Fabio Baronio
email: baronio@ing.unibs.it

Matteo Conforti

Costantino De Angelis

Daniele Modotto

Stefan Wabnitz

Marco Andreana

Alessandro Tonello

Philippe Leproux

Vincent Couderc

Second and third order susceptibilities mixing for supercontinuum generation and shaping

Keywords: nonlinear optics, supercontinuum, microstructured optical fiber

We report our recent development to model quadratic and cubic ultra-broadband nonlinear dynamics in photonic devices, by means of a nonlinear singleenvelope equation. We present the case of generation of tunable visible light from large band conversion, in a quadratic crystal, of the infrared continuum from standard photonic crystal fibers. Moreover, we show the study of a visible supercontinuum generation, initiated by second-harmonic generation, in a quadratic poled germanium-doped microstructured fiber.

Introduction

There is considerable interest in developing robust and compact optical frequency sources in the ultraviolet (UV), visible, infrared (IR) and midinfrared (MIR, i.e. from 2 -12µm) spectral regions for spectroscopy, coherence tomography, frequency metrology, few-cycle pulse generation, optical sensing, and material processing. Nonlinear optical processes are usually required to generate frequencies in regions not directly accessible through well-developed broadband laser gain media. Spectral generation of new frequency components is a inherent feature of nonlinear optics, and has been studied intensively since the early 1960s [START_REF] Bloembergen | Nonlinear optics: past, present, and future[END_REF]. In this context, several promising methods have been considered, including optical parametric oscillators (OPOs) [START_REF] Vodopyanov | Optical parametric oscillation in quasi-phase-matched GaAs[END_REF][START_REF] Phillips | Stability of the singly resonant optical parametric oscillator[END_REF], difference frequency generation (DFG) [START_REF] Levi | Difference frequency generation of 8-m radiation in orientation-patterned GaAs[END_REF][START_REF] Maddaloni | Mid-infrared fibre-based optical comb[END_REF][START_REF] Junginger | Single cycle multiterahertz transients with peak fields above 10MV/cm[END_REF], and supercontinuum (SC) generation [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF][START_REF] Price | Mid-IR Supercontinuum Generation From Nonsilica Microstructured Optical Fibers[END_REF][START_REF] Langrock | Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides[END_REF][START_REF] Schibli | Optical frequency comb with sub-mHz linewidth and > 10 W average power[END_REF].

The particular process known as SC generation occurs when narrow-band incident pulses undergo extreme nonlinear spectral broadening to yield a broadband spectrally continuous output. First reported in bulk glasses [START_REF] Alfano | Observation of selfphase modulation and small-scale filaments in crystals and glasses[END_REF], SC has since been the subject of numerous investigations in a wide variety of cubic nonlinear media, including solids, organic and inorganic liquids, gases, and various type of waveguides. The advent of a new class of optical waveguides in the form of a photonic crystal fiber (PCF) in the late 1990s has attracted widespread interest throughout the scientific community, and led to a revolution in the generation of ultrabroadband high brightness spectra through SC generation [START_REF] Ranka | Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm[END_REF][START_REF] Knight | Photonic crystal fibres[END_REF][START_REF] St | Photonic crystal fibres[END_REF]. SC PCF generation has already found numerous applications in such diverse fields as spectroscopy, pulse compression, and the design of tunable ultrafast femtosecond laser sources. In a telecommunications context, the spectral slicing of broadband SC spectra has also been proposed as a simple way to create multiwavelength optical sources for dense wavelength division multiplexing applications. SC PCFs are designed to provide a flat spectrum over a wide wavelength range, but the produced spectrum is directly dependent on parameters such as fiber material and structure, fiber length, pump power, pump wavelength.

In spite of the great flexibility in generating such a wide range of wavelengths, it is not already demonstrated the possibility to synthetise a well defined amplitude and phase profile of the SC spectrum directly by designing the fiber or other system parameters.

A general and extremely flexible option for reaching wavelengths which are difficult to obtain by employing cubic nonlinearities, or for shaping the spectrum of PCF sources, could be provided by exploiting a mixture of quadratic and cubic effects. In fact, quadratic nonlinear materials can be properly engineered to generate, with high-efficiency, pulses with transform limited duration, predetermined spectral and temporal shapes, and synchronization [START_REF] Conforti | From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion[END_REF][START_REF] Sapaev | Femtosecond pulse synthesis by efficient second-harmonic generation in engineered quasi phase matching gratings[END_REF][START_REF] Conforti | Spectral shaping of femtoseond pulses in aperiodic quasi-phasematched gratings[END_REF]. Examples of the synthesis of super-Gaussian pulses and of pairs of phase-coherent picosecond pulses with a given carrier-frequency detuning, by designing aperiodically poled quadratic ferroelectric domains, were recently given in Ref. [START_REF] Marangoni | Synthesis of picosecond pulses by spectral compression and shaping of femtosecond pulses in engineered quadratic nonlinear media[END_REF][START_REF] Marangoni | Observation of spectral drift in engineered quadratic nonlinear media[END_REF].

