1. Introduction

It is well known that the spectrum of incoherent continuous-wave (cw) light propagating inside an optical fiber is subject to broadening through optical nonlinearities [1]. As early as the 1970s the mechanism of spectral broadening was identified as nonlinear coupling between multiple waves of different optical frequencies [2]. But a rigorous treatment of this subject matter is complicated by both the statistical nature of incoherent cw light and the complexity of various wave-mixing processes that must be considered. To date, only a handful of analyses have been undertaken.

In previous studies the important role of dispersion in spectral broadening of incoherent light was not fully treated. In such studies, dispersion was regarded as a phase-detuning term that merely acts to retard spectral broadening. Accordingly, such studies predict that the spectral width of an incoherent signal injected into a long span of fiber increases monotonically as a function of propagation distance indefinitely. As will be shown in the present study, the prediction of monotonic spectral broadening stems from mathematical approximations that were necessary in the course of analytical derivations conducted in previous studies. The present study clearly demonstrates that broadening does not continue indefinitely in the presence of group velocity dispersion. Moreover, under a wide range of conditions, initial spectral broadening may be followed by one or more propagation intervals over which spectral recompression occurs.

In 1991 Manassah analyzed spectral broadening of incoherent light in optical fibers [2] for which stochastic properties were treated implicitly. In this study, the use of a simplified nonlinear Schrodinger equation in which fiber gain/loss and dispersion are omitted, on the one hand, enabled the derivation of an analytical expression for spectral broadening, but on the other hand, failed to explain the role of dispersion. In another study, Bouteiller [3] treated the problem of incoherent spectral broadening in the context of a Raman fiber laser. In this paper, Bouteiller tackled the problem in the frequency domain, where spectral broadening is explained through the interaction of numerous four-wave mixing quartets distributed over the entire optical spectrum. But neither the stochastic nature of incoherent light nor the complex dynamics of phase evolution was fully analyzed since the author adopted a rather fictitious relationship between the phase and the amplitude of each spectral component, which was necessary to simplify the numerical simulation. Later, based on the same approach, Babin, et al. adopted Zakharov’s kinetic equation [4,5] and conducted a more comprehensive analysis of spectral broadening in a fiber Raman laser [6–8]. But the role of dispersion was not fully analyzed because the instantaneous electrical field was replaced by a time-averaged quantity. Such an approach averages out dispersion effects and also does not provide an appropriate treatment of the role of dispersion in detuning the phase-sensitive parametric mixing process. Lastly, reference [9] reported notable discrepancies between experimental observations of incoherent spectral broadening and that predicted by theory in [6–8].

In the present study, we begin by introducing a fully stochastic model for incoherent spectral broadening in Section 2, where the role of dispersion in spectral broadening of incoherent light is examined through numerical study. In section 3, based on the differential equation that forms the basis of the stochastic model, we derive a simplified analytical expression that provides a great deal of insight into the role of dispersion with regard to the direction of energy flow in the frequency domain, and provides a viable alternative to the computationally intensive fully stochastic simulation that can be applied to a wide range real-world system architectures. Finally, conclusions follow in Section 4.

2. Numerical stochastic study of the role of dispersion in spectral broadening

We introduce a stochastic model for spectral broadening of incoherent cw light propagating in an optical fiber. The complete details of the rigorous stochastic modeling methodology used herein is presented elsewhere [10]. In this report, we introduce only the final differential equation model for brevity.

Familiar examples of incoherent cw signals include thermal radiation and amplified spontaneous emission. A typical fiber laser, whose gain bandwidth might encompass 10⁵ closely spaced longitudinal modes, and whose round-trip optical path length is continuously subjected to random detuning due to thermal and mechanical disturbances also may generate such an incoherent cw output signal.

