1. INTRODUCTION

The dynamics of supercontinuum (SC) generation in nonlinear fibers have been extensively studied under various conditions, including short or long pump pulses, anomalous or normal dispersion regimes [1], and the emphasis has now shifted toward optimizing SC characteristics for specific applications. Although the SC ultrabroad bandwidth is evidently an important parameter to be considered for any given application, the spectral and temporal coherence characteristics are also of primary interest [1]. Indeed, the coherence properties can significantly affect the performance of the SC as a radiation source and it is therefore paramount to evaluate the coherence properties of SC light in a systematic way. In this regard, several studies have investigated the coherence properties of SC light both theoretically [2, 3, 4, 5, 6] and experimentally [7, 8, 9, 10, 11, 12, 13] for various pumping regimes and states of polarization. In all these studies the coherence characteristics have typically been quantified through the normalized degree of spectral coherence originally introduced in the SC context by Dudley and Coen [2]. This spectral coherence function has been widely used because it directly corresponds to the experimentally measurable fringe visibility of a polychromatic delayed interferometer [14]. From an application point of view this particular description of SC coherence characterizes the shot-to-shot stability whose knowledge can be of crucial interest for applications, such as the realization of precise frequency combs. However, this spectral coherence function does not appear to be directly related to the fundamental second-order two-time and two-frequency correlation functions employed in the formal theory of coherence of nonstationary light [15] and which has been shown to provide valuable insight into the nonlinear propagation of short pulses [16].

In a recent paper [17], using the two-time and two- frequency correlation functions of second-order coherence theory of nonstationary light we have shown that, in the long-pulse regime, SC light contains distinct quasi-coherent and quasi-stationary contributions and that these contributions are characterized in the correlation functions by a quasi- coherent square and a quasi-stationary line, respectively. Here, we extend our previous results by exploring numerically more extensively the input parameter space, focusing on how the cross-spectral density and the mutual coherence functions are affected by a change in the SC dynamics. A major result is that, independently of the pump pulse parameters, the two-time and two-frequency correlation functions can indeed be decomposed into quasi-coherent and quasi-stationary contributions and that the relative magnitude of these two contributions depends on the pump pulse parameters. Another significant result is the establishment of a connection between the Dudley–Coen spectral coherence function and the correlation functions of second-order coherence theory of nonstationary light. Specifically, we show that the Dudley–Coen coherence function is directly related to the coherent part of the cross-spectral density but does not carry any information about the quasi-stationary part. We also address the experimental measurement of the two-time and two-frequency correlation functions, which allows for a complete characterization of the temporal and spectral coherence properties of SC light.

The paper is organized as follows. In Section 2 we introduce the standard definitions of the two-time and two-frequency correlation functions, as well as the definitions of the corresponding degrees of coherence. Using numerical simulations, we subsequently examine in detail in Section 3 the different regimes of coherence observed for various pumping conditions. Section 4 discusses possible experimental arrangements for the full characterization of the SC coherence properties. A connection between the new definitions of the SC coherence and the original definition as given by Dudley and Coen is also addressed in Section 4.

2. SECOND-ORDER COHERENCE OF NONSTATIONARY LIGHT

We begin by introducing the correlation functions and measures that are traditionally used in the formalism of second-order coherence of nonstationary light. The temporal coherence is characterized by the two-time mutual coherence function (MCF) $\mathrm{\Gamma }(t_1,t_2)$ defined as

The normalized form of the MCF then is given by

Correspondingly in the frequency domain, one can construct the two-frequency cross-spectral density function (CSD) $W({\omega }_1,{\omega }_2)$, defined as

The mean spectrum corresponds to $S(\omega )=W(\overline{\omega },\mathrm{\Delta }\omega =0)$ and the normalized form of the CSD is given by

The spectral width (FWHM) of the normalized CSD function $|\mu (\overline{\omega },\mathrm{\Delta }\omega )|$ in the $\mathrm{\Delta }\omega$ direction measures the spectral coherence bandwidth at frequency $\overline{\omega }$. The overall degree of spectral coherence $\overline{\mu }$ can also be evaluated by integrating the CSD over the $(\overline{\omega },\mathrm{\Delta }\omega )$ plane [18]

The overall degree of spectral (or temporal) coherence is bounded by the interval $0\le {\overline{\mu }}^2({\overline{\gamma }}^2)\le 1$, where the lower and upper limits indicate incoherence and full coherence, respectively. All the above definitions form a complete set of measures that are traditionally used to describe the temporal (spectral) coherence properties of nonstationary light [15].