In this paper, we report our recent developments in the study of broadband quadratic and cubic interactions. The paper is organized as follows: in Section 2 we introduce a single-envelope model to capture ultra-broadband interactions in quadratic and cubic nonlinear media. Our equation, besides providing a simple and powerful tool for analytical treatment due to its sim-plicity, can be easily solved with a modest computational effort. The only alternative way to numerically describe ultrabroadband quadratic and cubic phenomena is to solve directly Maxwell equations in the time domain, with an immense computational burden. Section 3 reports the case of tunable light sources from large band conversion of PCF continuum by quadratic crystals. Section 4 considers the SC visible generation, initiated by second-harmonic generation, in quadratic poled germanium-doped PCFs. Conclusions appear in Section 5.

Single-envelope model for broadband interactions

Starting from Maxwell equations (written in MKS units), neglecting transverse dimensions (i.e considering the propagation of plane waves in bulk media or bound modes in waveguides), we can write the 1+1D wave equation for the electric field E(z, t) [START_REF] Brabec | Nonlinear optical pulse propagation in the single-cycle regime[END_REF]:

∂ 2 E(z, t) ∂z 2 - 1 c 2 ∂ 2 ∂t 2 +∞ -∞ E(z, t ′ )ε(t -t ′ )dt ′ = 1 ε 0 c 2 ∂ 2 ∂t 2 P N L (z, t). (1)
By defining the Fourier transform F[E](ω) = Ê(ω) = +∞ -∞ E(t)e -iωt dt, we can write (1) in frequency domain:

∂ 2 Ê(z, ω) ∂z 2 + ω 2 c 2 ε(ω) Ê(z, ω) = - ω 2 ε 0 c 2 PNL (z, ω), ( 2
)
where c is the velocity of light in vacuum, ε 0 is the vacuum dielectric permittivity, ε(ω) = 1 + χ(ω), χ(ω) is the linear electric susceptibility and k(ω) = (ω/c) ε(ω) is the propagation constant. In the case of waveguides, k(ω) is the wavenumber of the propagating mode. We now factor out the fast dependence of the propagation coordinate from the electric field, for all frequencies: Ê(z, ω) = Û (z, ω) exp[-ik(ω)z] [START_REF] Brabec | Intense few-cycle laser fields: Frontiers of nonlinear optics[END_REF]. This definition amounts to write the electric field as the product of a spatial carrier wave and a slowly varying envelope. Since we remove the exact propagation constant at every frequency, we can avoid to make a requirement on the bandwidth of the pulses. The wave equation for the field Û reads as:

∂ 2 Û (z, ω) ∂z 2 -2ik(ω) ∂ Û (z, ω) ∂z = - ω 2 ε 0 c 2 PNL (z, ω)e ik(ω)z . (3)
We make the slowly evolving wave approximation (SEWA) [START_REF] Brabec | Intense few-cycle laser fields: Frontiers of nonlinear optics[END_REF], that is

|∂ z Û | << 2k(ω)| Û |
, and thus we can write

∂ Û (z, ω) ∂z = -i ω 2 2ε 0 c 2 k(ω) PNL (z, ω)e ik(ω)z , (4)
and from the definition of Û , we obtain the equation for the electric field:

∂ Ê(z, ω) ∂z + ik(ω) Ê(z, ω) = -i ω 2ε 0 cn(ω) PNL (z, ω). (5)
This equation has been found by an heuristic method, and has been termed

Forward Maxwell Equation (FME) [START_REF] Housakou | Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers[END_REF]. Once specified the form of nonlinearity, we can directly solve Eq.( 5) in frequency domain with a standard split-step Fourier method. When backward waves can be neglected, this equation is equivalent to Maxwell equations [START_REF] Housakou | Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers[END_REF][START_REF] Kinsler | Optical pulse propagation with minimal approximations[END_REF], but the numerical solution is much more efficient.

However, we can simplify further Eq. ( 5) by properly defining an envelope. We consider the electric field E and the nonlinear polarization P N L as the product of a complex envelope and a carrier wave:

E(z, t) = 1 2 A(z, t)e iω 0 t-iβ 0 z + c.c., (6)
P N L (z, t) = 1 2 A p (z, t)e iω 0 t-iβ 0 z + c.c, (7)
that in frequency domain reads:

Ê(z, ω) = 1 2 Â(z, ω -ω 0 )e -iβ 0 z + 1 2 Â * (z, -ω -ω 0 )e iβ 0 z , PNL (z, ω) = 1 2 Âp (z, ω -ω 0 )e -iβ 0 z + 1 2 Â * p (z, -ω -ω 0 )e iβ 0 z ,
where ω 0 is a reference frequency,

β 0 = Re[k(ω 0 )].
Particular care must be devoted to the definition of the complex envelope, since we do not want to impose any limitation to the frequency extent of the signals. For nonlinear media, a proper definition of the envelope is crucial [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF][START_REF] Conforti | Utra-broadband optical phenomena in quadratic nonlinear media[END_REF]. As usual in the theory of modulation [START_REF] Haykin | Communication System[END_REF], we define the analytic representation of the electric field:

Ẽ(z, t) = E(z, t) + iH[E](z, t), (8)
where

H[E](z, t) = 1 π p.v. +∞ -∞ E(z, t ′ ) t -t ′ dt ′ (9)
is the Hilbert transform of the electric field (p.v. indicates the Cauchy principal value of the integral). The Fourier transform of the analytic signal reads:

Ê(z, ω) =              2 Ê(z, ω) if ω > 0 Ê(z, 0) if ω = 0 0 if ω < 0 , (10)
that is a signal that contains only the positive frequency content of the electric field. Due to reality of E(z, t), its Fourier transform has Hermitian symmetry, so that only the positive (or the negative) frequencies carry information, and we can write:

Ê(z, ω) = 1 2 Ê(z, ω) + 1 2 Ê * (z, -ω), (11)
we can define the complex electric field envelope (6) as:

A(z, t) = Ẽ(z, t)e -iω 0 t+iβ 0 z , (12)
i.e. the inverse Fourier transform of the positive frequency content of E shifted towards the low frequency part of the spectrum by an amount ω 0 . No approximations on the frequency extent of the envelope has been done, and

supp{ Â(z, ω)} = (-ω 0 , +∞).
The substitution of expressions of Ê(z, ω) and PNL (z, ω) in Eq. ( 5) and the Taylor-expansion of k(ω) about ω 0 yields:

∂ Â(z, Ω) ∂z + i ∞ m=1 k m m! Ω m Â(z, Ω) = -i ω 2n(ω)cε 0 Âp (z, Ω), (13)
where Ω = ω -ω 0 and k m = ∂ m k ∂ω m (ω 0 ). In order to obtain a time domain equation we have to perform another approximation, i.e. we have to impose that n(ω) is nearly constant in the frequency band of interest. Since we want to take into account ultra-broad spectra, we require the validity of this condition on a band whose order of magnitude is the same of the carrier frequency, that is ∂n (ω) ∂ω | ω 0 ω 0 << n(ω 0 ). Far from resonances, this requirement is fulfilled in the majority of parametric processes in which all waves propagate in the same direction. By inverse Fourier transform we obtain from ( 13)

∂A(z, t) ∂z + iD ′ A(z, t) = -i ω 0 2n 0 cε 0 1 - i ω 0 ∂ ∂t A p (z, t), (14)
where we have defined the dispersive operator

D ′ = ∞ m=1 1 m! k m (-i ∂ ∂t ) m .
As a last step, we consider quadratic and cubic nonlinearities, in particular the following nonlinear polarization:

P N L (z, t) = P (2) N L (z, t) + P (3) N L (z, t) (15) = ε 0 χ (2) E(z, t) 2 + ε 0 χ (3) E(z, t) R(t -t ′ )|E(z, t ′ )| 2 dt ′ . ( 16
)
where the causal R(t) = (1 -f r )δ(t) + f r h r (t) includes both instantaneous electronic and delayed Raman contributions [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. The integrals are in the range (-∞, +∞). As to quadratic nonlinearity, P

N L (z, t) = ε 0 χ (2) E(z, t) 2 = ε 0 χ (2) 4 A 2 e 2iω 0 t-2iβ 0 z + A * 2 e -2iω 0 t+2iβ 0 z + 2|A| 2 . (2)
It is worth noting that, due to the definition of A, the first (second) term in the square brackets contains only positive (negative) frequencies, whereas the third has both. By going through the steps ( 8)-( 12) we can define the quadratic nonlinear polarization envelope [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF][START_REF] Conforti | Utra-broadband optical phenomena in quadratic nonlinear media[END_REF]:

A (2) p (z, t) = P (2) N L (z, t)e -iω 0 t+iβ 0 z = ε 0 χ (2) 2 A 2 e iω 0 t-iβ 0 z + |A| 2 + iH[|A 2 |] e -iω 0 t+iβ 0 z (17)
Before inserting Eq. ( 17) into Eq. ( 14), the term |A| 2 in Eqs. [START_REF] Conforti | Spectral shaping of femtoseond pulses in aperiodic quasi-phasematched gratings[END_REF] deserves further comments, since it is centered around zero in frequency domain. In particular to obtain the nonlinear polarization envelope in Eq. ( 17)

we had to filter out the negative frequency components of PNL (ω), as done for Ê(ω). We note however that (i) Â(z, ω -ω 0 ) does not contain negative frequency by definition, (ii) P N L is a small perturbation to the linear polarization and (iii) negative frequencies cannot be phase-matched. It follows that the task of filtering the negative frequency components of |A| 2 can be left to the propagation equation instead of having it explicitly in the definition of A (2) p (z, t). In other words, when inserting Eq. ( 17) into Eq. ( 14), we can write:

|A| 2 + iH[|A 2 |] ≈ 2|A| 2 .
We have checked numerically the good accuracy of this approximation. Even if this approximation in not necessary in the numerical solution (it is straightforward to calculate the exact nonlinear polarization envelope in frequency domain), it is suitable to obtain a simple and manageable model for further analytical investigations [START_REF] Conforti | Theory and experiments on multistep parametric processes in nonlinear optics[END_REF].

