Olivia Zurita-Miranda
email: olivia.zurita-miranda@u-bordeaux.fr

Coralie Fourcade-Dutin

Frederic Fauquet

Frederic Darracq

Jean-Paul Guillet

Patrick Mounaix

Herve Maillotte

Damien Bigourd
email: damien.bigourd@u-bordeaux.fr

Tunable ultra-fast infrared generation in a

INTRODUCTIONs

Ultra-fast optical parametric amplification (OPA) is a popularly well-established method in a large range of configurations to obtain interesting properties such as a very wide-tunability, an ultra-broad gain-bandwidth for ultra-short pulses and a very-high temporal contrast [START_REF] Musheghyan | Tunable, few-cycle, CEP-stable mid-IR optical parametric amplifier for strong field applications[END_REF][START_REF] Shao | Broadbandwidth high-temporal-contrast carrier-envelope-phase-stabilized laser seed for 100 PW lasers[END_REF]. The OPA process was performed in different materials, mainly by exploiting the second-order susceptibility of bulk non-linear crystals, but also in isotropic media where their third-order susceptibility is utilized [START_REF] Rubino | Generation of broadly tunable sub-30-fs infrared pulses by four-wave optical parametric amplification[END_REF]. In both cases, the interaction geometry is essential to get the optimal phase matching that allows efficient energy conversion from the pump to the targeted signal. To support the nonlinear process at very high power, the amplifier size can also be adjusted to decrease the pump intensity. Similarly, pulses are stretched to decrease the peak power at the system input while the targeted pulse is compressed after the amplification. This scheme, known as optical parametric chirped pulse amplification, guarantees the possibility of attaining a very high energy and/or high peak power [START_REF] Bromage | Technology development for ultra-intense all-OPCPA systems[END_REF]. An alternative procedure for ultra-fast OPA is to achieve the process in guided systems due to the significant advantages of the guiding geometry. Besides its robustness and compactness, the optical interaction between the involved pulses is more efficient thanks to the beam collinearity, excellent spatial beam profile and superposition that can be directly delivered within the waveguide. Particularly, we investigated this process in single-mode solid-core photonic crystal fibers in which the generation and the amplification were achieved from a Four Wave Mixing (FWM) mechanism. In this case, the phase matching is given by the nonlinear phase of the pump and the overall (chromatic and geometric) dispersion of the fiber [START_REF] Bigourd | High gain fiber optical parametric chirped pulse amplification of femtosecond pulses at 1 µm[END_REF]. As previously explained in bulk materials, the power scalability can be performed by increasing the fiber core diameter while keeping the suitable dispersion profile [START_REF] Morin | μJ-level Raman-assisted fiber optical parametric chirped-pulse amplification[END_REF][START_REF] Bigourd | Parametric gain shaping from a structured pump pulse[END_REF] or by stretching the involved pulses [START_REF] Bigourd | Ultra-broadband fiber optical parametric amplifier pumped by chirped pulses[END_REF][START_REF] Fourcade-Dutin | Ultra-broad band fiber optical parametric amplifier pumped by chirped pulses Part II : Sub-30 fs pulse amplification at high gain[END_REF][START_REF] Fu | Normal-dispersion fiber optical parametric chirped-pulse amplification[END_REF]. Another method is to perform the OPA process in a gas-filled waveguide in which the optical damage threshold is much higher than in silica. In this case, the dispersion can be controlled by the pressure in real time, allowing a tunable phase-matched process. Few groups focused their research on the amplification in gas filled hollow-core capillaries (HCC), with a large core, in the visible [START_REF] Ciriolo | Generation of ultrashort pulses by four wave mixing in a gas-filled hollow core fiber[END_REF], ultra-violet [START_REF] Belli | Highly efficient deep UV generation by four-wave mixing in gas-filled hollow-core photonic crystal fiber[END_REF][START_REF] Durfee | Intense 8 fs pulse generation in the deep ultraviolet[END_REF] and infrared ranges [START_REF] Faccio | Optical amplification in the near-infrared in gas-filled hollow-core fibers[END_REF][START_REF] Kida | Optical parametric amplification of a supercontinuum in a gas[END_REF][START_REF] Fourcade-Dutin | Generation of ultra-short pulse in the mid-infrared in a hollow-core fiber[END_REF]. In most cases, the

THEORETICAL BACKGROUND

The general principle of a degenerated FWM relies on the interaction of a strong pump pulse (p), a weak injected signal (s), and a generated idler (i) in a material exhibiting a third order nonlinearity 𝜒 (3) . For an HCC system based on a FWM process, the nonlinearity arises from the gas confined in the waveguide. The pump and signal pulses are coupled into the HCC and a third wave (the idler) is generated during the propagation in the gas. The energy is transferred to the idler and signal from the pump under specific conditions depending on parameters such as the gas pressure, gas species, capillary properties and pump features. In order to enhance the energy conversion efficiency and induce a spectral tunability, the pump and the signal pulses will be chirped. Thus, different spectral components from the two pulses will temporally overlap. To quantify the contribution of all these parameters, we first look at the key ingredients required for the FWM and introduce known equations that are useful for the discussion.

A. Phase-matching and gain

To achieve an efficient FWM process, one of the key requirements is that all the involved waves satisfyies a phase-matching condition , which is maintained over a specific spectral bandwidth

NL        (1)
where ∆β is the linear phase related to the gas pressure and waveguide properties. It is calculated from the sum of the propagation constant  of all the waves; i.e., =s+i-2p. NL is the nonlinear phase induced from the pump pulse. Assuming no pump depletion, the expression of this nonlinear term is given by 2 2 .

p NL P eff n P Ac        (2)
with p the angular frequency at the pump wavelength, c the light speed, Aeff the effective mode area and n2 the gas nonlinear index, which depends on the species and the pressure, Pp the pump power.

