1. Introduction

Parametric amplification can occur when a strong pump with frequency ω_p co-propagates in an optical fiber with a weaker signal with frequency ω_s . By degenerate four-wave mixing (FWM), induced by the cubic nonlinearity of the glass, energy from the pump is transferred to the signal and, because of energy conservation, to an idler with frequency ω_i =2ω_p-ω_s as well [1, 2]. A key advantage of parametric amplification is its wide bandwidth, which is not limited by the properties of erbium, as in erbium doped fiber amplifiers, or by the properties of Raman phonons in glass, such as in Raman amplifiers [1, 2, 3]. For parametric amplification to occur efficiently a phase matching condition needs to be satisfied that depends on the propagation constants of the fields as

where, β_p,s,i are the propagation constants of the pump, signal and idler.

It has been established that even in nominally uniform fibers, the parameters can vary slightly as a function of position, which in turn causes the zero-dispersion wavelength to vary with position [4, 5, 6, 7, 8, 9, 10]. This causes the propagation constants β_p,s,i to vary with position, and hence also the phase matching parameter Δβ (1), and thus the gain. These variations in the fiber parameters can occur over long length scales, of the order of 100–1000 m [4, 5, 6, 7, 9, 10], and short length scales, of the order of 0.1–1 m [4, 8], sometimes in the same fiber [11].

The effect of dispersion fluctuations on parametric gain was studied systematically before by Karlsson [4] and by Abdullaev et al. [12]. Abdullaev et al. restrict their work to fibers with periodically varying quadratic dispersion, and treat this case perturbatively. Karlsson considers the fiber dispersion to be derived from a stochastic process, but his subsequent analysis is flawed [4]. Here we follow Karlsson in that we model the dispersion as a stochastic process, but we solve the ensuing equations numerically.

2. Propagation in the presence of FWM

Here we briefly review light propagation in the presence of FWM. Following Agrawal and Karlsson [1, 4], we consider CW pump, signal and idler fields, all polarized in the same direction. The wave equation can then be solved in a scalar form for the discrete frequencies ω_p,s,i separately, ignoring weaker fields at other frequencies. In the undepleted pump approximation, we then find for the pump amplitude A_p ,

with the exponential factor corresponding to self-phase modulation. Following Agrawal [1], the total pump power P ₀=|A_p |². The signal and idler amplitudes, A_s, A_i satisfy the set of linear differential equations [4]

where, Δk=γP ₀-Δβ/2, A is the column vector with elements A_s and A*_i, and γ=ωn ⁽²⁾/(cA_eff ) is the effective nonlinearity of the fiber. Here n ⁽²⁾ is the nonlinear refractive index, A _eff the effective area of the modes, and ω the frequency of the light. In the treatment below we take γ to be the same for the three frequencies ω_p,s,i .

Though here we study the case in which Δk varies randomly with position, let us first consider constant dispersion. We then obtain the propagation matrix M that describes how the amplitudes of the signal and idler evolve with position

where $\alpha =\sqrt{{\left(\gamma P_0\right)}^2-{\left(\Delta k\right)}^2}$. Matrix M is written in terms of hyperbolic functions, which is appropriate for α ²>0, and the gain therefore grows exponentially. Otherwise, when α ²<0, sinusoidal functions must be used, and the gain is expected to be small. In this work we take the initial conditions A_i (0)=0 and A_s (0)=1. In the absence of dispersion fluctuations, and when α ²>0, the signal gain for a system of length L is thus

which peaks when Δk=0, or Δβ=2γP ₀.

 

Fig. 1. (a) Example of dispersion fluctuations according to Model I. (b) Same as (a), but for Model II.

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3. Stochastic models for dispersion variations

In the presence of dispersion fluctuations, the phase mismatch in the fiber, Δβ, is taken to be a stochastic variable and therefore so is Δk. We write Δk as Δk=Δ+δk where Δ is the average value of Δk, but δk varies randomly. Following Karlsson [4] we choose δk at a given position to follow the Gaussian distribution

where σ is the standard deviation of Δk, and the average vanishes since it is already given by Δ. The autocorrelation of the dispersion variations, describing the spatial distribution of Δk, is taken to have the form [4]

where L_c is the correlation length, the typical length scale over which the fluctuations occur.

Assumptions (6) and (7) do not uniquely define the stochastic process. Since the detailed properties of the dispersion fluctuations are not known, we construct two different models that are both consistent with (6) and (7), and use these to find the dependence of the results on the features of the models, which are described in detail below. Briefly, in Model I the fluctuations are piecewise constant, whereas in Model II, which is Gaussian to all orders, the fluctuations vary continuously. Figure 1 shows the examples of Model I and II for dispersion variations.