As to cubic nonlinearity, P

N L (z, t) = ε 0 χ (3) E(z, t) R(t -t ′ )|E(z, t ′ )| 2 dt ′ (3)
and, after some algebra through the steps ( 8)-( 12), we obtain

A (3) p (z, t) = ε 0 χ (3) 2 A(z, t) dt ′ R(t ′ )|A(z, t -t ′ )| 2 + + ε 0 χ (3) 4 A * (z, t) dt ′ R(t ′ )A 2 (z, t -t ′ )e -2iω 0 t ′ + + ε 0 χ (3) 4 Ae 2iω 0 t-2iβ 0 z dt ′ R(t ′ )A 2 (z, t -t ′ )e -2iω 0 t ′ . ( 19
)
.

Examples of numerical solutions of Eqs.( 14)-( 17) (but neglecting the Raman contribution) for describing harmonic wave and supercontinuum generation in quadratic and cubic nonlinear crystals were reported in Ref. [START_REF] Kozlov | Harmonic and supercontinuum generation in quadratic and cubic nonlinear optical media[END_REF]. The method reported above can be extended to the case of several propagating modes [START_REF] Phillips | Supercontinuum generation in quasi-phasematched waveguides[END_REF] and different electric field polarizations [START_REF] Conforti | Modelling of ultrabroadband and single-cycle phenomena in anisotropic quadratic crystals[END_REF].

Infrared PCF supercontinuum and quadratic crystals

As mentioned, supercontinuum light generation in PCF has been developed very successfully during recent decades as a powerful concept for wideband fiber light sources. It relies on cubic nonlinear effects such as four-wave mixing, soliton propagation, cross-phase modulation and Raman conversion.

Efficient supercontinuum light generation requires a suitable design of the dispersion relationship of the fiber. Microstructured PCFs offer new and flexible ways to manipulate the light generation process. Special glasses with high nonlinearity can also be used in the fiber core region [START_REF] Moghaddam | Flatly broadened supercontinuum generation in nonlinear fibers using a mode locked bismuth oxide based erbium doped fiber laser[END_REF]. By changing the structure of the cladding, the linear and nonlinear properties of the PCF can be adapted for pumping with different sources in order to generate the targeted wavelength ranges. In silica PCFs, numerous studies have demonstrated SC generation covering a wavelength range from UV (around 400nm)

to near-IR (2.2µm) [START_REF] Lesvigne | Visible supercontinuum generation controlled by intermodal four-wave mixing in microstructured fiber[END_REF]. By replacing the silica with other glasses such as tellurite or heavy oxide SF-57 glasses [START_REF] Buczynski | Supercontinuum generation in photonic crystal fibers with nanoporous core made of soft glass[END_REF], the SC spectrum is shifted towards the higher infrared wavelengths up to about 4µm.

In spite of the great flexibility in generating such a wide range of wavelengths, the possibility to achieve a specific wavelength spectrum directly by designing the fiber or other system parameters is not already demonstrated.

A single-pass frequency doubler device may convert the SC broadband infrared radiation into visible light with a tunable carrier wavelength, desired temporal durations, and good-efficiency conversion [START_REF] Conforti | From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion[END_REF][START_REF] Sapaev | Femtosecond pulse synthesis by efficient second-harmonic generation in engineered quasi phase matching gratings[END_REF][START_REF] Conforti | Spectral shaping of femtoseond pulses in aperiodic quasi-phasematched gratings[END_REF]. To this end, we can expoit the nonlinear envelope technique, introduced in Sec. 2, to simulate the dynamics of a SC PCF infrared radiation injected in a quadratic crystal and to guide experiments of frequency conversion.

For a demonstration experiment, the setup depicted in Fig. 1 is employed.

A passively Q-switched Nd:YAG laser delivering 800ps pulses at 1064nm is used as a pump source. The repetition rate and the peak power were 8kHz and 10kW , respectively. Two half wave plates and a polarizer allowed to control the power and polarization direction of pump radiation. The pump is coupled into a PCF by an aspheric lens with a coupling efficiency of 75%.

For fiber supercontinuum generation, we used a PCF with a core diameter of 4.2µm whereas the hole diameter and the hole-to-hole spacing are 2.5µm and 4µm respectively (Fig. 2). The zero dispersion wavelength for the fundamental guided mode is located close to 990nm (Fig. 2), and the mode effective area is about 3.6µm 2 at 1064 nm. The nonlinear refrative index is 2.7•10 -20 m 2 W -1 . By coupling the pump radiation into a 8m long fiber in the anomalous dispersion regime, we generated a broadband spectrum mainly extending in the infrared domain, induced by the combination of modulational instability, solitonic propagation and cross-phase modulation effects. Fig. 3 reports a typical experimental SC spectrum at the output of the PCF, characterized by an optical spectrum analyzer (OSA), and the spectrum obtained through numerical simulations, averaged over 30 measurements to mimic the integration time of the OSA. After the propagation through a spectral filter (FEL1000) in order to remove any noise in the 400nm-800 nm band, the SC extended in the IR range (1064 -2220nm) with an average power of 50mW .

