Sonia Boscolo

Frederic Chaussard

Esben Andresen

Hervé Rigneault

Christophe Finot
email: christophe.finot@u-bourgogne.fr

Impact of initial pulse shape on the nonlinear spectral compression in optical fibre

Keywords: Nonlinear spectral compression, nonlinear fiber optics, pulse shaping

Introduction

Self-phase modulation (SPM) in optical fibre is ordinarily associated with spectral broadening of an ultra-short optical pulse. However, for appropriate initial conditions of the input pulse, SPM can result in significant spectral compression [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF]. Indeed, SPM causes spectral compression or broadening depending on the initial frequency modulation (chirp) of the pulse electric field. Specifically, a pulse with a negative chirp, such as that imparted by an anomalously dispersive element, is compressed by the effects of SPM [START_REF] Zohrabian | Compression of the spectrum of picosecond ultrashort pulses[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF][START_REF] Planas | Spectral narrowing in the propagation of chirped pulses in single-mode fibers[END_REF]. This method of spectral compression has been implemented using various types of fibres [START_REF] Washburn | Transform-limited spectral compression due to selfphase modulation in fibers[END_REF][START_REF] Sidorov-Biryukov | Spectral narrowing of chirp-free light pulses in anomalously dispersive, highly nonlinear photonic-crystal fibers[END_REF][START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF][START_REF] Andresen | Spectral compression of femtosecond pulses in photonic crystal fibers[END_REF] and has also been studied in nonlinear waveguides [START_REF] Mei | Chirp-free Spectral Compression of Parabolic Pulses in Silicon nitride Channel Waveguides[END_REF]. It is suitable for a very large range of wavelengths including Ti:sapphire wavelengths [START_REF] Washburn | Transform-limited spectral compression due to selfphase modulation in fibers[END_REF][START_REF] Andresen | Spectral compression of femtosecond pulses in photonic crystal fibers[END_REF], the widely used 1-m [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF][START_REF] Rusu | All-fiber picosecond laser source based on nonlinear spectral compression[END_REF][START_REF] Limpert | SPM-induced spectral compression of picosecond pulses in a single-mode Yb-doped fiber amplifier[END_REF] and 1.55-m [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF] windows and the emerging 2-m band [START_REF] Bao | Spectral compression of a dispersion-managed mode-locked Tm:fiber laser at 1.9 um[END_REF].

The process can also sustain simultaneous amplification of the pulse [START_REF] Rusu | All-fiber picosecond laser source based on nonlinear spectral compression[END_REF][START_REF] Limpert | SPM-induced spectral compression of picosecond pulses in a single-mode Yb-doped fiber amplifier[END_REF][START_REF] Bao | Spectral compression of a dispersion-managed mode-locked Tm:fiber laser at 1.9 um[END_REF][START_REF] Wang | Spectral and temporal breathing selfsimilar evolution in a fiber amplifier for low-noise transform-limited pulse generation[END_REF], thereby providing an attractive solution to convert ultra-short pulses delivered by femtosecond oscillators into high-power, near-transform-limited picosecond pulses, and to counteract the spectrum expansion that usually occurs with the direct amplification of picosecond structures.

Most of the theoretical and experimental works to date have considered usual initial pulse profiles such as Gaussian or hyperbolic secant pulses [START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF][START_REF] Washburn | Transform-limited spectral compression due to selfphase modulation in fibers[END_REF][START_REF] Andresen | Spectral compression of femtosecond pulses in photonic crystal fibers[END_REF][START_REF] Rusu | All-fiber picosecond laser source based on nonlinear spectral compression[END_REF][START_REF] Limpert | SPM-induced spectral compression of picosecond pulses in a single-mode Yb-doped fiber amplifier[END_REF][START_REF] Wang | Spectral and temporal breathing selfsimilar evolution in a fiber amplifier for low-noise transform-limited pulse generation[END_REF]. However, several recent studies have demonstrated that the use of pre-shaped input pulses with a parabolic waveform can achieve spectral compression to the Fourier transform limit owing to the fact that for such pulses the cancellation of the linear and nonlinear phases can be made complete [START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude shaped pulses with negative chirp[END_REF][START_REF] Finot | Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime[END_REF][START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF]. The purpose of this paper is to provide insight into the influence of the input temporal intensity profile on the spectral dynamics of negatively chirped pulses that occurs upon nonlinear propagation in a fibre.

In particular, we emphasize that spectral narrowing does not occur for any initial pulse shape, and that there are significant differences between the propagation of parabolic, Gaussian, super-Gaussian or triangular pulses, which we elucidate with the aid of a time-frequency analysis. We show that initially parabolic-shaped pulses provide the best results in terms of quality of the compressed spectrum. Initially Gaussian pulses undergo efficient spectral compression, but the resulting spectrum exhibits residual pedestals stemming from uncompensated higher-order phase.

On the contrary, super-Gaussian pulses do not experience any spectral narrowing, and their spectral extent does not change significantly upon propagation. In the case of triangular pulses, we observe a spectral interference phenomenon similar to Fresnel bi-prism interference. Simple analytic formulae are presented, which can predict the evolution of the spectral extent of the different input pulses along the fibre. We also discuss the impact of realistic initial conditions and system parameters on the spectral dynamics, and show that the interference pattern observed with triangular pulses is strongly affected by deviations of the initial waveform from the ideal shape.

Ideal nonlinear propagation of differently shaped pulses

Situation under investigation

In this section, we study the propagation in a nonlinear optical fibre of initially perfect pulse waveforms with a negative linear chirp, given by

  2 ( ) ( ) exp / 2 in in t P I t i A t   . Here, Iin(t)
is the temporal intensity profile of the pulse, P is the pulse peak power, and A is the chirp coefficient. We assume that these pulses have been temporally stretched in an anomalously dispersive medium so as to acquire a parabolic temporal phase over far-field evolution. For the purpose of illustration, we consider pulses at the wavelength 1550 nm, with a full-width at half maximum (FWHM) duration Tfwhm = 100 ps (after dispersive temporal broadening) and A = -16.7 10 -3 rad.ps -2 , yielding a spectral FWHM bandwidth fwhm,0  |A| Tfwhm in the far-field regime.

Four ideal and symmetric pulse shapes are investigated: a parabolic pulse with   , TS = Tfwhm / 2(ln 2) 1/8 , and TT = Tfwhm. The peak power is set to 12.5 W for the parabolic, Gaussian and super-Gaussian pulses and to 40 W for the triangular pulse. Figure 1 shows the temporal and spectral intensity profiles of the initial pulses.

Our numerical simulations of pulse propagation in the fibre are based on the standard nonlinear Schrödinger equation (NLSE) for the pulse envelope [START_REF]Nonlinear Fiber Optics, Fourth Edition[END_REF]:

2 2 / 2 0 z tt i            ( 1
)
where z is the propagation distance, t is the reduced time, β2 is the group-velocity dispersion (GVD) parameter, and γ is the coefficient of cubic nonlinearity of the fibre. This equation neglects the effect of fibre loss, as well as higher-order linear and nonlinear effects. Although these effects can have noticeable impact on pulses shorter than 1ps, here we neglect them as the leading-order behavior is well approximated by Eq. ( 1). We consider here a 500m-long highly nonlinear fibre (HNLF) with the Kerr coefficient = 10 W -1 km -1 and a low GVD coefficient of β2 = 1 ps 2 km -1 .

With such parameters that are typical of various demonstrations of spectral compression due to SPM in fibre [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF], the nonlinearity-dominant regime of propagation is applicable [START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF]. In this regime, the dispersion term in Eq. ( 1) plays a relatively minor role and can be neglected.

Accordingly, the temporal intensity profile of the pulse does not change along the fibre length, whereas SPM gives rise to a chirp NL(t) proportional to the temporal gradient of the intensity profile, so that after a propagated distance z, NL(z,t) = - P z t Iin. The temporal gradients of the various pulse intensity profiles are plotted in Fig. 1(a2). Note that the nonlinear propagation problem being studied can be conveniently normalized by introducing a normalized distance through the nonlinear length 1 /  P associated with the pulse at the entrance of the fibre [START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF][START_REF]Nonlinear Fiber Optics, Fourth Edition[END_REF].