In the context of SC generation the coherence properties have been customarily examined in the light of the complex spectral coherence function introduced by Dudley and Coen and defined as

The degree of coherence as obtained from Eq. (11) is also bounded by the interval $0\le |g_{12}^{(1)}|\le 1$, with the limits 0 and 1 indicating incoherent and coherent SC, respectively [2, 3]. We point out that, in principle, one could similarly define a temporal coherence function $g_{12}^{(1)}(t)$ by replacing $\tilde{E}(\omega )$ with $E(t)$ in Eq. (10) and calculate an overall temporal degree of coherence by weighting with the mean intensity in the temporal domain.

3. COHERENCE PROPERTIES OF SUPERCONTINUUM

In this section we apply the standard theory of second-order coherence of nonstationary fields to characterize the coherence properties of SC light generated in a nonlinear fiber under various pumping conditions. In order to construct the two-time and two-frequency correlation functions defined in the previous section, we generate statistical ensembles of SC realizations using the generalized nonlinear Schrödinger equation, which is known to correctly reproduce the nonlinear propagation of broadband pulses [1]:

In order to investigate the statistical properties of the correlation functions, input shot noise was included through the addition of a noise seed of one photon per mode with random phase in each spectral discretization bin. We simulate the propagation of $T_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}=200\mathrm{fs}$ pulses in a $20\mathrm{cm}$ long fiber with zero-dispersion at around $1000\mathrm{nm}$ for various levels of input peak power. The pump pulses at $1060\mathrm{nm}$ lie in the anomalous dispersion regime, which is known to lead to the formation of the broadest SC spectra and corresponds to typical experimental input conditions for SC generation around $1\mathrm{\mu m}$ [20]. Of course, we have carefully checked that the results presented below are qualitatively the same for other pump wavelengths and noise implementation. The nonlinear coefficient of the fiber is $\gamma =0.01{\mathrm{W}}^{-1}{\mathrm{m}}^{-1}$ and the dispersion coefficients up to the 10th order used in the simulations are given in Table 1.

For a given initial peak power $P_P$, we construct the mutual coherence function $\mathrm{\Gamma }(\overline{t},\mathrm{\Delta }t)$ and CSD function $W(\overline{\omega },\mathrm{\Delta }\omega )$ by simulating an ensemble of 1000 realizations with different random noise seeds. The corresponding overall degrees of coherence plotted in Fig. 1 show that $\overline{\mu }$ strongly depends on the input pulse peak power. Specifically, for low input peak power values the overall degree of coherence $\overline{\mu }$ is close to unity and the generated SC is fully coherent. As the peak power is increased beyond about $8\mathrm{kW}$, $\overline{\mu }$ drops significantly and for input peak powers exceeding $12\mathrm{kW}$ the SC becomes quasi-incoherent. It is interesting to note that the SC coherence degree exhibits a quasi-binary behavior in the sense that the range of power for which the SC is partially coherent is relatively narrow.

For comparison, we also plot in Fig. 1 the degree of coherence $|g_{12}^{(1)}|$ as calculated from the Dudley–Coen spectral coherence function. We find that these two measures of coherence are in good agreement over the whole range of peak powers investigated and, in fact, they are identical for the quasi- coherent SC cases while the $\overline{\mu }$ value slightly exceeds that of $|g_{12}^{(1)}|$ in the incoherent regime. Simulations over a much wider range of input conditions and fiber parameters confirm the general good correspondence observed between $\overline{\mu }$ and $|g_{12}^{(1)}|$.

To get more insight into the coherence properties of SC radiation we examine in detail three different cases corresponding to quasi-coherent (case A), partially coherent (case B), and quasi-incoherent SC (case C) as indicated by the arrows in Fig. 1, with the overall degrees of coherence $\overline{\mu }$ equal to 0.95, 0.55, and 0.15, respectively. We plot for each case in Fig. 2 the normalized CSD (left subfigures) and MCF (right subfigures) in average and difference coordinates constructed from the realization ensembles. As could be expected from the variations of the overall degree of coherence seen in Fig. 1, the two-frequency and two-time correlation functions strongly depend on the input pulse peak power. More specifically, for low peak power and quasi-coherent SC (in the sense of the overall degree of coherence) both the CSD and MCF consist of a $45°$ tilted square with a quasi-constant value close to unity and extending over the entire SC bandwidth and temporal profile. This means that all the spectral components and all the temporal components are fully correlated and this region of high correlations can be referred to as quasi-coherence square [17].

