1. Introduction

Since its first experimental observation in air-silica microstructured fibers [1], supercontinuum (SC) generation in optical fibers has been a subject of intense investigation. During last years, this process of dramatic spectral broadening has been studied considering all pumping configurations, from the femtosecond to the continuous wave (CW) regime [2,3]. As a general rule, a large spectral broadening process is usually obtained by setting the pump frequency in the anomalous dispersion regime and close to the zero dispersion wavelength (ZDW) of the optical fiber. The physical mechanisms underlying SC generation are rather well understood and, in particular, coherent soliton structures are known to deeply affect the dynamics of the spectral broadening process [2,3]. The high nonlinearity of photonic crystal fibers (PCF) and the ability to readily engineer their dispersion characteristics permitted to control some of the key properties of the generated SC spectra [4-8]. Consequently, efforts are now focusing on developing and customizing SC spectra required for various applications such as frequency metrology, nonlinear microscopy, wavelength-division multiplexing. In particular, the emphasis is now shifting to CW pumping of nonlinear optical fibers leading to the highest spectral power and smoothest supercontinua [9-11].

The process of SC generation in a PCF that exhibits two ZDWs has also been the subject of extensive investigations [9-12]. The general physical picture is that modulational instability of the CW leads to the generation of a train of soliton-like pulses, which are known to emit (Cherenkov) radiation in the form of spectrally shifted dispersive waves [13]. The SC spectrum results to be essentially bounded by the dispersive waves, while the fundamental solitons exhibit a Raman self-frequency shift, which is eventually compensated in the vicinity of the second ZDW [4,9-13].

In a recent work we have studied theoretically the highly nonlinear regime of SC generation in a PCF whose dispersion curve exhibits two ZDWs [14]. In this incoherent regime [2,3], rapid temporal fluctuations of the field prevent the formation of robust coherent structures and solitons do not play any significant role in the spectral broadening process. We provided in Ref.[14] a thermodynamic interpretation of this incoherent regime of SC generation on the basis of wave turbulence theory [15-18]. The kinetic wave theory has been essentially developed with the aim of describing fully developed turbulence in a dissipative system driven far from equilibrium by an external source, as it occurs, e.g., in some laser systems [19]. However, the kinetic wave equations also describe the nonequilibrium evolution of random nonlinear waves in a conservative and reversible (hamiltonian) system. In analogy with kinetic gas theory, a closed system of incoherent nonlinear waves is expected to exhibit a thermalization process, which is characterized by an irreversible evolution of the system towards a thermodynamic equilibrium state. The kinetic wave theory describes the essential properties of this irreversible evolution towards the Rayleigh-Jeans equilibrium distribution.

In the context of nonlinear optics, the dynamics of wave thermalization has been studied in several circumstances [20-26] (for a simple introduction to optical wave thermalization see, e.g., Ref.[20]). In particular, in the recent work [14], we showed that the phenomenon of spectral broadening inherent to SC generation may be described as a natural thermalization of the optical field. More precisely, we showed that the saturation of SC spectral broadening may be ascribed to the natural tendency of the optical field to reach an equilibrium state, i.e., the state that realizes the maximum of nonequilibrium entropy. Accordingly, once the field has reached its thermodynamic equilibrium state, the spectral broadening process saturates and the spectrum of the optical field no longer evolves during propagation [14].

In this work we report an experimental study of the highly incoherent regime of SC generation and we confirm that this regime may be described as a thermalization process [14]. An experimental procedure based on a cutback of the PCF has allowed us to perform detailed measurements of the evolution of the spectrum of the field during its propagation. The experimental results of SC generation are in good agreement with the numerical simulations and thus provide some signatures of the process of optical wave thermalization. We remark in this respect that, to our knowledge, it is the first time that experimental spectra are directly compared with the Rayleigh-Jeans equilibrium distribution over the large spectral bandwidths considered here (~150 THz). Our study confirms that the thermalization process is characterized by the development of a double-peak spectrum, as predicted by the kinetic wave theory [14]. This peculiar property of the equilibrium spectrum has been demonstrated by means of accurate numerical simulations, which are found in quantitative agreement with the equilibrium spectrum predicted theoretically, without any adjustable parameters. Furthermore, in Ref. [14], the numerical simulations revealed that the Raman scattering is responsible for the generation of a kind of spectral incoherent soliton in the normal dispersion regime [27]. We confirm this prediction through the analysis of the spectrum of the incoherent soliton. In particular, a numerical spectrogram analysis confirms that the incoherent structure is confined in the spectral domain and not in the temporal domain, consistently with the properties of the spectral incoherent soliton [27]. Besides its fundamental interest, let us remark that the experiment reported here has been realized by using a simple commercial high-power nanosecond optical source, a feature that may provide useful insight for future developments of SC sources.

