on the ber tilts and the focal distances need to be limited to some milliradians and tens of micrometers, respectively. On the microlens array alignment, the requirements are less severe, but still on the order of a few micrometers on the lateral alignment, and only ∼ 0.1 mm on the focal distance.

Most importantly, the rotation of the microlens array with respect to the ber array needs to be matched to less than 0.2 mrad.

Furthermore, disparities in the spectral properties of the beams have been modeled. It has been shown that especially for low numbers of emitters, the losses in combining eciency signicantly depend on the number of bers, with higher losses for larger channel numbers. For the case of N = 61 bers, the residual phase error needs to be as low as λ/48 RMS in order to limit the decrease in combining eciency to one percentage point. Moreover, the rst and second order spectral phase mismatches should not exceed 10 % of the pulse duration or the square of this value, respectively. Last but not least, dierences in nonlinear phase should be limited to B-integral discrepancies of 0.32 rad.

These requirements pose high demands on the design of the CBC system. In order to meet the spatial accuracies stated above, a ber stacking approach has been implemented, where the ber endings are slid into high precision ferrules which are stacked into a V-shape holder. This approach yielded the requested precisions on the ber tilts and focal distances, whereas the errors on the lateral alignment of the bers are estimated to about 2.6 µm and will give rise to a decrease in combining eciency of about 4 %.

In the spectral domain, phase dierences of second order or higher are corrected for by matching the ber lengths of all the channels to ±3 cm. Path length dierences inferior to this value are adjusted with free space variable optical delay lines within a range of ∼ ±10 µm, which cancels any disparities of the rst order spectral phase. The remaining zeroth order is adjusted in real time using piezomechanical ber stretchers controlled by a phase matching algorithm.

When the system was operated in closed loop, a residual phase error of λ/48 RMS was measured on a single ber. Hence, the phase matching algorithm works well enough to avoid losses in combining eciency even in the upscaled version of our laser system. The combining eciency was measured to be η = 0.48, corresponding to 71 % of the eciency which could be reached in an ideal system, and to 86 % of the maximum eciency simulated by including the misalignments present in our setup. Furthermore, we measured a power stability of the combined beam of 1.7 % RMS and an M 2 of 1.10 on both axis. Finally, the combined beam was temporally compressed, yielding a nearly Fourier transform limited pulse width of 216 fs, and a time-bandwidth product of 0.53.

In a last series of measurements, we studied the impact of nonlinear eects on the combining eciency. Indeed, for B-integrals between 5 rad and 13.5 rad, η drops from 0.45 to 0.37. This decrease can be attributed to an increase of the discrepancies among the B-integrals of the dierent channels, which is a consequence of an increase of nonlinearities. However, at the intended working point of our laser system at B = 5 rad, the combining eciency is only slightly lower than in linear regime, whereas the temporal pulse prole starts to get degraded by the appearance of secondary pulses. Hence, this is the limiting factor of a further increase in peak power of our seven ber laser setup, just as it is the case for conventional systems.

However, the coherent beam combining architecture allows to overcome this limitation by including additional ber ampliers in the beam array. Indeed, the upscaling of the current seven ber setup to the nal 61 ber amplier CBC system is an essential part of the XCAN project.

In anticipation of this increase in channel number the seven ber setup has been designed such that it can easily be expanded to host all 61 bers. The implementation of this upscaled version is currently being realized. First photos taken during the assembly are shown in g. 5.1.

The immediate next step within the XCAN project will be the nishing of this setup and subsequently its characterization. The main dierence between this larger setup and the seven ber one will be that in the latter case, it was possible to manipulate each individual ber in order to obtain best results. In the 61 ber setup, this will not be possible due to the large number of bers and the inaccessibility of the completely assembled ber array. It is therefore necessary to perform the initial alignment as carefully as possible in order to minimize alignment errors.

Afterwards, no correction of misalignments will be possible, and the system will be characterized by the error distribution present on every parameter.

The nished 61 ber demonstrator could be involved in dierent tasks. First, XCAN is part of the 4F-project ("Filière Française de Fibres optiques pour les lasers de l'industrie du Futur" -French sector of optical bers for the industrial lasers of the future), which aims at the development of new photonic crystal bers for use in scientic and industrial applications. As the performance of the XCAN laser system highly depends on the ber ampliers in use, it would be instructive to test newly developed bers in the XCAN setup.

Second, it is intended to implement a post-compression setup at the output of the current laser system. Indeed, reducing the pulse duration from the current ∼ 220 fs to about 30 fs or even 3 fs would adapt the laser system to elds of applications such as high harmonics generation. The post-compression will be performed by passing the compressed laser pulses through a gas-lled capillary, so that the pulse spectrum gets signicantly broadened by self-phase modulation. In a subsequent compression stage the accumulated second order spectral phase is removed, so that the pulse duration is reduced close to the new Fourier transform limit. For the post-compression to 3 fs, a stabilization of the carrier-envelope phase of the combined pulses would need to be implemented in order to ensure that the maxima of the carrier wave and the envelope are well synchronized.