We considered a compact SC system with a wide wavelength range, which can be easily manipulated externally.

The SC is then focused onto a 2cm long quadratic biaxial LBO crystal, at At φ = 0, a 100nm width spectrum in the visible domain is generated. Fig. 6 also shows the theoretical contour plots of the normalised parametric gain for second harmonic generation varying the crystal azimuthal angle (the curves are calculated from Sellmeier equations for LBO at temperature T = 27 o C).

Reverse engineering [START_REF] Conforti | From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion[END_REF] through the nonlinear envelope equation could allow one to choose different crystals permitting the generation of waves in different regions of the spectrum and/or with desired temporal durations. As an example, the combination of a PCF designed for broadband SC generation in the visible and a KDP crystal would lead to the generation of fields in the UV spectrum.

Visible supercontinuum in optically poled PCFs

For many biomedical applications, it would be highly desirable to have a fiber-based broadband, high-brightness source of light in the visible range of the spectrum. An approach to generate visible optical SC is to use a tapered PCF, where the zero-dispersion wavelength, initially located close to the pump wavelength at, e.g., 800nm, is gradually shifted towards shorter wavelengths [START_REF] Travers | SNon-linear applications of microstructured optical fibres[END_REF]. As we saw in the previous section, a different approach to visible SC generation may involves the frequency doubling of the infrared SC by means of a quadratic crystal. Alternatively, visible SC may be directly generated in the fiber itself by exploiting its cubic and quadratic nonlinear response.

Although the second-order order susceptibility is not present in silica because of centrosimmetry, several experiments have indeed reported SHG in germanium-doped silica fibers, due to either dipolar effects at the corecladding interface [START_REF] Fujii | Sum-frequency light generation in optical fibers[END_REF], or to a cascading process involving the third-order nonlinearity mediated creation, from an initial doping-induced weak SHG, of a static electric field which induces a grating for quasi-phase-matching [START_REF] Osterberg | Experimental studies on efficient frequency doubling in glass optical fibers[END_REF]. In all of these experiments, the SHG efficiency was not sufficient to lead to visible SC generation when the fiber was pumped by an infrared beam. In 2004, a method for visible SC generation was investigated by several research teams [START_REF] Champert | White-light supercontinuum generation in normally dispersive optical fiber using original multi-wavelength pumping system[END_REF]. The method consists of the double excitation of a PCF by launching an IR radiation (1064nm) and its second harmonic (532nm), the latter being generated through a nonlinear crystal. A spectral broadening is obtained in the IR domain by means of solitonic effects, whereas the visible spectrum is induced by the cross-phase modulation (XPM) of the IR solitons on the harmonic pump wave. In that way, a significant visible spectrum enlargement can be obtained for any value of the secondharmonic (SH) power, even weak, its spectral width being determined by the IR radiations velocities. It was also demonstrated that the dual pumping process writes a permanent grating in the germanium-doped fiber, so that efficient broadband SHG can be obtained directly from the poled fiber when using the IR pump only [START_REF] Tombelaine | Optical poling in germanium-doped microstructured optical fiber for visible supercontinuum generation[END_REF]. Recent experiments have investigated the limitations due to the Raman effect in the self-poling process occurring in a germanium-doped standard single-mode fiber [START_REF] Couderc | Unprecedented Raman cascading and four-wave mixing from secondharmonic generation in optical fiber[END_REF].

Here, we consider and model the case of SC visible generation, initiated by second-harmonic generation, in quadratic poled germanium-doped PCFs [START_REF] Tombelaine | Optical poling in germanium-doped microstructured optical fiber for visible supercontinuum generation[END_REF].

A passively Q-switched Nd:YAG laser delivering 650ps pulses at 1064nm is used as a pump source. The repetition rate and the peak power were 12.5kHz and 2.5kW , respectively. The pump is coupled into a 4m-long germanium-doped PCF. For fiber supercontinuum generation, we used a PCF with a core diameter of 1.8µm whereas the average hole diameter is 2.1µm and the pitch is 2.3µm (Fig. 7). The zero dispersion wavelength for the fundamental guided mode is located close to 875nm (Fig. 7), and the mode effective area is about 2.4µm 2 at 1064 nm. The nonlinear refrative index is In addition to the output spectra, the numerical simulations also provide the temporal intensity profile of the output fields. For example, Fig. 10 reports a portion of typical temporal profiles of signals generated in the visible range, which exhibit fast sub-picosecond fluctuations. The inset in Fig. 10 shows with a dashed red curve the bandpass filter used to slice the visible spectrum.