The resulting temporal chirp of the pulse thus evolves longitudinally as (t,z) = A t + NL(z,t) = A t - P z t Iin, which translates in the frequency domain to a modification of the pulse spectrum.

The illustration of the impact of the pulse shape on these changes in the spectrum lies at the heart of the present study.

Longitudinal evolution of the pulse spectra

First we look into the evolution of the spectra of the different incident pulses in the HNLF. Figure 2 summarizes the results and highlights striking differences among the various evolutions. The spectra of the parabolic and Gaussian pulses undergo significant narrowing in the fibre (panels a and b), and subsequently re-broaden with further propagation. The super-Gaussian and triangular pulses exhibit very different spectral dynamics and do not experience appreciable spectral compression. The extent of the central part of the spectrum of the super-Gaussian pulse does not change much over the fibre length. On the contrary, the spectrum of the triangular pulse shows strong oscillations before splitting into two symmetric parts that move away from each other during propagation. The very different behavior of the various waveforms is also reflected in Fig. 3, where we plotted the longitudinal evolution of the spectral brilliance of the pulse at its central frequency (panel (a))

and of the Strehl ratio (panel (b)), defined as the ratio of the spectral peak power of the actual pulse and the spectral peak power obtained assuming a flat temporal phase of the pulse [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF][START_REF]Etude des effets combinés de la diffraction et des aberrations géométriques sur l'image d'un point lumineux[END_REF][START_REF] Schimpf | Self-phase modulation compensated by positive dispersion in chirped-pulse systems[END_REF]. It is seen that the parabolic waveform leads to optimum spectral compression, with the spectral brilliance featuring more than thirty-fold increase with respect to the value at the fiber entrance after a propagation distance zP = 334 m. The Strehl ratio reaches 1 at zP, which pinpoints a Fourier transform-limited pulse. The spectral compression performance is degraded for the Gaussian waveform, which achieves a fourteen-fold enhancement in spectral brilliance at the propagation distance zG = 310 m. The maximum Strehl ratio is 0.24 for this pulse shape, indicating an imperfect compression where the presence of uncompensated temporal chirp of the pulse leads to the appearance of side lobes in the pulse spectrum [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF][START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF]. On the contrary, the spectral brilliances of the triangular and super-Gaussian pulses do not rise greater than five times their initial values,

showing that these waveforms do not undergo efficient spectral compression. The corresponding Strehl ratios are very far from optimum values, highlighting the highly chirped nature of the waveforms. These results are confirmed by the evolution of the spectral compression ratio Crms (panel (c)), defined in terms of root-mean-square (rms) spectral width of the pulse: while Crms attains a value well above 10 for the parabolic pulse, it does not exceed 2 for the other waveforms being studied. The orange diamonds in subplot (c) represent the results of the exact analytical calculation of the rms spectral width

Nonlinear dynamics of the various pulse waveforms

We now illustrate the observed spectral dynamics of the nonlinear propagation of the various input pulse waveforms in the fibre based on a time-frequency representation of the pulses [START_REF] Treacy | Measurement and Interpretation of Dynamic Spectrograms of Picosecond Light Pulses[END_REF][START_REF] Azana | Time-frequency (Wigner) analysis of linear and nonlinear pulse propagation in optical fibers[END_REF].

Indeed, the use of spectrograms (such as those that can be recorded with cross-correlation frequency-resolved optical gating devices [START_REF] Linden | XFROG -A new method for amplitude and phase characterization of weak ultrashort pulses[END_REF]) enables us to provide some simple and intuitive, approximate analytical laws to describe the dynamics. Owing to the linear initial chirp, the common initial state for all waveforms features a distribution of the pulse energy along a line with the negative slope coefficient A.

Parabolic and Gaussian shapes

We first describe the dynamics experienced by an incident parabolic pulse in the fibre (Fig. 4). This is a textbook example to understand the process of spectral compression by SPM and has been already the subject of several investigations [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF][START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF][START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude shaped pulses with negative chirp[END_REF][START_REF] Cheng | Transform-Limited Spectral Compression by Using Negatively Chirped Parabolic Pulse in All-Normal Dispersion Photonic Crystal Fibers[END_REF]. The temporal gradient of the initial pulse intensity profile (indicated by white arrows in Fig. 4(a1)) provides the rate at which the instantaneous frequency of the pulse evolves along the fibre, z NL. In the case of an inverted parabola, the chirp generated by SPM has a perfectly linear temporal variation: NL(t,z) = 2  P z t / TP 2 . Hence, the temporal variation of the overall chirp of the pulse remains linear upon propagation, with a slope coefficient continuously increasing from negative to positive values: 2 . The slope coefficient becomes zero at the propagation distance zP = -A TP 2 / 2  P, hence the pulse is transform-limited (i.e. the Strehl ratio equals 1) at such distance, and the pulse spectrum is significantly narrowed (Fig. 4(b3)) with negligible pedestalsnotice that the Fourier transform of a parabolic pulse is a first-order Bessel function of the first kind (sinc-like shape) [START_REF] Finot | Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime[END_REF]. Further, the monotonic temporal variation of the chirp at any distance in the fibre inhibits the development of noticeable oscillations in the spectrum, which would otherwise result from interference between different pulse parts having the same instantaneous frequency. We note that for a linearly chirped parabolic pulse with a relatively large chirp coefficient, the temporal shape is imaged in the spectral domain. This can be explained using the stationary phase method, i.e., the cancellation of oscillating contributions with rapidly varying phase. The zero-crossing spectral (half) width can be then evaluated by P = |CP| TP. The optical spectra in Fig. 4(b) as well as the results plotted with red dashed lines in Fig. 2(a) show the excellent agreement of the pulse spectral extent foretold by this formula with the results of numerical simulations. The fact that the parabolic pulse spectrum preserves a compact nature with very reduced wings along propagation in the fibre makes it possible for the large rms spectral narrowing shown in Fig. 3(c).

CP(z) = A + 2  P z / TP
It is also apparent from Figs. 3 and4 that the spectral evolution of the parabolic pulse is perfectly symmetric with respect to the point zP of spectral focusing [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF]. Based on the above considerations, the evolution of the spectral compression ratio of the pulse can be estimated by using the formula:

A / |CP(z)| = 1 / (1 + 2  P z / A TP 2
) (cyan circles in Fig. 3(c)). However, this function clearly diverges as z approaches zP; accordingly, it overestimates the value of the compression ratio in the vicinity of zP. A more accurate description of the evolution of the compression ratio can be obtained by drawing an analogy between the linear chirp generated by the SPM of a parabolic pulse in the temporal domain and the linear chirp imparted onto a pulse by GVD in the spectral domain [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF][START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF][START_REF] Finot | Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime[END_REF]. Indeed, based on the existing literature on dispersive pulse propagation, we can say that the evolution of the rms spectral width of a parabolic pulse is described by the square root of a non-trivial second-order polynomial [START_REF] Anderson | Analytic study of pulse broadening in dispersive optical fibers[END_REF]. Figure 3(c) highlights the outstanding agreement between this theoretical prediction (blue diamonds) and the numerical simulation results. to the parabolic pulse case: CG(z) = A + 2  P z / TG 2 . However, the deviations from linearity of the chirp generated by SPM in the pulse edges are responsible for several differences between the Gaussian and parabolic spectral dynamics, which can be elucidated with the aid of the spectrograms shown in Fig. 5. We can first notice that as the influence of SPM on the low-intensity wings of the pulse is small, the small portion of energy contained in the wings is not redistributed towards the center of the spectrum. Consequently, while the central part of the spectrum experiences significant and continuous narrowing during an initial stage of evolution, the overall extent of the spectrum does not change appreciably over the fibre length (panels (b)). This explains why the rms spectral compression factor reaches only moderate values in spite of a large increase in brilliance of the central region of the spectrum (Fig. 3). It is noteworthy that the propagation distance at which Crms is maximal (zs = 440 m) significantly differs from the distance zG of optimum compression in terms of Strehl ratio. The exact formula for the rms spectral width of the pulse derived in the Appendix shows that Crms cannot exceed the value 1.69 even for very high values of the initial chirp coefficient A. In light of these observations, we can conclude that the rms compression factor does not provide a good metric to quantify the narrowing of a Gaussian pulse spectrum.