We start with the nonlinear Schrödinger equation in frequency domain [1,4,6,10]:

Mathematically, the parameter $A_j\left(z\right)$, in conjunction with an appropriate probability distribution function (PDF) over the complex plane for $A_j(0)$, provides a simple means to represent the stochastic properties of incoherent light in the frequency domain. Earlier studies involving stochastic modeling of nonlinear effects for incoherent light include references [11,12]. Stochastic modeling of noise in the picosecond or nanosecond pulses was studied intensively in references [13,14]. However, unlike these early studies, we derived a truly rigorous frequency-domain stochastic model for cw incoherent light, as described in reference [10]. A very detailed explanation of such stochastic model in frequency domain is presented in reference [10], along with model validation via comparison of numerical simulations and carefully conducted experimental measurements. The functional form of PDF is [10]

Based on the fully stochastic model, we then turned to the question of the role of the dispersion in spectral broadening. Based on the literature published to date, one might conclude that the spectral broadening is a monotonic phenomenon in which the spectral distribution continuously broadens as it propagates through the fiber. All of the published analytical treatments of incoherent cw wave mixing predict this same result, including [2,3,6–8]. In marked contrast, numerical simulations conducted in this study indicate (1) that fiber dispersion causes spectral broadening to eventually slow down as light propagates furtherdown the fiber, and (2) under some circumstances, eventually recompression of the spectrum is observed to a significant extent. The later effect in particular is certainly counter-intuitive.

As a measure of spectral broadening we use root-mean-square (rms) spectral bandwidth $\mathrm{\Omega }\triangleq 2{\left({\displaystyle \sum p_j{\left({\omega }_j-{\omega }_c\right)}^2}/{\displaystyle \sum p_j}\right)}^{1/2}$, where $p_j$ is the power of j^th spectral component. Our use of the rms spectral width in lieu of the conventional full-width half-maximum spectral width is motivated by the fact that the rms spectral width better accounts for broadening in the form of spectral wings. We will also refer to the spectral broadening factor, Θ, which is defined as the rms output spectral width divided by the rms input spectral width.

To investigate the role of fiber dispersion, we simulated a passive polarization maintaining fiber (Nufern PM980) with a 6.1 µm core diameter and 0.14 core NA, having a background loss of 1.0 dB/km. The input light was assumed to have a 0.20 nm spectral width (rms) with a Gaussian spectral distribution. The center wavelength was assumed to be 1130 nm. A nonlinear coefficient of γ = 4.0 W⁻¹ km⁻¹ was used and the fiber dispersion was calculated to be $D=-\left(2\pi c{\beta }_2\right)/{\lambda }_c^2$ = −17 ps nm⁻¹ km⁻¹, where ${\lambda }_c$ is the center wavelength.

Figure 1 shows the output spectra for different propagation distances at an input power of 2 W. Interestingly, the spectrum quickly broadens during propagation through the first 160 m of fiber, after which the rms spectral width begins to shrink slowly. For example, note that the spectral width after a propagation distance of 400 m is considerably smaller than at a propagation distance of 320 m. Such behavior is illustrated more clearly in Fig. 2 , in which the rms spectral broadening factor Θ is plotted as a function of propagation distance (at a number of different input powers). Note that Θ is larger for larger input power. This trend is universal regardless of input power. The right-hand portion of Fig. 2 shows the spectral broadening factor as a function of input power upon passage through a span of fiber of a specific length (80, 200, and 400 m). The power dependence of the spectral broadening factor is markedly different for different fiber lengths. For example, the spectral broadening factor appears to plateau at a value of ~5 in the 200 and 400-m-long fiber spans once the input power exceeds ~3 W. The same is not true of the 80 m long fiber. It is apparent that net amount ofspectral broadening incurred during transmission through a span of fiber is neither a simple function of fiber length nor input power.

 

Fig. 1 Normalized output spectra for various fiber lengths from a fully stochastic numerical simulation. The curves correspond to different propagation distances as indicated. The number of trials in each ensemble average was 1000, and the spectral resolution was 0.02 nm

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Fig. 2 Fully stochastic numerical simulation results. Left: spectral broadening factor as a function of fiber length, for various input powers (numerical annotations indicate power in Watts). Right: spectral broadening factor for passage through 80, 200, and 400 m long spans of fiber, as a function of input power.