Moreover, it is also considered to increase the combining eciency by implementing one or two phase plates, as has been demonstrated in [START_REF] Dortz | Mise en phase active de bres laser en régime femtoseconde par méthode interférométrique[END_REF].

A further upscaling of the performances of the laser system is currently not scheduled, but would be easily possible. Measures which could be taken for this purpose involve increasing the number of channels by including additional rings of bers in the hexagonal array, or by increasing the average output power by adding a second pump diode to each amplier. Moreover, it could be studied whether it is possible to implement a tiled aperture conguration with rod-type bers, which could considerably increase both the peak and average powers of the laser system. Finally, new ber architectures are also investigated in order to enable a backwards pumping scheme in tiled aperture conguration. Due to the close proximity of the bers in the laser head, this is technically challenging. However, there exist promising approaches which consist in putting small dichroic mirrors at the output of the ampliers in order to couple the pump beam in the ber.

An implementation of this technique could reduce the nonlinear eects in the system and thus increase the maximum possible peak powers.

Such an upscaling would approach the system to the requirements imposed by applications such as particle acceleration, and could thus pave the way to a new generation of high peak power and high average power ultrafast laser systems.

Title : Highly Scalable Femtosecond Coherent Beam Combining System of High Power Fiber Amplifiers

Keywords : Coherent beam combining, femtosecond lasers, fiber amplifiers Abstract : Future applications of high power ultrafast laser systems require simultaneously high average and peak powers. A technique which has proved to be capable of meeting these demands is coherent beam combining (CBC). In this technique, the beam is spatially split prior to amplification, and coherently recombined in one single beam afterwards. In order to achieve an efficient recombination, the spatial and spectral properties of all beams need to be perfectly matched. For applications such as particle acceleration, the coherent combining of several thousands of fibers needs to be considered. It is thus necessary to investigate highly scalable CBC architectures. The XCAN project aims at a first demonstration of such a scalable setup by coherently combining 61 fiber amplifiers. In order to study the scientific and technical challenges of such a system, a downscaled version consisting of seven fibers has been implemented. The design and characterization of this prototype is the subject of this thesis. As a starting point, numerical simulations have been performed in order to estimate the maximum tolerable mismatches between the spatial and spectral properties of the beams. Based on this modeling work, a seven fiber CBC system has been assembled and characterized. The obtained results are very promising and imply that our setup is well suited for the accommodation of all 61 fibers of the final XCAN demonstrator.

Figure 1 . 1 :

 11 Figure 1.1: Illustration of a laser pulse train with characteristic timescales.

Figure 1 . 2 :

 12 Figure 1.2: Schematic of a Gaussian beam.

Figure 1 . 3 :

 13 Figure 1.3: A Gaussian (red) and a top-hat (black) beam with the same 1/e 2 beam width.

  Frequency chirp ϕ2.

Figure 1 . 4 :

 14 Figure 1.4: Inuence of dierent components of the spectral phase on the electric eld in the time domain.

Figure 1 . 5 :

 15 Figure 1.5: Schematic illustrating the light guidance due to total internal reection in a step-index optical ber. Picture taken from [28].

Figure 1 . 6 :

 16 Figure 1.6: Working principle of a double-clad ber. Picture taken from [31].

Figure 1 . 7 :

 17 Figure 1.7: Picture of the cross sections of (a) a polarization-maintaining ber, taken from [40],

  Schematic of the main transitions. (b) Absorption (solid) and emission (dotted) cross sections, taken from [41].

Figure 1 . 8 :

 18 Figure 1.8: Spectroscopic properties of an Yb-doped ber amplier.

  (a) Pump power. (b) Signal power.

Figure 1 .

 1 Figure 1.10: (q) Pump and (b) signal power as a function of position along the ber amplier for both (1) forward and (3) backward pumping. The case of bidirectional pumping, a combination of the two other cases, is shown as well (2). The gure shows the qualitative behavior in both cases, since the given dimensions do not apply to our specic setup. Pictures taken from[START_REF] Parvin | Small signal gain and saturation intensity of a Yb:Silica ber MOPA system[END_REF].

Figure 1 . 11 :

 111 Figure 1.11: Working principle of chirped pulse amplication.

Figure 2 . 1 :

 21 Figure 2.1: General setup of a CBC system with active phase control.

Figure 2 . 2 :

 22 Figure 2.2: Schematic of dierent CBC architectures, with (a) and (b) a lled aperture conguration, and (c) a tiled aperture.

  2.3b. Since only completely lled hexagons make sens, this geometry gives rise to odd numbers of combined channels such as N = 7, 19 or 61. This number is related to the normalized diameter N D of the hexagon by N = 1 + 3 4 (N 2 D -1).

N = 7 , N D = 3 N

 73 = 19, N D = 5 N = 37, N D = 7 N = 61, N D = 9 (b) Hexagonal "honeycomb" array with corresponding number of beams N and normalized diameter ND. The green line indicates the shape of the actual seven ber aperture, while the blue line shows its approximation by a circle.

Figure 2 . 3

 23 Figure 2.3

  Lineout of the near eld with characteristic length scales.

  Lineout of the corresponding far eld, where the impact of ΠL is not shown.(c) Fourier transform of a hexagonal aperture. Note the logarithmic color scale.