2. Experimental configuration

Before entering into details, let us briefly recall the conditions required for an optical field to exhibit a thermalization process through SC generation [14]. A key assumption of wave turbulence theory is to model the optical wave as a random (incoherent) field characterized by a stationary statistics, i.e., the time correlation of the field thus needs to be much smaller than the typical duration of the optical pulse (t_c ≪ T₀). In practice, this means that SC should be initiated by long pulses and high-power sources. Under these conditions, the optical field is expected to exhibit a turbulent dynamics, whose rapid temporal fluctuations of the field prevent the generation of coherent soliton structures. Furthermore, we showed in Ref.[14] that a PCF whose dispersion curve exhibits two ZDWs accelerates the process of wave thermalization, while a unique ZDW may not arrest the spectral broadening of the field. Indeed, in the spirit of the kinetic theory, a ZDW manifests itself in frequency space as a potential-well in which quasi-particles tend to accumulate [14]. In the absence of the second ZDW, quasi-particles are no longer trapped and are thus free to migrate in frequency space: The thermalization process slows down significantly and the saturation of spectral broadening may not occur (as illustrated in Fig. 3(a) of Ref.[14]).

More specifically, the numerical simulations reported in Ref.[14] indicate that wave thermalization through SC generation may be studied with ~1 kW pump power quasi-CW optical sources. Even though SC generation in PCF has been recently demonstrated with an industrial class of 400 W CW fiber laser [11], we present here a rather simple experimental setup using a low-cost and extremely compact commercial high-power sub-nanosecond pump. Indeed, our passively Q-switched Nd:YAG laser delivers 660 ps pulses at 1064 nm, with an average power about 70 mW and a pulse repetition rate of 7.7 kHz, corresponding to a pulse energy of 9 μJ and a peak power up to 14 kW. Assuming typical values of the coupling efficiency into a PCF of 10~40%, this high-power quasi-CW pump ensures that the incoherent regime of SC generation may be investigated with pump powers exceeding 1 kW.

 

Fig. 1. (a) Calculated dispersion curve with the two zero dispersion wavelengths located at 1033 and 1209 nm. The scanning electron microscope image of the fiber cross-section is shown in the inset. (b) Calculated modulational instability gain (m^-1) bands as a function of pump wavelength for the dispersion curve (a) and for an input peak power of 3.5 kW.

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We have chosen a PCF with closely spaced zero dispersion wavelengths around the pump wavelength, in order to produce a spectral broadening in a limited spectral bandwidth and within a short PCF length, a feature which also inhibits detrimental effects inherent to fiber losses. Moreover, the reason that motivated the choice of these fibers relies on the fact that they exhibit low-values of the second-order dispersion coefficient. This leads to an enhancement of the Modulational Instability (MI) gain that should accelerate the development of the incoherent regime of SC generation. The fiber used in this work is a 40 m-long PCF manufactured by IRCICA. The scanning electron microscope image of the fiber cross-section is illustrated in Fig. 1(a). The dispersion profile of the fundamental mode was calculated using a commercial fully vectorial mode solver [see Fig. 1(a)]. The two ZDWs have been estimated to 1033 and 1209 nm. However, this numerical method is very sensitive to variations in the PCF structure. Consequently, these values of the ZDWs have been confirmed by simple experimental measurements based on the process of soliton self-frequency shift compensation using a femtosecond optical source [28]. By switching the fiber input with the output, our experimental method has revealed some variations about ±10 nm around the calculated position of the ZDWs, due to possible dispersion fluctuations along the 40-m-long fiber segment. At the pump wavelength, our PCF exhibits a low anomalous dispersion value of ~1.5 ps/km/nm and its nonlinear parameter was calculated to be γ = 21 W^-1.km^-1. Fiber losses were also measured to be ~40 dB/km around the pump wavelength, except at 1380 nm due to the OH absorption peak [29], with losses up to 240 dB/km.

Let us remark that PCFs with closely spaced ZDWs have been recently shown to lead to the formation of two sets of MI gain bandwidths [30]. Our fiber parameters and the input peak power injected (up to 3.5 kW) allow us to reach particular conditions that lead to an overlapping of the two sets of MI gain-bands, regardless of the pump-wavelength [see Fig. 1(b)]. Note that this phenomenology regarding the overlapping of the two sets of MI bands may be captured by simply introducing fourth-order dispersion effects in the standard MI analysis [31]. As will be discussed below, the MI gain bands are very sensitive to small variations of the PCF dispersion curve. More generally, our MI analysis reveals that PCFs featured by closely spaced ZDWs normally lead to high and large MI gain bands, a property which is important for a rapid development of the incoherent regime of SC generation.

