Parametric amplification in presence of dispersion fluctuations

Parametric amplification in presence of dispersionfluctuations

Mitra Farahmand1,2 and Martijn de Sterke1

1Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS) and School of Physics,University of Sydney 2006, Australia

2Optical Fibre Technology Centre, Australian Photonics CRC, University of Sydney 2006,Australia

[email protected]

Abstract: Parametric amplification in fibers with dispersion fluctuations isanalyzed. The fluctuations are modelled as a stochastic process, with theirsize at a given position modelled as a Gaussian, and the autocorrelationdecreasing exponentially. Two models are studied: in one the dispersion ispiecewise constant, while in the other it is continuous. We find that the am-plification does not depend on the models’ details and that only fluctuationswith long correlation lengths affect the amplification significantly.

© 2004 Optical Society of America

OCIS codes: (060.4370) Nonlinear optics, fibers; (190.4380) Nonlinear optics, four-wavemixing

References and links1. G. P.Agrawal, Nonlinear Fiber Optics (Academic Press, 1995)2. S. Kinoshita et al. Eds., Optical amplifiers and their application (Optical Society of America, Washington, DC

1999).3. M-C Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L.G. Kazovsky, “200-nm-bandwidth fiber optical amplifier

combining parametric an Raman gain,” J. Lightwave Technol. 19, 977-980 (2001).4. M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am.

B, 15, 2269-2275 (1998). It is noted that in the Appendix in this paper the matrices G and H are implicitly, andincorrectly, assumed to commute.

5. I. Brener, P.P. Mitra, D.D. Lee, and D.J. Thomson, “High-resolution zero-dispersion wavelength mapping insingle-mode fiber,” Opt. Lett. 23, 1520-1522 (1998).

6. J.S. Pereira et al, “Measurement of zero-dispersion wavelength using a novel method based on four-wavemixing,” in Optical Fiber Communications Conference, Vol. 2, 1998 OSA Technical Digest Series (Optical So-ciety of America, Washington, D.C., 1998), pp. 345-346.

7. M. Eiselt, R.M. Jopson, and R.H. Stolen, “Non-destructive position-resolved measurement of the zero-dispersionwavelength in an optical fiber,” J. Lightwave Technol. 15, 135-142 (1997).

8. N. Kuwaki and M. Ohashi, “Evolution of longitudinal chromatic dispersion,” J. Lightwave Technol. 8, 1476-1480(1990).

9. M. Gonzalez-Herraez, P. Corredera, M.L. Hernanz, and J.A. Mendez, “Retrieval of the zero-dispersion wave-length map of an optical fiber from measurement of its continuous wave four-wave mixing efficiency,” Opt. Lett.27, 1546-1548 (2002).

10. J.M. Chavez-Boggio, P. Dainese, and H.L. Fragnito, “Performance of a two-pump fiber optical parametric am-plifier in a 10 Gb/s×64 channel dense wavelength division multiplexing system,” Opt. Commun. 218, 303-310(2003).

11. J.M. Chavez Boggia, S. Tenenbaum, and H.L. Fragnito, “Amplification of broadband noise pumped by two lasersin optical fibers,” J. Opt. Soc. Am. B 18, 1428-1435 (2001).

12. F. Kh. Abdullaev, S.A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers withvariable dispersion,” Phys. Lett. A 220, 213-218 (1996).

13. A. Papoulis, Probability, random variables, and stochastic processes (McGraw-Hill, New York, 1965).14. P.K.A. Wai and C.R. Menyuk, “Anisotropic diffusion of the state of polarization in optical fibers with randomly

varying birefringence,” Opt. Lett. 24, 2493-2495 (1995).