Conclusions

We have presented a nonlinear single-envelope equation able to model quadratic and cubic ultra-broadband nonlinear dynamics in photonic devices.

Owing to a proper formal definition of the complex envelope, it is possible to treat pulses of arbitrary frequency content. A proper definition of the envelope is crucial for second-order nonlinearities, due to the generation of frequency components around zero. Previous works [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF][START_REF] Brabec | Intense few-cycle laser fields: Frontiers of nonlinear optics[END_REF], more focused on cubic nonlinearities, did not consider this aspect. We have shown the case of the generation of tunable visible light from large band conversion, in a LBO quadratic crystal, of the infrared continuum from standard photonic crystal fibers. Moreover, we have presented the modeling of a visible supercontinuum Here, the input peak power is 1.25kW .

  the temperature T = 27 o C. As an example, we show the case of light propagation in the principal plane XY of the biaxial crystal with the azimuthal angle φ varying from 0 o to 5 o . The spectra were subsequently characterized by the OSA. Fig. 4 reports a typical experimental spectrum at the output of the quadratic LBO crystal, and the spectrum obtained through broadband nu-merical simulations. In fact, second harmonic generation (SHG) takes place in the quadratic crystal, generating wavelengths in the visible band. Fig. 5 reports typical temporal profiles of signals generated in the visible range. The wavelengths of the waves can be tuned by simply varying the azimuthal angle of the LBO crystal. Fig. 6 reports experimental visible spectra at the output of the quadratic LBO crystal, at different azimuthal crystal angle φ.

2. 6 •

 6 10 -20 m 2 W -1 . The PCF has been previously poled by both green (532nm) and IR (1064nm) radiation beating. In the first 45cm of the PCF, a quadratic square spatial modulation is estimated, with 0.001pm/V magnitude and Λ = 25.8µm grating period, respectively. By coupling the pump radiation at 1064nm into the poled PCF, a broadband IR SC is produced by Raman self-frequency shifting solitons accompanied by a lower amplitude, visible and symmetric XPM-induced SC, extending over about 200nm around 532nm. Fig.8reports a typical experimental spectrum at the output of the PCF, characterized by the OSA. Fig.9reports the spectrum obtained through numerical simulations. Fig.9illustrates the numerical spectra at different lengths of the PCF, from 10cm up to 1m, for an input pump pulse peak power of 1.25kW . At 10cm, MI-induced sidebands appear next to the IR pump, and a narrowband SH wave is generated at 532nm. At 20cm, the pump spectrum broadens considerably owing to multiple four-wave mixing, and correspondingly the spectrum around 532nm also broadens in a rather symmetric fashion. At 40cm, the IR spectrum broadening is enhanced by Raman scattering towards the longer wavelengths; this also leads to an asymmetric broadening of the visible spectrum up to about 800nm. Since the poling and the associated quadratic susceptibility stops at 45cm, for longer fiber lengths there is no further conversion from the IR to the visible and the main effect is the continuous broadening towards longer wavelength of the IR SC owing to the Raman induced soliton self-frequency shift.

Fig. 1 .

 1 Fig. 1. Experimental setup. Nd:YAG laser pump delivering 800ps pulses, at 1064nm, 8kHz repetition rate; L1, L2, L3 lenses; λ/2 half wave plate; P polarizer; F, filter; photonic crystal fiber (PCF); quadratic biaxial LBO crystal; optical spectrum analyzer (OSA).

Fig. 2 .

 2 Fig. 2. Cross-sectional scanning electron microscope image of PCF and calculated group velocity dispersion parameter D.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Typical SC spectrum at the output of the PCF. Top, OSA meausurement, bottom, numerical trace. Here, input peak power of 7.8kW .Fig. 4. Spectrum at the output of the LBO crystal. Top, OSA meausurement, bottom, numerical result. The azimuthal angle φ = 0 o . Fig. 5. Typical numerical temporal profiles filtered at 700nm at the output of the LBO crystal.

Fig. 6 .

 6 Fig. 6. Top, OSA meausurement of visible spectra at the output of the LBO crystal at different crystal azimuthal angles (φ = 0 o , blue line; φ = 3 o , red line; φ = 4.5 o , light blue line). Bottom, parametric gain for second harmonic generation in LBO as a function of generated wavelength and crystal azimuthal angle.

Fig. 7 .

 7 Fig. 7. Cross-sectional scanning electron microscope image of the germanium-doped PCF and calculated group velocity dispersion parameter D.

Fig. 8 .

 8 Fig. 8. OSA measurement of typical spectrum at the output of the germanium-doped PCF. Here, input peak power of 2.5kW .

Fig. 9 .

 9 Fig. 9. Numerical trace of typical spectrum at different lenghts of the germanium-doped PCF. Here, input peak power of 1.25kW .

Fig. 10 .

 10 Fig. 10. Portion of numerical temporal profiles filtered around 532nm at the output of germanium-doped PCF. Dashed red curve, bandpass filter used to slice the visible spectrum. Here, input peak power of 1.25kW .