The second requirement is directly linked to the photon energy conservation law; i.e 2p=s+i, where s and i are the angular frequencies of the signal and idler respectively. This means the signal and idler waves are symmetrically positioned around the pump; i.e s-p=p-i. Expanding each propagation constant  in a Taylor series around p, one can demonstrate that odd terms (mostly the third order dispersion term at the pump frequency) have no influence on the phase matching for a continuous pump wave [START_REF] Vanvincq | Ultrabroadband fiber optical parametric amplifiers pumped by chirped pulses. Part 1: analytical model[END_REF].

In the undepleted regime and for a monochromatic pump wave, the FWM is well analytically described to understand the behavior of the process [START_REF] Marhic | Fiber Optical Parametric Amplifiers, Oscillators and Related Devices[END_REF][START_REF] Hansryd | Fiberbased optical parametric amplifiers and their applications[END_REF]. When only a single weak wave is injected in the HCC with the power Ps(0), the power Ps, can be expressed as a function of the HCC length in an ingenuous form as :

2 2 .. ( ) (0) 1 .sinh( ) .. p p s s eff nP P L P g L A c g                      (3)
With parametric gain g given by:

2 2 2 2
.

2

p P eff n g P Ac                          (4)
From Eq. 3 and 4, we can estimate the output power of the signal in the undepleted regime given that Ps is much weaker than Pp.

B. Specific parameter for FWM in HCC

The group-velocity-dispersion 2 of the HCC is an important parameter required to tune the phase matching condition of the FWM (Eq. 1). It mostly depends on the waveguide core radius (𝜙), and refractive index of the gas, ngas, which changes with the pressure Pgas and its species. ngas can be calculated from the Sellmeier equation [START_REF] Börzsönyi | Dispersion measurement of inert gases and gas mixtures at 800 nm[END_REF]. The modal propagation constant of a gas filled HCC is analytically calculated from the Marcatili and Schmelzer's model [START_REF] Marcatilli | Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers[END_REF] as a function of the wavelength,

    2 2 2 , 2 2 , 2 ,
, gas gas gas mn mn u P P

                n 4
(5) um, n is the n th zero of the m th -order Bessel function of the first kind. For example, u11=2.404 for the fundamental mode HE11. From Eq. 5, 2 can be calculated for HE11 and Fig. 1.a shows typical profiles when a HCC is filled with Argon for a core diameter of 150 µm. When the pressure increases, the dispersion curve is modified and the Zero Dispersion Wavelength (ZDW, 2=0 ps 2 /m) shifts to a longer wavelength [START_REF] Travers | Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers [Invited[END_REF]. The ZDW also changes with the core radius and the type of gas (Fig. 1.b). For example, the ZDW is set to a longer wavelength for xenon gas instead of argon for a constant core size and a fixed pressure of 1 bar. In all cases, the ZDW shifts to a lower wavelength as the core diameter decreases. The value and the sign of  are very important to predict and tune the FWM bands since it directly affects the phase-matching (Eq.1).

In particular, when the pump lies in the normal dispersion regime, the FWM spectral lobes can be considerably shifted away from the pump wavelength, but typically at the expense of the spectral bandwidth [START_REF] Harvey | Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber[END_REF]. Therefore, it is highly desirable to adjust the ZDW by changing not only the pressure, (Fig. 1.a) but also the core radius of the HCC and gas species (Fig. 1.b). However, the power is also one important constraint when choosing the size of the HCC. For example, to avoid any damage of the cladding or some spurious nonlinearities in the HCC, a large core radius is preferably chosen. Therefore we decided to take a radius x of 75 µm to fit with our experimental situation dealing with ultra-short pulse delivered by a Ti:Sa chain (800nm, 100fs, 1 mJ) to avoid optical damage or spurious nonlinearity. From Eq. 1-5, the gain, i.e G=Ps(L)/Ps(0), is calculated (Fig. 1.c) for a pump continuous wave at 800 nm with a peak power of 1.6 GW for several argon pressures. The spectral gain G curve is plotted as a function of the frequency offset from the pump; i.e s-p. In these cases, n2= 8.45 x10 -24 m 2 W -1 at 1 bar, and Aeff=1.5. x 2 . Clearly, the phase matching is tuned from the visible to the near infrared with a gain up to 15 dB, and the spectrum shifts away from the pump wavelength when the pressure increases (Fig. 1.c). It should be noted that, at this pressure range, the FWM process is only phasematched when all the waves propagate in the fundamental mode.

C. Specific parameters for ultra-fast FWM-OPA in HCC

Since the involved pulses have an ultra-short duration in the range of 100 fs, the group delay mismatch (GDM) between the pump and the signal/idler is firstly calculated for an argon filled HCC with x =75 µm (Fig. 1.d). As expected from the low refractive index, the GDM is relatively low for the gas pressure range. As an illustration, by considering pump and signal pulses whose central wavelengths are 800 nm and 530 nm respectively, the two pulses separate from each other at a rate ~17.7 fs.m -1 for a pressure around 1.2 bar. This means the pump and signal pulses with a duration of 100 fs do not overlap after propagation of ~5.6 m (walk-off length, LWL). Similarly, the corresponding idler at ~1630 nm (Fig. 1.c), separates from the pump at a rate of ~69 fs.m -1 , corresponding to LWL~1.4 m for a pulse duration of 100 fs.

Besides the GDM, spectral bandwidth is an important factor to consider in an ultra-fast amplifier. Indeed, as we aim to amplify a broadband signal, the phase-matched process should occur over a large spectral range [START_REF] Fourcade-Dutin | Ultra-broad band fiber optical parametric amplifier pumped by chirped pulses Part II : Sub-30 fs pulse amplification at high gain[END_REF] by taking benefit from the pump bandwidth.

