Four-wave mixing in fibers with random birefringence

Four-wave mixing in fibers with randombirefringence

C. J. McKinstrie, 1 H. Kogelnik,1 R. M. Jopson,1

S. Radic2 and A. V. Kanaev3

1Bell Laboratories, Lucent Technologies, Holmdel, NJ 077332Department of Electrical and Computer Engineering, University of California at San Diego,

La Jolla, CA 920933Department of Physics and Optical Science, University of North Carolina,

Charlotte, NC 28223

[email protected]

Abstract: Parametric amplification is made possible by four-wave mixing.In low-birefringence fibers the birefringence axes and strength vary ran-domly with distance. Light-wave propagation in such fibers is governed bythe Manakov equation. In this paper the Manakov equation is used to studydegenerate and nondegenerate four-wave mixing. The effects of linear andnonlinear wavenumber mismatches, and nonlinear polarization rotation,are included in the analysis. Formulas are derived for the initial quadraticgrowth of the idler power, and the subsequent exponential growth of thesignal and idler powers (which continues until pump depletion occurs).These formulas are valid for arbitrary pump and signal polarizations.

© 2004 Optical Society of America

OCIS codes: (060.2320) Fiber Amplifiers and Oscillators; (060.4370) Nonlinear Optics inFibers; (190.2620) Frequency Conversion; (190.5040) Phase Conjugation.

References and links1. K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, “CW three-wave mixing in single-mode optical

fibers,” J. Appl. Phys.49, 5098–5106 (1978).2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE

J. Quantum Electron.18, 1062–1072 (1982).3. J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with

49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett.13, 194–196 (2001).4. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar and C. Headley,

“Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett.14, 1406–1408 (2002).

5. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, “Fiber-based optical parametric amplifiersand their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002) and references therein.

6. C. J. McKinstrie, S. Radic and A. R. Chraplyvy, “Parametricamplifiers driven by two pump waves,” IEEE J. Sel.Top. Quantum Electron.8, 538–547 and 956 (2002) and references therein.

7. C. J. McKinstrie, S. Radic and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringentfibers,” Opt. Express.11, 2619–2633 (2003) and references therein.

8. C. J. McKinstrie, S. Radic and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringentfibers,” J. Opt. Soc. Am. B20, 1437–1446 (2003).

9. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992).

10. R. M. Jopson and R. E. Tench, “Polarisation-independentphase conjugation of lightwave signals,” Electron. Lett.29, 2216–2217 (1993).

11. K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonalpump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2033#3822 - $15.00 US Received 13 February 2004; revised 28 April 2004; accepted 28 April 2004

mailto:[email protected]

12. K. K. Y. Wong, M. E. Marhic, K. Uesaka and L. G. Kazovsky, “Polarization-independent two-pump fiber opticalparametric amplifier,” IEEE Photon. Technol. Lett.14, 911–913 (2002).

13. S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin and G. P. Agrawal, “Record performance of a parametricamplifier constructed with highly-nonlinear fiber,” Electron. Lett.39, 838–839 (2003).

14. T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiberoptical parametric amplification without idler spectral broadening,” IEEE Photon. Technol. Lett.15, 1573–1575(2003).

15. S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar and C. Headley, “Polarization-dependent paramet-ric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta,Georgia, 23–28 March 2003, paper ThK3.

16. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys.JETP38, 248–253 (1974).

17. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron.23, 174–176 (1987).

18. P. K. A. Wai, C. R. Menyuk and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt.Lett. 16, 1231–1233 (1991).

19. S. G. Evanglides, L. F. Mollenauer, J. P. Gordon and N. S. Bergano, “Polarization multiplexing with solitons,”J. Lightwave Technol.,10, 28–35 (1992).

20. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers withrandomly varying birefringence,” J. Lightwave Technol.14, 148–157 (1996).

21. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,”J. Opt. Soc. Am. B13, 2006–2011 (1996).

22. H. Kogelnik, R. M. Jopson and L. E. Nelson, “Polarization-mode dispersion,” inOptical Fiber Telecommunica-tions IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego, 2002), pp. 725–861.

23. M. Karlsson, J. Brentel and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installedfibers,” J. Lightwave Technol.18, 941–951 (2000).

24. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat.Acad. Sci.97, 4541–4550 (2000).

25. L. F. Mollenauer, J. P. Gordon and F. Heismann, “Polarization scattering by soliton–soliton collisions,” Opt. Lett.20, 2060–2062 (1995).

26. D. Wang and C. R. Menyuk, “Reduced model of the evolution ofthe polarization states in wavelength-division-multiplexed channels,” Opt. Lett.23, 1677–1679 (1998).

27. R. W. Boyd,Nonlinear Optics (Academic, San Diego, 1992), Sections 1.4, 2.3 and 6.5.28. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,”

Proc. IRE44, 904–913 (1956).29. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE

45, 1012–1013 (1957).30. M. Yu, C. J. McKinstrie and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion

regime,” Phys. Rev. E48, 2178–2186 (1993).31. C. J. McKinstrie, X. D. Cao and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B10,

1856–1869 (1993) and references therein.32. A. Carena, V. Curri, R. Gaudino, P. Poggiolini and S. Benedetto, “On the joint effects of fiber parametric gain

and birefringence and their influence on ASE noise,” J. Lightwave Technol.16, 1149–1157 (1998).33. Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and

wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003,paper TuP3.

34. K. Inoue, “Tunable and selective wavelength conversionusing fiber four-wave mixing with two pump lights,”IEEE Photon. Technol. Lett.6, 1451–1453 (1994).

35. T. Tanemura, C. S. Goh, K. Kikuchi and S. Y. Set, “Widely tunable wavelength conversion using nondegeneratefiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communica-tions, Rimini, Italy, 21–25 September 2003, paper We3.7.3.

36. G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt.Soc. Am. B9, 1047–1061 (1992).

1. Introduction

In recent years there has been a resurgence of interest in theparametric amplification (PA)of optical signals [1, 2]. Because of recent improvements inhighly-nonlinear fibers, it is nowa routine matter to produce four-wave-mixing (FWM) gains higher than 40 dB [3, 4] overbandwidths broader than 20 nm [4]. Such performance makes possible wavelength conversion


and impairment reduction by phase conjugation in wavelength-division-multiplexed (WDM)communication systems. PA driven by one pump wave (degenerate FWM) was reviewed byHansryd [5] and PA driven by two pump waves (nondegenerate FWM) was reviewed by McK-instrie [6, 7]. In this paper the latter process is studied indetail. (For completeness, the formerprocess is discussed briefly.)

PA is driven most strongly when the pumps have parallel polarization vectors. However, thesignal gain depends sensitively on the input signal polarization: It is maximal when the signal ispolarized parallel to the pumps and is minimal when the signal is polarized perpendicular to thepumps [1, 2]. Because transmission fibers are not polarization maintaining, practical amplifiersmust operate on signals with arbitrary polarizations.