Figure 2 . 4 :

 24 Figure 2.4: Lineout of the (a) near and (b) far eld of a tiled aperture conguration. (c) shows the two-dimensional Fourier transform of a hexagonal aperture function.

Figure 2 .

 2 Figure 2.5: (a) Schematic illustrating the phase measurement setup, adapted from [72]. For the sake of clarity, the tilt angle α of the reference beam is heavily exaggerated, and the imaging system in front of the camera is not represented. (b) Resulting interference fringes on the camera for N = 7 beams.

  Perfect overlap and good contrast not possible due to dierent chirps.

Figure 2 . 6 :

 26 Figure 2.6: (a) -(c) Interference fringe pattern and corresponding time-frequency mapping for dierent delays between two chirped pulses. (d) Situation for two pulses with dierent chirps.

Figure 2 . 7 :

 27 Figure 2.7: Schematic illustrating the dierent delays of the pulses due to the tilt angle of the reference beam, adapted from [73].

Figure 2 . 8 :

 28 Figure 2.8: Schematic illustrating the phase measurement based on correlation calculations. The recorded camera signal S is shown in blue and the reference patterns R i in red. The measured phase is the one used to calculate the reference pattern yielding the highest correlation coecient C. Picture taken from [20].

Figure 3 . 1 :

 31 Figure 3.1: Schematic of the simulated setup.

(3. 3 )

 3 which thus describes a hexagonal array of N truncated Gaussian beams as shown in g. 3.2a. It is thus equivalent to the expression given in eq. (2.2) (section 2.2.1), and was preferred here in order to simplify the implementation in the numerical simulations. (a) Near eld. (b) Far eld. (c) Vertical lineout of the far eld with r ml indicating the relative position of the rst minimum.

Figure 3 . 2 :

 32 Figure 3.2: Near eld and far eld for N = 61 beams. The peak intensities have been normalized to one and the color scale in (b) has been adjusted such that the secondary lobes become visible.

Figure 3 . 3 :

 33 Figure 3.3: (a) Truncation eciency and (b) recovered near elds as a function of aperture radius r for N = 61 beams. Note that the output elds are scaled arbitrarily.

  of the microlens ll factor τ r . The results for N = 61 beams are shown in g. 3.4b, and are highly similar to those obtained with only seven beams. The optimal ll factor is thus given independently from the number of bers by τ r = 0.93, and the corresponding seven-ber elds are shown in g. 3.4a (center). For this case, the microlens transmission is given by t µ-lens = 0.91 and the overall combining eciency reaches its optimum value of η ideal = 0.67. τ r = 0.4 τ r = 0.9 τ r = 1.3 (a) (left) Near elds and (right) far elds for N = 7 beams and dierent τr. (b) Combination eciency η as a function of microlens ll factor τr for N = 61 beams.

Figure 3 . 4 :

 34 Figure 3.4: Microlens ll factor τ r .

Figure 3 .

 3 5b shows a typical eciency curve, in gray for seven bers and in black for 61, obtained here for errors on the lateral alignment of the bers. It can be seen that the eciencies themselves vary only slightly with the ber number, which is why (a) Example far eld for N = 61 beams and ∆x = 5 µm (b) Example eciency curve for errors on the alignment of (gray) seven and (black) 61 bers. (c) (red) Gaussian probability distribution with standard deviation σ = ∆x = 1 µm and example histograms for (gray) 61 and (black) seven random variables drawn from this distribution.

Figure 3 . 5 in

 35 Figure 3.5

Figure 3 . 6 :

 36 Figure 3.6: Fiber tilt.

Figure 3 . 7 :

 37 Figure 3.7: Fiber displacement.

Figure 3 . 8 :

 38 Figure 3.8: Focal errors.

Figure 3 . 9 :

 39 Figure 3.9: Output power.

+ y 2 w 2 0 .( 3 . 16 )

 0316 3.10a. The amplitude of the beam is thus reduced by a factor p x = | p cos θ|, so thatE = p x exp -x2This expression resembles the one given above for disparities in the output power. However, the Schematic of a polarization vector which is turned by an angle θ with respect to the main polarization p, and its projection on p. ; (b) Eciency losses due to dierences in polarization.

Figure 3 . 10 :

 310 Figure 3.10: Beam polarization.

  Two example near elds corresponding to this case are shown in g. 3.11a (top left and right) for = 0.6 and = 1.4, respectively. As consequence of such elliptic beams, the microlens clipping is reduced on the short diagonal of the beam ellipse, but increased on the long one. Since the microlenses have a hexagonal shape, the clipping will be less important if the long diagonal of the ellipse is oriented (a) (top) Near elds and (bottom) far elds for an ellipticity of (left) = 0.6 and (right) = 1.4 common to all beams. The color scale of the far elds is the same as in g. 3.2b. (b) Eciency losses due to an ellipticity common to all beams.(c) Eciency losses due to random errors on the ellipticity of each beam, where a global ellipticity of = 1.07 was applied.