The previously described analytical model (section 2.A) does not fully explain the ultra-fast FWM since it does not include the contribution of the pump bandwidth. Each pump spectral component impacts the FWM process depending on the odd terms of the dispersion [START_REF] Vanvincq | Ultrabroadband fiber optical parametric amplifiers pumped by chirped pulses. Part 1: analytical model[END_REF][START_REF] Fourcade-Dutin | Temporal distribution of the parametric spectral gain in a photonic crystal fiber pumped by chirped pulses[END_REF]. A qualitative approach is to calculate the coherence length CL=1/, from a tunable continuous pump wave (Fig. 2.a) with a wavelength scanned across the bandwidth of the broadband pump (e.g 750-850 nm). As an example, Figure. For the ultra-fast FWM, the situation is more complicated. For a Fourier transform limited pump pulse, all the FWM bands temporally overlap within the walk-off length. However, in all cases, the pump owns a chirp since the pulse propagates in a weakly dispersive nonlinear medium creating a self-phase modulated spectrum. Alternatively, the pump chirp can be set at the input of the HCC to decrease the power. In both cases, the FWM bands generated from the chirped pump pulse were temporally delayed inducing a spectro-temporal distribution [START_REF] Fourcade-Dutin | Modulation Instability in a dispersion oscillating fiber pumped by a broad band pulse[END_REF].

SIMULATIONS AND DISCUSSION

To explore accurately the role of the parameters on the parametric conversion and spectral bands, numerical simulations were performed by integrating the generalized nonlinear Schrödinger equation (NLSE) with the split-step Fourier method. ) 𝐹(|𝐴| 2 𝐴)(Ω) [START_REF] Morin | μJ-level Raman-assisted fiber optical parametric chirped-pulse amplification[END_REF] where F stands for the Fourier transform.

In the following configuration, we use 2 15 points and a temporal resolution of 1.3 fs. This led to a total window of ~±21 ps and a frequency span of ~± 374 THz. Before performing the numerical simulations, we checked the analytical calculation to ensure that the high order modes do not contribute to the generation of FWM in our configuration. Therefore, we only considered the propagation of the fundamental mode. T is the time in a reference frame moving at the pump group velocity. A is the initial total complex electric field in the frequency domain at the angular frequency .  was calculated from Equation ( 5) and 0, 1 correspond to the zero and first coefficients of the Taylor expansion at the pump frequency. As the FWM process was investigated in an atomic gas, no delayed Raman effect was included in Eq. ( 6). The injected pump pulse has a central wavelength of 800 nm (p=375 THz) with a duration of ~120 fs at Full-Width at Half Maximum (FWHM) at the Fourier transform limit. The pump energy is 200 µJ leading to a peak power of ~1.6 GW. Fig. 3.a shows the evolution of the total simulated spectrum along a HCC for a 1.2 bar pressure and a total length of 60 cm. The pulsed pump was injected with a random noise initial condition by adding half a photon per temporal mode. As expected, the spontaneous FWM side lobes became stronger during the propagation. The general shape of this band represents the gain spectrum in the unsaturation regime but it does not provide the absolute gain value since no input signal was injected. Selected output spectra are also shown in Figure . 3.b, for L=10 cm, 35 cm, and 60 cm. Two weak lobes appeared at a frequency offset of ~± 225 THz from the pump that corresponds to ~500 and ~2000 nm. At the output, the bandwidth was wider in the anti-Stokes side as the consequence of the high order term [START_REF] Vanvincq | Ultrabroadband fiber optical parametric amplifiers pumped by chirped pulses. Part 1: analytical model[END_REF][START_REF] Fourcade-Dutin | Modulation Instability in a dispersion oscillating fiber pumped by a broad band pulse[END_REF] For comparison, we calculated the gain profile from the analytical equations (Eq. 1-5) with a CW pump at 800 nm. The spectral position of the FWM side lobes is in relatively good agreement with the pulsed case computed with Eq. 6 that takes into account the full bandwidth of the pump and other nonlinearities.

The main difference arouses in the instance of a broad pump pulse spectrum (pulsed case in the simulation), in which the spectrum bandwidth of the pump and FWM bands increased and were modulated. The modulations were mainly due to the self-phase modulation (SPM) process acquired on the pump pulse. To limit the SPM contribution and increase the pulse energy, the pump pulse was stretched at the HCC input while maintaining a constant peak power. Fig. 4 shows the spectra when a positively chirped pulse (CP, grey curve) or a Fourier transform limited pulse (TLP, black curve) propagates in a 60 cm long HCC with a pressure of 1.2 bar. The spectral and temporal profiles at z=60 cm are displayed in Fig. 4. a and b. The chirped pulse was stretched to 200 fs by applying a second order phase equals to 5807 fs 2 . In both cases, the generation of the FWM bands clearly appeared at ~30 cm (Fig. 4. c and e) and the side lobes were centered at a pump frequency offset of ~225 THz. However, the spectral bands are strongly modulated for the case with the chirped pulse. It probably results from the interference between the FWM bands generated at different delays from the modulated pump spectrum. From the spectrogram displaying the spectral evolution with time (Fig. 4 d andf), it can be noted that each instantaneous pump frequency generates a well-defined band at a specific time. In both cases, the two spectral bands are asymmetric and the bandwidth is wider in the anti-Stokes side.