PA can also be driven by pump waves with perpendicular polarization vectors. In (idealized)isotropic fibers, the signal gain associated with perpendicular pumps does not depend on the sig-nal polarization. However, real fibers are birefringent. Inbirefringent polarization-maintainingfibers (which we refer to as fibers with constant birefringence) the signal gain does depend onthe signal polarization [8]. Inoue [9] analyzed PA in birefringent non-polarization-maintainingfibers (which we refer to as fibers with random birefringence). In such fibers the random re-orientation of the birefringence axes washes out whatever signal-polarization dependence isassociated with birefringence: If the input polarizationsof the pumps are perpendicular, thesignal gain does not depend on the input signal polarization. (It was for such fibers that theuse of perpendicular pumps was first proposed.) Inoue’s predictions were verified by exper-iments with long (20 Km) dispersion-shifted fibers [10, 11].Recent experiments were madewith highly-nonlinear fibers [12, 13, 14].

The question of what model to use for highly-nonlinear fibersremains open. Some short (0.1–0.3 Km) sections of highly-nonlinear fiber behave like polarization-maintaining fibers and pro-duce polarization-dependent gain [15]. However, high-gain experiments are made with longerfibers (1–3 Km). Such fibers are long enough to change the wave polarizations, but are notlong enough to randomize them completely. It is reasonable to assume that the properties ofPA in Km-long highly-nonlinear fibers are intermediate between the properties predicted bythe constant-birefringence model [8] and the random-birefringence model [9]. In this paper theconsequences of the latter model are studied in detail. Inoue’s analysis of PA is simplified bythe use of the Manakov equation [16], and is extended by the inclusion of the effects of linearand nonlinear wavenumber mismatches, and nonlinear polarization rotation. Not only are for-mulas derived for the initial quadratic growth of the idler power, but formulas are also derivedfor the subsequent exponential growth of the signal and idler powers (which continues untilpump depletion and nonlinear detuning occur). These formulas are valid for arbitrary pumppolarizations.

This paper is organized as follows: In Section 2 the coherently-coupled nonlinearSchrodinger (NS) equations that govern light-wave propagationin fibers (one equation foreach polarization component of the wave) are stated [17]. These equations are the basis forderivations of the incoherently-coupled NS equations thatgovern light propagation in fiberswith constant (high) birefringence [17] and random birefringence [18, 19, 20, 21]. The latterequations are referred to, collectively, as the Manakov equation [16]. In Section 3 the Manakovequation is used to study the nonlinear polarization rotation of waves with incommensuratefrequencies (which do not exchange energy). In subsequent sections the Manakov equation isused to study the interaction of waves with commensurate frequencies (which exchange energyand experience polarization rotation simultaneously). Section 4 pertains to degenerate FWM,whereas Section 5 pertains to nondegenerate FWM. Finally, inSection 6 the main results ofthis paper are summarized.


2. Governing equations

Let Ex andEy denote the electric-field components of a light wave, measured relative to thebirefringence axes of a polarization-maintaining fiber. Itis convenient to measure (angular)frequencies relative to a reference frequencyω0 and wavenumbers relative to the associatedreference wavenumberk0 = [βx(ω0)+ βy(ω0)]/2, where eachβ (ω0) is a natural wavenumber(dispersion function), and to write the field components as

Ex(t,z) = Ax(t,z)exp[i(k0z−ω0t)]+ c.c., (1)

Ey(t,z) = Ay(t,z)exp[i(k0z−ω0t)]+ c.c. (2)

The evolution of the wave amplitudes (polarization components)Ax andAy is governed by thecoherently-coupled NS equations

−i∂zAx = βx(i∂t)Ax + γ(|Ax|2 +2|Ay|

2/3)Ax + γA2yA∗

x/3, (3)

−i∂zAy = βy(i∂t)Ay + γ(2|Ax|2/3+ |Ay|

2)Ay + γA2xA∗

y/3, (4)

whereβx andβy are modified dispersion functions andγ is the nonlinearity coefficient [17].In the frequency domain eachβ (ω) = ∑∞

n=0β (n)(ω0)ωn/n! − k0 is a Taylor expansion of adispersion function about the reference frequency. In the time domain the frequency differenceω is replaced by the time derivativei∂t .

According to Eqs. (3) and (4), the wave amplitudes are subject to (dispersive) wavenumbershifts of opposite sign. One can make the associated phase shifts explicit by definingAx =Bx exp(iδk0z) and Ay = By exp(−iδk0z), whereδk0 = [βx(ω0)− βy(ω0)]/2. By substitutingthese definitions in Eqs. (3) and (4), one finds that the coherent-coupling terms are multipliedby the phase factors exp(∓4iδk0z). Typical fibers have differential indices of refractionδn inthe range 10−7–10−5 [22]. The wavenumber shiftδk0 = ω0δn/c and the beat length 2π/δk0 =λ0/δn. For typical fibers the beat lengths are in the range 0.15–15.0 m. These beat lengths areall much shorter than the parametric gain length, which is typically of order 100 m. Thus, inthe context of PA, the coherent coupling terms oscillate rapidly and can be neglected: PA in apolarization-maintaining fiber with constant (high) birefringence (δn ≫ 10−7) is governed bythe incoherently-coupled NS equations

−i∂zBx = βx(i∂t)Bx + γ(|Bx|2 +2|By|

2/3)Bx, (5)

−i∂zBy = βy(i∂t)By + γ(2|Bx|2/3+ |By|

2)By, (6)

where eachβ (ω) = ∑∞n=1β (n)(ω0)ωn/n! represents the higher-order (convection and disper-

sion) terms in the Taylor expansion of a dispersion function[17]. One often makes the simpli-

fying assumption thatβ (n)x (ω0) = β (n)

y (ω0) for n ≥ 2. In a fiber with constant birefringence,the polarization components convect at different speeds, but experience similar dispersion. Thecross-phase modulation (CPM) coefficient differs from the self-phase modulation (SPM) coef-ficient by a factor of 2/3 and the coherent-coupling terms areabsent. The consequences of theconstant-birefringence model were studied in [8].

