Decoherence due to XPM-assisted Raman amplification for polarization or wavelength offset pulses in all-normal dispersion supercontinuum generation

Decoherence due to XPM-assisted Raman amplification for polarization or wavelength offset pulses in all-normal dispersion supercontinuum generation James S. Feehan1,* AND Jonathan H. V. Price2 Scottish Universities Physics Alliance, Department of Physics, University of Strathclyde, GlasgowG4 0NG, UK Optoelectronics Research Centre, University of Southampton, Hampshire, SO17 1BJ, UK *Corresponding author: james.feehan@strath.ac.uk


INTRODUCTION
Supercontinuum generation in all-normal dispersion (ANDi) specialty and photonic crystal fibers (PCFs) [1][2][3] has gathered significant interest due to ANDi fibers having a unique combination of both tight mode confinement and a low, flat, and single-signed dispersion profile. This permits a monotonic positive chirp across the entire final supercontinuum spectrum following its initial generation through the optical wavebreaking process [4,5]. ANDi fibers therefore provide access to broadband, dissipative nonlinear propagation solutions that maintain several desirable features of the input driving pulses, such as a single-peaked time-domain distribution and smoothly varying phase [6,7]. Such fibers have become a popular platform for generating high-brightness pulses that are compressible to the single-cycle limit [8].
Recent numerical and experimental investigations have revealed that an almost perfect output spectral coherence can be achieved using pump pulses with durations up to 1.5 ps [9], ensuring excellent shot-to-shot amplitude and phase stability.
This broadens the types of seed lasers that can be used when compared with pumping in the anomalous dispersion regime, which typically requires pulses shorter than ∼ 150 fs to mitigate the influence of quantum noise [10]. The complexity and cost of coherent supercontinuum sources can therefore be reduced when using ANDi fiber. Applications include optical coherence tomography [11,12], as well as spectroscopy and frequency metrology [13,14] with possible extension to gigahertz repetition rates [15]. Additionally, with the recent development of ANDi fibers with dispersion minima and low loss at wavelengths around 1.5 − 2 µm [16][17][18], the possibilities now extend to the mid-infrared [17,19], opening new opportunities for research in molecular fingerprinting [20] and silicon photonics [21].
For pulse durations above ∼ 1.5 ps, Raman gain coupled with the characteristic steep-edged, flat-top spectra in the ANDi regime [3,7] leads to the amplification of out-of-band quantum noise [9]. This is distinct from coherent energy transfer between pulse-seeded spectral components, which is the desired process in both the ANDi regime and in soliton self-frequency

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shifting in the anomalous dispersion regime [10,22,23]. Noise amplification is accelerated by the parametric amplification of higher-order Stokes and anti-Stokes waves through four-wave mixing (FWM) under non-phase-matched conditions [24][25][26] when the coherent self phase modulation (SPM) and optical wavebreaking processes fail to dominate the pulse propagation, and when the competing Raman-assisted parametric FWM process is not effectively suppressed [9]. Hence, the competition between optical wavebreaking and Raman-assisted parametric FWM is of central importance to the rate of decoherence of multi-picosecond pulses in ANDi fibers [9,27]. Developing a more complete understanding of the processes that limit coherent spectral broadening in the ANDi regime has therefore remained at the forefront of recent research [9,28,29].
Nonlinear coupling between optical signals through crossphase modulation (XPM) is a commonly used technique with applications in all-optical switching [30], multiplexing [31], and modelocking [32,33]. However, its influence on nonlinear dynamics in ANDi fiber has remained largely unexplored. Recent contributions have focused on the role of XPM in vector modulation instability (MI) [28], building on previous studies that introduced this effect using normal-dispersion pumping in step-index silica fibers [34,35].
Here, we investigate vector propagation effects in ANDi PCFs and show how they can couple with the Raman amplification of out-of-band quantum noise and pulse decoherence. Vector contributions to Raman-based decoherence have not previously been studied. We investigate the output coherence of a low-energy probe pulse that propagates with a high-energy, incoherent supercontinuum-generating pulse as a function of temporal walkoff due to differing wavelengths or polarization state, thereby generalizing previous scalar reports [9]. We focus on a central wavelength of 1040 nm, pulse energies around 60 nJ, and peak powers up to 15 kW. These pulse parameters approximate those output by commonly used sources such as femto-and picosecond Yb-fiber lasers operating at megahertz repetition rates. Additionally, non-polarization-maintaining (PM) and PM fibers are considered by adjusting the group velocity mismatch (GVM) between the pump and probe over the range of 0 to ±10 ps/m [36,37].