From the earlier description of the spontaneous FWM, several features were obtained, such as the center of the sidebands and their bandwidths. The next step is to include a broadband signal to either amplify an ultra-short pulse or to generate a tunable infrared idler pulse from the visible signal. Consequently, a signal pulse is now injected with a central wavelength of 530 nm to generate an idler at 1630 nm. The spectral bandwidth of the signal is 82 nm at FWHM (Figure 5.a). Its peak power is set at 264 W. A number of configurations are reviewed and summarized in Table 1. In case I.a and I.b, the signal pulse is positively chirped with a spectral phase equals to s=97 fs 2 or 159 fs 2 , respectively. In this configuration, the signal pulse is either shorter (81 fs) or longer (132 fs) than the Fourier transform limited pump pulse (Figure 5.b). In both cases, the peak power is identical. In case II, the configuration is such that the pump and signal pulses both temporally overlap in the HCC (Figure 5.c). Thus, the two pulses are linearly chirped to reach a similar pulse duration equals to 188 fs (case II.a) or 395 fs (case II.b) . Figure 6 shows the conversion efficiency, i.e Ps(z=L)/Pp(0) and Pi(z=L)/Pp(0), as the function of the pressure. For each case, the maximum efficiency is reached at a pressure between 1 and 1.25 bar and is higher in the idler side compared to the signal part. This optimal pressure range at this spectral range is expected from the calculated maximum CL (Figure 2.a). The conversion efficiency value is relatively high due to the injection of a strong signal power. It means that the process begins to deplete the pump pulses. To evaluate this effect, we plot the spectral and temporal profile evolutions as a function of the propagation length for the case I.b and II.a at 1 bar (Fig7.a and b). Clearly, the optical beating between the signal and the pump is observed, which lead to the FWM observed in the spectral domain. In the two cases, the FWM spectral bandwidth is wider in the Anti-Stokes side. For the case I.a and I.b, the conversion efficiency (Fig. 6.a) reached ~8-12% for the signal or idler. The efficiency is slightly higher when the complete signal pulse interacts with the pump (case I.b). The spectral profile may also change with the delay, T, between the pump and the injected signal. For example, Figure 8.a shows several output spectra in the case II.b when a relative delay of +0 or +30, -70 fs is added at the HCC input. While the signal and pump spectra are not significantly affected by the delay, the idler spectrum can be tuned. In addition to the GDM between the involved pulses, the chirps impact directly the instantaneous signal amplification and the idler generation. Since the signal is chirped, the lowest frequency is in the leading edge and the central wavelength (550 nm) is centered at the null delay. However, at 1.2 bar the highest CL is achieved at ~510 nm (Fig. 2.a) and thus a negative delay improves the interaction of this spectral range with the maximum pump peak power (in the simplest case, set at a null delay). Moreover, as the exact value of the instantaneous pump frequency within the bandwidth influences the coherence length (Fig. 2), the delay can also be adjusted to overlap together the optimal pump and signal spectral range. In our condition, the situation is more complex since the pump spectrum is also modulated due to the SPM effect and therefore, the maximum peak power is not necessarily centered to the central frequency of the pump spectrum. This behavior can also be explained through the spectrogram calculated at the HCC output (Fig. 8.b). For T=0 fs and at the HCC output, the maximum of the FWM bands appear at ~10 fs from the center of the pump pulse where a portion of the pump pulse starts to deplete. When a delay is adjusted at the input, the overlap between the FWM bands and the pump occurs at another time. For T=+30 fs, the output FWM bands are marginally delayed and the idler generation is a slightly less efficient (Fig. 8.a) since the signal pulse does not overlap with the maximum of the pump Gaussian profile at the HCC input. Alternatively, the temporal shift induced by the GDM can be compensated with a negative input delay. For T=-70 fs, the output pulses are centered at 0 fs; i.e., at the center of the pump pulse. It means that the signal and idler pulses overlap with the maximum of the pump peak power at the end of the HCC (Fig. 8.b) and start to deplete the center of the pump profile. In this case, the idler is generated further away from the pump, at ~1.6 µm, since the chirped signal part is preferably amplified at ~510 nm (Fig. 8.a).

Table 1. Second order spectral phases and pulse durations of the pump and signal for the cases under study

EXPERIMENTAL IMPLEMENTATION

In this section, we present an experimental implementation to highlight the generation and tunability of the infrared idler. Figure 9 shows the experimental setup. The HCC is located inside a metal chamber enclosed by two 5mm thick quartz windows. A pressure manometer is attached to the chamber while the argon gas inflow is adjusted by input and output controllers. The laser beam is generated from a Ti: Sapphire source emitting pulses at a central wavelength of 800 nm and a repetition rate of 1 kHz. The pulse duration is 120 fs at FWHM (Fig. 10). The beam is splitted into two paths with a 90/10 beam splitter: the stronger part of the beam is used to pump the HCC, while the weaker part of the beam is utilized for the signal generation. The signal is generated by focusing the weak beam (~5 µJ) into a 5mm thick sapphire plate to generate a filtered continuum whose spectral width extends from 450-600 nm (Fig. 11.a) . A delay line is included in the signal path to ensure the temporal overlap between the pump and signal pulses. Then, the pump and signal beams are collinearly superposed with a cold mirror and focused with two plano-convex lenses towards the 30 cm long HCC with a core diameter of 150 µm. At the chamber output, the pump and signal energies were ~170 µJ and 0.539 nJ, respectively. At the input of the HCC, the pump pulse can also propagate through a dispersive block of N-F2 glass to stretch the pulse to 180 fs. For high pump power, the beam starts to self-focus in the dispersive block even if the pulse is stretched. Therefore, we fixed the peak power to 1.1 GW for the stretched pulses and 1.6 GW for the unstretched pump. The autocorrelation traces of the pump are shown in Fig. 10. At the HCC output, the beam is split into three paths thanks to short and long-pass filters and the spectra are recorded simultaneously (pump, signal, and idler). The air was removed from the chamber, and subsequently filled with argon gas to 2 bar. Next, the pressure value was progressively reduced to 0.3 bar at which we observed the optimal generation of the idler in the stretched pump case. This is surprisingly different from the expected pressure (~1.2 bar) obtained from the simulation. We believe this difference is due to other nonlinear effects. Particularly, high order modes are excited during the propagation. Therefore, the pump beam is also selffocused and spatial oscillations can occur during the propagation [START_REF] Tempea | Theory of self-focusing in a hollow waveguide[END_REF]. This might lead to local intensity increases and a change of the nonlinearity conditions. As the continuum is also chirped from the nonlinear generation mechanism in the sapphire plate, the temporal overlap between the pump and the signal is adjusted for each pressure step to ensure the superposition of the specific spectral components to amplify with the pump pulse. For each step, the signal pump and idler properties were recorded with the same procedure. Figure 11 shows an example of the signal, idler, and pump spectra recorded at the HCC output and with a specific delay of 19.2 fs. The signal is centered at ~575 nm while the idler was generated at 1250 nm (Fig. 2). As the delay is scanned, different parts of the chirped signal interact with the chirped pump pulse (see Fig. 8.b). Consequently, the maxima of the amplified signal and generated idler were reached at different wavelengths (Fig. 12.a and b) for each delay. By changing the delay, a tunable infrared idler pulse of 1.2 to 1.5 µm was generated with a quasi-linear rate of ~13.3 nm/fs (Fig. 12. b) which depends on the signal chirp value. For the signal tunability (Fig. 12. a), the wavelength of maximum gain (red line) is linear with the delay, but on two distinct smaller zones. The same maximum can be obtained with two different delays. This is because the signal spectrum is strongly modulated when the pump pulse is chirped while the idler spectrum exhibits a single maximum (Fig. 13). This difference in the FWM band spectrum is also observed in the numerical simulations (Fig. 4.a).