Fibers with low birefringence (δn ∼ 10−7) have cross-sections that are almost circular. Be-cause it is difficult to manufacture such cross-sections reproducibly, the orientation of the bire-fringence axes and the value of the differential index of refraction vary randomly with distance.One models such fibers as sequences of concatenated sections. Equations (3) and (4) apply toeach section individually. Between sections one changes the orientation of the birefringenceaxes and the value of the birefringence parameterδk0. In a frame rotating with the birefrin-gence axes of the sections, the first change causes the Stokesvector of a (virtual) reference


wave (with frequencyω0) to rotate about the 3-axis (in Stokes space), whereas the secondchange causes it to rotate about the 1-axis. The combined effect of many such rotations movesthe Stokes vector randomly over the entire Poincare sphere and causes the associated Jonesvector to change randomly. By averaging Eqs. (3) and (4) overthe Poincare sphere, one findsthat PA in a non-polarization-maintaining fiber with variable (low) birefringence is governedby the incoherently-coupled NS equations

−i∂zAξ = β (i∂t)Aξ +(8γ/9)(|Aξ |2 + |Aη |

2)Aξ , (7)

−i∂zAη = β (i∂t)Aη +(8γ/9)(|Aξ |2 + |Aη |

2)Aη , (8)

whereβ (ω) = ∑∞n=2β (n)(ω0)ωn/n! represents the higher-order (dispersion) terms in the Taylor

expansion of the (common) dispersion function [18, 19, 20, 21]. The subscriptsξ andη denotepolarization components measured relative to basis vectors that track the polarization of thereference wave (one basis vector remains parallel to the Jones vector of the reference wave andthe other remains perpendicular to it.) In a fiber with randombirefringence both polarizationcomponents convect at the same speed and experience the samedispersion. [There are no con-vection terms in Eqs. (7) and (8) becauset represents the retarded time.] The SPM and CPMcoefficients are the same and the coherent-coupling terms are absent.

In the linear regime, the rotating-frame Stokes vector of a monochromatic wave remains con-stant while the laboratory-frame Stokes vector rotates rapidly and randomly about the Poincaresphere. According to Eqs. (7) and (8), every frequency component of a multichromatic wavebehaves in the same way. This behavior allows the fiber to distinguish the input polarizations ofthe pumps and signal. In reality, polarization-mode dispersion (PMD) causes the relative orien-tation of the Stokes vectors of waves with different frequencies to change [22]. Let~a1 and~a2

denote the Stokes vectors of waves with frequenciesω1 andω2, respectively, and let〈〉 denotesan ensemble average. Then the polarization correlation

〈~a1 ·~a2〉 = 〈~a1 ·~a2〉in exp(−δω2〈δτ2〉/3), (9)

whereδω is the frequency differenceω2−ω1 and〈δτ2〉 is the mean-square differential groupdelay of the fiber section [23]. For typical parameters, the polarization-decorrelation lengthpredicted by Eq. (9) is of order 1 Km. This prediction is consistent with the results of recentexperiments with a 1 Km-long fiber [13], which exhibited a strong dependence on the inputpolarizations: Equations (7) and (8) constitute a reasonable model for these and similar exper-iments. However, for experiments with very-long fibers or pumps with very-large frequencydifferences, a different model is required.

By introducing the Jones vector|A〉 = [Aξ ,Aη ]T , one can rewrite Eqs. (7) and (8) in thecompact form

−i∂z|A〉 = β (i∂t)|A〉+ γ〈A|A〉|A〉, (10)

whereγ = 8γ/9 and〈 | 〉 denotes an inner (dot) product. Henceforth, for simplicityof notation,the overbar onγ will be omitted and the factor of 8/9 will be implied. Equation (10) is theManakov equation [16].

The Jones-vector and Stokes-vector (Poincare-sphere) formalisms, and many mathematicalidentities that relate them, were described in detail by Gordon and Kogelnik [24]. We will usethese notations and results throughout this paper, withoutfurther comment.

3. Waves with incommensurate frequencies

Consider the interaction of two waves with different frequencies (ω1 andω2). By substitutingthe ansatz

|A〉 = exp(−iω1t)|A1〉+exp(−iω2t)|A2〉 (11)


in Eq. (10) and collecting terms of like frequency, one finds that

D|A1〉 = iβ1|A1〉+ iγ[(P1 +P2)|A1〉+ 〈A2|A1〉|A2〉], (12)

D|A2〉 = iβ2|A2〉+ iγ[〈A1|A2〉|A1〉+(P1 +P2)|A2〉], (13)

whereD = d/dz, j = 1 or 2, the wavenumberβ j = β (ω j) and the powerPj = 〈A j|A j〉. By usingthe identity|A〉〈A| = (PI +~a ·~σ)/2, where~σ is the Pauli spin operator and~a = 〈A|~σ |A〉 is theStokes (polarization) vector, one finds that

D|A j〉 = iH j|A j〉, (14)

where the (hermitian) operator

H j = [β j + γ(Pj +3Pk/2)]I + γ~ak ·~σ/2 (15)

andk 6= j. In Eq. (15) the termγPj represents SPM, whereas the termγPk represents CPM. Byusing the identity~a ·~σ |A〉 = P|A〉, one can rewrite operator (15) in the equivalent form

H j = [β j + γ(Pj +3Pk)/2]I + γ~at ·~σ/2, (16)

where the total Stokes vector~at =~a1 +~a2.It follows from Eqs. (14) and (16) that

D〈A j|Ak〉 = i〈A j|(Hk −H j)|A j〉 (17)

= i[βk −β j + γ(Pj −Pk)]〈A j|Ak〉. (18)

Although the vectors|A1〉 and |A2〉 evolve (in manners to be determined), the magnitude oftheir inner product〈A1|A2〉 is constant, as are their powersP1 andP2.

It also follows from Eqs. (14) and (16) that

D〈A j|~σ |A j〉 = i〈A j|[~σ ,H j]|A j〉, (19)

where[~σ ,H j] denotes the commutator~σH j −H j~σ . By using the identities[~σ , I] = 0 and[~σ ,~a ·~σ ] = 2i~a×~σ , one finds that[~σ ,H j] = −iγ~σ ×~at and, hence, that

D~a j =~a j ×~at . (20)

Had we used operator (15) rather than operator (16), we wouldhave obtained the standardequationD~a j =~a j ×~ak [25, 26], which manifests the fact that (nonlinear) polarization rotationis produced by CPM, but not by SPM. It follows from Eq. (20) that

D~at = 0, (21)

D(~a j ·~ak) = 0. (22)

In Stokes space each individual Stokes vector rotates aboutthe total Stokes vector, which isconstant. As in other rigid vector rotations, the lengths and relative orientations of the vectorsare constant. Equation (22) is consistent with Eq. (18) and the identity|〈A j|Ak〉|

2 = (PjPk +~a j ·~ak)/2.