Figure 3 . 11 :

 311 Figure 3.11: Beam ellipticity.

3 . 1 ,Figure 3 . 12 :

 31312 Figure 3.12: Microlens displacement.

  where the distances from the ideal position ∆x rot are given as a function of the rotation angle ϕ rot , yielding∆x rot = x f (1 -cos ϕ rot ) -y f (1 -sin ϕ rot ) (3.18)∆y rot = x f (1 -sin ϕ rot ) + y f (1 -cos ϕ rot ),

Figure 3 . 13 :

 313 Figure 3.13: Microlens rotation.

(3. 22 )

 22 If these two are well matched, only a small residual parabolic phase remains on each beam, given by ∆p pd dp f (a) Schematic illustrating a pitch dierence between the ber and microlens array. The dotted red lines are the individual wavefronts, which can be approximated and partially compentated for by a parabolic phase (solid line) after suitable phase oset compensation (dashed lines). (b) Eciency losses due to a pitch dierence for an array of (black) 61 and (gray) 7 bers, (top) without and (bottom) with correction of second order phase. Note the dierent scaling of the x-axis. (c) Far elds (top) without and (bottom) with correction of second order phase for ∆p = 3.5 µm and 40 µm, respectively.

Figure 3 . 14 :

 314 Figure 3.14: Microlens pitch.

ϕ

  res = k(z -z l ) = π λ ∆p pf (x -x l ) 2 ,(3.23)and leading to a small defocalization of each individual beam. This explains the excess of power in the side lobes of the corresponding far eld (g. 3.14c (bottom)) which are typically an indication of a defocalization. The resulting losses in combining eciency are shown in g. 3.14b (bottom).

  3.15a). As consequence of a focal error, the side lobes of the far eld become more prominent, as shown in g. 3.15c and 3.15d for an error in focal distance of 0.6 mm and 1.0 mm, respectively. As for a focal error of a single ber, the change in microlens ll factor due to a change in the focal distance and the corresponding losses in combining eciency are negligible (cf. section 3.3.1 on page 52). The curve shown in g. 3.15b indicates that the error on the focal distance of the microlens array should not exceed 0.13 mm, which corresponds to 0.25 % of its focal length, and to 19 % of the beam's Rayleigh range. ∆f (a) Schematic of an error in the focal distance of the microlens array. (b) Eciency losses due to an error in the focal distance of the microlens array (c) Far eld resulting from a focal error of ∆f = 0.6 mm (d) Far eld resulting from a focal error of ∆f = 1.0 mm

Figure 3 . 15 :

 315 Figure 3.15: Microlens focal distance.

Figure 3 . 16 :

 316 Figure 3.16: Microlens tilt.

Figure 3 .

 3 Figure 3.17c shows the corresponding losses in microlens transmission t µ-lens , far eld eciency η FF and overall eciency η = η FF × t µ-lens .

  (a) Example near eld with g = 800 µm. (b) Corresponding far eld.

  (a) λ = 1032 nm (b) λ = 1033.5 nm (c) λ = 1035 nm

Figure 3 . 18 :

 318 Figure 3.18: Example far elds for three dierent wavelengths from the center of the spectrum to its edge, calculated with a Gaussian spectral envelope and rst order phase of ∆ϕ 1 = 250 fs.

  (a) Overall far eld. (b) λ = 1038 nm (c) λ = 1040 nm

Figure 3 . 19 :

 319 Figure 3.19: Inhomogeneous spectral content of the far eld for large phase coecients. The red plus marks the position of the main lobe of the overall far eld.

Figure 3 .

 3 19b depicts the far eld calculated for a wavelength of λ = 1038 nm, where the intensity maximum is clearly shifted below and to the left of the position of the main lobe. Hence,

  one with the same intensity FWHM and |A sG (∆ω)| 2 = exp -(∆ω) 4 δω 4

  spectrum with the same intensity FWHM. (b) Corresponding eciency curves.

Figure 3 . 20 :

 320 Figure 3.20: Impact of the spectral envelope on the combining eciency.

  Figure 3.21: Example (a) spectra and (b) temporal pulse proles of (gray) an ideal combined beam and (black) a combined beam exhibiting a spectral phase of ∆ϕ 1 = 250 fs.

Figure 3 .

 3 Figure 3.22: (left tiles) Area of the main lobe for ideally combined beams, and (right tiles) corresponding area for incoherently combined beams, for (a) seven and (b) 61 beams. The ratio between the power contained in the red circles of the incoherently combined beams and the overall power of these beams denes the lower limit of the combining eciency.

Figure 3 . 23 :

 323 Figure 3.23: Phase oset ∆ϕ 0 .

Figure 3 . 24 :

 324 Figure 3.24: Delay ∆ϕ 1 .

Figure 3 . 25 :

 325 Figure 3.25: Group delay dispersion ∆ϕ 2 .

Figure 3 . 26 :

 326 Figure 3.26: Third order dispersion ∆ϕ 3 .

(3. 39 )

 39 Furthermore, as the pulses are subject to a strong temporal stretching in a CPA setup, the temporal pulse prole exhibits the same shape as the spectral one. It is therefore possible to substitute A(t) by A(∆ω) and P spec (t) by P spec (∆ω) = |A(∆ω)| 2 , where the correspondence between t and ∆ω is given by ∆ω(t) = t/ϕ 2 , as detailed in section 1.1.1. Hence, the nonlinear phase can be expressed as ϕ NL = B × |A(∆ω)| 2 , and the overall eld of one beam becomes E ϕ NL (x, y, ω) = E(x, y)A(∆ω) × exp iB|A(∆ω)| 2 . (3.40) Assuming that all beams exhibit the same spectrum, dierences in nonlinear phase are solely caused by disparities among the B-integrals of the beams. Those can arise due to dierences in the amplication conditions, such as slight mismatches in amplier length or variations in pump or seed signal power.