Our simple experimental setup uses a conventional half-wave plate-polarizer to modify the input power, while keeping the input linear polarization parallel to one of the PCF birefringent axes. The laser light is injected in the PCF with a 20x microscope objective. Input and output powers were measured with a thermal power meter and the coupling efficiency at the fiber input was determined to ~30%. Output spectra were measured at the PCF output with an optical spectrum analyzer (Anritsu MS9710B) and the spectral resolution was set to 0.5 nm.

3. Experimental results

The thermalization process is characterized by an irreversible evolution of the spectrum of the optical field towards an equilibrium state. This means that once the spectrum has reached equilibrium, it does not evolve during the propagation of the field. In order to provide experimental evidence of this relaxation process, it is important to record the spectrum of the field at different propagation lengths, while other experimental parameters (e.g., power) remain unchanged. To this end, we performed an experiment by cutting back the PCF length in few tens of cm increments. At each length, the spectrum and output power were measured.

The experimental results are reported in Fig. 2(a), which illustrates the evolution of the spectrum along the 40 m of the PCF for an input peak power of 3.5 kW. Each recorded spectrum does not correspond to a single shot spectrum, but to the sampling of a multitude of spectra and the average of a few of them for each sampling point. This is due to the fast pulse repetition rate of the source compared to both the integration time and the sweep time of our spectrum analyzer (video bandwidth set to 1 kHz or 100 Hz with 1001 sampling points). We should thus keep in mind that our measurement method leads to a “partial smoothing” of the SC spectra. No additional averaging was performed to smooth the optical spectrum.

We clearly observe in Fig. 2(a) an initial spectral broadening in the first meter of propagation in the PCF, which is due to the development of two sets of MI gain-bands. The first one generates frequencies around the pump frequency in the anomalous dispersion regime, while the second occurs 50 THz beyond the pump frequency in the normal dispersion regime. The development of the two sets of MI gain-bands then appears in disagreement with the theoretical MI analysis reported in Fig. 1(b). As discussed above, the stability analysis reveals a great sensitivity of the MI bands on slight modifications of the PCF dispersion curve. For instance, by means of a slight shift of the dispersion curve towards higher dispersion values [i.e., a vertical shift of the dispersion curve in Fig. 1(a)], one recovers a pair of well-separated MI gain bands. We thus attribute the two sets of MI bands reported in the experiment to dispersion fluctuations of the PCF fiber. Although such dispersion fluctuations affect the initial propagation [see Fig. 2(a)], they are essentially averaged out through the whole 40 m propagation length. We also note that the stability analysis reported through Fig. 1(b) is confirmed by the numerical simulations of the GNLSE [see Figs. 3(a-c)].

It is interesting to remark in Fig. 2(a) that the process of spectral broadening saturates in the high-frequency part of the spectrum. Also note that for large propagation distances, the SC edge exhibits a slight depletion. We have verified by numerical simulation that such a depletion is due to the Raman effect, which leads to a transfer of power towards the low frequency components (see Sec. 4). We remark that the saturation of spectral broadening does not result from fiber losses, which are estimated to ~2dB for the 40-m-long PCF segment. In the next Sections we shall see that such a saturation of spectral broadening may be regarded as a signature of the thermalization process of the optical field. In particular, we shall see that the high-frequency edge of the saturated SC spectrum (located at +70 THz from the pump frequency, i.e. at 850 nm) is in agreement with the numerical simulations and the equilibrium spectrum predicted by the kinetic wave theory.

Let us underline that, because of the stimulated Raman scattering effect, the saturation of spectral broadening cannot occur in the low-frequency part of the spectrum. The Raman effect is indeed responsible for a permanent transfer of power towards the low-frequency components of the field. As discussed in details in Ref.[27], this prevents, in principle, the establishment of a thermodynamic equilibrium state. However, we showed in our previous work [14] that the Raman effect is responsible for the generation of a kind of spectral incoherent solitons in the low-frequency part of the SC spectrum (normal dispersion regime). The spectral soliton moves away from the main SC spectrum, which reduces the power in the low-frequency edge of the SC spectrum. The Raman effect thus becomes almost inefficient, thus leading to a saturation of the spectral broadening in the low-frequency edge of the main SC spectrum. This scenario is clearly visible in the numerical simulation reported in Fig. 2(a) of Ref.[14]. In the experimental results reported above through Fig. 2(a), the separation of the spectral soliton from the main SC spectrum is not apparent. Note however that the frequency-shift of a spectral-hump may be identified without ambiguity in Fig. 2(a). This aspect becomes clear through the comparison of the spectra at 5 m and 40 m of propagation in the PCF, as remarkably illustrated in Fig. 2(b). One should also consider that the limited spectral window of our analyzer (600-1750 nm) prevents a complete observation of the frequency shift. As will be discussed in more details in Sec. 6, the properties of the spectral hump identified experimentally are reminiscent of those of spectral incoherent solitons [27].