During the initial stage of spectral compression, the chirp has a monotonic temporal variation across the pulse so that no interference effects are observed in the compressed spectrum (Fig. 5(b2)). The chirp becomes zero in the central part of the pulse (i.e. CG = 0) at the propagation distance -A TG 2 / 2  P = 240 m. However this distance does not correspond to the optimum compression point zG: shortly after this distance, the chirp develops a non-monotonic temporal variation that leads to interference between different pulse parts having the same frequency. Such a constructive interaction has a beneficial impact on the spectral compression process as it contributes to further narrowing of the spectrum and increase of the Strehl ratio, but this comes at the expense of an oscillatory structure of the spectrum (Fig. 5(b3)) [START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF]. For propagation in the fibre beyond zG, as a result of the increase of the chirp slope near the pulse center, the spectrum broadens and the range of frequencies that can interfere becomes larger, thereby leading to the buildup of strong and detrimental oscillations in the pulse wings together with strong side lobes in the resulting spectrum (Fig. 2(b) and panels 4 of Fig. 5). The white arrows have a length proportional to the temporal gradient of the pulse intensity profile.

Super-Gaussian shape

Now we explain why the nonlinear dynamics experienced by an incident super-Gaussian pulse in the fibre does not show any significant spectral compression. When considering the temporal chirp generated by SPM of a super-Gaussian pulse (Fig. 1(b)), we may notice that contrary to a parabolic or a Gaussian pulse, SPM will only induce a chirp in the steep edges of the pulse. Indeed, as the central part of the pulse is almost flat, i.e. it has a very low temporal gradient it will not contribute to create new frequencies in the pulse. Consequently, the main changes during propagation occur in the temporal wings of the pulse, where the energy is redistributed towards the lower frequencies, as it can be observed in Fig. 6. During an initial propagation stage (panels 2), this energy redistribution produces oscillations in the outer region of the temporal plateau of the pulse and, concomitantly, large spikes appear in the pulse spectrum, which thus develops a 'batman'-like shape. Then, as the chirp develops further, the spectral region in which interference occurs progressively extends to the whole spectrum. At the propagation distance z = 125 m (panels 3), the energy initially contained in the pulse edges is pushed towards the central frequency, hence the rms spectral width of the pulse is at its minimum. For longer propagation in the fibre (panels 4), the chirp induced in the edges of the pulse becomes larger than the initial extent of the spectrum, thus engendering the development of side lobes in the spectrum. As the new frequencies so generated do not interfere with any other parts of the pulse, these side lobes are free from interference patterns.

We can derive simple rules to characterize these dynamics. If we assume that the plateau of the super-Gaussian waveform (where the SPM is negligible) extends between ±30% of the FWHM duration, then the initial chirp profile will remain unchanged in this region, which in turn will result into a spectral breadth between ±S, where S = 0.3 |A| Tfwhm. These frequencies will represent the boundaries of the region of spectrum where most of the energy is present (red dashed lines Fig. 2(c) and green dashed lines in Fig. 6). Further, if we estimate that the temporal gradient of the pulse profile is achieved at a time position close to half of the FWHM pulse duration, then . The predictions provided by this simple formula are in qualitative agreement with the results of numerical simulations (dotted red lines in Fig. 2(c) and dotted green lines in panels 4 of Fig. 6), and explain the linear longitudinal evolution of the side lobes that are generated in the pulse spectrum. For propagation distances z < zS1, where zS1 can be estimated as the distance where S1+ = S1-= 0, i.e., zS1  120 m, interference phenomena do not affect the whole central region of the spectrum but are limited to frequencies above S1or below S1+. Shortly after zS1, the interference occurring at the central frequency engenders an increase of the rms spectral compression factor. For distances z > zS2, where zS2 can be estimated as the distance where ± S = S1±, i.e. zS2  192 m, side lobes appear in the spectrum and, accordingly, the rms compression ratio progressively decreases to values below 1 (Fig. 3(c)). The red arrows represent the frequency shift at t = Tfwhm/2.

Triangular shape

The observed spectral dynamics for an initial triangular pulse (Fig. 2(d)) are very different from the previously studied cases: after an initial stage of spectral narrowing in which strong oscillations appear in the spectrum, the spectrum splits symmetrically into two parts whose frequency spacing increases linearly over the fibre length. Once again, a spatio-temporal representation of the pulse provides a pedagogical way to visualize and understand these spectral dynamics. Indeed, the temporal intensity profile of a triangular pulse has a constant gradient, which translates into a constant and distinct (opposite sign) frequency chirp induced by SPM onto the leading (downshifted) and trailing (up-shifted) edges of the pulse:

 P z / TT, so that the overall chirp of the pulse evolves according to (t,z) = A t ±  P z / TT. We can therefore estimate the frequency boundaries T1 and T2 of each half of the pulse as 1 T   = (A TT +  P z / TT ) and 2 T   =  P z / TT, (the subscript + andbeing used for the leading and trailing edges respectively).

The different stages of spectral evolution are illustrated in Fig. 7. During an initial stage, the interference that develops in the pulse region containing most of the energy results into strong oscillations in the central part of the spectrum. Then, at the distance zT1 = -A T 2 T / 2  P where T1-= T2+ = -A TT / 2 (panels 2), the leading and trailing edges of the pulse overlap and the extent of the spectrum is reduced approximately twofold. At this distance, the spectrum exhibits very fast oscillations with a spectral interfringe of 1/TT, which can be inferred from the temporal spacing between the two pulse parts at the same frequency.

At the propagation distance zT2 = 2 zT1 (panels 4), T1-= T1+ and the spectrum has the same full width T2+ -T2as the initial spectrum, but now the negative frequencies are all contained in the leading edge of the pulse while the trailing edge contains only positive frequencies. The interference pattern is now much lessened. As the pulse further propagates in the nonlinear fiber (i.e., for z > zT2), the spectral brilliance at the central frequency drops down to zero (Fig. 3(a)), and the two parts of the spectrum separate in frequency, as demonstrated in [START_REF] Latkin | Doubling of optical signals using triangular pulses[END_REF][START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping[END_REF] where the spectral doubling of an optical signal by the SPM or XPM phase shift generated by a triangular pulse has been investigated. A natural extension of this analysis would include the use of saw-tooth pulses, where the asymmetry of the intensity profile would cause one portion of the spectrum to evolve at a higher rate than the other [START_REF] Parmigiani | An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper[END_REF]. We would like to emphasize that the spectral dynamics of the triangular pulse can be interpreted in terms of the well-known Fresnel bi-prism interference [START_REF]OEuvres complètes d'Augustin Fresnel[END_REF], which is widely taught in university textbooks [START_REF] Hecht | Optics, 4th[END_REF][START_REF] Born | Principles of Optics[END_REF]. Indeed, taking advantage of the concept of space-time duality [START_REF] Tournois | Sur l'analogie optique de certains montages électroniques: Formation d'images temporelles de signaux électriques[END_REF][START_REF] Salem | Application of space-time duality to ultrahigh-speed optical signal processing[END_REF][START_REF] Kolner | Space-time duality and the theory of temporal imaging[END_REF] and extending it to the frequency domain [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF], the spectral interference pattern associated with the triangular pulse dynamics can be viewed as the spectral analogue of the spatial interference pattern that is observed after a bi-prism: similarly to the timedomain counterpart of the bi-prism experiment described in Ref. [START_REF] Chaussard | Two-wave interferences space-time duality: Young slits, Fresnel biprism and Billet bilens[END_REF], the spectral interference pattern is localized with a bright fringe at the central frequency (see Fig. 2(c)).