FFigure 1 :

 1 Figure 1: Experimental setup. Nd:YAG laser pump delivering 800ps pulses, at 1064nm, 8kHz repetition rate; L1, L2, L3 lenses; λ/2 half wave plate; P polarizer; F, filter; photonic crystal fiber (PCF); quadratic biaxial LBO crystal; optical spectrum analyzer (OSA).

Figure 2 :

 2 Figure 2: Cross-sectional scanning electron microscope image of PCF and calculated group velocity dispersion parameter D.

Figure 3 :

 3 Figure 3: Typical SC spectrum at the output of the PCF. Top, OSA meausurement, bottom, numerical trace. Here, input peak power of 7.8kW .

Figure 4 :

 4 Figure 4: Spectrum at the output of the LBO crystal. Top, OSA meausurement, bottom, numerical result. The azimuthal angle φ = 0 o .

Figure 5 :

 5 Figure 5: Typical numerical temporal profiles filtered at 700nm at the output of the LBO crystal.

Figure 6 :

 6 Figure 6: Top, OSA meausurement of visible spectra at the output of the LBO crystal at different crystal azimuthal angles (φ = 0 o , blue line; φ = 3 o , red line; φ = 4.5 o , light blue line). Bottom, parametric gain for second harmonic generation in LBO as a function of generated wavelength and crystal azimuthal angle.

Figure 7 :

 7 Figure 7: Cross-sectional scanning electron microscope image of the germanium-doped PCF and calculated group velocity dispersion parameter D

Figure 8 :

 8 Figure8: OSA measurement of typical spectrum at the output of the germanium-doped PCF. Here, the input peak power is 2.5kW .

Figure 9 :

 9 Figure 9: Numerical trace of typical spectrum at different lenghts of the germanium-doped PCF. Here, the input peak power is 1.25kW .

generation, initiated by second-harmonic generation, in a quadratic poled germanium-doped microstructured fiber.

Acknowledgments

The present research is supported in Brescia by the Italian Ministry of


F Baronio
email: baronio@ing.unibs.it

M Conforti

C De Angelis

M Andreana

A Tonello

V Couderc

Tunable light sources from photonic crystal fibers and quadratic crystals

Keywords: Optical sources and standards, Laser optical system: design and operation, Nonlinear Optics PACS: 42.72.-g, 42.60.-v, 42.65.-k

We consider the conversion of broadband infrared light, generated from photonic crystal fibers, into narrow-band visible pulses, through frequency conversion in quadratic biaxial crystals. We exploit the nonlinear envelope technique predict cubic and quadratic ultrabroadband dynamics and to guide the experiments. The combination of photonic crystal fibers and quadratic crystals provides the possibility to achieve low-cost laser-light sources with a great flexibility of tuning the spectral properties and covering wavelengths from UV to the infrared range. a) and numerical b) spectral traces of visible light generated through photonic crystal fibers and quadratic crystals

Introduction

The availability of light sources with specific and tunable spectral properties is of interest in many fields of physics, from spectroscopy [1,2,3] to spectral imaging [4], from optical sensing to material processing [5,6]. Solid state lasers [7,8,9] may offer the chance for tuning the laser wavelength in a limited range. Other more complex lasers, such as dye lasers, can also provide wider wavelength tuning ranges. The generation of complex wavelength distributions, however, would require the superposition of several light sources.

Nowadays, the development of modern supercontinuum (SC) fiber light sources offers novel opportunities for tunable light sources and complex wavelength distributions [10,11,12]. SC light sources are designed to provide a flat spectrum over a wide wavelength range, but limited options for spectral shaping are related to the dependence of the produced spectrum on parameters such as fiber material and structure, fiber length, pump power, pump wavelength. SC light sources are compact and can generate various wavelength distributions with high spatial coherence and high spectral intensity. They are adapted for waveguide coupling and focusing to small spot sizes.

A general and extremely flexible option for shaping the spectrum of such light sources could be provided by combination with quadratic nonlinear crystals [13,14]. Moreover, a comprehensive framework to study the nonlinear evolution of ultrabroadband pulses in quadratic nonlinear media has been recently discovered [15,16].

In this Letter, we consider the conversion of broadband infrared light, generated from photonic crystal fibers (PCFs), into narrow-spectrum pulses, thorough frequency conversion in quadratic crystals. We exploit the nonlinear envelope technique [15,16] to simulate the cubic and quadratic ultrabroadband dynamics and to guide the experiments. The combination of PCFs and quadratic crystals could provide the possibility to achieve low cost laser-light sources with a great flexibility in tuning the spectral properties and covering wavelengths form UV to the infrared range.