It is instructive to consider the phase and polarization shifts separately. Let|A j〉 =exp(iθ j)|B j〉, where the phase shift

θ j(z) = β jz+ γ∫ z

0[Pj(z

′)+3Pk(z′)]dz′/2. (23)


(For waves with incommensurate frequencies, the powers areconstants. However the transfor-mation is also valid for variable powers.) Then

D|B j〉 = iHt |B j〉, (24)

where the (common) operatorHt = γ~at ·~σ/2. SinceHt is hermitian, there exists a (common)unitary operatorUt such that

|B j(z)〉 = Ut(z)|B j(0)〉. (25)

Furthermore, sinceHt is a constant (matrix) operator with zero trace,Ut = exp(iHtz) is uni-modular (has unit determinant). One can make the transformations associated with Eqs. (24)and (25) in either order. It follows from the preceding observations that

|A j(z)〉 = exp(iHtz+ iθ j)|A j(0)〉. (26)

Equation (26) has the canonical form of a unitary transformation (a unimodular transformationcombined with a phase shift). As predicted by Eq. (20), the two waves are subject to the samepolarization rotation (which depends on the powers and polarizations). They are also subject todifferent phase shifts (which depend on the powers, but not the polarizations).

It is easy to extend the results of this section to a collection of n waves with incommensuratefrequencies (no linear combination of frequencies, with integer coefficients, equals zero): Theone indexk is replaced by a sum over then−1 indicesk 6= j and the two-vector sum in~at isreplaced by ann-vector sum.

4. Degenerate four-wave mixing

Now consider waves with commensurate frequencies (some linear combinations of frequencies,with integer coefficients, equal zero). Not only does nonlinearity allow such waves to modifytheir phases and polarizations, it also allows them to exchange power.

Degenerate FWM involves three waves with frequencies that satisfy the matching condition2ω2 = ω3 +ω1. By substituting the ansatz

|A〉 =3

∑j=1

exp(−iω jt)|A j〉 (27)

in the Manakov equation (10) and collecting terms of like frequency, one finds that

D|A1〉 = iH1|A1〉+ iγ〈A3|A2〉|A2〉, (28)

D|A2〉 = iH2|A2〉+ iγ(〈A2|A3〉|A1〉+ 〈A2|A1〉|A3〉). (29)

Consistent with Eq. (16) and the discussion at the end of Section 3,

H j = [β j + γ(Pj +3∑Pk)/2]I + γ~at ·~σ/2, (30)

where the summation involves the two indicesk 6= j and~at = ~a1 +~a2 +~a3. One can deducethe equation for wave 3 from the equation for wave 1 by interchanging the subscripts 1 and 3.Henceforth, when equations for waves 1 and 2 are cited, the associated equation for wave 3 iscited implicitly.

It follows from Eqs. (28) and (29) that

D〈A1|A1〉 = iγ(〈A3|A2〉〈A1|A2〉−〈A2|A3〉〈A2|A1〉), (31)

D〈A2|A2〉 = 2iγ(〈A2|A3〉〈A2|A1〉−〈A3|A2〉〈A1|A2〉). (32)


By combining Eqs. (31) and (32), one finds that

D(P1−P3) = 0, (33)

D(P1 +P2 +P3) = 0. (34)

Equations (33) and (34) are exact consequences of Eqs. (28) and (29), which, in turn, are exactconsequences of the Manakov equation. They imply that the total power is constant, and thedifference between the signal and idler powers is constant.The first implication is true, but thesecond must be false because it contradicts the Manley–Rowe–Weiss (MRW) equations [28,29], upon which the photon interpretation of FWM is based: Forevery pair of pump photonsthat is destroyed, one signal photon and one idler photon arecreated. Since the signal andidler frequencies are different, the photon energieshω j are also different, so the signal andidler powers must increase at different rates. The Manakov equation is based on the envelopeapproximation that the relative frequenciesω j are much smaller than the reference (carrier)frequencyω0 and the concomitant approximation that the nonlinearity coefficient γ does notdepend onω j. Had we used the Maxwell equations to derive Eqs. (28) and (29) directly, thenonlinear terms would have included factors of 1+ ω j/ω0 [27]. We would have eliminatedthese factors from Eqs. (31) and (32) by rewriting them in terms of the photon fluxesFj =Pj/h(ω0 + ω j), and would have obtained the MRW equations [(33) and (34), with Pj replacedby Fj]. By combining the MRW equations with the frequency-matching condition, we wouldhave confirmed that the total power is conserved exactly. In typical experiments|ω j/ω0| ∼10−2. For such frequencies the envelope approximation is accurate and the distinction betweenpower and photon flux is (quantitatively) unimportant.


D〈A1|~σ |A1〉− i〈A1|[~σ ,Ht ]|A1〉 = iγ(〈A3|A2〉〈A1|~σ |A2〉−〈A2|A3〉〈A2|~σ |A1〉), (35)

D〈A2|~σ |A2〉− i〈A2|[~σ ,Ht ]|A2〉 = iγ(〈A2|A3〉〈A2|~σ |A1〉−〈A3|A2〉〈A1|~σ |A2〉

+ 〈A2|A1〉〈A2|~σ |A3〉−〈A1|A2〉〈A3|~σ |A2〉). (36)

By combining Eqs. (35) and (36) one finds that

D~at = 0. (37)

Even in the presence of FWM the total polarization vector is conserved. Equation (37) is anexact consequence of the Manakov equation. However, as discussed after Eq. (34), the Manakovequation does not account for the dependence ofγ on ω j. Had we retained this frequencydependence, we would have found that the power-weighted polarization vector is not conserved,but the photon-flux-weighted polarization vector is conserved exactly. One can derive equationsthat are similar to Eqs. (35) and (36) for any hermitian operator G. (The FWM terms alwayssum to zero.)

Now consider the initial evolution of degenerate FWM. In the first case two powerful waves(1 and 2) are launched into the fiber and a weak idler wave (3) isgenerated in the fiber byFWM. The linearized analysis of this process is based on the approximation that the idler doesnot affect the input waves, in which case

D|Al〉 = iHl |Al〉, (38)

D|A3〉 = iH3|A3〉+ iγ〈A1|A2〉|A2〉, (39)

wherel = 1 or 2. The inputs are subject to different phase shifts (which depend onP1 andP2)and the same polarization rotation (which depends on~at =~a1 +~a2), as discussed in Section 3.


The idler is subject to a phase shift (which depends onP1 andP2) and the same polarizationrotation as the inputs. In the notation of Section 3, let|A j〉 = exp(iHtz+ iθ j)|B j〉. Then

D|Bl〉 = 0, (40)

D|B3〉 = iγ〈B1|B2〉|B2〉exp[i(2θ2−θ3−θ1)]. (41)

It follows from Eq. (40) that the (transformed) input vectors are constant, as are the inputpowers (upon which the phase shifts depend), and it follows from Eq. (41) that the (originalor transformed) idler vector is parallel to the (original ortransformed) vector of input 2: Theinitial evolution of degenerate FWM does not depend on the (rigid) polarization rotation.