  (3.40) |A(∆ω)| 2 has to be replaced by P spec (∆ω) = |A(∆ω)| 2 -1 -∂|A(∆ω)| 2 ∂ω ∆ω. However, for the special case of the symmetric spectrum used in these simulations, ∂|A(∆ω)| 2 ∂ω = 0, so that P spec (∆ω) simplies to P spec (∆ω) = |A(∆ω)| 2 -1. The nal electric eld used in the simulations reads thus E ϕ NL (x, y, ∆ω) = E(x, y)A(∆ω) × exp(iP spec (∆ω)∆B).

(3. 41 )

 41 The corresponding eciency curves are shown in g. 3.27. For seven bers, the dierences in B-integral should be limited to ∆B = 0.46 rad, whereas for 61 bers, ∆B = 0.32 rad should not be exceeded. (a) Seven bers. (b) 61 bers.

Figure 3 . 27 :

 327 Figure 3.27: Nonliner phase ∆ϕ NL = ∆B.

Figure 4 . 2 :

 42 Figure 4.2: Schematic of the front end of our laser system.

Figure 4 . 3 :

 43 Figure 4.3: Photo of the front end.

Figure 4 . 4 :

 44 Figure 4.4: Photo of a phase and delay control element.

  (a) Output spectra of the front end operated at 55 MHz, whith attenuators in front of all preampliers (light blue), or in front of all but the LMA preampliers (dark blue). The shifted bandwidth is due to a dierent transmission of the pulse shaper. (b) Output spectrum of the front end operated at 2 MHz. (c) Distorted spectra of an uncoiled power amplier, recorded in the top and bottom of the output beam.

Figure 4 . 5 :

 45 Figure 4.5: Dierent spectra recorded in the front end.

Figure 4 . 6 :

 46 Figure 4.6: Photo of a combiner (silver box) and an MFA (black box) taped to a breadboard. On the left, one signal and two pump input bers are visible.

Figure 4 . 7 :

 47 Figure 4.7: Picture of the seven power ampliers in operation. Note the 90 • bending near the laser head. The green color is due to orescence in the visible wavelength range.

Figure 4 . 8 :

 48 photodiode

  (a) With everything operating normally. (b) Without air conditioning. (c) Without air conditioning and pre-amplier fans.

Figure 4 . 9 :

 49 Figure 4.9: Phase noise spectra of the front end for dierent measurement conditions.

  microlens array. Due to its large divergence, the pump of one ber impinges on several neighboring Mechanical drawing of a small size test version of our laser head, illustrating the working principle. The ber ampliers are not shown.(b) Photo of the laser head with ve bers inserted. The large spacing between the bers was chosen for preliminary tests.

Figure 4 . 10 :

 410 Figure 4.10: Laser head.

  (a) Photo of the laser head assembly. (b) Temperature without water cooling (Tmax = 79.1 • C). (c) Temperature with water cooling (Tmax = 48.3 • C).

Figure 4 .

 4 Figure 4.11: (a) Photo and (b), (c) false-color images of the laser head, the microlens array and the blocking plate, taken with an infrared camera. Note the dierent color scales.

  4.12a, and a photo of a ber ending in g. 4.12b. The design has been developed in close collaboration with Optical Engines, who also took over its manufacturing. Schematic of a ber inserted in the laser head (not to scale). Photo of a ber ending, taped on a white plastic block.

Figure 4 . 12 :

 412 Figure 4.12: Final ber ending design.

  (a) Incoherent far eld. (b) Near eld. (c) Near eld with interference fringes.

Figure 4 . 13 :

 413 Figure 4.13: Fields used for the alignment of the ber array and the characterization of the laser head.

  Distance larger than the microlens focal distance.

Figure 4 . 14 :

 414 Figure 4.14: Example of the shape of a single beam at the position of the far eld, while changing the distance between the microlens array and the laser head.

  datasheet, which entails a signicant increase of the optimum microlens focal length. Hence, in the experimental setup using the smaller focal length, the beams have not yet reached their optimum size when they hit the microlens array. This reduced beam width results in an enhancement of the side lobes of the far eld as shown in g. 4.15b and a reduction of the maximum possible combining eciency to η(τ r = 0.7) = 0.61.

  (a) Ideal far eld. (b) Far eld for a microlens ll factor of τr = 0.7.

Figure 4 . 15 :

 415 Figure 4.15: Simulated far elds.

  Near eld image of three lenses. (b) High precision photo of one lens. The cross is a leftover of a mask applied during the manufacturing process.