As an example, the wavelength at the second maximum is also plotted (Fig. 12.a, grey line). The maximum power is reached at about ~1250 nm (Fig. 12. c). We also reproduce the same experiment for the unchirped pump pulse (black curves). In this case, the wavelengths at maximum power change linearly with the delay (see Fig. 4.a) and the curve corresponds to one of the two preceding curves (Fig. 4.a red and gray curves). At the HCC output, the idler (at 1.25 µm) pulses were measured using an intensity autocorrelator (Fig. 14). Assuming a Gaussian temporal profile, the pulse duration is ~220 fs, at FWHM with pedestals at ~+/-600 fs due to an uncompensated spectral phase while the FT limited pulse duration is ~45 fs thanks to the generation of a broad bandwidth in the infrared. The pulse energy at 1250 nm is relatively weak (few nJ) and is consistent with the low gain value (<5 dB). Therefore, this scheme is suitable for use as a pre-amplifier to seed a subsequent ultra-fast amplifier. However, the low value can be significantly improved by increasing the HCC length to reach the multi-µJ level.

CONCLUSIONS

We investigated the tunability of infrared pulses generated by the FWM process in a gas filled HCC. We performed detailed analytical and numerical analyses of the process to understand the role of key parameters such as the gas pressure, the HCC length, GDM, pumpsignal frequency offset and delay, and the pump and signal chirps. At the HCC output, the femtosecond idler can be generated and tuned in the infrared region by controlling the pump-signal delay. An experiment was conducted to discuss the parameters and highlight the wide tunability from 1.2 to 1.5 µm. Ultra-short idler pulses are directly obtained at the HCC output and a duration of 45 fs can be reached if the phase is compensated. It is important to note that, the entire experiment was carried out with a 30 cm long HCC without the intention of optimizing conversion efficiency, but it can be implemented with a longer capillary (e.g. 60 cm) to generate higher energy pulses. This configuration is very promising for the development of reliable, ultra-fast sources of tunable pulses in near and mid-infrared.

Funding. ANR (ANR-17-EURE-0002, ANR-10-IDEX-03-02), and Bourgogne Franche-Comté council (456789).

Acknowledgments. O. Zurita-Miranda benefits of a CONACyT-GOBFrances (2018/710318). We also thank M.E. Ojo for helpful comments.

Disclosures. The authors declare no conflicts of interest.

Data Availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Fig . 1 a

 1 Fig .1 a) Dispersion curve of the HCC with a 150 µm core diameter and filled with argon at different pressure b) ZDW as a function of the core radius at a pressure of 1 bar for three gas species c) Gain spectrum as a function of the gas pressure for a pump power of 1.6 GW. The white line depicts a reference pressure for Fig.2, and Fig. 3. d) Group delay mismatch in a HCC with a core diameter of 150 µm and filled with different argon pressures. The red points are examples cited in the text.

  2.b displays a selection of signal bands at a pressure of 0.75 bar for several pump central wavelengths. The spectral bandwidth of the FWM increases when the broadband pump is used in comparison with the case where the pump is monochromatic. For example, in the two dimensional representation of the CL the FWM idler band expands from 1200 up to 2000 nm for the broadband pulse at a pressure of 0.75 bar, (line with grey shadow in Fig2a). At higher pressure, the FWM bands also became narrower and were shifted away from the pump spectrum since the pump lies in the normal dispersion for a pressure higher than 1.2 bar (Fig.1.a).

Fig. 2 a

 2 Fig. 2 a) Coherence length CL, for a tunable continuous pump wave at different gas pressures of Argon. The HCC has a core diameter of 150 µm. b) FWM band for a pressure of 0.75 bar and a tunable CW pump wave (750-850 nm).

  𝛽(Ω) -𝛽 0 -𝛽 1 . Ω)𝐴 ̂= 𝑖 [

Fig. 3 .

 3 Fig. 3. (a,c) Evolution of the spectrum along the HCC as the function of the frequency offset from the pump calculated from the NLSE with a pulsed pump and the analytical equations with a CW. (b, d) Selected spectrum at a length of 10, 35 and 60 cm. The pressure is 1.2 bar.