 

Fig. 2. (a) Experimental results using a logarithmic intensity scale (dB) to illustrate the spectral evolution as a function of propagation distance in our 40-m-long PCF, for an input peak power about 3.5 kW. (b) Experimental output spectra obtained after 5 and 40 m of propagation. (c) Experimental spectra recorded after 40 m propagation in the PCF, as a function of the input pulse peak power. (d) Experimental output spectra at P₀ = 0.5 kW and P₀ = 3.5 kW. The white dashed lines show the two zero dispersion wavelengths of the optical fiber. ‘S’ indicates the position of the spectral incoherent soliton. Note in Fig. 2(a) the saturation of the spectral broadening in the high-frequency edge of the SC spectrum. (The video bandwidth of the spectrum analyzer was set to 100 Hz.)

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In the next Section we shall see that the absence of a clear separation between the spectral incoherent soliton and the saturated SC spectrum is due to the envelope profile of the pulses delivered by our laser source. For this reason we are not able to clearly identify a saturation of the spectral broadening process in the low-frequency part of the spectrum. Nevertheless, one may discern in Fig. 2(a) a small trough on the low-frequency part of the spectrum, which is located at -75 THz from the pump (i.e. at 1450 nm). It is interesting to remark that the frequency of this small depression corresponds to the low-frequency edge of the equilibrium spectrum predicted by the numerical simulations and the kinetic wave theory (see Secs. 4-5). Also note that another slight trough is in evidence at -65 THz from the pump (i.e. at 1380 nm), which is due to the fiber losses associated to the OH absorption peak [29].

Before cutting-back the PCF, we also studied SC generation in our 40 m-long PCF as a function of the input peak power P₀. Figures 2(c) and (d) show the evolution of the spectrum obtained by varying P₀ from 0 to 3.5 kW. We remark in Fig. 2(c) that the high-frequency edge of the SC spectrum exhibits a spreading as the power is increased, as clearly illustrated by the comparison of the spectra at P₀ = 0.5 kW and 3.5 kW [see Fig. 2(d)]. Note that this feature was not observed in the saturation of the spectral broadening discussed in Fig. 2(a). We shall see in Sec. 5 that this distinguished feature results from two factors: (i) the thermal equilibrium spectrum depends on the power of the field, (ii) 40 m of propagation are not sufficient for the optical field to reach equilibrium for small powers (we recall that the nonlinear propagation length decreases with the power, L_nl = 1/γP [29]). Furthermore, let us remark in Fig. 2(c) that a spectral-hump is generated in the low-frequency part of the spectrum for input peak powers higher than 0.5 kW. As discussed above through Fig. 2(a), we shall see in Sec. 6 that this effect is reminiscent of the generation of a spectral incoherent soliton. In particular, this effect becomes apparent when one compares the spectra of the field at low- and at high-pump powers, as illustrated in Fig. 2(d) for P₀ = 0.5 kW and P₀ = 3.5 kW.

In the following Sections we shall analyze in more details the experimental spectra reported through Fig. 2, in particular by comparing them with the numerical simulations and the thermodynamic equilibrium spectra predicted by the kinetic wave theory.

4. Numerical simulations

We have performed numerical simulations of the Generalized NonLinear Schrödinger Equation (GNLSE), which is known to provide an accurate description of the propagation of the optical field envelope A(z,t) in a PCF [2,3],

The higher-order time derivatives account for the dispersion curve of the PCF, which is characterized by the following expression of the linear dispersion relation of the GNLSE,

We note in particular that k”(ω) = ∂²k/∂ω² refers to the whole dispersion curve of the PCF [Fig. 1(a)], which was shown to play a key role in the kinetic wave theory [14]. The dispersion coefficients β_m have been calculated from the dispersion curve [Fig. 1(a)] by means of a polynomial fit up to the 12^th order: β₂ (s²/m) = -8.7648 × 10^-028 ; β₃ (s³/m) = 1.4296 × 10^-041 ; β₄ (s⁴/m) = 1.1613 × 10^-055 ; β₅ (s⁵/m) = -6.1660 × 10^-070 ; β₆ (s⁶/m) = 2.8689 × 10^-084 ; β₇ (s⁷/m) = -4.8289 × 10^-099 ; β₈ (s⁸/m) = -3.8989 × 10^-155 ; β₉ (s⁹/m) = -8.5981 × 10^-127 ; β₁₀ (s¹⁰/m) = 6.8595 × 10^-141 ; β₁₁ (s¹¹/m) = 3.2009 × 10^-155 ; β₁₂ (s¹²/m) = -3.8263 × 10^-169.

The GNLSE (1) describes self-steepening effects through the optical shock term, i.e., the term which is proportional to i τ_s ∂/∂t. This time derivative term accounts for the dispersion of the nonlinearity [29,32]. The GNLSE also describes the instantaneous (Kerr) and delayed (Raman) nonlinear effects described by the response function R(t) = (1-f_R) δ(t) + f_R h_R(t). Finally, the last term accounts for the PCF losses, the symbol $\widehat{\alpha }$ A meaning that the Fourier’s transform is α(ω) A(z,ω), α(ω) being the losses of the optical field vs frequency measured experimentally. For a complete discussion of the different terms of Eq.(1), we refer the reader to Refs. [2].