Supercontinuum fiber light source and quadratic crystals

Supercontinuum light generation has been developed very successfully during recent decades as a powerful concept for wideband fiber light sources [17]. It relies on third order nonlinear effects such as four-wave mixing, solitons and cross-phase modulation. Efficient supercontinuum light generation requires well-defined dispersion conditions. Microstructured PCFs offered new and flexible ways to manipulate the light generation process. Special glasses with high nonlinearity can be used in the fiber core region. By changing the structure of the cladding, the linear and nonlinear properties of the PCF can be adapted for pumping with different sources in order to generate the targeted wavelength ranges. In silica PCFs, numerous studies have demonstrated SC generation covering a wavelength range from UV (around 350nm) to near-IR. By replacing the silica with other glasses such as tellurite or SF6 material, the SC spectrum is shifted towards the higher infrared wavelengths up to about 4µm. In spite of the great flexibility in generating such a wide range of wavelengths, it is not already demonstrated the possibility to achieve a specific wavelength spectrum directly by designing the fiber or other system parameters.

A single-pass frequency doubler device may convert the SC broadband infrared radiation into narrow spectrum visible pulses with a tunable carrier wavelength, desired temporal durations, and good-efficiency conversion. We exploited the nonlinear envelope technique to simulate cubic and quadratic ultrabroadband dynamics and to guide the experiments [15,16].

For a demonstration experiment the setup depicted in Fig. 1 is employed. A passively Q-switched Nd:YAG laser delivering 800ps pulses at 1064nm is used as a pump source. The repetition rate and the peak power were 8kHz and 10kW , respectively. Two half wave plates and a polarizer allowed to control the power and polarization direction of pump radiation. The pump is coupled into a PCF by an aspheric lens with a coupling efficiency of 75%. For fiber supercontinuum generation, we used a PCF with a multi-component glass structure in the core in order to increase the nonlinearity of the waveguide [18]. By coupling the pump radiation into a 8m long fiber, we generated a broadband spectrum induced by the combination of four-wave mixing, solitonic and cross-phase modulation effects. We considered a compact SC system with a wide wavelength range, which then can be easily manipulated externally. The SC radiation is then focused onto a 2cm long quadratic biaxial LBO crystal [START_REF] Dmitriev | Handbook of Nonlinear Optical Crystals[END_REF]. As an example, we show the case of light propagation in the principal plane XY of the biaxial crystal with the azimuthal angle φ varying from 0 o to 20 o . The spectra were subsequently characterized by an optical spectrum analyzer (OSA). simulations, averaged over 30 measurements to mimic the integration time of the OSA. The SC extended in the IR range (1064 -2220nm) for an average power of 50mW . Fig. 3a reports a typical experimental spectrum at the output of the quadratic LBO crystal. Fig. 3b reports the spectrum obtained through broadband numerical simulations [15,16]. We observed the generation of visible signals at 600nm and 700nm. In fact, second harmonic generation for type I processes takes place in the quadratic crystals [START_REF] Conforti | [END_REF]. phenomena. We have considered the conversion of broadband infrared light, generated from a photonic crystal fiber, into narrow spectrum visible pulses, through parametric frequency conversion in a quadratic LBO biaxial crystal. The combination of photonic crystal fibers and quadratic crystals would provide the possibility to achieve low-cost laser-light sources with a great flexibility in shaping the spectral and temporal properties and covering wavelengths form UV to the infrared range. These optical sources could be of interest in spectroscopy, spectral imaging, optical sensing and material processing.

Results and discussion

Conclusion In this Letter, we have shown the potential of the nonlinear envelpe technique in describing ultrabroadband

Figure 2

 2 Figure 2 Typical SC spectrum at the output of the PCF. a) OSA meausurement, b) numerical trace.

Fig. 2a reportsFigure 3

 3 Fig.2areports a typical experimental SC spectrum at the output of the PCF, characterized by the OSA. Fig.2breports the spectrum obtained through standard numerical

Fig. 4

 4 reports typical temporal profiles of signals generated in the visible range: trains of 200f s pulses. The use of LBO crystals permits the generation of two visible waves spatially and temporally overlapped, but spectrally separated. The carrier wavelengths of the two waves can be tuned simply varyng the azimuthal angle of the LBO crystal. Fig. 5a reports experimental visible spectra at the output of the quadratic LBO crystal, at different azimuthal crystal angle φ. Fig. 5b shows the contour plots of the normalised parametric gain for second harmonic generation varying the crystal azimuthal angle (the curves are calculated from Sellmeier equations for LBO at temperature T = 27 o [19]). The use of the nonlinear envelope technique, thorough reverse engineering [13], would allow to choose different crystals permitting the generation of waves in different regions of the spectrum and/or with desired temporal durations. As an example, the combination of a PCF designed for broadband SC generation in the visible and a KDP crystal would lead to the generation of fields in the UV spectrum.

Figure 4

 4 Figure 4 Typical numerical temporal profiles at 600nm (green line) and 700nm (red line) at the output of the LBO crystal.

Figure 5 a

 5 Figure 5 a) OSA meausurement of visible spectra at the output of the LBO crystal at different crystal azimuthal angles. b) Parametric gain for second harmonic generation in LBO as a function of generated wavelength and crystal azimuthal angle.

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Acknowledgements The present research is supported by PRIN-MIUR grant n o 2009P 3K72Z and by Fondazione CARIPLO grant n o 2010 -0595.