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Referring back to the left-hand side of Fig. 2, it appears that in the limit of long fiber length, spectral broadening may exhibit asymptotic steady state behavior. There is in fact one experimental result previously reported in the literature [15] from 2006 in which a similar trend of spectral broadening was observed. In this paper the authors did not posit any plausible explanation of such behavior, and there has been no subsequent analysis to date. Another interesting observation from the left-hand portion of Fig. 2, pertaining to cases where overshoot is observed, is that the extent of overshoot (i.e., the peak value for spectral broadening) scales with input power.

In order to see the effect of dispersion quantitatively, we performed the numerical simulation with a series of different dispersion values as shown in Fig. 3 . For this calculation, the input power was set to 2 W, and dispersion was the only simulation parameter varied (20, −40, −60, −80, and −100 ps nm⁻¹ km⁻¹). The role of dispersion in spectral broadening is clearly illustrated in this figure. For each of the five curves, the initial rate of broadening appears to be the same. But beyond a propagation distance of 30 m, the extent to which further spectral broadening occurs is highly dependent on fiber dispersion. The maximum value of Θ scales inversely with the dispersion, and each curve appears to converge to an asymptotic spectral broadening ratio whose value depends on fiber dispersion. Spectral broadening also appears to exhibit damped oscillations in which the damping coefficient scales with fiber dispersion. We also confirmed that the spectral broadening depends only on the absolute value of dispersion; upon changing the sign of D, identical spectral broadening is observed.

 

Fig. 3 Fully stochastic numerical simulation results. Spectral broadening factor Θ as a function of propagation distance, for various fiber dispersion values. Simulated with input power of 2 W.

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Confirmation of the slow down and subsequent reversal of spectral broadening provides strong motivation to seek an analytical expression from which $p_j(z)$, the power contained in the j^th Fourier component as a function of z, can be more easily calculated. In addition to providing an alternative to computationally intensive fully stochastic simulations, the mathematical form of such an analytical expression will presumably provide more physical insight into the underlying mechanisms for such spectral broadening behavior.

3. Analytical study of the role of dispersion in spectral broadening

3.1. Analytical derivation of expression for calculation of $p_j(z)$

Solving Eq. (1) analytically is not straightforward. Reference [7] solves Eq. (1) specifically in the context of Raman fiber laser, under a specific set of operating conditions, employing a turbulence wave-kinetic equation. However, this method is not suitable for the far more general range of applications considered here, for the following reasons. (1) The analytical derivation in [7] relies on a specific relationship between the intracavity power and the laser cavity parameters. (2) The derivation requires the assumption of nearly constant intracavity power, and is further restricted to the special case of very low cavity loss/gain (i.e. where both cavity mirrors have nearly 100% reflectivity). (3) Because the effects of dispersion are averaged early in the derivation, the final spectral broadening formula predicts a monotonic increase of the spectral bandwidth as a function of fiber length. In contrast, in our analysis we generalize the mathematical treatment of incoherent spectral broadening to encompass a much wider range of applications, including passive fibers, fiber amplifiers, and fiber lasers, in which the optical power may change significantly during the course of propagation. Accordingly, we cannot employ the wave-kinetic equation technique.

References [4,5] provide an excellent technique for determination of the fourth-order moment of a zero-mean Gaussian random variable in terms of the sixth-order moment, which again can be approximated a sum of second-order moment terms. Although we can adopt this technique, we cannot adopt the whole wave-kinetic equation method for the reasons described above.

We multiply Eq. (1) by$A_j^{*}$ and taking the ensemble-average of both sides of the equation. This results in

We now investigate the condition required for use of quasi-Gaussian moment expansion theory, which was used for deriving Eq. (5). Such an approximation is possible only when the random processes of the different spectral components are very weakly correlated through nonlinear mixing. Qualitatively speaking, the implication of such a small nonlinearity is a small change in the amplitude of the spectral power density. But the actual applicability of quasi-Gaussian moment expansion depends on the quantitative extent of such nonlinear mixing interactions in practice. After rather a lengthy derivation starting from Eq. (5), we obtained a sufficient condition for using the quasi-Gaussian moment approximation (For detailed derivation, see reference [10].) The result is

Here, we define the quantity $\sigma \triangleq 2\sqrt{2}\gamma P_0{L}_{eff}$ for future reference. In a strict sense Eq. (7) quantitatively limits the validity condition for applying the quasi-Gaussian moment expansion theorem. But given that Eq. (7) represents a sufficient but not necessary condition, there might be parameter space outside of that specified by Eq. (7) for which the derived analytical expression is valid. This will be investigated momentarily.