In Model I, illustrated in Fig. 1(a), the phase mismatch is constant over finite intervals, while changing discontinuously between them. The distribution (6) gives Δk in each interval, while the dispersion jumps follow a Poisson distribution with average distance L_c between events. Thus, the distribution of the segment lengths L_s follows the exponentially decreasing function

This is so since if on the interval between z ₀ and z ₀+z ₁ the dispersion changes, the dispersion at these positions is uncorrelated. In contrast if the dispersion does not change, the probability of which is exp(-z ₁/L_c ), then the dispersion at z ₀ and z ₀+z ₁ is perfectly correlated. We generate the values of δk and of the length segments numerically using the LAPACK Auxiliary Routine Version 2 Random Number Generator. Thus, δk(z)=σ×m, where m is the output of the normal distribution [0,1] of the random number generator, and L=-L_c ×lnm, where m now is the output of the uniform distribution [0,1]. Once the dispersion is defined, the propagation matrix M from Eq. (4) is used to calculate the fields in each segment. The total gain follows by multiplying the propagation matrices of the individual segments.

In Model II, shown in Fig. 1(b), the dispersion varies continuously, which is likely to be more realistic [5, 6, 7, 9, 10]. It is constructed as follows: the Gaussian distribution (6) is used to give δk(z ₀) at the beginning of the fiber. Then δk(z ₁), at z ₁>z ₀, is given by the conditional distribution

where r is the correlation coefficient between z=z ₀ and z=z ₁. When r=0, distribution (9) reduces to (6), which is independent of δk(z ₀). When r=1, then δk(z ₀)=δk(z ₁). For 0<r<1, δk(z ₀) and δk(z ₁) are partly correlated. Now r(z)≡C(z)/σ2, and so r(z)=exp(-|z|/L_c ) from (7). Thus δk(z) is generated by taking small steps Δz and applying (9). Since our r(z) has the property r(z ₁)r(z ₂)=r(z ₁+z ₂), the result does not depend on the step size, as required. The resulting stochastic process is Gaussian to all orders, as can be ascertained, for example, by evaluating 4th order correlations [13].

We generate the numerical values of δk(z) from (9) by $\delta k\left(z_1\right)=r\times \delta k\left(z_0\right)+\sigma \sqrt{\left(1-r^2\right)}\times m$, where m is the output of the normal distribution [0,1] of the same random number generator used for Model I and where r=exp(-|z ₁-z ₀|/L_c ). In our implementation, we sample Δk along the length with the step size of 0.004L_c or 0.001L, whichever is smaller. Then we use the fourth order Runge Kutta integration method to integrate the coupled equations (3). Note that in Model I, δk is constant in each segment of average length L_c so that (4) can be used, whereas in Model II the calculation is done for at least 250 steps per correlation length. Therefore, the speed of computation for Model II is smaller than for Model I.

3.1. Numerical results

In Fig. 2 we present the numerical results for the amplification according to Model I and II. The data points show the gain at the end of the fiber, averaged over an ensemble of N=900,000 realizations for Model I, indicated in red, and N=5,000 realizations for Model II, indicated in green. The ensemble is smaller for Model II than for Model I since, as discussed in Section 3, the speed of calculation is lower in this case. Since the gain is typically measured in decibells, the average gain can be computed in two ways. We can either calculate the gain in decibells for each realization and then average these, or first average over the realizations and then express this average in decibells. The first of these corresponds to the average that would be measured if the ensemble of fibers were connected in series; the second would be obtained if the ensemble were connected in parallel. Here we choose the former since it corresponds to situation we wish to describe. We also calculate the standard deviation σg of the gain distribution, which is given by the error bars in the graphs. The upward error corresponds to the width following from Model I, whereas the lower bar corresponds to that following from Model II. Note finally that the accuracy of the average gain is given by σ_a=σ_g /√N, which is below 0.14 dB for Model II, and smaller for Model I.

Since the typical gain of interest is approximately 30 dB, we present the results, which are summarized in Fig. 2, for γP ₀ L=4, corresponding by Eq. (4) to a maximum gain of cosh² γP ₀ L=28.7 dB when Δ=0. However, other values of γP ₀ L lead to the same conclusions. In Figs. 2 we show the average gain versus the dimensionless quantity γP ₀ L_c , for fixed values of Δ/γP ₀ and σ/γP ₀. It is noted that from the governing Eqs. (3), the results only depend on these ratios. Therefore we immediately see that the effect of the fluctuations decreases when γP ₀ increases, i.e., for increasing pump power or fiber nonlinearity. We also see that the effect of the correlation length of a fiber depends on how it compares with the gain length 1/(γP ₀). We also show the amplification results for two extreme cases that are calculated easily: (1) σ=0, i.e., no fluctuations, and (2) L_c ≫L. When σ=0 then the “ideal gain” is simply given by Eq. (5). In Figs. 2 we show this result by a solid line. When L_c ≫L, then Δk does not vary over the length of fiber and so the average gain is given by the expectation value of (5) subject to distribution (6)

As discussed when α ²<0, the hyperbolic functions need to be replaced by sinusoidal functions. The result of Eq. (10) is given by the dotted line in Figs. 2.