For increased peak power (case B), both the CSD and MCF also include a square region of relatively high correlations that spans the whole SC bandwidth, but we observe the emergence of a distinct thin line along the $\overline{\omega }$ ($\overline{t}$) direction with an almost constant width in the $\mathrm{\Delta }\omega$ ($\mathrm{\Delta }t$) direction. We do note some residual variations in the average frequency (time) direction but these changes are not significant compared to the variations along the difference frequency (time) coordinate and this narrow line whose characteristics are nearly independent of average frequency (time) thus indicates a quasi-stationary contribution [17]. We note that the appearance of a quasi- stationary contribution is combined with a degradation of the correlations within the coherent square compared to the case of low input power. In particular, careful inspection of the coherent square in the frequency domain reveals that high correlations are observed in several areas around the pump initial center frequency (at $283\mathrm{THz}$) and near the edges of the SC spectrum. Correspondingly in the time domain, high correlations are located mostly in areas around the leading and trailing edges of the temporal profile. For even larger peak powers [see case C in Figs. 2e, 2f] the area of the coherence square is significantly reduced and localized around the pump initial center frequency for the CSD and at the pulse leading edge for the MCF. In fact, and although it is not shown here, the CSD and MCF eventually reduce to quasi-stationary lines for very large input peak powers.

Numerical simulations over a wide range of input conditions/fiber parameters show that the signature of quasi- coherent and quasi-stationary contributions in the correlation functions is a general feature of SC light. It is not our intention to illustrate here every possible case, and we rather show in Fig. 3 and Fig. 4 how the pump center affects the MCF and CSD. For example, for a pump pulse initially tuned to the normal dispersion regime, the MCF and CSD also evolve from a quasi-coherent square toward a quasi-stationary part as the initial peak power is increased (Fig. 8). The major difference with the case of anomalous pumping lies in the much higher level of peak power beyond which the overall coherence starts degrading and the quasi-stationary contribution manifests. Increasing the input pulse width affects the CSD and MCF essentially in the same way as when the input power is increased, i.e., the overall degree of coherence decreases and the CSD/MCF evolve from a quasi-coherent square for short durations [Figs. 4a, 4b] toward a dominant quasi-stationary line for long durations [Fig. 4c, 4d].

We next focus our discussion on the dynamics related to the three cases corresponding to quasi-coherent (case A), partially coherent (case B), and quasi-incoherent SC (case C) as indicated by the arrows in Fig. 1. The transition from a coherence square to a quasi-stationary line observed in the correlation functions when the peak power is increased can be associated with the particular nonlinear dynamics that are nvolved in the development of the SC spectrum. In the anomalous dispersion regime, the different mechanisms leading to the SC formation can generally be described with respect to the input soliton number $N=\sqrt{\gamma P_0{T}_0^2/|{\beta }_2|}$, where $T_0=T_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}/1.763$. For a low input soliton number (as is the case for A, where $N=10$), the spectral broadening is a self-seeded process dominated by soliton dynamics [1]. Specifically, the evolution of the SC in this case consists of an initial stage of higher-order soliton compression followed by fission into fundamental $N=1$ soliton propagating in the anomalous dispersion regime and redshifted by stimu lated Raman scattering [1]. Simultaneously to the soliton fission process, blueshifted dispersive wave components are generated in the normal dispersion regime [1]. During the initial stage of compression the pulse maintains a fixed spectral (and temporal) phase relationship, which persists after the breakup into individual solitons. The localized soliton structures are coupled with the dispersive waves via cross-phase modulation [21, 22], which explains the quasi-perfect spectral (temporal) correlations observed across the entire SC spectrum (pulse).