Figure 4 . 16 :

 416 Figure 4.16: Estimations of the gap between the microlenses.

Figure 4 . 17 :

 417 Figure 4.17: Schematic of the laser back end (not to scale). The red symbols indicate the polarization of the beam. The far eld is located at the spatial ltering aperture, and the combined beam is found by truncating it at its rst extinction.

  (a) Mount of the spatial ltering pinhole. The metallic adapters to the left can be used to connect the hoses for water cooling. The brownish color is due to excessive heating of an uncooled mount. (b) Unused pinhole with aperture diameter marked on the piece. (c) Filtering pinhole after use in the experiments. The impact of the six side lobes of the far eld are visible on the discolored area near the hole itself.

Figure 4 . 18 :

 418 Figure 4.18: Components used for the spatial ltering of the far eld.

Figure 4 . 19 :

 419 Figure 4.19: Photo of the compressor used in our laser system.

  photodiodes.

Figure 4 . 20 :

 420 Figure 4.20: Phase noise measurement.

Figure 4 .

 4 Figure 4.20b shows a typical result of a phase noise measurement, in this case taken between the two topmost bers of the hexagonal array. The open loop phase noise characterizes the impact of the noise present in the lab on the free-running ber ampliers. In closed loop, the active phase locking reduces the phase noise in the low frequency regime by up to ve orders of magnitude.

  Truncation eciency of (gray, dotted) an ideal far eld, (gray, dashed) a far eld with reduced microlens ll factor, and (black, solid) the measured far eld.

Figure 4 . 21 :

 421 Figure 4.21: Characterization of the experimental far eld.

  For comparison, also the curves of an ideal far eld, and of one with the reduced microlens ll factor τ r = 0.7 are shown. Clearly, the experimental curve is smoother than the theoretical ones, which indicates less clear extinctions between the dierent lobes, and a larger amount of power concentrated outside of the main lobe. These characteristics can also be seen in the lineouts of the far elds shown in g. 4.22, and are due to the misalignments discussed above.(a) Vertical lineouts.(b) Horizontal lineouts.

Figure 4 . 22 :

 422 Figure 4.22: Lineouts of (gray, dotted) an ideal far eld, (gray, dashed) a far eld with reduced microlens ll factor, and (black, solid) the measured far eld. The peak intensities of all lineouts are normalized to one.

Figure 4 . 23 :

 423 Figure 4.23: Power noise spectrum of the combined beam.

Figure 4 .

 4 Figure 4.24a shows the position of the ltering aperture in the experimental far eld for a truncation at the rst extinction. The ltered beam is thus equivalent to the combined beam, and the picture of it shown in g. 4.24b conrms the neat ltering and the very clean shape of the combined beam. Since this beam is the main lobe of the far eld, its shape can be described by an Airy pattern which is clipped at its rst zero. The very good agreement between the lineout of the measured far eld and the tted function is shown in g. 4.25a and conrmed by the correlation coecient of the two curves which equals ρ = 0.9999.

Figure 4 . 24 :

 424 Figure 4.24: Measured elds for spatial ltering at the rst extinction.

7

 7 can be found in the upper line of g. 4.27. The left image shows the complete far eld and where it will be truncated (white circle), the middle one illustrates an intermediate step in the beam propagation, and the right image displays the recovered near eld. (a) Truncated far eld. (b) During beam propagation.(c) Recovered near eld.

Figure 4 .

 4 Figure 4.27: (Top line) Simulation and (bottom line) measurement of the truncated beam (a) in the far eld, (b) during beam propagation, and (c) in the recovered near eld, for a large ltering aperture.

  (Blue) measured and (red) simulated autocorrelation trace. (b) Spectrum of the combined beam.

Figure 4 .

 4 Figure 4.28: Temporal compression of the combined beam.

  bining eciency has been measured by truncating the resulting far elds at their rst extinction. These measurements have been performed for the four pulse repetition rates of 2 MHz, 1.5 MHz, 1 MHz and 0.75 MHz. The corresponding far elds are shown in g. 4.31a, where a far eld recorded in linear regime has been included for comparison. It can be seen that the increasing amount of nonlinear eects leads to an increased speckle pattern around the main lobe. In consequence, the combining eciency decreases, as shown in g. 4.30.

Figure 4 . 30 :

 430 Figure 4.30: Combining eciencies as a function of pulse repetition rate. The dashed line indicates the theoretical lower limit.

Figure 4 .

 4 Figure 4.31: Comparison of the main results at increasing levels of nonlinearity. To the left, the dierent working points and the corresponding measured combining eciencies η are specied. Next to it, the recorded far elds are represented. The color scale is the same as in the other gures of this chapter. The remaining two columns show the autocorrelation traces and the spectra of the temporally compressed combined beams. An overall decrease in system performance can be observed for increasing B-integrals.

  (a) An artist's impression of me and my collegue working on a partially assembled 61 ber setup. (b) The rst 18 bers in operation.