Results with realistic conditions

In the previous section, we have based our analysis on initially perfectly shaped pulses propagating in a purely nonlinear fiber. However, except for Gaussian waveforms, such ideal temporal profiles may be hard to generate. The goal of this section is to study the impact of realistic initial conditions of the input pulses and system parameters on the pulse evolution in the fibre. To this end, we have generated the various input waveforms by the dispersive frequency-to-time mapping method that has been previously successfully used to synthesize high-quality parabolic pulses [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF][START_REF] Schimpf | Controllong the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum[END_REF][START_REF] Huh | Generation of high-quality parabolic pulses with optimized duration and energy by use of dispersive frequency-to-time mapping[END_REF].

Within this approach, pulses with the desired temporal shape can be generated by shaping the pulse spectrum to be of the target waveform, followed by linear propagation in a second-order dispersive medium. In the far-field regime, the quadratic spectral phase imparted by dispersion onto the pulse translates into a quadratic temporal phase and, thus, the resulting temporal waveform is a scaled replica of the spectrum [START_REF] Huh | Generation of high-quality parabolic pulses with optimized duration and energy by use of dispersive frequency-to-time mapping[END_REF]. The small distortions of the temporal waveform generated by this approach lead to a degraded temporal gradient compared to the ideal case: the imperfect pulse shapes resulting from realistic conditions are presented as the dashed lines in Fig. 1 and may turn into phase fluctuations through SPM, as for instance has been pointed out in previous studies of chirped pulse amplification. Figure 8 summarizes the results obtained by this approach and accounting for the GVD of the HNLF. Given the strongly nonlinear regime of propagation, the fibre dispersion does not have a major impact on the spectral dynamics of the pulses. The use of a realistic initial condition of the input pulse does not bring about any significant qualitative difference in spectral dynamics with respect to the ideal case as for the super-Gaussian waveform is concerned. The dynamics of the parabolic shaped pulse are also qualitatively well reproduced with a realistic initial condition. However, the maximum spectral brilliance of the pulse is reduced to nearly a third of the value achieved in the ideal case, and the longitudinal evolution of the brilliance is not any longer symmetric with respect to the point of spectral focusing (blue dashed line in Fig. 3(a)). These observations are consistent with previous experiments where the maximum Strehl ratio measured did not exceed 0.6 [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. Conversely, in the case of the triangular input pulse the dynamics are visibly impaired by the use of a realistic initial condition (Fig. 8(d)). Once again, the use of spectrograms helps us elucidate the origin of the sensitivity of the triangular waveform to deviations from ideal shape. Indeed, while the chirp generated by SPM of an ideal triangular pulse has a discontinuity at the pulse center, the use of a realistic initial pulse shape results into a continuous temporal variation of the chirp, which now features a linear variation in the vicinity of t = 0 between the two constant frequency offsets induced by SPM (Figs. 1 and9).

As a result of this, interference now develops between three different temporal portions of the pulse (panels 2 of Fig. 9), leading to a narrower central peak in the pulse spectrum with increased brilliance (Fig. 3(a)). Further, Fig. 9 highlights that for propagation in the fibre beyond zT2 when the two portions of the split spectrum separate in frequency in the ideal case, now energy remains always present in the central part of the spectrum, and as the temporal chirp near the pulse center evolves linearly, it contributes to the development of strong oscillations across the whole spectrum.

Conclusion

We have studied the influence of the temporal intensity profile of the initial pulse on the process of spectral compression that occurs upon nonlinear propagation in an optical fibre. We have shown that necessary condition for nonlinear narrowing of the spectrum is that the input waveform can be fitted by a parabolic profile in its central part. Accordingly, neither super-Gaussian nor triangular pulses experience spectral narrowing while their nonlinear dynamics exhibit distinctly different features. The use of spectrograms helped us explain and predict in a simple manner the main trends observable in the evolution of the various incident waveforms. We have also emphasized that triangular pulses may lead to a spectral interference pattern very similar to the spatial interference that is observed in a Fresnel bi-prism experiment.

We believe that our results, obtained in the context of fibre optics, can be readily extended to highly nonlinear waveguides [START_REF] Mei | Chirp-free Spectral Compression of Parabolic Pulses in Silicon nitride Channel Waveguides[END_REF] to achieve on-the-chip functions. Moreover, we expect that the present study can support a better understanding of the practical tolerance of multistage fibre amplifier architectures to pulse degradation factors [START_REF] Korobko | Multistage fiber preamplifier comprising spectral compression for generation of high-energy laser pulses[END_REF] as well as provide a deeper insight into the in-cavity dynamics of mode-locked fibre lasers [START_REF] Boscolo | Amplifier similariton fiber laser with nonlinear spectral compression[END_REF][START_REF] Lan | Enhanced spectral breathing for sub-25 fs pulse generation in a Yb-fiber laser[END_REF]. The proposed approach relying on time-frequency analysis can be naturally extended to the scenarios in which the temporal chirp and/or additional phase corrections [START_REF] Boscolo | Enhanced nonlinear spectral compression in fibre by external sinusoidal phase modulation[END_REF] are applied to the propagating pulse by a an external phase modulator [START_REF] Plansinis | Spectral changes induced by a phase modulator acting as a time lens[END_REF] or through cross-phase modulation [START_REF] Guo | Spectral narrowing of negatively chirped femtosecond pulse by crossphase modulation in a single-mode optical fiber[END_REF]. Lastly, we would like to note that recent experiments have shown that spectral compression is also relevant to quantum light [START_REF] Karpiński | Bandwidth manipulation of quantum light by an electro-optic time lens[END_REF].

Following the calculation described by S.C. Pinault and M. J. Potasek in Ref. [START_REF] Pinault | Frequency broadening by self-phase modulation in optical fibers[END_REF], we derive an exact formula for the rms spectral width of an initially linearly chirped Gaussian pulse after undergoing SPM. The rms spectral width   is defined as

where ' and * denote derivation and complex conjugation, respectively.

In the case of a chirped Gaussian pulse , where m Pz   is the maximum nonlinear phase shift occurring at t = 0, the closed-form evaluation of the integrals above yields

where 0   is the initial rms spectral width of the chirped pulse.

Equation (3) can be used to draw some interesting conclusions. The kind of linear variation of   over the fibre length depends on the initial chirp imposed on the pulse: while in the case of a positive chirp A > 0, the rms bandwidth of the pulse broadens monotonically with propagation distance, when A < 0, the bandwidth first narrows, reaching a minimum at the propagation distance 2 [START_REF] Tournois | Sur l'analogie optique de certains montages électroniques: Formation d'images temporelles de signaux électriques[END_REF] 42

G AT z P    (4)
Using this result in Eq. ( 3), we obtain a maximum rms spectral compression ratio given by  

We can see that for large values of the product A TG, (Crms)max approaches a constant value  

Figure 1 :

 1 Figure 1: (a) Temporal intensity profiles (subplot 1) and temporal intensity gradients (subplot 2), and (b) spectral intensity profiles of parabolic (blue), Gaussian (red), fourth-order super-Gaussian (orange) and triangular (violet) pulses at the entrance of the fibre. The ideal waveforms are plotted with solid lines while the results obtained through frequency-to-time shaping are plotted with dotted lines (see section 4). The vertical orange dashed lines in panels a delimit the region where the intensity distribution of the super-Gaussian pulse can be reasonably considered as flat.

Figure 2 :

 2 Figure 2: Longitudinal evolution of the spectra of ideal (a) parabolic, (b) Gaussian, (c) fourth-order super-Gaussian, and (d) triangular incident pulses. The green dashed horizontal lines indicate the various distances at which the spectrograms of Figs. 4, 5, 6, 7 are taken. The red dashed and dotted lines represent analytical predictions based on a simplified approach.

Figure 3 :

 3 Figure 3: Longitudinal evolution of (a) the spectral brilliance at the central frequency, (b) the Strehl ratio, and (c) the rms spectral width for parabolic, Gaussian, fourth-order super-Gaussian and triangular incident pulses (the same color code as in Figure 1 is used). The dotted curves in subplot (a) represent the results obtained with non-ideal input waves (see section 4). The blue diamonds and cyan circles in subplot (c) represent the results of a theoretical fit to the evolution of the parabolic pulse properties and the predictions from an approximate analytical model, respectively.