Increasing the peak power correspondingly increases the input soliton number ($N\approx 13$ for case B), which affects the initial stage of higher-soliton compression due to the amplification of spectral sidebands outside the pulse bandwidth through modulation instability (MI). In the time domain, MI manifests itself by the appearance of a modulation on top of the input pulse and whose contrast increases with propagation in accordance with the exponential amplification of the spectral sidebands. In contrast to soliton dynamics, MI is initially seeded from noise so that the spectral (temporal) phase and amplitude fluctuates substantially across the components associated with MI, both in a single realization and between realizations. The competition between MI and soliton fission thus degrades the correlations compared to the deterministic case where only soliton fission is present. High correlations are still observed in the vicinity of the pump spectral components, as these essentially correspond to the self-seeded compression dynamics, which are not significantly perturbed by MI. It is precisely these spectral components that are located at the pulse leading edge and appear to be highly correlated in the two-time correlation function. In addition, as the solitons are ejected from the pump before the MI sidebands have been significantly amplified, the spectral components of the solitons and their corresponding trapped dispersive waves remain also highly correlated, which in the time domain is manifested at the trailing edge of the SC pulse. On the other hand, the spectral components that are initially amplified by MI are uncorrelated because of the random noise, which results in the emergence of a quasi-stationary contribution.

For large input peak powers corresponding to a large input soliton number ($N\approx 19$ for case C) noise-seeded MI dynamics are primarily responsible for the initial phase of spectral broadening. In the time domain the temporal subpeaks of the resulting high-contrast modulated pulse evolve into fundamental solitons that separate from the broad temporal background under the effect of higher-order dispersion and stimulated Raman scattering. Because of the random noise, the propagation stage at which the high-contrast modulated pulse has developed from MI fluctuates from realization to realization, resulting in strong variations of the output spectral (temporal) phase and amplitude in the ensemble. These shot-to-shot fluctuations mean that all the frequency (temporal) components are essentially uncorrelated, except for a small coherence square localized around the pump center frequency (leading edge of the SC pulse) and which corresponds to a weak residual self-phase modulation process that occurs in the early propagation stage.

It is clear from the definition of the CSD and MCF that when the overall degree of coherence $\overline{\mu }(\overline{\gamma })=1$ the CSD and MCF are completely separable in the $({\omega }_1,{\omega }_2)$ plane. If $\overline{\mu }(\overline{\gamma })=0$ the CSD and MCF are stationary and only depend on the difference coordinate. As seen from the simulated results of Fig. 1, these fully coherent and fully incoherent limits are asymptotically approached when the peak power is gradually increased from a low to a large value. The numerical results also suggest that the correlation functions can be approximated as a sum of these limit cases, i.e., the sum of a quasi-coherent and quasi- stationary contribution represented by a square region and a line in the $(\overline{\omega },\mathrm{\Delta }\omega )$ plane, respectively. Furthermore, it can also be seen from Fig. 2 that the relative magnitudes of these two contributions depend on the nonlinear dynamics involved in the SC generation process, and, hence, on the initial peak power. Of course, this partition of the correlation functions is an approximation; nevertheless, it allows to gain considerable insight into the coherence properties of SC and, in particular, it provides a path to their experimental characterization as addressed in the next section. Using such a decomposition, we can write the normalized two-frequency correlation functions as

Correspondingly, we can separate the spectrum (temporal intensity) into quasi-coherent and quasi-stationary parts. Indeed, the spectrum (temporal intensity) corresponds to the diagonal of the CSD (MCF) and using the partition of the CSD (MCF) into quasi-coherent and quasi-stationary parts, we can also write the spectrum (temporal intensity) as $S(\omega )=W(\omega ,\omega )=S_c(\omega )+S_q(\omega )$ ($I(t)=\mathrm{\Gamma }(t,t)=I_c(t)+I_q(t)$). The quasi-coherent contribution of the spectrum $S_c(\omega )=|{\tilde{E}}_c(\omega ){|}^2$ (temporal intensity $I_c(t)=|E_c(t){|}^2$) can then be retrieved from the CSD (MCF) using the $45°$ cross sections of the quasi-coherent region. The quasi-stationary part is simply obtained from $S_q=S(\omega )-S_c(\omega )$ ($I_q=I(t)-I_c(t)$). The quasi-coherent and quasi-stationary contributions are illustrated in Fig. 5 for the three cases analyzed in Fig. 2. We can see how the increase in the peak power gradually reduces the overall coherent contribution and limits the region of high coherence around the pump components.

It is also instructive to consider the evolution dynamics of the two contributions. We show in Fig. 6 how the quasi- coherent and quasi-stationary contributions develop on propagation with respect to the mean spectrum evolution. In particular, the false color representation clearly illustrates that the initial stage of self-phase modulation is mostly coherent, whereas the emergence of MI components outside the spectrally broadened pump is associated with the development of the quasi-stationary contribution. It is interesting to note that the coherent contribution formed during the initial evolution stage is only affected on the high-frequency side with further propagation beyond $5\mathrm{cm}$.