The effect of dispersion in spectral broadening can be further clarified by considering following: Since the fiber dispersion term can be represented through Eq. (6) $\mathrm{\Delta }{k}_{jlmn}={\beta }_2\left({\omega }_j-{\omega }_m\right)\left({\omega }_l-{\omega }_m\right)$, we can approximate the average impact of fiber dispersion in terms of a wavelength-averaged phase mismatch of the form, $\mathrm{\Delta }{k}_{avg}={\beta }_2{\overline{\mathrm{\Omega }}}^2$, where $\overline{\mathrm{\Omega }}$ is the average root-mean-squared full optical bandwidth over $z^{\prime }\in \left[0,z\right]$. We can then calculate the z-averaged and wavelength-averaged “dispersion detuning factor” κ as follows:

In order to study the behavior of spectral broadening, we consider a quantity defined by${\rho }_{pk}(z)=p_{pk}\left(z\right)e^{\alpha z}$ where $p_{pk}\left(z\right)$ represents the spectral peak power. Reference [10] demonstrates that, over a vast majority of the parameter space corresponding to real-world applications, reducing ${\rho }_{pk}(z)$ implies broadening spectrum and vice versa. We can then derive an algebraic expression for the effect of dispersion from Eq. (5) [10]:

Let us now define a quantity that we refer to as the “nonlinear wave mixing strength”, Λ:

The latter effect is the most distinguishable difference between coherent four-wave mixing, where energy transfer among different frequency components exhibits undamped oscillations, and incoherent wave-mixing, in which the rate of energy transfer eventually decays to zero (via under-damped, critically damped, or over-damped oscillations). Having said that, a quantitative description of such behavior would require that we derive an explicit analytical expression for the parameter $\overline{\mathrm{\Omega }}$, which is beyond the scope of this paper.

3.2. Validation of the analytical solution for $p_j(z)$

We now compare the analytical formula derived above with the results obtained via fully stochastic numerical simulation. Here we describe one representative set of calculations in which we compared the results obtained from the fully stochastic numerical simulation with those obtained from the analytical formula derived above. In this representative calculation, we repeat the previous fully stochastic simulation (same fiber and fiber length) at an input power of 0.4 W. We also assume the same background loss (1.0 dB/km), nonlinear coefficient (γ = 4.0 W⁻¹ km⁻¹) and fiber dispersion (D = −17 ps nm⁻¹ km⁻¹). The input signal was assumed to have a Gaussian spectrum with an rms spectral width of 0.10 nm. For the fully stochastic simulation, we calculated an ensemble average of 1000 randomly initialized trials. The spectral resolution of the calculation was 0.02 nm.

Figure 4 shows a comparison of calculated spectral broadening factor as a function of propagation distance. Referring to results of the fully stochastic simulation (red curve), at a propagation distance of 400 m the spectral width roughly doubles, and a systematic error of order 10% is observed for the analytical calculation (blue curve). Referring back to Eq. (7), we find that the validity condition for the analytical derivation is clearly violated (σ = 1.8). Nonetheless, the level of agreement between the fully stochastic simulation and the numerical evaluation is relatively good. The two curves are nearly identical up to a propagation distance of ~190 m, after which systematic deviation is observed. At a propagation distance of 190 m, σ has a value of 0.9. Thus we find that in this particular case, the validity criterion given by Eq. (15) is quite conservative.

 

Fig. 4 The spectral broadening factor as a function of propagation distance predicted by the fully stochastic simulation (red) and the analytical formula for ensemble-averaged spectral power density (blue).