Let us remark that, due to the huge numerical time consuming, no averaging over the realizations was performed, i.e., the numerical spectra reported here correspond to a single realization of the initial noise. However, we underline that in the incoherent regime of SC generation investigated here, the output spectra do not depend significantly on the input small random noise superposed on the initial condition (pulse or CW). In order to quantitatively compare the numerical spectra with the theoretical and experimental spectra, we thus followed the smoothing procedure outlined in Ref.[12], in which the Savitzky-Golay smoothing filter (low-pass filter) was used. Moreover, we verified that the spontaneous Raman scattering effect does not affect the highly incoherent regime of SC generation considered here. Finally, let us note that the numerical simulations with a CW input were realized by using 2¹⁴ points with a temporal resolution of 1.95 fs, a time window of 32 ps and a spectral resolution of 31 GHz. The simulations with an initial input pulse refer to 2¹⁷ points with a temporal resolution of 1.9 fs, a time window of 250 ps and a spectral resolution of 4 GHz. In both cases, the longitudinal spatial discretization was 10 μm. Note that, by neglecting the Raman and loss terms, Eq.(1) exhibits a Hamiltonian structure. We verified that the Hamiltonian is conserved by the numerical scheme up to 10^-5.

 

Fig. 3. (a) Numerical simulations of the GNLSE Eq.(1) with the dispersion curve of Fig. 1(a) and for an input power of 3.5 kW. The simulations have been realized with an initial Gaussian pulse of 60 ps, i.e. ~10 times shorter than the experimental pulses. (b) Numerical spectrum for both propagation lengths 5 and 40 m. (c-d) Same as in (a-b), except that the initial condition refers to a continuous wave and that the numerical simulation neglects the Raman, shock and loss terms, i.e., Eq.(1) for τ_s = $\widehat{\alpha }$ = f_R = 0. The white dashed lines show both fiber ZDWs. ‘S’ indicates the position of the spectral incoherent soliton. Note the development of the double peak structure in the evolution of the spectrum, a feature which constitutes a key signature of wave thermalization (see Sec. 5).

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Figure 3(a) shows the evolution of the spectrum of the field obtained by numerically solving Eq.(1) for the dispersion curve reported in Fig. 1(a). The initial condition refers to a 60-ps (FWHM) Gaussian pulse with a peak power of 3.5 kW. This value of the pulse width is significantly smaller than that of the laser source employed in the experiment, i.e., 660 ps. We have not been able to perform numerical simulations with such large pulse widths because of the huge time-consuming calculations that they involve. In this respect, let us stress the impact of the limited pulse duration of the envelope pulse profile. Indeed, in Ref.[14], the numerical simulations were performed by starting from a CW with a superposed small random noise to seed the MI. The comparison of the above Fig. 3(a) with Fig. 2(a) of Ref.[14] reveals the essential impact of the envelope pulse profile on the dynamics of the spectral incoherent soliton. For the CW case, the frequency shift inherent to the spectral incoherent soliton leads to a complete separation of the soliton with respect to the saturated SC spectrum (see Ref.[14]). We have verified that such a separation also occurs with the dispersion curve of Fig. 1(a), as illustrated in Fig. 6(c) below. Conversely, the envelope profile of the short pulse considered in Fig. 3(a) prevents the formation of a spectral soliton. Recalling that the experimental pulses are 10 times longer than those of Fig. 3(a), the experimental conditions refer to an intermediary situation, in which the separation of the soliton begins to take place [see Fig. 2(b)]. As a matter of fact, the dynamics responsible for the generation of a spectral soliton from a broad initial spectrum constitutes a difficult theoretical problem, which is presently under investigation by exploiting the kinetic equation derived in Ref.[27].

Let us now discuss the saturation of the SC spectral broadening in the high frequency edge of the spectrum. For this purpose, we reported in Fig. 3(c-d) the same numerical simulation of Fig. 3(a-b), but starting with a CW and neglecting the loss, Raman and shock terms, i.e., Eq.(1) with $\widehat{\alpha }$ = τ_s = f_R = 0. This simplified Schrödinger equation will be referred to the NonLinear Schrödinger Equation (NLSE) in the following. This equation corresponds to the conservative limit considered in Ref.[14], in which SC thermalization was clearly identified. Such a thermalization process is confirmed by the simulation of Fig. 3(c-d), where a clear saturation of spectral broadening occurs in both the high- and low-frequency edges of the SC spectrum, i.e., at +70 THz and -75 THz from the pump frequency. Let us remark that the value of the high-frequency edge is in good agreement with that obtained in the simulations of the GNLSE [Fig. 3(a)], as well as in the experimental spectra [Fig. 2(a)]. Note however in Fig. 3(a) that the high-frequency edge of the SC spectrum is not truly invariant during the field propagation, a feature which is also visible in the experimental results [Fig. 2(a)]. Such a spectral depletion is merely due to the Raman effect, as revealed by the numerical simulations of the NLSE that neglects Raman scattering: a comparison of Figs. 3(a) and 3(c) shows that the Raman effect prevents the establishment of a truly stationary state in the high-frequency edge of the SC spectrum.