While in experiments the choice of seed laser will cause specific parameters such as peak power thresholds and fiber lengths to vary from those presented in this work, the main conclusions will apply to seeding with titanium sapphire lasers as well as Yb-fiber lasers, and to mid-infrared sources used with chalcogenide ANDi fiber [17]. The findings may also be applicable to picosecond and femtosecond fiber amplifier and oscillator design, especially where optical wavebreaking is suppressed and the critical power for Raman amplification is exceeded [38].
The model is detailed in Section 2 with supplementary material in Appendix A. In Section 3, we show how XPM and Raman amplification couple to cause decoherence between co-propagating pulses. Orthogonally polarized pulses are considered in Section 3.A. We investigate the probe coherence as the pump transitions from coherent to incoherent dynamics (Section 3.A.1) and show that the probe decoherence can occur faster than the pump (Section 3.A.2). Co-polarized pulses with different central wavelengths are then studied in Section 3.B. Section 4 concludes the paper.

NUMERICAL METHODS
To model the influence of XPM on the coherence, we used the pump-probe approach outlined in Fig. 1. We consider transform-limited Gaussian input pulses for both the pump and probe, with a temporal field envelope given by (T is the retarded time in the co-moving frame of reference traveling at the group velocity of the pump pulse central wavelength, and T 0 is the full width at half maximum pulse duration and is equal for the pump and probe unless otherwise specified.) Peak powers were on the order of kiloWatts and Watts for the pump and probe, respectively, so they experienced different degrees of spectral broadening as demonstrated by the example of orthogonally polarized pump and probe in a birefringent PCF shown in the right-hand plot of Fig. 1. Our choice of pulse parameters was determined by the inequalities shown by Eqs. (1) and (2). These ensured that any nonlinear propagation effects and resulting decoherence experienced by the probe were driven entirely by the pump, and that the fiber length was sufficient for the pump to undergo incoherent spectral broadening. In Eq. (1), L N = 1/(γ P 0 ) is the nonlinear length, γ = kn 2 /A eff parameterizes the fiber nonlinear response, and P 0 is the pulse peak power: Here, L WB ≈ 1.1 √ L N L D is the wavebreaking length [6], and L D = T 2 0 /|β 2 | is the dispersion length. The Raman length is defined as L R = 1/(g s P 0 ) [9], where g s is the mixed parametric-Raman peak gain coefficient [24][25][26], given by Eq. (3): (3) is the linear phase mismatch between the seed pulse and (anti)-Stokes wave normalized to the nonlinear contribution to the mismatch, is the complex Raman susceptibility [9]. The Raman effect was included using the experimentally measured silica response [39] via a fractional contribution to the nonlinear response of f R = 0.18 and a maximumχ (3) R = −1.38i at a frequency detuning of R /(2π ) = 13.2 THz.
The vector generalized nonlinear Schrödinger equation (GNLSE) included polarization dynamics and interactions between pulses resulting from XPM, and is detailed in appendix A. The GNLSE was integrated using the Runge-Kutta fourth-order interaction picture method (RK4IP) [40] along with the conservation quantity error method (CQEM, [41]) adaptive step sizing technique for a negligible integration error [42]. The CQEM ensured that the total photon number over both the pump and probe pulses was conserved (accounting for loss) so that energy transfer between the pump and probe was permitted.
We considered a hexagonal-lattice silica ANDi PCF with pitch and relative hole size of = 1.7 µm and d / = 0.3, giving max(D(λ)) = −31 ps/(nm km) and γ = 0.027 rad(Wm) −1 at 1040 nm. (The wavelength dependence of γ was also included.) The dispersion profile ( Fig. 2) was calculated using an analytical approach [43,44]. Fiber loss was included using a measured loss spectrum for standard singlemode silica fibers digitized from Ref. [45] (Fig. 2). To simplify the calculations, D(λ) and A eff (and hence γ ) were the same for both birefringence axes. For orthogonally polarized pump and probe pulses, the polarization-dependent GVM was included using β 1 = β 1 pump − β 1probe , where β 1 is the inverse group velocity. The GVM for co-polarized pump and probe pulses was determined by the fiber dispersion. The XPM contribution was also modified depending on the relative polarization by modifying XPM relative to SPM by XPM = 2/3 or 2 for orthogonal and parallel polarization states, respectively [46] [Appendix A, Eq. (A3)]. This method of incorporating XPM and the polarization GVM has been used to accurately model vector nonlinear propagation [47]. Spontaneous Raman scattering noise was also included [10,48].