Figure 4 :

 4 Figure 4: Spectro-temporal representation of a parabolic pulse at different propagation distances in the fibre: z = 0, z = 165 m, z = zP, and z = 500 m (panels 1 to 4, respectively). Subplots (a) and (b) show the spectrograms and the spectral intensity profiles, respectively. The white dashed lines represent the slope coefficient of the pulse chirp at the fibre input CP(0), while the slope coefficients CP(z) at the particular nonzero z are plotted with red dashed lines. The green dashed lines indicate the zero-crossing points of the pulse spectrum ±P as obtained by assuming a strictly parabolic shape of the spectrum. The white arrows have a length proportional to the temporal gradient of the pulse intensity profile.

Figure 5 :

 5 Figure 5: Spectro-temporal representation of a Gaussian pulse at different propagation distances in the fibre: z = 0, z = 93 m, z = zG, and z = 500 m (panels 1 to 4, respectively). Subplots (a) and (b) show the spectrograms and the spectral intensity profiles, respectively. The white dashed lines represent the slope of the chirp in the central part of the pulse at the fibre input CG(0), while the slope coefficients CG(z) at the particular nonzero z are plotted with red dashed lines.

Figure 6 :

 6 Figure 6: Spectro-temporal representation of a fourth-order super-Gaussian pulse at different propagation distances in the fibre: z = 0, z = 62 m, z = zS1, and z = 300 m (panels 1 to 4, respectively). Subplots (a) and (b) show the spectrograms and the spectral intensity profiles, respectively. The white dashed lines represent the slope coefficient of the pulse chirp at the fibre input. The green dashed and dotted lines represent the theoretical predictions for ±fS1 and fS2±, respectively. The white arrows have a length proportional to the temporal gradient of the pulse intensity profile.

Figure 7 :

 7 Figure 7: Spectro-temporal representation of a triangular pulse at different propagation distances in the fibre: z = 0, z = 65 m, z = zT1, z = zT2, and z = 500 m (panels 1 to 5, respectively). Subplots (a) and (b) show the spectrograms and the spectral intensity profiles, respectively. The white dashed lines represent the slope coefficient of the pulse chirp at the fibre input. The green dashed lines represent the theoretical predictions for T1± and T2± , respectively. The white arrows have a length proportional to the temporal gradient of the pulse intensity profile. The red arrows represent the frequency shifts experienced by the leading and trailing edges of the pulse γ P z / TT .

Figure 8 :

 8 Figure 8: Longitudinal evolution of the spectra of frequency-to-time shaped (a) parabolic, (b) Gaussian, (c) fourthorder super-Gaussian, and (d) triangular incident pulses. The red dashed and dotted lines represent analytical predictions based on a simplified approach.

Figure 9 :

 9 Figure 9: Spectro-temporal representation of a triangular pulse at different propagation distances in the fibre: z = 0, z = 65 m, z = zT1, z = zT2, and z = 500 m (panels 1 to 5, respectively). Subplots (a) and (b) show the spectrograms and the spectral intensity profiles, respectively.

  input pulse parameters.

   is the Fourier transform of the pulse envelope () t  . Using Fourier-transform and convolution theorems in a combination with integration by parts,2

			22   				2		22 22 ( ) ( ) ( ) ( ) dd dd    2                  	2	(1)
	where ()
									  can be expressed in terms
	of () t 	as [21]
				2 	2 22 '( ) *( ) ( ) t dt '( ) ( ) t t dt t dt t dt           	2

given in the Appendix and based on Ref.[START_REF] Pinault | Frequency broadening by self-phase modulation in optical fibers[END_REF]. The dashed vertical lines indicate the distances zS and zP of maximum Strehl ratio for the super-Gaussian and parabolic pulses, respectively.

Acknowledgements

We acknowledge financial support by the Conseil Regional de Bourgogne (Pari Photcom) and the Agence Nationale de la Recherche (Labex ACTION program ANR-11-LABX-01-01).


Cécile Crosnier De Bellaistre

Alain Aspect

Antoine Georges

Laurent Sanchez-Palencia

Effect of a bias field on disordered waveguides: Universal scaling of conductance and application to ultracold atoms

Introduction.-Anderson localization in unbiased disordered materials is signaled by the exponential suppression of diffusion and conductance [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF][START_REF] Lee | Disordered electronic systems[END_REF][START_REF] Abrahams | 50 Years of Anderson Localization[END_REF]. The connection between the two is firmly established by linearresponse theory and the Einstein-Sutherland relation [START_REF] Rammer | Quantum Transport Theory[END_REF][START_REF] Akkermans | Physics of Electrons and Photons[END_REF]. Hence, localized wave packets and transmission coefficients are characterized by the same exponential decay with distance. Bias fields induce a strong nonlinear response, which significantly affects localization and questions this relation. For a weak bias field, algebraic (rather than exponential) localization has been established in previous numerical [START_REF] Soukoulis | Localization in one-dimensional disordered systems in the presence of an electric field[END_REF][START_REF] Bentosela | Stark-Wannier states in disordered systems[END_REF] and analytical [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF][START_REF] Delyon | From powerlocalized to extended states in a class of one-dimensional disordered systems[END_REF] work. More precisely, Ref. [START_REF] Delyon | From powerlocalized to extended states in a class of one-dimensional disordered systems[END_REF] presented a rigorous proof that the eigenstates become extended beyond a critical value of a dimensionless parameter α, which characterizes the ratio of the bias force (opposite gradient of the bias field) to the disorder (see the precise definition of α below). It is qualitatively consistent with a diagrammatic calculation of the asymptotic density of an expanding wave packet [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF], yielding the average density n(x) ∼ 1/x β dens in the direction of the bias force, with β dens = 1+(1-α) 2 /8α for α < 1 and where the overline denotes disorder averaging. For α > 1, the asymptotic density is not normalizable, hence signaling a delocalization transition at α = 1. In contrast, numerical evidence was provided in Ref. [START_REF] Soukoulis | Localization in one-dimensional disordered systems in the presence of an electric field[END_REF] that the transmission coefficient for the Kronig-Penney lattice model decays algebraically for arbitrary large bias force, yielding exp ln T ∼ 1/x βtr , and is thus unaffected by the delocalization transition. Moreover, the exponent β tr 1/2α is not trivially related to the exponent β dens . These apparently contradicting behaviors raise several questions. First, it is unclear whether the result of Ref. [START_REF] Soukoulis | Localization in one-dimensional disordered systems in the presence of an electric field[END_REF] is universal or model dependent. In particular, the behavior of the characteristic quantity exp ln T in continuous-space models as considered in Ref. [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF] is unknown. Second, available numerics do not provide the behavior of the average transmission T , which would be more directly comparable to the average density n. Third, the behavior of physical quantities, in particular those directly related to the average transmission, such as Landauer conductances, remains unclear. These questions have applications in mesoscopic physics, including the propagation of microwaves in inhomogeneous disorder [START_REF] Xu | Light transmission properties in inhomogeneouslydisordered random media[END_REF] and the electric response to a bias field in disordered carbon nanotubes [START_REF] Hong | A flexible approach to mobility[END_REF] or silicon nanowires [START_REF] Hasan | A review on electronic and optical properties of silicon nanowire and its different growth techniques[END_REF], for instance. It also applies to ultracold atoms where Anderson localization can be studied quantitatively [START_REF] Sanchez-Palencia | Anderson localization of expanding Bose-Einstein condensates in random potentials[END_REF][START_REF] Shapiro | Expansion of a Bose-Einstein condensate in the presence of disorder[END_REF][START_REF] Kuhn | Coherent matter wave transport in speckle potentials[END_REF][START_REF] Piraud | Localization of a matter wave packet in a disordered potential[END_REF][START_REF] Billy | Direct observation of Anderson localization of matter waves in a controlled disorder[END_REF][START_REF] Roati | Anderson localization of a non-interacting Bose-Einstein condensate[END_REF][START_REF] Kondov | Three-dimensional Anderson localization of ultracold matter[END_REF][START_REF] Jendrzejewski | Three-dimensional localization of ultracold atoms in an optical disordered potential[END_REF][START_REF] Semeghini | Measurement of the mobility edge for 3D Anderson localization[END_REF][START_REF] Chabé | Experimental observation of the Anderson metal-insulator transition with atomic matter waves[END_REF][START_REF] Lopez | Experimental test of universality of the Anderson transition[END_REF] (for reviews, see also Refs. [START_REF] Modugno | Anderson localization in Bose-Einstein condensates[END_REF][START_REF] Sanchez-Palencia | Disordered quantum gases under control[END_REF]). In those systems, a bias field can easily be turned on, and both conductance [START_REF] Brantut | Conduction of ultracold fermions through a mesoscopic channel[END_REF][START_REF] Stadler | Observing the drop of resistance in the flow of a superfluid Fermi gas[END_REF][START_REF] Krinner | Superfluidity with disorder in a thin film of quantum gas[END_REF] and expansion dynamics [START_REF] Billy | Direct observation of Anderson localization of matter waves in a controlled disorder[END_REF][START_REF] Roati | Anderson localization of a non-interacting Bose-Einstein condensate[END_REF][START_REF] Kondov | Three-dimensional Anderson localization of ultracold matter[END_REF][START_REF] Jendrzejewski | Three-dimensional localization of ultracold atoms in an optical disordered potential[END_REF] are accessible.