4. DISCUSSION

From an experimental perspective, the direct measurement of the two-frequency and two-time correlation functions is not straightforward, as it would require either direct characterization of individual realizations (pulses) or slicing the spectrum (temporal intensity) into small frequency (time) intervals and subsequently measure the correlations with detectors having bandwidths of several hundreds of terahertz. Yet, our numerical results open a new avenue for the experimental characterization of the CSD and MCF through their decomposition into quasi-coherent and quasi-stationary contributions.

We first consider the measurement of the coherent con tribution of the unnormalized CSD $W_c$. As indicated by the square shape of $W_c$, it is quasi-separabale in $({\omega }_1,{\omega }_2)$ and can therefore be approximated by $W_c({\omega }_1,{\omega }_2)\approx 〈{\tilde{E}}^{*}({\omega }_1)〉〈\tilde{E}({\omega }_2)〉$. Similarly, for the normalized form we have ${\mu }_c({\omega }_1,{\omega }_2)\approx [〈{\tilde{E}}^{*}({\omega }_1)〉/\sqrt{S({\omega }_1)}][〈\tilde{E}({\omega }_2)〉/\sqrt{S({\omega }_2)}]$. Significantly, this particular formulation of the coherent part of the CSD can be directly connected with the spectral coherence function $g_{12}^{(1)}(\omega )$, which is readily accessible experimentally. Indeed, the measured fringe visibility of the polychromatic interferogram obtained from the interference of consecutive SC pulses is precisely equal to the modulus of the spectral coherence function $g_{12}^{(1)}(\omega )$ [14]. In what follows, we denote this measured fringe visibility by $g_m(\omega )$. In order to highlight the link between ${\mu }_c({\omega }_1,{\omega }_2)$ and $g_m(\omega )$ we begin by reformulating the Dudley–Coen spectral coherence function. Considering an ensemble of N SC realizations and performing the summation over the various SC pairs, the modulus of $g_{12}^{(1)}(\omega )$ can be approximately rewritten as (see Appendix A for more details)

From this alternative formulation we can interpret $g_m(\omega )$ as the modulus of the mean field squared normalized to the mean spectrum. Hence, the coherent part of the normalized CSD can simply be interpreted as

The quasi-stationary contribution of the CSD (MCF) is directly linked to the quasi-stationary part $I_q(t)$ ($S_q(\omega )$) of the mean temporal intensity (mean spectrum) by Fourier transformation

Similarly to the CSD, the MCF can be reconstructed from its two distinct contributions. The coherent part $I_c(t)$ of the temporal intensity is obtained from $I_c(t)=I(t)-I_q(t)$, which allows to construct $|{\gamma }_c(t_1,t_2)|=[\sqrt{I_c(t_1)}/\sqrt{I(t_1)}][\sqrt{I_c(t_2)}/\sqrt{I(t_2)}]$. On the other hand, the quasi-stationary part ${\gamma }_q(\mathrm{\Delta }t)$ is the inverse Fourier transform of $S_q(\omega )$. In some cases it may also be interesting to know the phase of the CSD and MCF. Although the phase information is lost in the interferometric measurement of the coherent part of the CSD, it can still be recovered in principle with standard phase retrieval algorithms, since the CSD and MCF form a Fourier transform pair according to Eq. (9). The phase of the quasi-stationary contribution is directly obtained from Eqs. (18, 19). Therefore, with two sets of measurements one is able to characterize all the second-order coherence properties of SC light, as is summarized in Fig. 8. Of course, it is important to remember that the suggested experimental implementation assumes that the complex degree of coherence can be written as a sum of a coherent and a quasi-stationary contribution as indicated by the wide range of parameters explored.

Following this approach, simulated examples of reconstruction of the normalized CSD function are shown in Fig. 9 for the cases of partially coherent and quasi-incoherent SC, which include both a quasi-coherent and a quasi-stationary contribution (case A only involves a coherent contribution that is retrieved from $g_m(\omega )$). The quasi-coherent contribution is reconstructed from the measurable spectral coherence function $g_m(\omega )$, while the quasi-stationary part is obtained by Fourier transformation of the average temporal intensity that is spectrally filtered around the pump. For the partially coherent and quasi-incoherent case a 10th-order super- Gaussian filter with $50\mathrm{THz}$ and $40\mathrm{THz}$ FWHM, respectively, was used. We can see how the CSDs reconstructed from experimentally measurable quantities in both cases nicely reproduce the main features of the CSDs that are directly calculated from the simulated realizations, showing that, in principle, the second-order coherence properties of SC light can be fully characterized experimentally.