Shot noise was included in the simulations using the standard one photon per mode approach [10]. Although noise levels above the quantum limit cause a more rapid reduction in coherence [9] and have been shown to influence the propagation dynamics [29,49], our focus was to explore the combined effect of XPM and Raman amplification, so we assume quantum noise limited seed lasers. We consider pulse durations between 100 fs and 7 ps, implying the use of mode-locked lasers, which have excellent noise properties [50]. Pulse peak powers were restricted to < 100 kW so we did not expect to see incoherent optical wavebreaking or incoherent cloud formation [9] over the meter-length fibers used. The generality of our work is therefore not reduced by considering only quantum noise limited seed pulses for the pulse energies and peak powers investigated.
The shot-to-shot output spectral stability was calculated over an ensemble of independently generated pairs of spectra using the modulus of the first-order degree of coherence and its average (denoted by angular brackets), defined by Eqs. (4) and (5), respectively [51,52]. These equations were applied to the pump and probe ensembles separately: Each simulation comprised 24 individual shots (276 pairs of spectra, sufficient for the averaging [9,10]). Equations (4) and (5) are defined over the interval [0; 1], with a value of 1 indicating perfect coherence.
The fiber length over which |g (1) 12 | decreases to 0.9 is the coherence length, L c . Its dependence on the pulse duration is expressed in Eq. (6), and while an approximate analytic formula exists [9], we circumvent the need for numerical fitting by calculating |g (1) 12 | directly as a function of fiber length during the simulations: Alternatively, the coherence length can also be expressed as

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Although the ANDi regime does not permit solitons, the soliton number [Eq. (7)] is a useful parameter for interpreting the results:

DECOHERENCE DUE TO XPM-ASSISTED RAMAN AMPLIFICATION
We first explore how the coherence of an orthogonally polarized probe is affected by XPM-assisted Raman amplification (Section 3.A) and then show the effect of the competition between the coherent optical wavebreaking process with Raman-assisted parametric FWM in the pump (Section 3.A.1). We then show how the probe energy affects the probe coherence (Section 3.A.2). The second mode of decoherence with copolarized but offset wavelength pulses is studied in Section 3.B. The XPM-assisted decoherence process generally follows a four-step pattern: 1. The pump pulse undergoes a combination of FWM and Raman scattering, amplifying Stokes and anti-Stokes waves that are seeded by quantum noise and therefore have stochastic amplitude and phase profiles. This essentially scalar process degrades the pump coherence following the pattern identified in Ref. [9]. 2. The pump Stokes and anti-Stokes waves interfere with the main part of the pump to produce stochastic time-domain intensity modulations that are amplified with propagation distance. 3. While the pump and probe overlap in the time domain, XPM transfers the stochastic Raman-amplified intensity modulations of the pump pulse to a stochastic phase modulation in the probe. 4. This phase modulation drives Raman-assisted parametric FWM in the probe, which develops noise-seeded, incoherent Stokes and anti-Stokes spectral sidebands, degrading the probe coherence.
A. Cross-Polarization Decoherence Figure 3 shows the pump and probe at the output of a 1.3 m length of the ANDi PCF (representative of lengths used in recent studies of supercontinuum decoherence in ANDi fiber [9,[27][28][29]). Here, E pump = 60 nJ (N pump = 770), E probe = 5 pJ, and β 1 = 0 ps/m. The pump has undergone significant spectral broadening as described in step 1 above and in scalar reports of Raman-induced decoherence in ANDi fiber. Strong Stokes and anti-Stokes bands are visible in the ensemble average as a shallow spectral modulation with a frequency corresponding to the Raman gain peak at 13.2 THz. The time-domain modulations outlined in step 2 are visible on the left-hand side, and the coherence of the spectral components resulting from the noise-seeded broadening is low. The probe ensemble (second row) shows spectral broadening through XPM-assisted Raman amplification outlined in steps 3 and 4, which results in very low probe coherence. It is not possible to characterize this effect using a scalar model alone: the output probe ensemble without the XPM term was always coherent. XPM dominates the probe propagation, as the spectral broadening is driven externally (as opposed to SPM in the pump propagation), leading the initially identical pump and probe spectra to evolve differently. The one feature common to the pump and probe is a pair of Raman peaks (first and second Stokes, highlighted by the black arrows), with a spacing equal to the frequency of the main peak in the Raman response of silica (13.2 THz). However, these features also differ in detail, with those in the probe spectrum having narrower bandwidths and shorter central wavelengths.