Here, we study the transmission of a continuous onedimensional disordered waveguide subjected to arbitrary disorder and bias field. The statistical distribution of transmission is written in a universal form. It is characterized by a unique parameter, the length of the waveguide expressed in a rescaled metrics, which encapsulates all microscopic features of the medium and bias field. For white-noise disorder and a uniform bias force, we derive analytically the relations exp(ln T ) ∼ 1/x 1/2α and T ∼ 1/x 1/8α for arbitrary strength of the bias force. We also perform numerical calculations for various models, and obtain excellent agreement with the analytical results. The different power law obtained for T in compar-

k(0) k(L) ΔV -Fx μ-ΔV/2 x L 0 μ+ΔV/2 Figure 1.
Transmission and conductance of a matter waveguide of length L (black rectangle) in the presence of a bias force F (x) and disorder (random green line). The bias field is shown for a constant force (red line). The incident and reflected wave vectors are k(0) and the transmitted one is k(L). The Landauer conductance is measured from the discharge between two reservoirs (left and right blue boxes) with average chemical potential µ and infinitesimal potential difference ∆V .

ison to that of n found in Ref. [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF] is attributed to the different boundary conditions, whose role is enhanced by the long-range algebraic tails of the eigenstates in the presence of a bias field. Application to conductance measurements is discussed. In particular, we include important realistic features, which significantly affect the conductance as measured in electronic or ultracold atomic systems, such as finite disorder correlations, nonuniform bias forces, and finite temperatures.

Statistical distribution of transmission.-To start with, consider the transmission of a coherent wave in a disordered material of length L in the presence of a bias force F (x) (see the dashed black rectangle in Fig. 1). In the following, we adopt the language of quantum matter waves for concreteness. The results are, however, directly applicable to other types of waves, such as linearly polarized microwaves or sound waves since the stationary states of all are governed by similar secondorder space differential equations. The disordered potential V (x) is homogeneous and Gaussian. Its average is zero and its two-point correlation function reads C(x) ≡ V (x + x)V (x ). The latter may model a theoretical white-noise disorder, with C(x) = U R δ(x) and U R the disorder strength, or a more realistic correlated disorder, where C(x) is some function decaying on the typical length scale σ R .

To compute the statistical distribution P (T, x) of the transmission coefficient T at the distance x, we use the transfer-matrix approach, here generalized to include a, possibly inhomogeneous, bias field. In brief (see details of the derivation in the Supplemental Material), P (T, x) is governed by the Fokker-Planck equation

-(x) ∂P ∂x = ∂ T 2 P ∂T + ∂ 2 ∂T 2 T 2 (1 -T )P , (1)
with

-(x) 2 2 K(x)/m C[2k(x)], m the particle mass, E ≥ 0 the energy, K(x) ≡ E + x 0 dx F (x )
, and the initial condition P (T, x = 0) = δ(T -1). Equation ( 1) has a straightforward physical interpretation. We find the same equation as for unbiased disordered systems, except that the backscattering mean free path -(x) must be computed at the effective semiclassical kinetic energy K(x) and is thus position dependent. The validity of Eq. ( 1) relies on the sole assumptions that the disorder is weak, i.e., -(x) 1/k(x), σ R with k(x) = 2mK(x)/ the local wave vector, and that the work of the bias force is negligible on the disorder correlation length, i.e.,

F (x)σ R K(x), 2 k(x)/2m∂ k ln C[2k(x)].
Note that for a positive bias force F and a bounded disorder power spectrum C, both the backscattering mean free path -(x) and the wave vector k(x) increase with the distance x. Hence the validity conditions of Eq. ( 1) are always fulfilled, at least in the asymptotic limit x → ∞.

The quantity -(x) provides the natural metrics in the disordered material in the presence of the bias field, and, using the inhomogeneous dimensionless coordinate

s(x) = x 0 dx -(x ) , (2)
it disappears from Eq. ( 1). The same rescaling holds for inhomogeneous disorder (see, for instance, Ref. [START_REF] Xu | Light transmission properties in inhomogeneouslydisordered random media[END_REF]). Equation ( 1) thus admits the analytic solution

P (T, L) = 2 e -s(L)/4 √ πs(L) 3/2 T 2 ∞ cosh -1 √ 1/T dy y e -y 2 /s(L) cosh 2 y-1/T , (3)
i.e., the same as for unbiased, homogeneous disorder with L/ -rescaled to s(L) (see, for instance, Ref. [START_REF] Abrikosov | The paradox with the static conductivity of a one-dimensional metal[END_REF]). Note that in this universal form all the microscopic features of the medium, such as disorder correlations and bias field, are fully encapsulated into the definition of the metrics (2). Formula (3) is rigorous and allows us to compute exactly the various disorder-averaged quantities relevant to different questions.

Algebraic localization.-Let us start with the logarithm of the transmission. Due to its self-averaging character, it represents the typical transmission and it is the quantity that is usually computed numerically, as in Ref. [START_REF] Soukoulis | Localization in one-dimensional disordered systems in the presence of an electric field[END_REF], for instance. Using Eq. ( 3), we find ln T (L) = -s(L). This formula, together with Eq. ( 2), exactly matches the heuristic formula proposed in Ref. [START_REF] Soukoulis | Localization in one-dimensional disordered systems in the presence of an electric field[END_REF] to interpret numerical results for the specific Kronig-Penney model. Our analysis justifies this formula on rigorous grounds and generalizes it to any model of disorder and bias field. For white-noise disorder and uniform bias force, where C(2k) = U R and K(x) = E + F x, we find

s(L) = L 0 dx mU R 2 2 (E + F x) = 1 2α ln 1 + F L E , (4)
where α = 2 F /mU R is the relative strength of the bias In order to compare the behavior of the transmission to the result of Ref. [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF] for the expanding wave packet, we now compute the average transmission. This quantity is also the one that determines various physically relevant quantities, such as the Landauer conductance (see below). Using Eq. (3), we find the exact formula

T (L) = 4 e -s(L)/4 √ πs(L) 3/2 ∞ 0 dy y 2 e -y 2 /s(L) cosh(y) . (5)
Again, it is in excellent agreement with exact numerical calculations [see the solid blue squares and the solid line on Fig. 2(a)]. For white-noise disorder and uniform bias force, we find the asymptotic behavior T (L) ∼ 1/L 1/8α , up to logarithmic corrections. The difference of the scalings exp ln T (L) ∼ 1/L 1/2α and T (L) ∼ 1/L 1/8α originates from the well known large fluctuations associated with the statistical distribution (3) [START_REF] Beenakker | Random-matrix theory of quantum transport[END_REF].