By integrating Eq. (41) one finds that

P3(z) = Γ(z)P2|〈A1|A2〉|2, (42)

where the spatial growth factorΓ(z) = γ2sin2(kz)/k2 (43)

and the FWM wavenumber (mismatch)

k = [β1−2β2 +β3 + γ(2P2−P1)]/2. (44)

For reference, Eq. (42) can be rewritten in the Stokes-vector form

P3(z) = Γ(z)P1P22 (1+~e1 ·~e2)/2, (45)

where~e1 and~e2 are unit vectors parallel to~a1 and~a2, respectively.Whenk = 0 the idler power increases quadratically with distance. When k 6= 0 the idler power

is a periodic function of distance, with period and maxima that are inversely proportional tok.The nonlinear contribution tok depends on the input powers, but not the input polarizations:Formula (44) for vector FWM is the same as the formula for scalar FWM [30, 31]. However,the idler power produced by parallel inputs in a fiber with random birefringence is lower thanthat produced in a polarization-maintaining fiber by a factor of (9/8)2 ≈ 1.3 (1.0 dB). As thealignment between the input polarizations decreases, so also does the output idler power. Inparticular, if the input polarizations are perpendicular,no idler is produced.

Table 1. Properties of degenerate FWM driven by two input waves

|E3〉 P3

|E1〉 ‖ |E2〉 |E2〉 1|E1〉 ⊥ |E2〉 — 0

random random 1/2

The symbols‖ and⊥ should be interpreted in the sense of Jones vectors: The rows of thetable apply to parallel and perpendicular linearly-polarized waves, co- and counter-rotatingcircularly polarized waves or any other pair of aligned and orthogonal waves.

The properties of degenerate FWM are summarized in Table 1. Each row pertains to a par-ticular combination of input polarizations, which is illustrated in Figure 1. The first columndescribes the output idler polarization and the second column describes the FWM efficiency(normalized to a maximal efficiency of 1). The random-polarization results were obtained byaveraging the right side of Eq. (45) over the Poincare sphere (〈~e1 ·~e2〉 = 0).


ω||

⊥

12

3

ω||

⊥

1 2

3

Fig. 1. Polarization diagrams for degenerate FWM driven by two input waves. The symbols‖ and⊥ should be interpreted in the sense of Jones vectors, and the symbol◦ signifies thatno idler is produced. Figures 1a and 1b correspond to rows 1 and 2 of Table 1, respectively.

In the second case a weak signal wave (1) and a powerful pump wave (2) are launched intothe fiber and a weak idler wave (3) is generated. This process is often called modulationalinstability (MI). Since the idler power quickly becomes comparable to the signal power, theeffects of the idler on the signal cannot be neglected, in which case

D|A1〉 = iH1|A1〉+ iγ〈A3|A2〉|A2〉, (46)

D|A2〉 = iH2|A2〉, (47)

D|A3〉 = iH3|A3〉+ iγ〈A1|A2〉|A2〉. (48)

The pump is subject to a phase shift (which depends onP2), but no polarization rotation (be-cause~at = ~a2 and |A2〉 is an eigenvector of~a2 ·~σ ). The signal and idler are subject to thesame phase shift (which depends onP2) and polarization rotation (which depends on~a2). Let|A j〉 = exp(iH2z)|B j〉. Then

D|B1〉 = i(β1−β2 + γP2)|B1〉+ iγ〈B3|B2〉|B2〉, (49)

D|B2〉 = 0, (50)

D|B3〉 = i(β3−β2 + γP2)|B3〉+ iγ〈B1|B2〉|B2〉. (51)

It follows from Eqs. (49)–(51) that the (transformed) pump vector is constant, and the initialevolution of MI does not depend on the polarization rotation.

By defining the wavenumber shiftδk1 = β1−β2 + γP2 and the operatorD1 = D− iδk1, andmaking similar definitions for wave 3, one can rewrite Eqs. (49) and (51) in the compact form

D1|B1〉 = iγ〈B3|B2〉|B2〉, (52)

D∗3〈B3| = −iγ〈B2|B1〉〈B2|. (53)


(D∗3D1I − γ2P2|B2〉〈B2|)|B1〉 = 0. (54)

Let |E‖〉 and|E⊥〉 be unit vectors parallel and perpendicular to|B2〉, and (since wave 1 is the

signal) let|B1〉/P1/21 = S‖|E‖〉+ S⊥|E⊥〉. Then, by using this decomposition, one can rewrite

Eq. (54) in the matrix form[

D∗3D1− γ2P2

2 00 D∗

3D1

][

S‖S⊥

]

= 0. (55)


The parallel part of Eq. (55) represents coupled signal–idler evolution with the characteristicMI wavenumbers

k± = (δk1−δk3)/2± [(δk1 +δk3)2/4− γ2P2

2 ]1/2, (56)

which are the roots (eigenvalues) of the characteristic equation(k−δk1)(k +δk3)+ γ2P22 = 0.

Because the nonlinear coupling term in Eq. (56) is negative,the MI is potentially unstable[Im(k) 6= 0]. By defining the linear wavenumber mismatchδβ = (β1−2β2+β3)/2, one obtainsthe growth-rate formula

κ = [γ2P22 − (δβ + γP2)

2]1/2. (57)

Notice that the wavenumber mismatchδβ +γP2 is theP1 → 0 limit of the mismatch (44). As thesignal and idler propagate, their powers increase exponentially with distance. Formulas for the(parallel) signal and idler powers are stated in [6]. When themismatch is optimal (δβ = −γP2)the maximal growth rateκ = γP2. The signal and idler wavenumbers arek1 = β1−δβ +γP2 andk3 = β3− δβ + γP2, respectively. [For each wavenumber, one contributionγP2/2 comes fromthe diagonal term inH2 and the other contributionγP2/2 comes from the identity exp(iγ~a2 ·~σz/2)|E‖〉 = exp(iγP2z/2)|E‖〉.] These results are the standard results for scalar FWM drivenby one pump [2], with the nonlinearity coefficient reduced bya factor of 9/8. The associatedreduction in growth rate was observed in numerical simulations of noise amplification by MI infibers with random birefringence [32]. For short distances (before the idler power is comparableto the signal power), the idler produced by a pump and a weak signal should grow in the sameway as an idler produced by a pump and a strong signal [Eq. (42)]. The results described in thisparagraph are consistent with the first row of Table 1. (See Figures 1a and 2a.)

ω

12

3||

⊥

ω

1 2

3||

⊥

Fig. 2. Eigenpolarizations of MI. The dashed lines denote sidebands thatpropagate inde-pendently.

The perpendicular part of Eq. (55) represents stable signaland idler propagation with thewavenumbersk1 = β1 + γP2 andk3 = β3 + γP2, respectively. [For eachk j, a contributionγP2

comes fromδk j, a contributionγP2/2 comes from the diagonal term inH2 and the oppositecontribution−γP2/2 comes from the identity exp(iγ~a2 ·~σz/2)|E⊥〉= exp(−iγP2z/2)|E⊥〉.] Thepump modifies the signal and idler wavenumbers, but does not couple their evolution, so the(perpendicular) signal power is constant and no idler is produced. The results described in thisparagraph are consistent with the second row of Table 1. (SeeFigures 1b and 2b.)