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 136#3200 - $15.00 US Received 21 October 2003; revised 22 December 2003; accepted 23 December 2003

mailto:[email protected]

1. Introduction

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

�β = 2βp −βs −βi , (1)

where, βp,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 withposition [4, 5, 6, 7, 8, 9, 10]. This causes the propagation constants βp,s,i to vary with position,and hence also the phase matching parameter ∆β (1), and thus the gain. These variations in thefiber 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 beforeby Karlsson [4] and by Abdullaev et al. [12]. Abdullaev et al. restrict their work to fibers withperiodically varying quadratic dispersion, and treat this case perturbatively. Karlsson considersthe 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 wesolve the ensuing equations numerically.

2. Propagation in the presence of FWM

Here we briefly review light propagation in the presence of FWM. Following Agrawal andKarlsson [1, 4], we consider CW pump, signal and idler fields, all polarized in the same direc-tion. The wave equation can then be solved in a scalar form for the discrete frequencies ωp,s,i

separately, ignoring weaker fields at other frequencies. In the undepleted pump approximation,we then find for the pump amplitude Ap,

Ap =√

P0 exp(iγP0z) , (2)

with the exponential factor corresponding to self-phase modulation. Following Agrawal [1], thetotal pump power P0 = |Ap|2. The signal and idler amplitudes, As, Ai satisfy the set of lineardifferential equations [4]

dAdz

=(

i�k iγP0

−iγP0 −i�k

)A. (3)

where, �k = γP0 −�β/2, A is the column vector with elements As and A∗i , and γ =

ωn(2)/(cAeff) is the effective nonlinearity of the fiber. Here n(2) is the nonlinear refractive in-dex, Aeff the effective area of the modes, and ω the frequency of the light. In the treatmentbelow we take γ to be the same for the three frequencies ωp,s,i.

Though here we study the case in which �k varies randomly with position, let us first con-sider constant dispersion. We then obtain the propagation matrix M that describes how theamplitudes of the signal and idler evolve with position

A(z) = MA(0) ≡ coshα z+ i�k

α sinhα z iγP0α sinhα z

− iγP0α sinhα z coshα z− i�k

α sinhα z

A(0), (4)


0 5 10 15 20

4

2

−2

−4

0

(a)4

2

0

−2

−420151050z/Lc z/Lc

(b)

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

where α =√

(γP0)2 − (�k)2. Matrix M is written in terms of hyperbolic functions, which isappropriate 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 theinitial conditions Ai(0) = 0 and As(0) = 1. In the absence of dispersion fluctuations, and whenα 2 > 0, the signal gain for a system of length L is thus

G = cosh2 α L+(�k)2

α 2 sinh2 α L, (5)

which peaks when ∆k = 0, or ∆β = 2γP0.

3. Stochastic models for dispersion variations

In the presence of dispersion fluctuations, the phase mismatch in the fiber, ∆β , is taken to be astochastic variable and therefore so is ∆k. We write ∆k as ∆k = ∆+δk where ∆ is the averagevalue of ∆k, but δk varies randomly. Following Karlsson [4] we choose δk at a given positionto follow the Gaussian distribution

fG(δk) =1

σ√

2πexp

[−1

2

(δkσ2

)2]

, (6)

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

C(z) = σ2 exp(−|z|/Lc) , (7)

where Lc 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 thatare both consistent with (6) and (7), and use these to find the dependence of the results on thefeatures of the models, which are described in detail below. Briefly, in Model I the fluctuationsare piecewise constant, whereas in Model II, which is Gaussian to all orders, the fluctuationsvary 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, whilechanging discontinuously between them. The distribution (6) gives ∆k in each interval, while


the dispersion jumps follow a Poisson distribution with average distance Lc between events.Thus, the distribution of the segment lengths Ls follows the exponentially decreasing function

P(Ls) ∝ e−Ls/Lc . (8)

This is so since if on the interval between z0 and z0 +z1 the dispersion changes, the dispersion atthese positions is uncorrelated. In contrast if the dispersion does not change, the probability ofwhich is exp(−z1/Lc), then the dispersion at z0 and z0 + z1 is perfectly correlated. We generatethe values of δk and of the length segments numerically using the LAPACK Auxiliary RoutineVersion 2 Random Number Generator. Thus, δk(z) = σ × m, where m is the output of thenormal distribution [0,1] of the random number generator, and L = −Lc × lnm, where m nowis the output of the uniform distribution [0,1]. Once the dispersion is defined, the propagationmatrix M from Eq. (4) is used to calculate the fields in each segment. The total gain follows bymultiplying the propagation matrices of the individual segments.