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Figures 5 , 6 show the calculated spectral distribution generated by the fully stochastic and analytical calculations, at 40-m intervals along the 400-m-long segment of fiber, on both linear and logarithmic scales. Figure 5 shows excellent agreement between the two sets of calculated spectra when plotted on a linear scale, which again confirms that the validity of our analytical derivation far exceeds the conservative criterion based on Eq. (7). Figure 6, in which the data are plotted on a logarithmic scale, shows some discrepancy with regard to power in the spectral wings. Recall that Ω is the rms spectral width, which makes Ω sensitive to errors in the spectral wings; because the analytical derivation under-predicts power in the spectral wings (Fig. 6), it underestimates the spectral broadening factor (Fig. 4).

 

Fig. 5 Comparison of linear spectra taken at 40 m intervals of propagation distance. Left: Fully stochastic simulation. Right: Numerical evaluation of the analytical derivation.

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Fig. 6 Comparison of logarithmic spectra taken at 40 m intervals of propagation distance. Left: Fully stochastic simulation. Right: Numerical evaluation of the analytical derivation.

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We examined the validity criterion $\sigma =2\sqrt{2}\gamma P_0{L}_{eff}\ll 1$ by comparing the fully stochastic simulation and numerical evaluation of the analytical derivation for a wide range of cases having different values of σ. Given that$L_{eff}=\left(1-e^{-\alpha L}\right)/\alpha$, there are four parameters that determine σ, namely, the nonlinear coefficient γ, input power $P_0$, fiber length L, and gain/loss coefficient α. If the inequality of Eq. (7) closely represents the necessary condition for validity, there is a well-defined manifold (surface) in 4-D parameter space (γ,$P_0$,L,α) defined by an equation $\sigma \left(=2\sqrt{2}\gamma P_0\left(1-e^{-\alpha L}\right)/\alpha \right)={\sigma }_0$, where ${\sigma }_0$ is a constant within which the spectral broadening effects predicted by the fully stochastic and analytical methods are nearly identical. To the extent that this is true, one can verify in quantitative terms the “tightness” of the inequality derivation.

For this set of calculations, we assumed the same nominal values of $\gamma =4{\text{ W}}^{-1}{\text{ km}}^{-1}$, $P_0=0.4\text{ W}$, $L=400\text{ m}$, and $\alpha =1\text{ dB/km}$, for which $\sigma =1.8$. We then constructed a series of parametric plots in which each of the four parameters γ, $P_0$, L, and α was varied such that σ covered the range $0.1\le \sigma \le 3.5$. This required that the three parameters γ, $P_0$, and L be varied from 10% to 200% of their nominal values, and that α be varied from –14 dB/km (gain) to 300 dB/km (loss).

Figure 7 shows the calculated spectral broadening factor Θ as a function of σ. Results from both the fully stochastic simulation (solid line) and the numerical evaluation of the analytical derivation using Eq. (5) (dashed line) are shown. In Fig. 8 this data is re-plotted in terms of the relative error of the analytical derivation for Θ. Regardless of which parameter is varied, a similar functional dependence for $\mathrm{\Theta }(\sigma )$ is observed, which strongly indicates that σ is truly a characteristic value for incoherent four-wave mixing. On the basis of Figs. 7 and 8, for practical applications the validity criterion of Eq. (7) should be revised to

 

Fig. 7 Spectral broadening factor Θ predicted by fully stochastic simulation (solid curves) and numerical evaluation of the analytical derivation (dotted curves).

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Fig. 8 Relative error in Θ for calculation based on analytical rate equation.

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Even at $\sigma =3.0$, where $\mathrm{\Theta }=3.5$, the relative error in Θ is less than 20%. These results clearly demonstrate the usefulness of the derived analytical solution, even at very large values of σ.

Note that the calculations for Figs. 7 and 8 were carried out assuming $D\approx -17psn{m}^{-1}k{m}^{-1}$. In a fiber with a much lower (higher) value of $\left|D\right|$, the RHS of the inequality of Eq. (12) presumably must be made significantly less than (greater than) unity to account for the stronger (weaker) four-wave mixing interaction; the effect of D on Θ can be seen for instance in Fig. 2, and is discussed extensively in Section 2. This clearly limits the generality of Eq. (12). On the other hand, fiber laser technology is such that high-power cw signals are typically encountered in the vicinity of 1.1 and 1.5 μm. For a typical fused silica single-mode fiber, in the vicinity of 1.1 μm $D\approx -20psn{m}^{-1}k{m}^{-1}$, and in the vicinity of 1.5 μm $D\approx 20psn{m}^{-1}k{m}^{-1}$. From the standpoint of many practical applications, Eq. (12) therefore provides an effective guideline for the applicability of the analytical result presented in Section 3.1.