Fig. 4 .

 4 Fig. 4. a-b) Output spectral and temporal profiles at a length of 60 cm when the pump pulse is at the Fourier transform limit (FLP, black line) and chirped (CP, grey line) for a constant peak power. The pressure is 1.2 bar. Spectrum as the function of the HCC length when the pump pulse is unchirped (c) and chirped (e). Corresponding spectrograms at L=60 cm are shown in (d) and (f).

Fig. 5 .

 5 Fig. 5. (a) Normalized spectral intensity of the pump (black line), and signal (blue line) pulses b-c) Temporal profiles used in cases I and II.

Fig. 6 .

 6 Fig. 6. (a) Signal and idler efficiency for case I, (b) Signal and idler efficiency for case II.

Fig. 7

 7 Fig.7 Spectral and temporal evolutions along the HCC, for case I.b (a) and II.a (b) at a pressure of 1.2 bar.

Fig. 8 .

 8 Fig. 8. (a) Output spectrum for selected delays between the pump and signal. (b) Spectrogram at the HCC input and output for selected delays. The input condition corresponds to Case II.b.

Fig. 9

 9 Fig. 9 Experimental set-up. M Mirror, BS Beam splitter, ND Absorptive neutral density filter, SL Spherical-plano Convex lens, DM Dichroic cold mirror, SF Sapphire plate, LF Long-pass filter, SF Short-pass filter, SP Spectrometer.

Fig. 10 .

 10 Fig.10. (a) Autocorrelation trace of the unchirped pump pulse (data points) with a Gaussian fit (blue lines), (b) Autocorrelation trace when the pump passes through a dispersive block of N-F2. The FTL AC (black curves) corresponds to the autocorrelation of the Fourier transform pulse calculated from the spectrum.

Fig. 11 .

 11 Fig. 11. Normalized spectrum of the input signal (grey line, the output signal (blue line), idler (green line), output pump for a pressure of 0.3 bar (red line) or under vacuum (grey line).

Fig. 12 .

 12 Fig. 12. (a-b) Maximum wavelength in the signal and idler output spectra as a function of the delay when the pump pulse is unstretched (black line) or stretched (red line). The gray line corresponds to a secondary maximum. c) Maximum power of the idler as a function of the delay.

Fig. 13 .

 13 Fig. 13. Spectrum as the function of the delay for the signal (a) and idler (b)for a chirped pump pulse.

Fig. 14 .

 14 Fig. 14. Experimental autocorrelation trace of the idler (circles) and its Gaussian fit (black line). The red line corresponds to the FT limited pulse autocorrelation.


Z Guédé

O Pantalé

S Caperaa

Integration of a Multi-Physics Model in a Reliability-Based Design Framework: Application to Power Converters

Keywords: multi-physics model, structural reliability analysis, FORM, SORM, Importance sampling, microelectronic package

Structures of power converters constitute very complex systems, the behaviour of which introduces several phenomenons (electric, thermal and mechanic). An accurate modelling of their behaviour, thus, requires a multi-physics model which makes it possible to take into account the physical phenomenon in consideration. To achieve this, a FEM code (e. g. MulPhyDo) adapted to power converters and dedicated to simulation of structure in a multi-physics context has been built in the laboratory. The aim of the present study is to include in the modelling the existence of randomness and uncertainties, particularly of interest in the case of power converters. In order to achieve this, the multi-physics FEM code is coupled with a program dedicated to structural reliability analysis, in an Oriented Object Programming context. In this way, it is possible to compute the failure reliability of the structure under consideration and the sensitivity of the reliability index to each random variable. An application example of a microelectronic package is presented.

Introduction

The design for reliability of structures such as microelectronic devices is a very challenging issue. In fact, microelectronics components experience a complex structural behaviour, which introduces several physical phenomenons (e. g. electric, thermal and mechanic). Moreover, their behaviour is affected by numerous uncertainties due to: natural variations of material properties and manufacturing process, unpredictable operating conditions (mission profile, environmental factors…) , and the lack of adequate model parameters for many of the materials being applied.

In order to predict the performance and the reliability of these devices more efficiently, it would be coherent to use realistic numerical simulations which, on one hand, take into account the couplings between the various existing physics, on the other hand, incorporate the uncertainties in the design parameters. In this way, it would be possible to assess the reliability of the given structure and to quantify the influence of the various input parameters on the structural reliability.

The aim of the present paper is to integrate a global reliability approach in a multi-physical finite element computation scheme. An illustration example of a solder joint low thermal fatigue reliability assessment is given.

Multi-physics modelling

The multi-physics modelling intends to take into account all existing physics, namely electric, thermal and mechanic. In this case, the structure is submitted to electrical loads and experiences a thermo-mechanical behaviour. The simulation procedure begins with an electrical analysis. The purpose of this analysis is to determine the field of electric potentials of the model when it is subjected to loadings and boundary conditions of electric nature (e. g. current density and potential). Once the electric analysis is finished, the heat loss by Joule effect in the circuit is calculated and introduced as loading into a thermal analysis. One can add if necessary the thermal boundary conditions (e. g. imposed temperature or heat transfer coefficient). At the end of the thermal analysis, the temperature field of the model is obtained. The temperature field then makes it possible to determine the differential expansions to which the package is subjected and, taking account of displacements and mechanical loads applied, the stress field can be computed through a mechanical analysis. The procedure for electro-thermo-mechanical analysis is shown on the Figure 1.

A finite element code devoted to multi-physics computing (e. g. MulPhyDo) has been built in the laboratory [START_REF] Menanteau | A coupled electro-thermomechanical fem code for large scale problems including multi-domain and multiple time-steps aspects[END_REF]. This code can perform weak couplings between electrical, thermal and mechanical physics, including floating subdomains and parallel architecture to gain computational time. Beside, it makes it possible to have different time-steps in the multi-physics integration.