The origin of these differing characteristics is illustrated in Fig. 4 for 90 cm of PCF and β 1 = 7 ps/m, so that the probe advances through the pump, but neither the pump nor probe becomes fully incoherent, so it is easier to distinguish each nonlinear process. XPM-assisted Raman amplification in the probe is seen directly: the Stokes and anti-Stokes waves in both pulses occur at the same delay (−4 ps). The weaker contribution of XPM to the probe versus SPM to the pump means that the chirp at −4 ps is lower for the probe. Coupled with the common Raman modulation frequency of R /(2π ) = 13.2 THz, this leads to the probe Raman bands having different central frequencies to the pump Raman bands. The Raman bands of the probe also have narrower bandwidths because the probe has undergone less broadening than the pump, and so the Raman bands are generated from a narrower initial spectrum. The XPM-assisted Raman amplification is purely a phase effect, evidenced by the chirp of each pump/probe Raman band, which closely matches that of the pump/probe pulses, showing that the probe Raman amplification occurs without transferring photons across polarization axes [46]. This is also confirmed by the pump and probe energies, which remain stable apart from loss.
The continuous Raman amplification seen in Fig. 4 would be truncated if, instead of being positive, β 1 becomes negative so as to minimize the temporal overlap of the probe with the pump Raman band at its leading edge. An asymmetry in probe  Figs. 3 and 4). In all cases, the pump Raman scattering was mainly confined to the pump leading edge, as the temporal walkoff between the pump Stokes and anti-Stokes waves was negligible due to the low fiber dispersion of −31 ps/(nm km).
The left-hand plot of Fig. 5 ( β 1 = 7 ps/m) is the same as Fig. 4 but for 130 cm of PCF instead of 90 cm. The longer interaction length and large temporal overlap has allowed the pump to continuously stimulate Raman amplification in the probe, causing its average coherence to degrade to |g (1) 12 | = 0.15.
Energy transfer from the probe to its incoherent Stokes waves was significant (highlighted in the figure by the red dashed lines). This changes for β 1 = −7 ps/m, shown in the central column. The lower probe group velocity moved the probe towards the trailing edge of the pump, so the same average temporal overlap as for β 1 = 7 ps/m resulted in significantly reduced XPM-assisted Raman amplification in the probe and a higher probe coherence of |g (1) 12 | = 0.6. It was possible to maintain a high probe coherence for a positive GVM only if the GVM was increased substantially. Using β 1 = 10 ps/m (third column, Fig. 5) maintained an average probe coherence of 0.6. For this GVM, the probe reached a delay approximating the pump FWHM duration after propagating L c pump = 72 cm, so temporal overlap of the pulses was small by the time that Raman scattering had become significant for the pump.

Competition between Optical Wavebreaking and Raman-Assisted Parametric FWM
The competition between optical wavebreaking and Raman-assisted parametric FWM determines whether the supercontinuum process in ANDi fiber is coherent or incoherent [9]. This competition is parameterized by the coherence length, L c [Eq. (6)], at which |g (1) 12 | decreases to 0.9. Since L WB increases if the pulse duration is increased, even if the peak power is held constant, this results in a reduced coherence length and a faster rate of decoherence. In this section, by adding XPM, we provide the first study of the probe coherence as the pump transitions from coherent to incoherent propagation dynamics.