Besides the saturation of the process of spectral broadening, wave thermalization is characterized by another important signature, i.e., the development of a double-peak in the evolution of the spectrum of the field. This constitutes a key property of the equilibrium spectrum predicted by the kinetic wave theory [14]. The double peak spectrum is clearly visible in the numerical simulations of the NLSE, as remarkably illustrated in Fig. 3(c-d). The development of the double peak is also apparent in the simulations of the GNLSE before 8 m of propagation [see Fig. 3(a-b)]. The next Section is devoted to a detailed analysis of this double-peak spectrum through a comparison of the experimental, numerical and theoretical results.

5. Comparison between experiment, simulations and kinetic theory

A thermodynamic description of the SC generation process was given in Ref.[14] on the basis of the wave turbulence theory. To apply the kinetic wave theory, the equation that governs the evolution of the optical field should exhibit a hamiltonian structure [15-18]. This essentially explains why the non-conservative Raman effect does not lead to a thermalization of the field, as discussed in details in Ref. [27]. Furthermore, for simplicity reasons, we also neglected the shock term in the kinetic theory presented in Ref.[14]. Actually, we recently extended the kinetic theory [14] by including the nontrivial shock term. However, the derivation of the kinetic equation is rather technical and will be the subject of a more detailed article [33].

On the basis of the so-called “random-phase approximation”, a kinetic equation may be derived from the NLSE [14]. In complete analogy with the celebrated Boltzmann’s equation in kinetic gas theory [34], the kinetic equation exhibits a H-theorem of entropy growth, d_zS ≥ 0, where S(z) = ∫ Log[n(z, ω)] dω denotes the nonequilibrium entropy, n(z, ω) being the averaged spectrum of the optical field [20-25]. The thermodynamic equilibrium spectrum may thus be calculated as in standard statistical mechanics. It refers to the Rayleigh-Jeans spectrum that realizes the maximum of nonequilibrium entropy, given the constraints on the conservation of the energy E, the momentum M, and the power P of the field [14],

where k(ω) refers to the dispersion relation (2), while T, λ, and μ refer to the Lagrangian’s multipliers associated to the conservation of (E, M, P). By analogy with thermodynamics, T and μ denote the temperature and the chemical potential of the optical field at equilibrium. The three unknown parameters (T, λ, μ) are calculated by substituting the equilibrium spectrum (3) into the definitions of E, M and P. One obtains an algebraic system of three equations for three unknown parameters, which has been solved numerically. Let us underline that we always found a unique triplet solution (T, λ, μ) for a given set (E, M, P), a feature which is consistent with the fact that a closed (hamiltonian) system should exhibit a unique thermodynamic equilibrium state.

 

Fig. 4. Comparison of the theoretical, numerical and experimental spectra in logarithmic scale. (a) Plot of the equilibrium spectrum n^eq(ω) given in Eq.(3) without adjustable parameters. (b) Spectrum obtained by solving numerically the NLSE, i.e., equation (1) without Raman, loss and shock terms (τ_s = $\widehat{\alpha }$ = f_R = 0) [see Fig. 3(e) at z = 40 m]. (c) Spectrum obtained by solving numerically the GNLSE equation (1) [see Fig. 3(a) at z = 8 m]. (d) Spectrum recorded in our experiment [see Fig. 2(a) at z = 8 m]. ‘S’ indicates the position of the spectral incoherent soliton. Note the good agreement of the frequencies of the spectral peaks. (e) Evolution of the nonequilibrium entropy during the propagation of the optical field corresponding to the simulation of the NLSE in (b): the process of entropy production saturates once the equilibrium state is reached, as described by the H-theorem of entropy growth.