Structural reliability methods

The purpose of structural reliability methods is to assess the probability of failure of a structure under a given criterion. These methods are based on [START_REF] Ditlevsen | Structural reliability methods[END_REF]:

The choice of a stochastic model for the set X of input random variables (i.e. marginal density and correlation matrix or better, the joint density function f X (x));

The definition of a failure scenario through the determination of a limit state function G(X) which splits the safe domain (defined by G(X) > 0) from the failure domain (defined by G(X) 0). The boundary defined by {X such that G(X) = 0} is called the limit state surface. The failure probability reads:

0 X X x ) x ( G f d f P (1)
This probability may be evaluated using the FORM and SORM (First and Second Order Reliability Method). FORM requires initially a transformation U = T(X), where U is a set of uncorrelated standard normal variables. The transformed limit state function is denoted by H(U). Then, the design point u*, which belongs to the limit state surface and lies closest to the origin, is obtain solving an optimization problem (e.g. u* =min u under the constraint H(U) 0). The distance from the origin to the design point is the reliability index denoted by . Finally, using a linearization of the limit state function at the design point and the circular property of the standard normal space, an analytic approximation of the failure probability is derived by: P f = where is the standard normal integral. FORM gives also the sensitivity of the reliability index to changes in any input variable (e. g. the direction cosines of the unit normal vector at the design point to the limit state surface). SORM approximation additionally uses the curvatures i of the limit state function at the design point and the failure probability is estimated by [START_REF] Breitung | Asymptotic approximations for multinormal integrals[END_REF]:

1 1 2 / 1 1 ) ( n i i f P ( 2
)
where n is the number of random input variables.

To check FORM and SORM approximations, one can use simulation methods such as importance sampling, which is a Monte-Carlo simulation targeted at the design point. In this case, the failure probability is estimated by:

K i i i n i f u u u H I K P 1 ) ( ) ( ) ( ) ( ) ( 0 ) ( 1 ( 3
)
where K is the number of samples, is termed "importance sampling" distribution and n is the multinormal distribution. This method allows to obtain accurate values of the failure probability for a relatively low number of samples (e. g. 1000 samples). These three methods are illustrated in Figure 2. In practice, the computation of the failure probability using the above methods is carried out by coupling the stochastic model (e. g. the probabilistic characterization of the input random variables) and a mechanical model (e. g. the procedure to compute the limit state function) [START_REF] Lemaire | Fiabilité des structures -couplage mécano-fiabiliste statique[END_REF]. The structural reliability methods are implemented in the software FERUM (Finite Element Reliability Using Matlab) [START_REF] Haukaas | A computer program for nonlinear finite element reliability analysis[END_REF] which is used in the present analysis.

FORM

) One of the main failure modes of microelectronic packaging is the low cycle thermal fatigue of the flip-chip solder joints. In fact, semiconductor devices experience temperature variations which are caused by heat dissipation from electrical circuits. Due to the CTE mismatch between the joined components, different elongation and contractions take place in the layer. Thus, the solder joint between the layers experiences cycling shear strain. The structure under consideration is an element of a PBGA package made up of one chip and one substrate connected by 4x4 solder bumps (Figure 3). We want to assess the thermal fatigue reliability of the package. We assume that the package is subjected to a variable heat flux dissipated by the electrical circuits as shown on Figure 4. This variation consists on the application in a periodic sequence of 90 s of three cycles a power loss variation of 280 W followed by three other cycles of power loss variation of 140 W. We chose three cycles for each amplitude because it is generally the minimum number of thermal cycling at which a stabilized stress-strain hysteresis loop, required to use a strain energy based fatigue model, can be expected. We assume also, that both the substrate and the chip outer surface are subjected to film conduction. Their respective heat transfer coefficients are denoted h s and h c and the ambient temperature is taken equal to 70°C. At this stage, it must be noted that the further data given to the various parameters, especially the scatter of the chosen random variables are not realistic. That why the present study should not be considered as an industrial application but only as an academic example.

( f P SORM 1 1 2 / 1 1 ) ( n i i f P i : curvature at u* U 2 failure domain u* U 1 importance sampling K i i i n f u u U H I K P 1 ) ( ) ( 0 ) ( 1 importance sampling distribution (c) (b) (a)

Package modelling

Noting that MulPhyDo FEM code described above (section 2.) cannot yet solving nonlinear problems such as the one considered in this study, we use the finite element code ABAQUS [START_REF] Abaqus | User's Manual[END_REF] and we will only perform a thermo-mechanical fatigue analysis. For computation time reasons, a 2D finite element analysis of the package is carried out. Due to the package symmetry, only a half model needs to be made. 2D-8 nodes quad elements have been used for all the parts of the structure. Figure 5 shows a view of the FE model.

The solder joints were modelled with a steady-state creep material behaviour using the Darveaux constitutive relation [START_REF] Darveaux | Constitutive relations for tin-based solder joint[END_REF]: where cr is the steady-state creep strain rate, A is a constant characteristic of the underlying micromechanism, prescribes the stress level at which the power law break down, is the applied stress, n is the stress exponent, Q is the activation energy, k is Boltzmann's constant and T is the absolute temperature. For 62Sn36Pb2Ag solder, the material constants are: A = 9. 10 5 s -1 ;

T k Q A n cr exp sinh (4)
2 10 7 . 8 MPa -1 ; 3 . 3 n ; 8110 k Q K.
We assume implicitly here that these parameters are temperature independent. The other components material were modelled with linear elastic properties, which are supposed not to be temperature dependent. The nominal values of the elastic and thermal properties of the applied materials are given in Table 1. The parts respective materials and section thicknesses are given in Table 2.