We calculated the average coherence for both the pump and the probe over a broad range of GVM, pump energy, and duration values while maintaining a fixed pump peak power of 15 kW (approximating values commonly used in experiments and simulations [3,7,9]). We parameterize the pump characteristics using N pump [Eq. (7)] and use a constant peak power to fix L N = 2.5 mm (γ P 0 = 405 rad/m). The pump duration  12 | is minimized for a given N pump . The purple region shows where L c probe is shorter than L c pump , and therefore where the decoherence for the probe occurs at a higher rate than the pump. Bottom: energy transferred from the pump and probe pulses to the phonon field as a function of propagation distance (N pump = 900 and β 1 = 0 ps/m). ranged from 0.1 ps to 7 ps and the pump energy from 1.6 nJ to 112 nJ, giving L R = 0.1 m while varying L WB from 0.04 m to 2.9 m to show the effect of both coherent and incoherent pump dynamics on the probe. The probe energy and duration were held constant at 5 pJ and 7 ps (P 0 ≈ 0.7 W), giving probe nonlinear and dispersion lengths of 61 m and 2.7 km, respectively, ensuring that Eq. (1) was fulfilled for the full parameter space. The results are shown in Fig. 6. Due to the PCF length of 1 m, the |g (1) 12 | = 0.9 contour in Fig. 6 gives the pairs of β 1 and N pump values resulting in L c probe = 1 m.

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The average pump coherence (Fig. 6, top left) had no GVM dependence because Eq. (1) was fulfilled. For clarity, we therefore show the average pump coherence for β 1 = 0 ps/m only. L c pump = 1 m for N pump = 400 (E pump ≈ 42 nJ, T 0 = 2.7 ps), shown by the purple dashed lines. The variation of L c pump ∝ L R /L WB versus pump parameters and fiber dispersion is also shown. The transition from optical wavebreaking to Raman-assisted parametric FWM occurs for N pump ≈ 400.
Our previous observation that the probe coherence is reduced more quickly for β 1 > 0 ps/m is shown here by the asymmetry of the |g (1) 12 | contours around β 1 = 0 ps/m (Fig. 6, top right). The minimum N pump that results in L c probe = 1 m occurs at β 1 = 5 ps/m. For large | β 1 | values, L c probe = 1 m is maintained only if the pump soliton order increases linearly with | β 1 |.
The asymmetry of the probe decoherence with GVM becomes less significant as N pump (and thus the pump decoherence rate) is increased. This is shown by the negative gradient of the red dashed line (Fig. 6, top right), which tracks the minimum probe coherence for each N pump , and happens because Raman amplification occurs at the leading edge of the pump first and then progresses through the peak towards the trailing edge [9], and this progression occurs earlier in the propagation as N pump is increased. Figure 6 shows that for some values of N pump and β 1 , the probe coherence degrades further than that of the pump despite the reduction in the pump influence on the probe by XPM = 2/3 [Eq. (A3)]. This is most pronounced when the probe energy is significantly smaller than the pump energy. For example, the average pump coherence is equal to 0.3 when N pump = 950, but the average probe coherence takes the same value between −2 ≤ β 1 ≤ 7 ps/m and 630 ≤ N pump (shown by the white contour), where E probe ≈ 10 −4 E pump . The region where the pump coherence exceeds the probe coherence is confined to smaller ranges of β 1 as N pump is decreased, until eventually the pump coherence is higher only when the probe pulse propagates faster than the pump pulse. These effects can be understood by examining the mixed parametric-Raman gain for the first Stokes waves of the pump and probe [9,[24][25][26], which take comparable values of g s P 0 = 9.6 m −1 and (2/3)g s P 0 = 6.4 m −1 for an average probe coherence of 0.3 at β 1 = 0 ps/m (where N pump ≈ 800, T 0 = 5.3 ps, and E pump = 85 nJ). The low probe energy combined with the comparable Raman gain and equal Stokes seed energy (set by the quantum noise) results in a higher fraction of the probe energy being transferred to the Stokes and anti-Stokes waves:

Accelerated Decoherence for Low-Energy Probe
We attempted to calculate the energy in the pump and probe Stokes and anti-Stokes waves directly, but it is often not possible to separate the characteristics of each nonlinear process in the noisy spectra accurately. The total energy transferred to the phonon field (i.e., the energy loss due to the frequency detuning between the pulses and the Stokes waves) is an equivalent metric, obtained using Eq. (8). E (z)/E input gives the fractional energy loss for the pulses at propagation distance z, α fiber (z) is the fractional energy loss from linear propagation in the fiber, and α P (z) is the fractional energy lost by the pulses to the phonon field through Raman amplification. This quantity is reliable because of the negligible error introduced by our chosen integration method, because SPM, self-steepening, and dispersive pulse broadening are lossless, and because of the higher gain for the Stokes waves in comparison with the anti-Stokes waves [9,[24][25][26], causing a net transfer of energy to the phonon field from the pump and probe pulses.