The first conclusion to be drawn from Figs 2 is that the results for Model I and Model II are very similar. We have found this to be true for all situations we have considered. Since our two quite different models evidently lead to the same results, the details of the models do not matter, and the results can thus be considered quite general. This conclusion is similar to that of Wai and Menyuk, who studied fibers with randomly varying birefringence [14].

Before considering the numerical results in detail we establish one more result. When L_c →0, so that the dispersion varies rapidly with position, we can use an argument that applies to both Models, but most directly to Model I. It is straightforward to see from Eqs. (3) that in the presence of dispersion fluctuations, the powers |A_p,i |² and its first derivative are both continuous, while the second derivative is discontinuous. Deviations between different realizations within a segment thus increase with $L_c^2$, since L_c is the average length of the segments. The average number of segments is L/L_c , and the total deviation in the gain between different realizations thus scales as $L_c^2$×L/L_c ∝L_c . Hence, when L_c →0, the effect of the fluctuations disappears. Thus in this limit result (5) applies (solid lines in Figs. 2) and the variations in the gain disappear, whereas when L_c →∞ we have result (10) (dotted line in Figs. 2).

We now discuss Figs. 2 in detail, ignoring the distinction between the models. We have identified three regimes: |Δ|≲γP ₀ (Figs. 2(a) and (b)), |Δ|≳γP ₀ (Figs. 2(c) and (d)), and |Δ|⋍γP ₀ (Figs. 2(e) and (f)). These correspond to frequencies that are tuned close to the gain peak, far from the gain peak, and tuned close to the edges of the region with exponential gain, respectively. In all three regimes the gain behaves as required when L_c →0 and L_c →∞. Figures 2(a) and (b) show that when |Δ|≲γP ₀, the gain decreases monotonically with L_c . We can understand this by considering the extreme case where Δ→0. Then the gain is maximum and fluctuations can only cause the gain to decrease. By increasing σ the gain decreases more strongly. In contrast, Figs. 2(c) and (d) show that when |Δ|≳γP ₀, the gain increases monotonically with L_c . This can be understood in a similar way as before: when σ=0, α ²<0, and no exponential significant gain arises. Fluctuations may cause some exponential gain to occur, thus increasing the average gain. Finally, Figs. 2(e) and (f) show results for |Δ|=γP ₀. Now the fluctuation may cause the average gain either to increase, or to decrease, and the dependence on L_c is not monotonic.

 

Fig. 2. Numerical results for the expectation value of the gain versus γP ₀ L_c , for γP ₀ L=4. The horizontal solid line refers to result (5) whereas the dotted line indicates result (10). The red coloring refers to Model I, whereas green refers to Model II. The error bars indicate the variance of the distribution. In (a) and (b) Δ/(γP ₀)=0.25; in (c) and (d) Δ/(γP ₀)=2.00; in (e) and (f) Δ/(γP ₀)=1.00. Further, in (a), (c) and (e) σ/(γP ₀)=0.50, while in (b), (d) and (f) σ/(γP ₀)=1.00.

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4. Discussion and conclusions

We have studied parametric amplification in the presence of dispersion fluctuations. To analyze the effect of fluctuations, we use two models, one piecewise constant and the other continuous. The numerical results show that the amplification does not depend on the details of the models. We also observe that fluctuations with short coherence length, γP ₀ L_c ≪1, do not affect the operation of the parametric amplifier, as can be understood using a simple scaling argument. This manifests itself in two ways. First, the gain in this limit approaches that in the absence of fluctuations. Secondly, in this limit the fluctuations in the gain rapidly vanish. Therefore, only dispersion fluctuations that vary on long length scales matter for the amplification, while the effect of short-length scale fluctuations decreases quickly with decreasing correlation length. As an example, consider a highly nonlinear fiber with γ=20 W^-1km^-1, L=400 m and P ₀=0.5 W, so that γP ₀ L=4, and the maximum gain is 28.7 dB, as in Section 3. Any fluctuations with σ≪γP ₀=2 km^-1 is likely to be negligible. For fluctuations for which this inequality is not satisfied, those with L_c ≈1m(so γP ₀ L_c =0.01) can likely still be ignored, while fluctuations with a correlation length of around 100 m (so γP ₀ L_c =1) need to be considered carefully.

The authors thank Dr. M. Karlsson for correspondence regarding his earlier work. This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program.