In Model II, shown in Fig. 1(b), the dispersion varies continuously, which is likely to bemore realistic [5, 6, 7, 9, 10]. It is constructed as follows: the Gaussian distribution (6) is usedto give δk(z0) at the beginning of the fiber. Then δk(z1), at z1 > z0, is given by the conditionaldistribution

f (δk(z1)|δk(z0)) =1

σ√

2π(1− r2)e− 1

2σ2(1−r2)(δk(z1)−rδk(z0))2

, (9)

where r is the correlation coefficient between z = z0 and z = z1. When r = 0, distribution (9)reduces to (6), which is independent of δk(z0). When r = 1, then δk(z0) = δk(z1). For 0 < r <1, δk(z0) and δk(z1) are partly correlated. Now r(z) ≡ C(z)/σ2, and so r(z) = exp(−|z|/Lc)from (7). Thus δk(z) is generated by taking small steps ∆z and applying (9). Since our r(z) hasthe property r(z1)r(z2) = r(z1 + z2), 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 δk(z1) = r×δk(z0)+σ√

(1− r2)×m, where m is the output of the normal distribution [0,1] of the same random number generatorused for Model I and where r = exp(−|z1 − z0|/Lc). In our implementation, we sample ∆kalong the length with the step size of 0.004Lc or 0.001L, whichever is smaller. Then we use thefourth order Runge Kutta integration method to integrate the coupled equations (3). Note that inModel I, δk is constant in each segment of average length Lc so that (4) can be used, whereasin Model II the calculation is done for at least 250 steps per correlation length. Therefore, thespeed 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 = 900,000realizations for Model I, indicated in red, and N = 5,000 realizations for Model II, indicatedin green. The ensemble is smaller for Model II than for Model I since, as discussed in Sec-tion 3, the speed of calculation is lower in this case. Since the gain is typically measured indecibells, the average gain can be computed in two ways. We can either calculate the gain indecibells for each realization and then average these, or first average over the realizations andthen express this average in decibells. The first of these corresponds to the average that wouldbe measured if the ensemble of fibers were connected in series; the second would be obtainedif the ensemble were connected in parallel. Here we choose the former since it corresponds tosituation we wish to describe. We also calculate the standard deviation σg of the gain distribu-tion, 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 σa = σg/

√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 γP0L = 4, corresponding by Eq. (4) to a maximum gain ofcosh2 γP0L = 28.7 dB when ∆ = 0. However, other values of γP0L lead to the same conclu-sions. In Figs. 2 we show the average gain versus the dimensionless quantity γP0Lc, for fixedvalues of ∆/γP0 and σ/γP0. It is noted that from the governing Eqs. (3), the results only dependon these ratios. Therefore we immediately see that the effect of the fluctuations decreases whenγP0 increases, i.e., for increasing pump power or fiber nonlinearity. We also see that the effectof the correlation length of a fiber depends on how it compares with the gain length 1/(γP0).We also show the amplification results for two extreme cases that are calculated easily: (1)σ = 0, i.e., no fluctuations, and (2) Lc � L. When σ = 0 then the “ideal gain” is simply givenby Eq. (5). In Figs. 2 we show this result by a solid line. When Lc � L, then ∆k does not varyover the length of fiber and so the average gain is given by the expectation value of (5) subjectto distribution (6)

G(Lc � L) =1√

2πσ

∫ +∞

−∞d(∆k)e−(∆k−∆)2/2σ2

log[cosh2 α L+(�k)2

α 2 sinh2 α L]. (10)

As discussed when α 2 < 0 , the hyperbolic functions need to be replaced by sinusoidal func-tions. 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 IIare very similar. We have found this to be true for all situations we have considered. Since ourtwo quite different models evidently lead to the same results, the details of the models do notmatter, and the results can thus be considered quite general. This conclusion is similar to thatof Wai and Menyuk, who studied fibers with randomly varying birefringence [14].