Figure 5 . 1 :

 51 Figure 5.1: Impressions of the 61 ber setup.

  du système.Les travaux de cette thèse s'inscrivent dans le cadre du deuxième point du projet, c'est-à-dire l'assemblage et la caractérisation du prototype à sept bres amplicatrices. En premier lieu, des simulations ont été eectuées an d'évaluer les désaccords entre les faisceaux qui peuvent être tolérés sans entraîner une dégradation importante de l'ecacité de combinaison. Ces résultats ont permis de dénir notamment les contraintes spatiales et spectrales à imposer au système expérimental.Celui-ci a ensuite été caractérisé. L'erreur résiduelle de phase sur une bre individuelle est inférieure à λ/38, ce qui indique une très bonne stabilité en phase. Une ecacité de combinaison de 48 % a été mesurée en régime linéaire, ce qui correspond à 86 % de ce qui est théoriquement accessible compte tenu des défauts de l'assemblage expérimental. De plus, une excellente qualité spatiale du faisceau recombiné a été mesurée, avec M 2 = 1.10 sur les deux axes. Finalement, l'impulsion combinée a été comprimée à une durée temporelle de 216 fs, ce qui est très proche de la limite par transformée de Fourier. En régime non-linéaire, les distorsions engendrées sur les sept faisceaux n'ont qu'un impact modéré sur l'ecacité de combinaison (qui se réduit à 45 % à un intégral B de 5 rad), démontrant la robustesse du système dans des conditions moins favorables à haute énergie. En conclusion, les résultats présentés ci-dessus montrent que le système de combinaison cohérente auparavant implémenté avec 19 bres passives est bien adapté au fonctionnement en régime actif. Le dispositif conçu, mis en place et caractérisé dans le cadre de cette thèse a donc parfaitement rempli son rôle de prototype exploratoire et est maintenant prêt à accueillir 61 bres, ouvrant ainsi la voie à une nouvelle génération de lasers à hautes puissances crête et moyenne. Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France Titre : Architecture évolutive de combinaison cohérente femtoseconde pour amplificateurs à fibre de puissance Mots clés : Combinaison cohérente de faisceaux, lasers femtoseconde, amplificateurs à fibre Résumé : Allier de fortes puissances moyennes et crêtes donne accès à un champ applicatif très large pour un système laser ultrarapide. Une technique qui s'est avérée capable de satisfaire ces exigences est la combinaison cohérente de faisceaux (CBC). Elle permet de séparer spatialement les faisceaux avant l'amplification pour les recombiner ensuite d'une manière cohérente en un unique faisceau. Afin d'obtenir une recombinaison efficace, les propriétés spatiales et spectrales de tous les faisceaux doivent être parfaitement en accord. Pour des applications comme l'accélération de particules, le recours à plusieurs milliers de fibres doit être envisagé. Il est donc nécessaire d'étudier des architectures CBC fortement évolutives en termes de canaux amplificateurs. Le projet XCAN vise à une première démonstration d'un tel système en réalisant la combinaison cohérente de 61 fibres amplificatrices. Afin d'étudier les défis scientifiques et techniques d'une telle architecture, une version de taille réduite comprenant sept fibres a été mise en place. La conception et la réalisation de ce prototype sont le sujet de cette thèse. Dans un premier temps, des simulations ont été effectués afin d'estimer les désaccords tolérables entre les propriétés spatiales et spectrales des différents faisceaux. Basé sur ce travail de modélisation, un système laser de combinaison cohérente de sept fibres a été ensuite assemblé et caractérisé. Les résultats obtenus sont très prometteurs et montrent que notre architecture est bien adaptée pour accueillir les 61 fibres du démonstrateur final XCAN.

Table 3 .

 3

	criterion	maximum	maximum	criticality
		absolute error	relative error
	ber displacement	1 µm	0.03 %	high
	ber tilt	2.5 mrad	8.9 %	medium
	focal errors	0.08 mm	12 %	medium
	beam polarization	7 •	-	medium
	beam ellipticity	0.12	-	low
	output power	25 %	-	low
	microlens rotation	0.16 mrad	-	high
	microlens pitch	10 µm	0.3 %	high
	microlens gap	30 µm	0.9 %	high
	microlens displacement	1.4 µm	0.04 %	medium
	microlens focal	0.13 mm	19 %	medium
	microlens tilt	1 •	-	medium

1: Recapitulation of the modeled misalignments, their maximum tolerable errors if eciency losses shall be limited to one percentage point, and how critical they will be in the alignment.

  3.23. It can be observed that the distribution of phase osets should not exceed π/19 rad or λ/38 RMS in case of CBC of seven bers, and π/24 rad or λ/48 RMS in case of 61 bers.

  Nonlinear Phase ϕ NL Last but not least, the beams can exhibit a nonlinear phase ϕ NL = k n 2 I(z, t) dz = γ NL P (z, t) dz This expression can be simplied by separating P (z, t) into a spatial and a normalized temporal part, so that P (z, t) = P peak (z)P spec (t), and ϕ NL (t) = γ NL P ( z, t) dz = P spec (t) × γ NL P peak (z) dz = P spec (t) × B.

	(3.37)
	as introduced in section 1.1.2. (3.38)
	Inserting this phase in the general expression of a beam exhibiting a nonlinear phase given in
	eq. (3.33) yields
	E

ϕ NL (x, y, t) = E(x, y)A(t) × exp (iP spec (t)B) .