The contribution to the phonon energy from the pump and probe is shown as a percentage of their respective input energies in the bottom row of Fig. 6 for N pump = 900 (T 0 = 6 ps, E pump ≈ 95 nJ) and β 1 = 0 ps/m. At the fiber output, the probe has contributed 2.2% of its energy to the phonon field for PCF lengths above 0.8 m, whereas the pump has contributed 1.6% in that section of the fiber, which confirms the proposed explanation for the relative coherences of the pump and probe. Equivalently, the average probe coherence is expected to be proportional to the probe energy as long as Eq. (1) is satisfied. This is confirmed by Fig. 7, which shows how |g 12,probe | depends on the ratio of probe energy to pump energy for a constant We note that the accelerated probe decoherence is important for a very broad range of GVM values, and occurs faster than the pump decoherence (top right plot in Fig. 6, purple shading) when N pump > 300 and when β 1 is between 0.3 and 8 ps/m. The corresponding birefringence of 9 × 10 −5 to 2.4 × 10 −3 covers both non-PM and PM fibers. The faster loss of coherence for low-energy signal propagating with an incoherently broadened high-energy pulse is therefore expected whenever it is polarized along the fiber fast (lower index) axis.
B. Co-Polarized, Cross-Wavelength Decoherence Figure 8 shows co-polarized pump and probe pulses in the same format as Figs. 4 and 5 and for the same fiber parameters. The pump central wavelength was constant at 1040 nm, but the probe was set to 1100 nm (top row of Fig. 8), and 1300 nm (bottom row) to explore how the fiber dispersion affects the probe decoherence. The pulse duration was 7 ps (probe bandwidths of 0.25 nm and 0.35 nm, respectively, and a pump bandwidth of 0.23 nm). Differences in the propagation for co-polarized pulses in comparison with cross-polarized pulses are due to the increased strength of XPM, which is three times higher compared with the cross-polarized case.
The GVM was just 2 ps/m for a probe central wavelength of 1100 nm, so its large overlap with the pump leading edge caused significant Raman amplification. The Raman gain for the pump was g s P 0 = 7 m −1 [L R = 14 cm, N pump = 770; see Eqs. (2) and (3)], but the gain for the probe was higher at approximately 15.3 m −1 because XPM = 2 for co-polarized pulses [Eq. (A3)] and because of the different dispersion values at the pump and probe wavelengths. The Raman gain for the 1300 nm probe (bottom row of Fig. 8) was 22.5 m −1 , but the effect is reduced by the larger GVM of ∼ 9 ps/m. The larger Raman gain means the probe coherence degraded faster than for the orthogonally polarized pump and probe in Section 3.A. In the top row of Fig. 8, the probe coherence degraded to |g (1) 12 | = 0.15 compared to |g (1) 12 | = 0.24 for the pump. L c pump = 72 cm, and L c probe was 71 cm and 83 cm for the two wavelengths, respectively (compare with equivalent average coherence values for cross-polarized probe pulses, where N pump = 770 and β 1 = 2 and 9 ps/m: 0.23 and 0.57, respectively). The fiber dispersion profile therefore has a similar influence on the probe coherence to the polarization GVM, except the probe Raman gain is higher because of the stronger influence of XPM.
For completeness, note that the most general case of XPMassisted decoherence occurs for cross-polarized pulses with different central wavelengths. For those conditions, the probe decoherence will be a function of the pulse polarization state as well as the fiber polarization GVM and dispersion. Although the size of this parameter space goes beyond the scope of the current work, the two cases outlined in Sections 3.A and 3.B

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provide insights that are useful for predicting the propagation dynamics in the more general case. For example, cross-polarized pump and probe pulses with respective central wavelengths of 1040 nm and 900 nm would have a GVM of −6.2 ps/m for the PCF used in this work, and so, in a weakly birefringent fiber, the probe would develop in a way similar to the β 1 = −7 ps/m simulation in Fig. 5. However, were the probe propagated with its polarization aligned to the fast (low-index) axis of a highly birefringent fiber instead, then this GVM would be counteracted, resulting in a more significant loss of coherence for the probe. Note that we have chosen wavelength offsets that are mismatched from the Raman gain peak at 13.2 THz to avoid coherently seeded Raman amplification.