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In Ref. [14] we discussed the generic properties of the Rayleigh-Jeans equilibrium spectrum (3). For a dispersion curve with two ZDWs, we showed that the equilibrium spectrum is always characterized by a double peak structure [14]. However, such a double peak structure was shown to emerge very slowly in the numerical simulations. More precisely, the propagation lengths required to reach equilibrium were extremely large, and the emergence of a double peak spectrum was not identified in the numerical simulations of Ref. [14]. It is interesting to note that the dispersion curve considered here exhibits two closely spaced ZDWs, a feature which highly accelerates the process of wave thermalization. This is remarkably illustrated in Fig. 3(c) in which the double peak structure is shown to clearly emerge in the simulations. In Fig. 4(a-b) we compare the equilibrium spectrum (3) with the spectra obtained by numerically solving the NLSE. Let us stress the quantitative agreement between the numerical simulations and the theoretical spectrum. We underline that such a good agreement has been obtained without adjustable parameters: the parameters (T, λ, μ) used to plot Eq.(3) in Fig. 4(a) were calculated from the three conserved quantities (E, M, P) of the kinetic wave equation (see Ref.[14]). We also reported in Fig. 4(e) the evolution of the nonequilibrium entropy of the optical field corresponding to the numerical simulation of Fig. 3(c) and 4(b). We may note that the process of entropy production saturates as the optical field approaches the equilibrium state, consistently with the H-theorem of entropy growth inherent to the kinetic wave equation.

Let us now compare the equilibrium spectrum (3) with the numerical simulations of the GNLSE and with the experimental results. First of all it is important to note that the signature of wave thermalization is clearly apparent in the simulations of the GNLSE, as witnessed by the development of the double peak spectrum at the beginning of the propagation [see Fig. 3(a-b) or Fig. 4(c)]. Let us remark that the frequency positions of the two peaks correspond to those obtained with the NLSE. However, as already discussed, the Raman effect prevents the establishment of a genuine equilibrium state beyond 8 m of propagation, i.e., just before the formation of the spectral incoherent soliton. We also identified one of the two spectral peaks in the experiment for a propagation distance between 3 and 8 m. This is illustrated in Fig. 4(d), which reveals a remarkable agreement between the low-frequency peak (located at -47 THz) and the corresponding theoretical frequency predicted by the equilibrium spectrum (3). Note however that the high-frequency peak is not visible in the experiment. We attribute this to the Raman effect, which naturally tends to transfer power from the high- to the low-frequency peak. This is clearly illustrated by a comparison of Figs. 3(a) and 3(c), which shows that the Raman effect alters the power distribution in the two spectral peaks.

 

Fig. 5. (a) Numerical simulations of the NLSE showing the evolution of the spectra of the field (in logarithmic intensity) as a function of the input peak power in our 40-m-long PCF. (b) Numerical output spectra recorded for input peak powers of 0.5 and 3.5 kW. The white dashed lines show both fiber ZDWs. (c) Comparison of the thermodynamic equilibrium spectra predicted by the kinetic theory [Eq.(3)] with the numerical spectra of the NLSE. We note an appreciable discrepancy between theory and numerics at small power, which is due to the short (effective) nonlinear propagation length. Conversely, at higher powers, 40 m of propagation becomes sufficient for the field to reach thermal equilibrium.

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Let us finally discuss the evolution of the spectrum of the field as a function of the power for a 40 m of propagation length. We report in Fig. 5(a) the results of the numerical simulations of the NLSE vs optical power. We remark that, contrary to Fig. 3(c) where the SC spectrum reaches a stationary state in the propagation, here the spectrum exhibits a notable spreading as the power is increased. Also note in Fig. 5(a) that, despite this spectral broadening, the frequency positions of the two spectral peaks do not vary with the power of the field. It is interesting to note that the equilibrium spectra predicted by the kinetic theory exhibit the same properties. This is illustrated in Fig. 5(c), which compares the equilibrium spectra predicted by Eq.(3) and the corresponding spectra obtained by numerically solving the NLSE.

One may notice an appreciable discrepancy between the theory and the simulations at small powers. This simply results from the fact that the effective nonlinear interaction length decreases with the power (Lnl = 1/λP), so that 40 m of propagation are no longer sufficient for the field to reach the equilibrium state. It results that, as the power increases, the double peak structure gets more pronounced in the numerical spectra, while the valley between the two peaks gets deeper, in agreement with the theoretical prediction (3). This is remarkably illustrated in Fig. 5(c), which shows that the numerical spectra evolve towards the theoretical spectrum (3) as the power is increased.

In the experimental results reported in Fig. 2(d), we remarked that the high-frequency edge of the SC spectrum exhibits a spreading as the power is increased. Let us underline that such a spreading is in good agreement with the numerical simulations and the theory discussed in Fig. 5 through the NLSE. This corroborates the fact that the behavior of the high-frequency edge of the SC spectrum reported experimentally is well described by the equilibrium spectrum predicted by the kinetic wave theory [Eq.(3)].