Table 3 .

 3 2: Recapitulation of the modeled spectral phase disparities, their maximum tolerances if eciency losses in average power shall be limited to one percentage point, and how critical they will be in the alignment. section 2.2.2, and the devices presented in section 4.1.

  4.2.

	Master Oscillator 225 fs, 55 MHz	SPM ber spectral broadening	Pulse Picker 55 MHz → 18 MHz	CFBG Stretcher 225 fs → 500 ps	Pre-Amp → 100 mW
	Pulse Shaper	Pre-Amp	Pulse Picker	Splitter 1x8	Pre-Amp
	spectral phase	→ 100 mW	not used		→ 100 mW
	compensation
	Pulse Picker 18 MHz → 2 MHz ... 50 kHz	∆ϕ phase and delay correction	LMA Pre-Amp → 100 mW	Tap for security	PCF Power Amplier → 25 W

Table 4 .

 4 1: Overview of the estimated errors of our laser head and the resulting losses in combining eciency.

	eciency loss threshold xed in chapter 3. Given that we inserted and removed all of the bers
	criterion	estimated errors	eciency losses
	microlens ll factor τ r	0.7	6 %
	microlens gap	150 µm	3 %
	polarization	< 7 •	negligible
	defocus ∆f	0.04 mm	negligible
	ber tilt ∆θ	0.7 mrad	0.5 %
	ber pitch ∆x	2.6 µm	4 %

* The beam radius at the microlens array w(z) is the width of a Gaussian beam, w(z) = w0 1 + (z/zR) 2 ≈ w0z/zR. The dierence in beam radius ∆w(z) resulting from a dierence in z is thus given by ∆w(z) = w0∆z/zR = w(z)∆z/z. For ∆z = ∆f = 0.5 mm and z = f = 50 mm this yields ∆w(z)/w(z) = ∆z/z = 0.01. Furthermore, since the microlens ll factor is given by τr = 2w(z)/d, it follows that ∆τr = 2∆w(z)/d = 2w(z)/d × ∆z/z = τr∆z/z, which means that τr changes in the same way as w(z).

* This denition is slightly dierent from the one given in section 1.1. However, we are interested in the ratio between two peak powers, for which both denitions yield the same results.

* Simulations taking into account dierent absolute values of B have been presented in[START_REF] Dortz | Mise en phase active de bres laser en régime femtoseconde par méthode interférométrique[END_REF]. Since the calculated losses in combining eciency remain independent of B, these results conrm that only ∆B needs to be considered.

* In the seven ber conguration, the is equivalent to a truncation just before the secondary lobes.

Acknowledgments

I am very lucky that I have been given the opportunity to perform my PhD studies on a cuttingedge research project at one of the most renowned educational institutions of France.

Philippe Adam from the Direction générale de l'armement (DGA, French ministry of defensethis thesis. Also, I'd like to thank the DGA for nancially supporting my work.

Besides, in every of the following subsections only the nal equation describing a certain misalignment will be given in two dimensions. Everywhere else, the y-dimension can be found completely analogous to the x-dimension and has therefore been omitted for the sake of readability.

For the same reason, also the index i has been omitted and the coordinates ((x -x f,i ), (y -y f,i ))

have been replaced by (x, y), even if the given elds describe only one beam, and not necessarily the central one. a Gaussian-like shape. Indeed, the lineout in g. 4.25b shows that the agreement between the measured beam shape and the Gaussian curve is slightly less good than the one obtained in the far eld, but still very high. This is conrmed by the correlation coecient of ρ = 0.9995. The combined beam was obtained by spatially ltering the far eld at the rst extinction next to the main lobe. It exhibits a power stability of 1.7 % RMS, and an excellent spatial beam quality of M 2 = 1.10 on both axes. Finally, the temporal compression of the combined beam resulted in an almost Fourier transform limited pulse with a duration of 216 fs.

Experimental Results in Nonlinear Regime

In order to evaluate the performance of our laser system in the presence of nonlinear eects, it was operated at a constant average power of 25.4 W per ber, but at reduced pulse repetition rates between 2 MHz and 0.75 MHz, corresponding to B-integrals between approx. 5 rad and 13 rad.

Among these values, B = 5 rad is the nominal working point of the laser system, and has been chosen since at this level of nonlinearity, a temporally compressed pulse of reasonable quality can be obtained. As this is hardly possible for higher B-integrals, the measurements under these conditions have been taken in order to evaluate the behavior of the CBC system under the impact of large amounts of nonlinear phase.

The results of these measurements are presented in this section. They include the characterization of both the residual phase noise and the combining eciency for dierent levels of nonlinearities. Furthermore, we study the power stability and the spatial quality of the combined beam in comparison to the linear regime. Finally, the combined beam was temporally compressed at each pulse repetition rate, revealing the impact of increasing B-integral on the temporal pulse shape.

Residual Phase Noise

The residual phase noise has been measured between dierent ber pairs for a pulse repetition rate of 2 MHz and 1 MHz using the same setup as presented in section 4.