In turn, it is remarkable that the scaling T (L) ∼ 1/L 1/8α differs from that found for the density profile of an expanding wave packet, n(x) 1/x 1+(1-α) 2 /8α [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF]. To make a direct comparison, we have computed the quantity T (L) using a different formalism than the transfermatrix approach, namely the diagrammatic approach used in Ref. [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF] to compute the quantity n(x) in the expansion scheme. The latter needs to be adapted to the transmission scheme we consider here. Indeed, all diagrams involving scattering outside the region [0, L] must be excluded. Taking this difference into account, the diagrammatic method allows us to recover the behavior predicted by the transfer-matrix approach. Physically, the strong difference between expansion and transmission can be understood as follows. Consider a particle initially at position x = 0 and look at the probability that it has been transmitted beyond x = L after infinite time. If the disorder is restricted to the space interval [0, L], this probability is given by the transmission coefficient T (L) and therefore decays as L -1/8α . In contrast, if the disorder extends over the full x line, it turns into ∞ L dx n(x), which decays as L -(1-α) 2 /8α . This slower decay is due to the presence of long-range algebraically localized eigenstates, centered beyond x = L, whose overlap with the initial wave function is significant, thus enhancing the probability of finding the particle at x > L. Note that this effect is expected to be less important when the eigenstates are more strongly localized, i.e., when α vanishes. This is consistent with the equality of the two exponents, (1 -α) 2 /8α 1/8α in the limit α → 0.

Experimental observation.-The transmission can be measured directly via the Landauer conductance [START_REF] Akkermans | Physics of Electrons and Photons[END_REF][START_REF] Landauer | Spatial variation of currents and fields due to localized scatterers in metallic conduction[END_REF][START_REF] Buttiker | Symmetry of electrical conduction[END_REF] in mesoscopic materials [START_REF] De Picciotto | Four-terminal resistance of a ballistic quantum wire[END_REF] or ultracold atoms [START_REF] Brantut | Conduction of ultracold fermions through a mesoscopic channel[END_REF][START_REF] Stadler | Observing the drop of resistance in the flow of a superfluid Fermi gas[END_REF][START_REF] Krinner | Superfluidity with disorder in a thin film of quantum gas[END_REF], for instance. While the discussion below is generic, we focus on ultracold atoms for the sake of concreteness. From a practical point of view, these systems offer key features, such as the possibility to control the disorder and to accommodate arbitrary large forces without damage. The Landauer conductance is defined as the ratio of the current I induced by the potential imbalance between two charge reservoirs to their potential difference ∆V (see Fig. 1) [START_REF] Buttiker | Symmetry of electrical conduction[END_REF]. In the simplest case where ∆V is measured inside the reservoirs (two-terminal scheme), it reads G 2 (L) = S 2π ∆V µ+∆V /2 µ-∆V /2 dE T (E, L), where µ ± ∆V /2 are the chemical potentials of the two reservoirs and S is the spin degeneracy. The various longitudinal energy levels give statistically independent contributions since they probe different Fourier components of the disorder, which are independent. Assuming that the potential difference ∆V is large compared to the longitudinal energy level spacing inside the guide but smaller than the energy variation scale of T (E, L), the finite potential difference ∆V may realize disorder averaging [START_REF] Mailly | Reduction of quantum noise in a GaAlAs/GaAs heterojunction by a magnetic field: An orthogonal-to-unitary Wigner statistics transition[END_REF][START_REF] Mailly | Sensitivity of quantum conductance fluctuations and of 1/f noise to time reversal symmetry[END_REF] and G 2 (L) T (µ, L). Moreover, in real experiments, the guide is not purely one dimensional and may contain tens of transverse modes, which realize further effective disorder averaging. For short guides, however, one might rather measure the typical conductance T typ ∼ exp ln(T ) .

Realistic models of disorder should include finite-range correlations. The metrics s(L) then reads

s(L) = L 0 dx m C[2k(x)] 2 2 (E + F x) , (6)
which increases as s(L) ∼ k(L) k(0) dk C(2k)/k.
Since any model of disorder with finite-range correlations has an integrable two-point correlation function C(k) in reciprocal space, the metrics s(L) saturates to a finite value when L → ∞ in the presence of finite-range correlations. Hence, the distribution P (T, L), and all disorder-average functions of T saturate to a nonzero value when L → ∞. This effect suppresses algebraic localization and entails that any finite-range correlations induce delocalization. Similarly, an increase of the bias force also entails delocalization. For instance, for white-noise disorder and F (x) ∼ x ε with ε > 0, we find s(L) ∼ const-1/L ε , which also saturates when L → ∞. Delocalization, as signaled by the saturation of T (L) and ln T (L), is confirmed by numerical calculations for either Gaussian correlations, Delocalization appears for any realistic model of disorder with finite-range correlations. Assuming that C(x) decays on the length scale σ R , the transmission crosses over from algebraic decay to saturation for a length L * given by ∆k(L * )σ R ∼ 1 with ∆k(L) = k(L) -k(0). In the two-Fermi terminal configuration of Fig. 1,k

C(x) = UR σR √ 2π exp -x 2 /2σ 2 R ,
(0) = k F = √ 2mk B θ F /
is the Fermi wave vector of the lefthand-side reservoir, with θ F the Fermi temperature and

k B the Boltzmann constant. Using ∆k(L) ∼ ∂ L k(0)×L ∼ mF L/ 2 k F , we find L * ∼ 2kBθF mσ 2 R F .
For mesoscopic channels designed in ultracold atomic systems [START_REF] Brantut | Conduction of ultracold fermions through a mesoscopic channel[END_REF][START_REF] Krinner | Superfluidity with disorder in a thin film of quantum gas[END_REF], where typically θ F ∼ 500 nK and σ R ∼ 0.5 µm, and assuming that the force results from gravity on 6 Li atoms, F 10 -25 N, we find L * ∼ 80 µm, which is of the order of magnitude of the experimentally realizable channel lengths. Note that the effective force can be re-duced by compensating gravity with a magnetic levitation field [START_REF] Jendrzejewski | Three-dimensional localization of ultracold atoms in an optical disordered potential[END_REF] or using a nonvertical geometry. Whatever their strength, these delocalization effects can, however, be incorporated into the metrics to test the universal relation [START_REF] Abrahams | 50 Years of Anderson Localization[END_REF]. Indeed, as shown on Fig. 2(a), we recover a universal behavior of the transmission by rescaling the Euclidean distance L to the metrics s(L) for correlated disorder, as well as nonuniform bias field [START_REF]We have checked that the distribution of transmission, Eq. (3), still[END_REF].

Another matter of concern is finite temperature, which is typically a fraction of the Fermi temperature in ultracold atoms. Using the Sommerfeld expansion of the current-potential characteristic function, we find

G 2 (L) = G 0 T [s(L)] + π 2 (k B θ) 2 6 A(L) , (7)
where, for a constant force F ,

A(L) = 1 F ∂ ∂µ 1 -(L) - 1 -(0) ∂T ∂s (8)
+ 1 F 2 1 -(L) - 1 -(0) 2 ∂ 2 T ∂s 2 .
For the parameters above and the typical disorder strength V R /k B ∼ 0.5 µK, we find that finite-temperature effects contribute the conductance G 2 from less than 2% for θ ∼ 0.1 θ F up to 15% for θ ∼ 0.3 θ F . Since the quantity A(L) is not a universal function of the rescaled length s(L), such finite-temperature effects break universal scaling. However, if the system is sufficiently large that -(L) -(0), then G 2 (L) becomes a pure function of s(L) and universal scaling is recovered.

Conclusion and outlook.-In summary, we have computed the statistical distribution of transmission in a disordered matter waveguide in the presence of a bias field. For white-noise disorder and a uniform bias force, we have shown analytically that the transmission decays algebraically, irrespective of the value of the force, in agreement with numerical calculations [START_REF] Soukoulis | Localization in one-dimensional disordered systems in the presence of an electric field[END_REF]. This behavior differs from what is predicted in the expansion of a wave packet, which features a delocalization transition [START_REF] Prigodin | One-dimensional disordered system in an electric field[END_REF] (see also Refs. [START_REF] Bentosela | Stark-Wannier states in disordered systems[END_REF][START_REF] Delyon | From powerlocalized to extended states in a class of one-dimensional disordered systems[END_REF]). This striking difference has been traced back to the long-range character of the algebraic decay of the localized eigenstates in the presence of the bias field. We have proposed a concrete observation of the transmission decay using ultracold atoms. While finite-range disorder correlations or nonuniform bias force entails systematic delocalization, we have shown that a universal behavior can be recovered using appropriate rescaling. We have also found that finite-temperature effects can be ignored for a long-enough waveguide.