Material Density

(kg/m 3 )

Young's Modulus (GPa)

Poisson coef.

Thermal expansion

(K -1 ) Conductivity (W.m -1 .K -1 )
Specific heat (J.kg -

Probabilistic model for input variables

In this example, we consider six random parameters, namely, both the substrate and the chip outer surfaces heat transfer coefficients, the thermal conductivity and the Young's modulus of the solder joint material, and both the substrate and the chip coefficients of thermal expansion (CTE). The other parameters, in particular the geometrical dimensions, are supposed to be deterministic. Note that, no uncertainty is introduced in the modelling parameters of the solder joint creep behaviour. The statistical properties of the random variables are given in Table 3

Low-cycles thermal fatigue modelling

The Darveaux energy-based fatigue model is used for determining the crack initiation in the structure [START_REF] Darveaux | Reliability of plastic ball grid array assembly[END_REF]:

e e e ave K ave V V W W W K N , 2 1 (5)
where, e W is the dissipated energy for one element, which is the area contained in the stress-strain hysteresis loop, e V is the associated element volume and ave W is an average dissipated energy computed by weighting the energy for each element with its volume and then normalize with the total volume. This fatigue model makes it possible to reduce the effect of geometric singularities. The respective values of the model parameters are . In this example, the crack is assumed to initiate in the periferic solder joint, connecting the chip to the ball solder via an intermetallic layer, since this is where the largest dissipated energy occurs (Figure 5). The linear Miner's rule is used to cumulate fatigue damage caused by both cycling heat flux variations of the sequence. Thus, denoting N 1 and N 2 the respective numbers of allowable cycles for both variations of the heat flux on the given sequence (Figure 4), the fatigue damage caused by one sequence reads:

2 1 1 1 N N D ( 6
)
Then the life-time of the structure is given by: N = 1/D and is expressed in terms of number of sequences.

Reliability analysis

Now, let us compute the thermal fatigue failure probability of the structure. The reliability design criterion is defined as follows: given a service life-time N 0 , the failure occurs when the package fails before reaching the given service life-time N 0 .

The limit state function is therefore express by:

0 N N G or equivalently by ) ln( ) ln( 0 N N G (7)
The second formulation of the limit state function is preferred since the applied logarithm smooth the limit state surface. In this study, the service life-time is given the value of 500 sequences. The relatively large discrepancy between the reliability index respective values by FORM and SORM, shows that the limit state surface would have strong curvatures. Concerning the sensitivity factors, their signs are coherent with the expected effect on the reliability of any increase of the associated random variable. In fact, any positive variation of the heat transfer coefficients improve the thermal resistance of the structure, which agree with the positive sign of their sensitivity factor. In other hand, any positive variation of the solder thermal conductivity or Young's modulus, whose sensitivity factors are negatives, increase the stress magnitude in the solder joint, and reduce logically its reliability. Surprisingly, the CTE of both substrate and chip have a negligible importance. This may be due to the fact that they were both taken as random variables. It would be instructive to consider only the discrepancy between them as a single random variable, which was not done in the present study. However, it should be emphasized that, since the obtained results have not been yet checked by a simulation method no definitive conclusion can be made. In fact, the limit state surface can have a complex form with several optimums due to the presence of strong non-linearity of the problem (e. g. creep behaviour of the solder material).

Conclusion

In the present work, global probabilistic analysis integration in a multi-physics model is presented. It aims at incorporating both the coupling between various physics in consideration and the existence of uncertainties in the parameters of the simulation. This approach can be of particular interest in the microelectronic industry whose component experience hardly predictable operating conditions and whose material model parameters present large scatter. In fact, it proposes a coherent analysis of such structures and its results (e. g. failure reliability, sensitivity factors) make it possible to better understand and quantify the various factors influencing the structural response through accurate numerical modelling. The academic illustration example presented in this paper prior to treat an industrial application, intends to show that it is possible to assess low cycle thermal fatigue reliability of solder joints.

Figure 1 :

 1 Figure 1: Overview of the electro-thermo-mechanical analysis.

Figure 2 :

 2 Figure 2: Failure probability approximation: (a) by FORM ; (b) by SORM ; (c) by importance sampling..

1 4

 1 Fatigue reliability-based assessment of a solder joint 4.1 Problem statement

Figure 3 :

 3 Figure 3: Schematic diagram of the PBGA.

Figure 4 :

 4 Figure 4: Heat flux variation in one sequence.

Figure 5 :

 5 Figure 5: (a) 2D-FE model of the package element; (b) dissipated strain energy in the package element after a sequence of heat flux.

Figure 6 :

 6 Figure 6: Sensitivity factors.

Table 1 :

 1 Nominal values of elastic and thermal material properties.

	1 .K -1 )

Table 2 :

 2 Package parts material and section thickness.

Table 3 :

 3 . Statistical properties of the random variables.

	Variable description	Distribution	Mean	Coef. of var.
	Heat transfer coef., h s	log-normal	20000	0.1
	Heat transfer coef., h c	log-normal	20000	0.1
	Solder joint thermal conductivity	log-normal	38	0.1
	Solder joint Young's modulus	log-normal	20	0.1
	Substrate CTE	log-normal	3.90 10 -6	0.1
	Chip CTE	log-normal	3.24 10 -6	0.1

  The failure probability is then computed by FORM and SORM, importance sampling results are not available yet, due to CPU reasons. The results are reported in Table4. The respective sensitivity factors of the random variables in the structural reliability are shown in the diagram in the Figure6.

	Reliability method	Reliability index	Failure probability
	FORM	1.88	3.005 10 -2
	SORM	2.41	7.98 10 -3

Table 4 :

 4 Reliability index and failure probability obtained by FORM and SORM analysis