CONCLUSION
We have studied a new mechanism through which the ANDi supercontinuum decoherence process can be transferred to a co-propagating probe signal via a combination of XPM and Raman amplification of out-of-band quantum noise. We have systematically characterized the mechanism using a vector GNLSE to investigate the decoherence of a low-energy probe that co-propagates with a high-energy, continuum-generating pump. Our work extends a scalar investigation that showed that Raman decoherence is highly significant for picosecond pulses in meter-length fibers [9] to include XPM-based nonlinear coupling.
We explored a broad range of pulse and fiber parameters such as fiber birefringence, pulse durations, peak powers, central wavelengths, and the assignment of the pump and probe pulses to both the fast and slow polarization axes. The pump pulses investigated had sufficient energy and duration to undergo substantial decoherence over fiber lengths of approximately 1 m if propagated on their own. For the probe, we chose pulse parameters that resulted in a negligible contribution to nonlinear and dispersive effects.N The main conclusion is that there is an asymmetry in the probe decoherence versus the sign of the GVM. This effect can be observed only using vector simulations, and occurs because the temporal overlap of the probe pulse with the Raman amplification in the pump leading edge is maximized when the probe has a higher group velocity. This is the case when an orthogonally polarized probe is propagated along the fast (lower index) axis of the fiber, or when a co-polarized probe has a longer central wavelength, and occurs in both PM and non-PM fibers. The polarization GVM and the fiber dispersion have equivalent roles in the probe decoherence for narrow-band input pulses, and so the PCF hole and pitch size as well as the birefringence will determine the probe coherence in general. The relative polarization state of the pump and probe was also found to be of central importance as it determines the strength of the XPM coupling between the pulses. We also observed that the probe decoherence can occur at a higher rate where the probe energy is sufficiently small, and that this happens for both orthogonally and co-polarized probes.

APPENDIX A. GENERALIZED NONLINEAR SCHRÖDINGER EQUATION
We simulated the nonlinear propagation of both the pump and probe using Eq. (A1). OperatorsD andN are responsible for linear and nonlinear effects acting on field envelope A(z, T): We took the standard approach of applying the fiber dispersion and loss in the frequency domain using the linear operator defined in Eq. (A2): where α( ) is the fiber loss spectrum, β 2 ( ) is the full fiber group velocity dispersion (its frequency dependence accounts for the higher orders), and = ω − ω 0 is the angular frequency grid centered on zero. The polarization GVM is accounted for using β 1 as described in Section 2 and, being the difference between the pump and probe group velocities, was included for orthogonally polarized probe pulses only to keep the pump pulse centered at the origin of the temporal grid. The fiber dispersion determined the GVM for co-polarized pulses.
Although the nonlinear operator is commonly defined in the time domain, we opted for the frequency domain definition given by Eq. (A3) to account for the frequency dependence of γ and because this definition is more readily incorporated into the CQEM [42]: For the probe propagation,N probe takes the same form as Eq. (A3), but A pump (z, T) and A probe (z, T) are swapped. F and F −1 denote the fast Fourier transform and its inverse, respectively. XPM is included using XPM |A(z, T)| 2 , where XPM = 2/3 or 2 for orthogonally polarized or co-polarized pulses, respectively.h R ( ) is the Raman response of the fiber, and R (z, T) accounts for spontaneous Raman scattering noise [10,48]. We neglected the degenerate FWM term given by A pump A 2 probe exp(−4i nz/λ) (or, for the probe A probe A 2 pump exp(4i nz/λ)) because we found that its contribution averaged to zero [46], where n > 3 × 10 −5 (corresponding to | β 1 | > 0.1 ps/m, which covers the majority of the GVM range explored), and also because our aim was to investigate the effects of XPM-assisted Raman amplification on the probe decoherence in isolation from other field coupling effects. Additionally, a Raman term involving the product of both A 1 and A 2 had a negligible effect for the pump and probe energies selected, and so was excluded to reduce computation time.