It is informative to compare Eq. (12) with the threshold formula for stimulated Raman scattering (SRS) in a single mode fiber (in which the signal and pump waves are both linearly polarized along the same axis). SRS is below threshold in the regime $P_o{L}_{eff}{g}_R/A_{eff}\le 16$ where $g_R$ is the Raman gain coefficient, and $L_{eff}=\left(1-e^{-\alpha L}\right)/\alpha$ [1]. Assuming the bandwidth of the incoherent cw signal is small relative to the gain bandwidth for SRS, significant spectral broadening of an incoherent cw signal will be observed at input powers below the SRS threshold provided the following criterion is met for the dimensionless ratio Ξ:

Note that Ξ depends only on the material properties of the fiber and the operating wavelength. For a typical fused silica fiber,

Equation (13) therefore predicts that substantial spectral broadening of an incoherent cw signal via four-wave mixing will occur well below the threshold power for SRS. In a fiber with a value of $\left|D\right|$ much lower (higher) than $20psn{m}^{-1}k{m}^{-1}$, the RHS of the inequality of Eq. (13) presumably must also be made significantly less than (greater than) unity to account for the stronger (weaker) four-wave mixing interaction. But given the numerical result shown in Eq. (14), we anticipate that only in the case of extremely high dispersion fibers would the condition $\mathrm{\Xi }>1$ ever be encountered.

4. Discussions and conclusions

We have studied the role of dispersion on the spectral broadening of incoherent cw light propagating in optical fibers. The effect of group velocity dispersion (${\beta }_2$) is to (1) reduce spectral broadening, (2) cause the rate of spectral broadening during propagation to gradually decrease, (3) eventually reverse sign in some cases (thereby creating a spectral recompression effect), and (4) ultimately cause the spectral broadening factor Θ to reach a final steady state value in the limit of large z.

In constructing the fully stochastic model, it became clear that the various physical processes described above could not be ascertained directly from the differential equation for each ensemble sample, for two reasons. The first reason is that group velocity dispersion and phase mismatch are highly coupled in Eq. (1), and phase mismatch is in turn strongly affected by the wave-mixing process. The second reason is that even if one were able to determine the role of dispersion directly from Eq. (1), which governs the behavior of a single ensemble trial, it is not trivial to ascertain the expected value for an ensemble average; extrapolation of information pertaining to a single statistical sample to information pertaining to ensemble averaged behavior is generally not possible. It was only the analytical derivation for spectral power density that allowed us to clearly define the role of group velocity dispersion in the spectral broadening of incoherent cw light.

In conclusion, we have conducted a systematic study of the role of fiber dispersion on the spectral broadening of incoherent cw light propagating in optical fibers. Using a fully stochastic numerical model, we built a numerical simulator for spectral broadening of incoherent cw light. Although the computational burden of the fully stochastic simulation is large, it does provide a reliable means of calculating the effects of spectral broadening to a high degree of accuracy. This allowed us to use the fully stochastic numerical simulator to empirically demonstrate the role of dispersion in spectral broadening of incoherent cw light under a wide range of conditions.

We also derived an analytical expression for the ensemble-averaged spectral power density assuming the applicability of the quasi-Gaussian moment expansion. The resulting analytical expression provided clear insight into the underlying physics of spectral broadening, especially with regard to the pronounced role of dispersion. This analytical expression indicates that the magnitude of spectral broadening as a function of propagation distance may exhibit over-damped or under-damped oscillations during propagation, depending on the range of input parameters under consideration. In portions of the fiber where $\mathrm{\Lambda }(z)$ < 0, compression, rather than spectral broadening, is observed. Such behavior is analogous to the periodic energy transfer observed in coherent four-wave mixing, except that in the case of incoherent four-wave mixing, this oscillatory energy transfer of energy between different wavelengths is subject to damping.