5. CONCLUSIONS

In conclusion, we have studied the coherence properties of supercontinuum light by applying the two-time and two- frequency correlations of second-order coherence theory. This approach allows for full characterization of SC coherence properties and makes it possible to define measures of coherence time, coherence bandwidth, and effective degree of coherence. Our analysis shows that independently of the pumping regime, the SC can be separated into distinct quasi- coherent and quasi-stationary contributions, the relative magnitudes of which depend on the pump pulse parameters. This decomposition can be measured separately and thereby opens a route for complete experimental characterization of SC coherence properties.

APPENDIX A

The measured spectral coherence function $g_{\mathit{m}}(\omega )$ can be written as

(A1)$$g_{\mathit{m}}(\omega )=|g_{12}(\omega )|.$$

It is straightforward to show that, in general, the imaginary part of $g_{12}(\omega )$ is negligible compared to its real part (except in the incoherent case but then both parts are essentially equal to zero). We can thus rewrite the measured spectral coherence function as

(A2)$$g_m(\omega )=\Re [g_{12}(\omega )]=\frac{〈{\tilde{E}}_i^{*}(\omega ){\tilde{E}}_j(\omega ){〉}_{i\ne j}}{〈|\tilde{E}(\omega ){|}^2〉}+\frac{〈{\tilde{E}}_j^{*}(\omega ){\tilde{E}}_i(\omega ){〉}_{i\ne j}}{〈|\tilde{E}(\omega ){|}^2〉}.$$

For a given set of N realizations and writing specifically the summation over all possible pairs in the numerator we have

(A3)$$g_m(\omega )=\frac{1}{n_{\text{pairs}}}\frac{\sum _{i\ne j}^N[{\tilde{E}}_i^{*}(\omega )E_j(\omega )+{\tilde{E}}_i(\omega ){\tilde{E}}_j^{*}(\omega )]}{〈|\tilde{E}(\omega ){|}^2〉}.$$

Using the binomial formula one obtains

(A4)$$g_m(\omega )=\frac{1}{n_{\text{pairs}}}\frac{|\sum _{i=1}^N{\tilde{E}}_i(\omega ){|}^2-\sum _{i=1}^N|{\tilde{E}}_i(\omega ){|}^2}{〈|\tilde{E}(\omega ){|}^2〉},$$

and it follows that

(A5)$$g_m(\omega )=\frac{1}{n_{\text{pairs}}}\frac{|N〈\tilde{E}(\omega )〉{|}^2-N〈|\tilde{E}(\omega ){|}^2〉}{〈|\tilde{E}(\omega ){|}^2〉}.$$

As the number of pairs corresponding to an ensemble of N realizations is $N^2-N$, we finally obtain for the measured spectral coherence function

(A6)$$g_m(\omega )=\frac{1}{N^2-N}\frac{N^2|〈\tilde{E}(\omega )〉{|}^2-N〈|\tilde{E}(\omega ){|}^2〉}{〈|\tilde{E}(\omega ){|}^2〉}.$$

APPENDIX B

The coherent part of the non-normalized form of the CSD is separabale in $({\omega }_1,{\omega }_2)$ and can be factored as

(B1)$$|W_c({\omega }_1,{\omega }_2)|\approx \sqrt{S({\omega }_1)}\sqrt{g_m({\omega }_1)}\sqrt{S({\omega }_2)}\sqrt{g_m({\omega }_2)}.$$

It then follows that

(B2)$${\overline{\mu }}^2=\frac{{\int }_0^{\infty }{\int }_0^{\infty }|W({\omega }_1,{\omega }_2){|}^2\mathrm{d}{\omega }_1\mathrm{d}{\omega }_2}{[{\int }_0^{\infty }S(\omega )\mathrm{d}\omega {]}^2}=\frac{{\int }_0^{\infty }{\int }_0^{\infty }|W_c({\omega }_1,{\omega }_2)+W_q({\omega }_1,{\omega }_2){|}^2\mathrm{d}{\omega }_1\mathrm{d}{\omega }_2}{[{\int }_0^{\infty }S(\omega )\mathrm{d}\omega {]}^2},$$

where we have separated the CSD into quasi-coherent and quasi-stationary parts. It is clear that the main contribution to ${\overline{\mu }}^2$ arises from the first term of the summation in Eq. (27), since the area of the quasi-stationary line is negligible as compared to the area of the coherent square. Hence, we can approximate