In neither case do the signal and idler vectors rotate, because they are both parallel or perpen-dicular to the pump vector. However, ifS‖ andS⊥ are both nonzero, polarization rotation occurs(in addition to the rotation associated withH2) because the polarization components grow atdifferent rates.

The effects of PMD on degenerate FWM, in a 2 Km-long fiber with a PMD coefficient of0.05 ps/(Km)1/2, were studied recently [33]. Most of the other fiber and pump parameters weretypical of current experiments. For signal and idler wavelengths that differed by 40 nm, PMD


reduced the signal gain from 29 to 24 dB, and for signal and idler wavelengths that differed by80 nm, PMD reduced the gain from 29 to 22 dB. These results are consistent with the assertion(made in Section 2) that the Manakov equation is a reasonable(first) model for FWM in fiberswith lengths of order 1 Km.

5. Nondegenerate four-wave mixing

Nondegenerate FWM involves four waves with frequencies thatsatisfy the matching conditionω2 +ω3 = ω4 +ω1. By substituting the ansatz

|A〉 =4

∑j=1

exp(−iω jt)|A j〉 (58)

in the Manakov equation (10) and collecting terms of like frequency, one finds that

D|A1〉 = iH1|A1〉+ iγ(〈A4|A2〉|A3〉+ 〈A4|A3〉|A2〉), (59)

D|A2〉 = iH2|A2〉+ iγ(〈A3|A4〉|A1〉+ 〈A3|A1〉|A4〉), (60)

whereH1 andH2 are defined by Eq. (30). For nondegenerate FWM the summation inEq. (30)involves the three indicesk 6= j and~at =~a1 +~a2 +~a3 +~a4. One can deduce the equations forwaves 3 and 4 from the equations for waves 2 and 1 by interchanging the subscripts 2 and 3, andthe subscripts 1 and 4, respectively. Henceforth, when equations for waves 1 and 2 are cited,the associated equations for waves 3 and 4 are cited implicitly.

It follows from Eqs. (59) and (60) that

D〈A1|A1〉 = iγ(〈A4|A2〉〈A1|A3〉+ 〈A4|A3〉〈A1|A2〉

− 〈A2|A4〉〈A3|A1〉−〈A3|A4〉〈A2|A1〉), (61)

D〈A2|A2〉 = iγ(〈A3|A4〉〈A2|A1〉+ 〈A3|A1〉〈A2|A4〉)

− 〈A4|A3〉〈A1|A2〉−〈A1|A3〉〈A4|A2〉). (62)

By combining Eqs. (61) and (62) one obtains the MRW equations

D(P1−P4) = 0, (63)

D(P2−P3) = 0, (64)

D(P1 +P2 +P3 +P4) = 0. (65)


D〈A1|~σ |A1〉− γ〈A1|~σ ×~at |A1〉 = iγ(〈A4|A2〉〈A1|~σ |A3〉+ 〈A4|A3〉〈A1|~σ |A2〉)

− 〈A2|A4〉〈A3|~σ |A1〉−〈A3|A4〉〈A2|~σ |A1〉), (66)

D〈A2|~σ |A2〉− γ〈A2|~σ ×~at |A2〉 = iγ(〈A3|A4〉〈A2|~σ |A1〉+ 〈A3|A1〉〈A2|~σ |A4〉)

− 〈A4|A3〉〈A1|~σ |A2〉−〈A1|A3〉〈A4|~σ |A2〉). (67)

By combining Eqs. (66) and (67) one finds that

D(〈A1|~σ |A1〉+ 〈A2|~σ |A2〉) = γ(〈A1|~σ ×~at |A1〉+ 〈A1|~σ ×~at |A1〉)

+ iγ(〈A4|A2〉〈A1|~σ |A3〉−〈A2|A4〉〈A3|~σ |A1〉

+ 〈A3|A1〉〈A2|~σ |A4〉−〈A1|A3〉〈A4|~σ |A2〉). (68)

By combining Eq. (68) with the associated equation for waves3 and 4, one finds that

D~at = 0. (69)


Even in the presence of FWM the total polarization vector is conserved.Now consider the initial evolution of nondegenerate FWM. In the first case three powerful

waves (1, 2 and 3) are launched into the fiber and a weak idler wave (4) is generated in the fiberby FWM. The linearized analysis of this process is based on theapproximation that the idlerdoes not affect the input waves, in which case

D|Al〉 = iHl |Al〉, (70)

D|A4〉 = iH4|A4〉+ iγ(〈A1|A2〉|A3〉+ 〈A1|A3〉|A2〉), (71)

wherel = 1, 2 or 3. The inputs are subject to different phase shifts (which depend onP1, P2 andP3) and the same polarization rotation (which depends on~at = ~a1 +~a2 +~a3), as discussed inSection 3. The idler is subject to a phase shift (which depends onP1, P2 andP3) and the samepolarization rotation as the inputs. In the notation of Section 3, let|A j〉 = exp(iHtz + iθ j)|B j〉.Then

D|Bl〉 = 0, (72)

D|B4〉 = iγ(〈B1|B2〉|B3〉+ 〈B1|B3〉|B2〉)exp[i(θ2 +θ3−θ4−θ1)]. (73)

It follows from Eq. (72) that the (transformed) input vectors are constant, as are the input powers(upon which the phase shifts depend), and it follows from Eq.(73) that the orientation of the(original or transformed) idler vector is determined by a fixed combination of the (original ortransformed) vectors of inputs 2 and 3: The initial evolution of nondegenerate FWM does notdepend on the (rigid) polarization rotation.

By integrating Eq. (73) one finds that

P4(z) = Γ(z)(P2|〈A3|A1〉|2 +P3|〈A1|A2〉|

2

+ 〈A2|A1〉〈A1|A3〉〈A3|A2〉+ 〈A3|A1〉〈A1|A2〉〈A2|A3〉), (74)

where the spatial growth factorΓ was defined in Eq. (43) and the FWM wavenumber (mis-match)

k = [β1−β2−β3 +β4 + γ(P2 +P3−P1)]/2. (75)

For reference, Eq. (74) can be rewritten in the Stokes-vector form

P4(z) = Γ(z)P1P2P3(3+2~e1 ·~e2 +~e2 ·~e3 +2~e3 ·~e1)/2, (76)

where each~el is a unit vector parallel to~al .Whenk = 0 the idler power increases quadratically with distance. When k 6= 0 the idler power

is a periodic function of distance, with period and maxima that are inversely proportional tok.The nonlinear contribution tok depends on the input powers, but not the input polarizations:Formula (75) for vector FWM is the same as the formula for scalar FWM [31]. However, theidler power produced by parallel inputs in a fiber with randombirefringence is lower than thatproduced in a polarization-maintaining fiber by a factor of(9/8)2 ≈ 1.3 (1.0 dB).