Before considering the numerical results in detail we establish one more result. When Lc → 0,so that the dispersion varies rapidly with position, we can use an argument that applies to bothModels, but most directly to Model I. It is straightforward to see from Eqs. (3) that in the pres-ence of dispersion fluctuations, the powers |Ap,i|2 and its first derivative are both continuous,while the second derivative is discontinuous. Deviations between different realizations within asegment thus increase with L2

c , since Lc is the average length of the segments. The average num-ber of segments is L/Lc, and the total deviation in the gain between different realizations thusscales as L2

c ×L/Lc ∝ Lc. Hence, when Lc → 0, the effect of the fluctuations disappears. Thusin this limit result (5) applies (solid lines in Figs. 2) and the variations in the gain disappear,whereas when Lc → ∞ we have result (10) (dotted line in Figs. 2).

We now discuss Figs. 2 in detail, ignoring the distinction between the models. We have iden-tified three regimes: |∆| � γP0 (Figs. 2(a) and (b)), |∆| � γP0 (Figs. 2(c) and (d)), and |∆| � γP0

(Figs. 2(e) and (f)). These correspond to frequencies that are tuned close to the gain peak, farfrom the gain peak, and tuned close to the edges of the region with exponential gain, respec-tively. In all three regimes the gain behaves as required when Lc → 0 and Lc → ∞. Figures 2(a)and (b) show that when |∆|� γP0, the gain decreases monotonically with Lc. We can understandthis by considering the extreme case where ∆ → 0. Then the gain is maximum and fluctuationscan only cause the gain to decrease. By increasing σ the gain decreases more strongly. In con-trast, Figs. 2(c) and (d) show that when |∆| � γP0, the gain increases monotonically with Lc.This can be understood in a similar way as before: when σ = 0, α 2 < 0, and no exponentialsignificant gain arises. Fluctuations may cause some exponential gain to occur, thus increasingthe average gain. Finally, Figs. 2(e) and (f) show results for |∆| = γP0. Now the fluctuationmay cause the average gain either to increase, or to decrease, and the dependence on Lc is not


Fig. 2. Numerical results for the expectation value of the gain versus γP0Lc, for γP0L = 4.The horizontal solid line refers to result (5) whereas the dotted line indicates result (10). Thered coloring refers to Model I, whereas green refers to Model II. The error bars indicatethe variance of the distribution. In (a) and (b) ∆/(γP0) = 0.25; in (c) and (d) ∆/(γP0) =2.00; in (e) and (f) ∆/(γP0) = 1.00. Further, in (a), (c) and (e) σ/(γP0) = 0.50, while in(b), (d) and (f) σ/(γP0) = 1.00.

monotonic.

4. Discussion and conclusions

We have studied parametric amplification in the presence of dispersion fluctuations. To analyzethe 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, γP0Lc � 1, do not affect theoperation 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 offluctuations. Secondly, in this limit the fluctuations in the gain rapidly vanish. Therefore, onlydispersion fluctuations that vary on long length scales matter for the amplification, while theeffect 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 P0 =0.5 W, so that γP0L = 4, and the maximum gain is 28.7 dB, as in Section 3. Any fluctuationswith σ � γP0 = 2 km−1 is likely to be negligible. For fluctuations for which this inequality isnot satisfied, those with Lc ≈ 1 m (so γP0Lc = 0.01) can likely still be ignored, while fluctuationswith a correlation length of around 100 m (so γP0Lc = 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 Centresof Excellence program.


Parametric amplification in presence of dispersion fluctuations

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