(B3)$${\overline{\mu }}^2\approx \frac{{\int }_{-\infty }^{\infty }{\int }_0^{\infty }|W_c(\overline{\omega },\mathrm{\Delta }\omega ){|}^2\mathrm{d}\overline{\omega }\mathrm{d}\mathrm{\Delta }\omega }{[{\int }_0^{\infty }S(\omega )\mathrm{d}\omega {]}^2}=\frac{{\int }_0^{\infty }{\int }_0^{\infty }{g}_m({\omega }_1)S({\omega }_1)g_m({\omega }_2)S({\omega }_2)\mathrm{d}{\omega }_1\mathrm{d}{\omega }_2}{[{\int }_0^{\infty }S(\omega )\mathrm{d}\omega {]}^2},$$

which finally leads to

(B4)$${\overline{\mu }}^2\approx [\frac{{\int }_0^{\infty }{g}_m(\omega )S(\omega )\mathrm{d}\omega }{{\int }_0^{\infty }S(\omega )\mathrm{d}\omega }{]}^2=|g_{12}^{(1)}{|}^2.$$

Since we have neglected the quasi-stationary contribution in the evaluation of $\overline{\mu }$ (which is not accounted for in $g_{12}^{(1)}(\omega )$), $\overline{\mu }$ is always slightly larger than $|g_{12}^{(1)}|$. This is also apparent in Fig. 1 for the incoherent SC cases, in which the quasi- stationary part dominates.

ACKNOWLEDGMENTS

This work was funded by the Academy of Finland (grants 134891, 130099, 134998, 128331, and 135030) and the Graduate School of Modern Optics and Photonics. Miro Erkintalo is acknowledged for useful discussions.

Table 1. Taylor-series Expansion Coefficients of the Fiber Dispersion

View Table

 

Fig. 1 Overall degrees of spectral coherence $\overline{\mu }$ (red diamonds) and $|g_{12}^{(1)}|$ (black circles) versus input pulse peak power. A, B, and C mark the three cases of quasi-coherent, partially coherent, and quasi-incoherent SC light investigated in detail.

Download Full Size | PDF

 

Fig. 2 Left: normalized cross-spectral density $|\overline{\mu }(\omega ,\mathrm{\Delta }\omega )|$ for (a) case A, (c) case B, and (e) case C. Right: normalized mutual coherence function $|\gamma (\overline{t},\mathrm{\Delta }t)|$ for (b) case A, (d) case B, and (f) case C. For clarity, the absolute frequency and times axes are indicated. Note the different color scale in (a) and (b) compared to (c)–(f).

Download Full Size | PDF

 

Fig. 3 Left: normalized cross-spectral density $|\overline{\mu }(\omega ,\mathrm{\Delta }\omega )|$ and right: normalized mutual coherence function $|\gamma (\overline{t},\mathrm{\Delta }t)|$ for $960\mathrm{nm}$ input pulses with (a),(b) $P_P=22\mathrm{kW}$ and (c),(d) $P_P=88\mathrm{kW}$.

Download Full Size | PDF

 

Fig. 4 Left: normalized cross-spectral density $|\overline{\mu }(\omega ,\mathrm{\Delta }\omega )|$ and right: normalized mutual coherence function $|\gamma (\overline{t},\mathrm{\Delta }t)|$ for (a),(b) $50\mathrm{fs}$ input pulses, (c),(d) $1\mathrm{ps}$ input pulses.

Download Full Size | PDF

 

Fig. 5 Left: mean spectra (black lines), coherent (red lines) and quasi-stationary (blue lines) spectral contributions for (a) case A, (c) case B, and (e) case C. Right: mean temporal intensity (black lines), coherent (red lines) and quasi-stationary (blue lines) intensity contributions for (b) case A, (d) case B, and (f) case C.

Download Full Size | PDF

 

Fig. 6 False color representation of the evolution of (a) mean spectrum, (b) quasi-coherent contribution, and (c) quasi- stationary contribution for case C ($P_P=22\mathrm{kW}$).