The properties of nondegenerate FWM are summarized in Table 2. Each row pertains to aparticular combination of input polarizations, which is illustrated in Figure 3. The first col-umn describes the output idler polarization and the second column describes the FWM powerefficiency (normalized to a maximal efficiency of 1). For the polarization combinations de-scribed in the second and third rows, in which the inputs 2 and3 are perpendicular, the idlerpower produced in a fiber with random birefringence is lower than that produced by parallelinputs in a polarization-maintaining fiber by a factor of 4×(9/8)2 ≈ 5.1 (7.0 dB). The random-polarization results were obtained by averaging the right side of Eq. (76) over the Poincaresphere (〈~e j ·~ek〉 = 0 whenk 6= j).


Table 2. Properties of nondegenerate FWM driven by three input waves

|E4〉 P4

|E1〉 ‖ |E2〉 ‖ |E3〉 |E2〉 1|E1〉 ‖ |E2〉 ⊥ |E3〉 |E3〉 1/4|E1〉 ‖ |E3〉 ⊥ |E2〉 |E2〉 1/4|E1〉 ⊥ |E2〉 ‖ |E3〉 — 0

random random 3/8

The symbols‖ and⊥ should be interpreted in the sense of Jones vectors: The rows of thetable apply to parallel and perpendicular linearly-polarized waves, co- and counter-rotatingcircularly polarized waves or any other pair of aligned and orthogonal waves.

12

34

ω||

⊥

12

34 ω

||

⊥

ω

1 2

3 4||

⊥

1 23

4 ω||

⊥

Fig. 3. Polarization diagrams for nondegenerate FWM driven by three input waves. Thesymbols‖ and⊥ should be interpreted in the sense of Jones vectors, and the symbol◦signifies that no idler is produced. Figures 3a–3d correspond to rows 1–4 of Table 2, re-spectively.

For the special case in whichk = 0, the results listed in Table 2 were obtained by Inoue[9], who analyzed FWM in a system of concatenated, birefringent fiber segments. By apply-ing the Jones-vector formalism to the Manakov equation (10), one simplifies the FWM analy-sis significantly. Inoue neglected the coherent-coupling termsγA2

yA∗x/3 andγA2

xA∗y/3 a priori.

These terms were retained in the derivation of the Manakov equation. When averaged overthe Poincare sphere, they contributed to the term−γ〈A|A〉|A〉/9, which reduced the nonlinearcoefficient in the Manakov equation fromγ to γ = 8γ/9. (They also eliminated the nonlinearpolarization rotation associated with SPM.) Inoue used hisresults for nondegenerate FWM toinfer the properties of degenerate FWM (|A3〉 = |A2〉). By doing so, one reproduces the resultsof Table 1. However, one overestimates the idler power by a factor of 4 [Eqs. (45) and (76)].


In the second case a weak signal wave (1) and two powerful pumpwaves (2 and 3) arelaunched into the fiber and a weak idler wave (4) is generated in the fiber by FWM. This processis often called phase conjugation (PC). Since the idler power quickly becomes comparable tothe signal power, the effects of the idler on the signal cannot be neglected, in which case

D|A1〉 = iH1|A1〉+ iγ(〈A4|A2〉|A3〉+ 〈A4|A3〉|A2〉), (77)

D|Al〉 = iHl |Al〉, (78)

D|A4〉 = iH4|A4〉+ iγ(〈A1|A2〉|A3〉+ 〈A1|A3〉|A2〉), (79)

wherel = 2 or 3. The pumps are subject to different phase shifts (whichdepend onP2 andP3)and the same polarization rotation (which depends on~at = ~a2 +~a3). The signal and idler aresubject to the same phase shift (which depends onP2 andP3) and polarization rotation (whichdepends on~at). Let |A1〉= exp(iH2z)|B1〉, |Al〉= exp(iHlz)|Bl〉 and|A4〉= exp(iH3z)|B4〉. Then

D|B1〉 = i(β1−β2 + γP2)|B1〉+ iγ(〈B4|B2〉|B3〉+ 〈B4|B3〉|B2〉), (80)

D|Bl〉 = 0, (81)

D|B4〉 = i(β4−β3 + γP3)|B4〉+ iγ(〈B1|B2〉|B3〉+ 〈B1|B3〉|B2〉). (82)

It follows from Eqs. (80)–(82) that the transformed pump vectors are constant, and the initialevolution of PC does not depend on the polarization rotation.


D1|B1〉 = iγ(〈B4|B2〉|B3〉+ 〈B4|B3〉|B2〉), (83)

D∗4〈B4| = −iγ(〈B2|B1〉〈B3|+ 〈B3|B1〉〈B2|). (84)


[D∗4D1I − γ2(P2|B3〉〈B3|+P3|B2〉〈B2|

+ 〈B2|B3〉|B2〉〈B3|+ 〈B3|B2〉|B3〉〈B2|)]|B1〉 = 0. (85)

Let |E‖〉 and |E⊥〉 be unit vectors parallel and perpendicular to|B2〉, and (with the subscript

3 omitted for brevity) let|B3〉/P1/23 = B‖|E‖〉 + B⊥|E⊥〉 and (since wave 1 is the signal)

|B1〉/P1/21 = S‖|E‖〉+S⊥|E⊥〉. Then

[

D∗4D1− γ2P2P3(1+3|B‖|

2) −2γ2P2P3B‖B∗⊥

−2γ2P2P3B∗‖B⊥ D∗

4D1− γ2P2P3|B⊥|2

][

S‖S⊥

]

= 0. (86)

It follows from Eq. (86) that the characteristic PC wavenumbers

k± = (δk1−δk4)/2± [(δk1 +δk4)2/4− γ2P2P3∆±]1/2, (87)

where∆± = (1±|B‖|)

2 (88)

are the normalized eigenvalues of the coupling matrix [associated with the projection operatorsin Eq. (85)]. Each eigenvalue (subscript+ or −) is associated with a second-order (differentialor polynomial) equation, to which both signs (non-subscript + and−) on the right side ofEq. (87) apply. Because∆± ≥ 0, the PC process is potentially unstable. By defining the linearwavenumber mismatchδβ = (β1−β2−β3 +β4)/2, one obtains from Eq. (87) the growth-rateformula

κ± ={

γ2P2P3∆±− [δβ + γ(P2 +P3)/2]2}1/2

. (89)


Notice that the wavenumber mismatchδβ + γ(P2 + P3)/2 is theP1 → 0 limit of the mismatch(75). It follows from Eqs. (86)–(88) that

|S⊥/S‖|2± = [(1−|B‖|)/(1+ |B‖|)]

±1 (90)

and, hence, (for normalized signal eigenvectors) that

|S‖|2± = (1±|B‖|)/2. (91)

The wavenumber mismatch depends on the pump powers, but not the pump polarizations. Incontrast, the determinant of the coupling matrix and the signal-eigenvector polarizations de-pend on the pump polarizations, but not the pump powers. It follows from Eqs. (83) and (84)that the idler eigenvectors are the same as signal eigenvectors (apart from the phase factorsexp[±i(δk4−δk2)z/2], which do not alter|I‖| and|I⊥|.) Thus, Eqs. (90) and (91) also apply tothe idler. Each signal-idler eigenmode evolves independently of the other, in the same way thatthe signal and idler evolve in scalar PC [6].