6. Spectrogram analysis of the spectral incoherent soliton

In this Section we analyze in more details the properties of the so-called “spectral incoherent soliton” that has been discussed through the experimental results (Sec. 3) or in the numerical simulations of the GNLSE (1) (Sec. 4). First of all, we report in Fig. 6(a) a zoom of the spectrum of the soliton recorded experimentally. It reveals that the spectrum exhibits pronounced intensity fluctuations, despite the fact that the spectrum analyzer implicitly computes an average over a multitude of spectra. This indicates that the optical field is expected to be highly delocalized in the temporal domain (a field that exhibits a stationary statistics in the temporal domain is characterized by a δ-correlated spectrum [35]). This actually refers to a distinguished property of spectral incoherent solitons. Contrary to conventional solitons, spectral incoherent solitons do not exhibit a confinement in the temporal domain, but exclusively in the frequency domain. We refer the reader to Ref.[27] for a detailed discussion of this aspect. Let us note that this kind of solitons was investigated in plasma physics to study Langmuir turbulence or stimulated Compton scattering [36-39]. A spectrally localized soliton that exhibits a periodic temporal shape was also reported in optics in a different context [40].

In order to further analyze the properties of the spectral soliton, we calculated the spectrogram [2] of the optical field obtained by numerically solving the GNLSE (1). The spectrogram has been computed with a single-shot numerical simulation, whose parameters refer to those discussed above through Fig. 3(a) (the initial condition refers to a Gaussian pulse of 60 ps and 3.5 kW peak power). Note that the spectrogram also gives information on the statistical properties of the incoherent field. We remark in particular in Fig. 6(b) that the spectrogram uniformly fills the time-frequency domain, a feature which confirms the property of stationary statistics within the whole optical pulse. Let us remark that the contribution of the spectral incoherent soliton may be easily identified and is characterized by a slow curvature in the frequency-temporal diagram. This merely reveals that the low-frequencies components were generated by the Raman effect before the high-frequencies components, so that they have cumulated a larger frequency-shift in the propagation. We underline that the curvature in the spectrogram is a consequence of the envelope pulse profile considered in the numerical simulations. This is illustrated by the fact that such a curvature in the spectrogram disappears when the numerical simulation is realized with an initial CW [see Fig. 6(c)]. As discussed above through Figs. 3, an initial CW leads to a complete separation of the spectral incoherent soliton from the main SC spectrum. The spectrogram analysis confirms this property, as illustrated in Fig. 6(c). It shows that the spectral incoherent soliton is characterized by a well-confined spectrum, whereas in the temporal domain the field exhibits a stationary statistics (it exhibits fluctuations at any time), consistently with the key property of spectral incoherent solitons [27]. According to this spectrogram analysis, we may state that the “spectral-hump” identified experimentally and numerically in the low-frequency part of the spectrum is reminiscent of a spectral incoherent soliton.

 

Fig. 6. (a) Experimental output spectrum obtained after 17 m of propagation in our PCF for an input peak power of 3.5 kW (the video bandwidth of the spectrum analyzer was set to 1 kHz). The left inset shows in particular the output spectrum of the spectral incoherent soliton, which reveals high intensity fluctuations. (b-c): Numerical spectrogram showing the temporal distribution of the spectral power after 17 m, obtained by solving the GNLSE, for a 60 ps input pulse (b), and a CW input (c). The white dashed lines show both fiber ZDWs. ‘S’ indicates the position of the spectral incoherent soliton. The reference pulse used to compute the spectrogram is a 20-fs sech pulse.

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7. Conclusion

In summary we have reported an experimental, numerical and theoretical study of the incoherent regime of SC generation in a two ZDWs PCF. An experimental cutback of the PCF has allowed us to perform detailed measurements of the evolution of the spectrum of the field in the propagation. The experimental results of SC generation are in agreement with the numerical simulations of the GNLSE and thus provide experimental evidence of the process of optical wave thermalization. In particular, the saturation of the high-frequency edge of the SC spectrum reported in the experiment has been found in good agreement with the numerical simulations (of both the GNLSE and NLSE) and with the theoretical predictions of the kinetic wave theory. Such a good agreement has been confirmed through the study of SC generation with increasing pump powers. Furthermore, the double-peak structure which characterizes the equilibrium spectrum has been found in quantitative agreement with the numerical simulations of the NLSE, without adjustable parameters. The development of such a spectral double peak is also clearly visible in the simulations of the GNLSE, despite the non-conservative nature of the Raman effect that prevents the establishment of an equilibrium state. We also identified the low-frequency peak of the equilibrium spectrum in the experimental results, while the high-frequency peak is masked by the Raman effect.

Besides wave thermalization, we showed experimentally that Raman scattering leads to the generation of a spectrally localized incoherent structure in the normal dispersion regime. A numerical spectrogram analysis revealed that the incoherent structure shares the same properties than spectral incoherent solitons, i.e., it is localized in the spectral domain but not in the temporal domain. The incoherent spectral-hump identified experimentally may thus be considered as being reminiscent of spectral incoherent solitons. In conclusion, the incoherent regime of SC generation seems to open new perspectives for the experimental study of incoherent nonlinear optics and for the experimental verification of kinetic theories inherent to fully developed turbulent behaviors.