Download Full Size | PDF

 

Fig. 7 Left: calculated cross-spectral density $|W(\overline{\omega },\mathrm{\Delta }\omega )|$ for (a) case A, (c) case B, and (e) case C. Right: coherent part of the cross-spectral density $|W(\overline{\omega },\mathrm{\Delta }\omega )|$ retrieved from $g_{12}^{(1)}(\omega )$ for (b) case A, (d) case B, and (f) case C.

Download Full Size | PDF

 

Fig. 8 Schematic for complete experimental characterization of the SC coherence properties. OSA, optical spectrum analyzer; FROG, frequency-resolved optical gating; MZI, Mach– Zehnder interferometer.

Download Full Size | PDF

 

Fig. 9 Reconstructed normalized CSD when measuring separately the quasi-coherent and quasi-stationary contributions for (b) case B and (d) case C. For comparison, the original CSD as directly numerically computed from the ensemble of simulation realizations is also shown for (a) case B and (c) case C.

Download Full Size | PDF

1. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]  

2. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]  

3. J. M. Dudley and S. Coen, “Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,” IEEE J. Sel. Top. Quantum Electron. 8, 651–659 (2002). [CrossRef]  

4. Z. Zhu and T. G. Brown, “Experimental studies of polarization properties of supercontinua generated in a birefringent photonic crystal fiber,” Opt. Express 12, 791–796 (2004). [CrossRef]   [PubMed]  

5. S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self- frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005). [CrossRef]  

6. S. M. Kobtsev and S. V. Smirnov, “Coherent properties of super-continuum containing clearly defined solitons,” Opt. Express 14, 3968–3980 (2006). [CrossRef]   [PubMed]  

7. X. Gu, M. Kimmel, E. Zeek, P. O’shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber supercontinuum,” Opt. Lett. 27, 1174–1176 (2002). [CrossRef]  

8. X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11, 2697–2703 (2003). [CrossRef]   [PubMed]  

9. Z. Zhu and T. G. Brown, “Polarization properties of supercontinuum spectra generated in birefringent photonic crystal fibers,” J. Opt. Soc. Am. B 21, 249–257 (2004). [CrossRef]  

10. F. Lu and W. H. Knox, “Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers,” Opt. Express 12, 347–353 (2004). [CrossRef]   [PubMed]  

11. I. Zeylikovich, V. Kartazaev, and R. R. Alfano, “Spectral, tem poral, and coherence properties of supercontinuum generation in microstructure fiber,” J. Opt. Soc. Am. B 22, 1453–1460 (2005). [CrossRef]  

12. F. Lu and W. H. Knox, “Generation, characterization, and application of broadband coherent femtosecond visible pulses in dispersion micromanaged holey fibers,” J. Opt. Soc. Am. B 23, 1221–1227 (2006). [CrossRef]  

13. D. Türke, S. Pricking, A. Husakou, J. Teipel, J. Herrmann, and H. Giessen, “Coherence of subsequent supercontinuum pulses generated in tapered fibers in the femtosecond regime,” Opt. Express 15, 2732–2741 (2007). [CrossRef]   [PubMed]  

14. M. Bellini and T. W. Hänsch, “Phase-locked white-light continuum pulses: toward a universal optical frequency-comb synthesizer,” Opt. Lett. 25, 1049–1051 (2000). [CrossRef]  

15. M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997). [CrossRef]  

16. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andr`es, “Pulseby-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express 18, 14979–14991 (2010). [CrossRef]   [PubMed]  

17. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010). [CrossRef]   [PubMed]  

18. C. Iaconis, V. Wong, and I. A. Walmsley, “Direct inter ferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998). [CrossRef]  

19. M. Erkintalo, B. Wetzel, G. Genty, and J. M. Dudley, “Limitations of the linear Raman gain approximation in modeling broadband nonlinear propagation in optical fibers,” Opt. Express 18, 25449–25460 (2010). [CrossRef]   [PubMed]  

20. G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007). [CrossRef]  

21. G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub $30\mathrm{fs}$ pulses,” Opt. Express 12, 4614–4624 (2004). [CrossRef]   [PubMed]  

22. A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photon. 1, 653–657 (2007). [CrossRef]  

23. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997). [CrossRef]  

24. I. A. Walmsley and C. Dorrer, “Characterization of ultra short electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009). [CrossRef]