Ultracold atoms also offer an interesting alternative to measure the transmission. The idea is to create a trapped Bose-Einstein condensate, which is less sensitive to finite temperatures than fermions, above a finite-size disordered region. Releasing the condensate from the trap as done in Refs. [START_REF] Billy | Direct observation of Anderson localization of matter waves in a controlled disorder[END_REF][START_REF] Roati | Anderson localization of a non-interacting Bose-Einstein condensate[END_REF][START_REF] Kondov | Three-dimensional Anderson localization of ultracold matter[END_REF][START_REF] Jendrzejewski | Three-dimensional localization of ultracold atoms in an optical disordered potential[END_REF], for instance, the atoms fall down in the gravity field, possibly partially compensated by a levitation magnetic field to control the force. The number of transmitted atoms is then proportional to the average transmission. The same experiment with the condensate created directly in an infinite-size disorder will in turn yield the asymptotic average density. This offers a single platform to compare directly the two quantities T (x) and n(x). More precisely, this scheme would provide the quantities T and n integrated over the energy distribution of the falling condensate [START_REF] Sanchez-Palencia | Anderson localization of expanding Bose-Einstein condensates in random potentials[END_REF][START_REF] Piraud | Localization of a matter wave packet in a disordered potential[END_REF]. This issue can, however, be circumvented by using an energy-selective radio-frequency transfer of atoms from an atomic state insensitive to the disorder to another state sensitive to the disorder [START_REF] Pezzé | Localized and extended states in a disordered trap[END_REF].

-Supplemental Material-Effect of a bias field on disordered waveguides: Universal scaling of conductance and application to ultracold atoms

In this supplemental material, we provide details about the transfer matrix formalism for an inhomogeneous medium and the derivation of the Fokker-Planck equation [Eq. (1) of the main paper].

Transmission and reflection coefficients.-To compute the transmission coefficient of a particle submitted to a bias force F (x) through a disordered sample in the space interval [0, L], it is convenient to define the semiclassical kinetic energy K(x) ≡ E + x 0 dx F (x ) and the associated wave vector k(x) = 2mK(x)/ . At any position x, we may write the particle wave function ψ(x) and its derivative ∂ x ψ(x) in the form ψ(x) ∂ x ψ(x) = e ik(x)x e -ik(x)x ik(x)e ik(x)x -ik(x)e -ik(x)x ψ + (x) ψ -(x) (S1)

A unique solution (ψ + (x), ψ -(x)) exists provided the determinant of the above matrix does not vanish, i.e., k(x) = 0. The particle flux, j(x) ≡ 2im (ψ * ∂ x ψ -ψ∂ x ψ * ) then reads j(x) = j + (x) + j -(x), where

Figure 2 .

 2 Figure 2. Analytical prediction vs numerical results for T (solid lines and solid symbols) and exp(ln T ) (dashed-dotted lines and open symbols), for α = 0.08 and 2mF/ 2 k 3 (0) = 0.01 in the following cases: uniform bias force and white-noise disorder (blue squares), uniform bias force and Gaussian correlated disorder with σRk(0) = 0.3 (magenta circles), harmonic bias force with 2mω/ k(0) 2 = 0.01 and uncorrelated disorder (brown diamonds). (a) Results plotted as a function of the metrics s(L) [for white noise and constant force, s(L) ∝ ln(1 + F L/E); see Eq. (4)]. (b) Results plotted as a function of 1 + F L/E.Inset: Probability distribution of ln T for F L/E = 28, as found from numerical simulations (histogram) compared to analytical prediction (line), for a constant force, with correlated (dark, magenta) and uncorrelated (light, blue) disorder.

  and a uniform bias force [see Fig. 2(b), magenta circles] or white-noise disorder and a slightly linearly increasing force F (x) = F + mω 2 x [see Fig. 2(b), brown diamonds].

j|r| 2 +

 2 ± (x) = ± k(x) m |ψ ± (x)| 2 (S2) are the right-moving (+) and left-moving (-) fluxes. The transmission coefficient is then defined as the ratio of rightmoving fluxes at the boundaries of the sample in the case where the incident flux is right-moving, i.e., j -(L) assumption of right-moving incident flux, the reflection coefficient is defined as the ratio of left-moving emergent flux and the right-moving incident flux, and reads R(L) ≡ |j -Scattering matrix.-Consider now a finite sample in the interval [x 1 , x 2 ], where 0 ≤ x 1 < x 2 ≤ L. We define the scattering matrix 1 )ψ-(x 1 ) k(x 2 )ψ + (x 2 ) = S k(x 1 )ψ + (x 1 ) k(x 2 )ψ -(x 2 ) . (S6) Particle-flux conservation between x 1 and x 2 imposes that the scattering matrix is unitary, S † S = 1. Moreover, time-reversal symmetry entails S * = S † . Those two relations lead to the usual relations |t| 2 = |r | 2 + |t| 2 = 1 t * r + r * t = 0. (S7) Straightforward calculations then lead to the usual relations for the transmission and reflection coefficients of samples between the points x 1 and x 2 , T = |t| 2 and R = |r| 2 = |r | 2 . (S8)

Acknowledgments.-We thank Boris Altshuler, Gilles Montambaux, Marie Piraud, and Christian Trefzger for useful discussions. This research was supported by the European Commission FET-Proactive QUIC (H2020 Grant No. 641122). It was performed using HPC resources from GENCI-CCRT/CINES (Grant No. c2015056853). Use of the computing facility cluster GM-PCS of the LUMAT federation (FR LUMAT 2764) is also acknowledged.

Transfer matrix.-We now define the transfer matrix T(x 2 , x 1 ) such that

Straightforward calculations yield T = 1/t * r /t -r/t 1/t (S10)

Transfer matrices can then be chained, i.e.,

Fokker-Planck equation.-Considering two samples in the intervals [0, x] and [x, x + ∆x] respectively, where the sample [0,x] has transmission coefficient T (x) and reflection coefficient R(x) = 1-T (x), and the sample [x, x+∆x] has transmission coefficient T ∆x (x) and reflection coefficient R ∆x (x), the transmission coefficient of the sample [0, x + ∆x] can be calculated from the product of the two transfer matrices. It yields

where θ ∆x (x) is the phase accumulated during one total internal reflection at point x, with r (x)r ∆x (x) = |r (x)r ∆x (x)| e iθ∆x(x) . Let us define the backscattering mean free path at kinetic energy K(x),

where C[2k(x)] is the disorder power spectrum. For weak disorder, i.e., -(x) λ(x), σ R , we may choose intermediate elementary lengths ∆x, such that λ(x), σ R ∆x -(x), where λ(x) = 2π/k(x). Since ∆x -(x), the nonvanishing value of the reflection coefficient R ∆x (x) results from typically less that one scattering and it may thus be computed in the single-scattering approximation, provided the local mean free path -(x) is well defined on the elementary cell ∆x. This implies that ∂ x -(x)∆x -(x), or equivalently it corresponds to assuming that the work of the force on the length of the elementary cell is small, F (x)∆x

1, where the overline denotes averaging over the disorder. Consequently, R ∆x (x) 1 and we may use the following expansion of ∆T (x) ≡ T (x + ∆x) -T (x):

The transmission coefficient is thus governed by a stochastic process when the system length x increases. The Kramers-Moyal expansion of the corresponding master equation for the probability distribution of the transmission coefficient at a given length, P (T, x), reads

Assuming that the quantity θ ∆x (x) is uniformly distributed on 2π and knowing R ∆x (x), the average on both quantities can be performed independently. We then find

M n = 0 for n ≥ 3. (S17)

The Kramers-Moyal expansion (S15) thus reduces to its first two moments, which yields the Fokker-Planck equation (1) of the main paper.