For reference, Eqs. (88) and (91), and its analog for the idler, can be rewritten, respectively,in the Stokes-vector forms

∆± = (3+~e2 ·~e3)/2± [2(1+~e2 ·~e3)]1/2, (92)

(~e1 ·~e2)± = ±[(1+~e2 ·~e3)/2]1/2, (93)

(~e4 ·~e2)± = ±[(1+~e2 ·~e3)/2]1/2. (94)

A complimentary analysis of PC, which is based on the matrix version of Eqs. (83) and (84), isdescribed in the Appendix.

First, consider the case in which the pumps are parallel (B‖ = 1 andB⊥ = 0). In this case thereis no pump-pump polarization rotation. The eigenvalues∆± = 4, 0. The+ root (4) is associatedwith signal and idler eigenvectors that are parallel to the (common) pump vector and, hence,do not rotate. The signal and idler wavenumbers arek1 = β1− δβ + 3γ(P2 + P3)/2 andk4 =β4−δβ +3γ(P2+P3)/2, respectively. When the mismatch is optimal [δβ =−γ(P2+P3)/2] themaximal growth rateκ = 2γ(P2P3)

1/2. Formulas for the signal and idler powers are stated in [6].These results are the standard parallel-polarization results [2], with the nonlinearity coefficientmodified by a factor of 8/9. The− root (0) is associated with signal and idler eigenvectors thatare perpendicular to the pump vector and, hence, do not rotate. They propagate independentlyand stably, with wavenumbersβ1+3γ(P2+P3)/2 andk4 = β4+3γ(P2+P3)/2, respectively. Forshort distances (before the idler power is comparable to thesignal power), the idler produced bytwo pumps and a weak signal should grow in the same way as an idler produced by two pumpsand a strong signal [Eq. (76)]. The results described in thisparagraph are consistent with thefirst and fourth rows of Table 2. (See Figures 3a, 3d and 4.)

12

34

ω||

⊥

1 23

4 ω||

⊥

Fig. 4. Eigenpolarizations of PC driven by parallel pumps. The dashedlines denote side-bands that propagate independently.


Second, consider the case in which the pumps are perpendicular (B‖ = 0 andB⊥ = 1). Inthis case there is also no pump-pump polarization rotation.The eigenvalues∆± = 1, 1 and thepropagation matrix in Eq. (86) is diagonal, with both elements equal toD∗

4D1 − γ2P2P3. Be-cause of this degeneracy, the signal gain does not depend on the signal polarization. It shouldbe re-emphasized that the term perpendicular is used in the sense of Jones vectors: The preced-ing results apply equally to pumps with perpendicular linear polarizations, circular polarizationswith opposite helicity or any other orthogonal polarizations. For some input signal polarizationsthe (original) signal and idler vectors will rotate, whereas for others they will not. The specialcases in which the signal is parallel or perpendicular to thepumps are illustrated in Figure 5.When the mismatch is optimal [δβ =−γ(P2+P3)/2] the maximal growth rateκ = γ(P2P3)

1/2.For the special case in which the waves are linearly polarized, these results are similar to thestandard perpendicular-polarization results [8]. (To be precise, the SPM coefficient, which de-termines the nonlinear wavenumber mismatch, changes from 1to 8/9 and the CPM coefficient,which determines the coupling strength, changes from 2/3 to8/9.) The results described in thisparagraph are consistent with the second and third rows of Table 2. (See Figures 3b, 3c and 5.)

12

34 ω

||

⊥

ω

1 2

3 4||

⊥

Fig. 5. Eigenpolarizations of PC driven by perpendicular pumps.

For fibers with random birefringence, the growth rate (or gain, in dB) associated with per-pendicular pumps is lower than the growth rate (or gain) associated with parallel pumps bya factor of 2. In contrast, for polarization-maintaining fibers with constant birefringence theperpendicular growth rate is lower than the parallel growthrate by a factor of 3 [8].

In the third case a weak signal wave (2) and two powerful pump waves (1 and 3) are launchedinto the fiber and a weak idler wave (4) is generated in the fiberby FWM [34, 35]. We call thisprocess Bragg scattering (BS). (The terminology comes froma spatial analog of this process[36], in which a wave with parallel wavevectork, that is incident on a grating with wavevec-tor −2k, is reflected as a wave with parallel wavevector−k.) Since the idler power quicklybecomes comparable to the signal power, the effects of the idler on the signal cannot be ne-glected, in which case

D|Al〉 = iHl |Al〉, (95)

D|A2〉 = iH2|A2〉+ iγ(〈A3|A4〉|A1〉+ 〈A3|A1〉|A4〉), (96)

D|A4〉 = iH4|A4〉+ iγ(〈A1|A2〉|A3〉+ 〈A1|A3〉|A2〉), (97)

wherel = 1 or 3. The pumps are subject to different phase shifts (whichdepend onP1 andP3)and the same polarization rotation (which depends on~at = ~a1 +~a3). The signal and idler aresubject to the same phase shift (which depends onP1 andP3) and polarization rotation (whichdepends on~at ). Let |Al〉 = exp(iHlz)|Bl〉, |A2〉 = exp(iH1z)|B2〉, and |A4〉 = exp(iH3z)|B4〉.Then

D|Bl〉 = 0, (98)


D|B2〉 = i(β2−β1 + γP1)|B2〉+ iγ(〈B3|B4〉|B1〉+ 〈B3|B1〉|B4〉), (99)

D|B4〉 = i(β4−β3 + γP3)|B4〉+ iγ(〈B1|B2〉|B3〉+ 〈B1|B3〉|B2〉). (100)

It follows from Eqs. (98)–(100) that the transformed pump vectors are constant and the initialevolution of BS does not depend on the polarization rotation.


D2|B2〉 = iγ(〈B3|B4〉|B1〉+ 〈B3|B1〉|B4〉), (101)

D4|B4〉 = iγ(〈B1|B2〉|B3〉+ 〈B1|B3〉|B2〉). (102)


[D4D2I + γ2(P3|B1〉〈B1|+ |〈B1|B3〉|2

+ 〈B3|B1〉|B3〉〈B1|+ 〈B1|B3〉|B1〉〈B3|)]|B2〉 = 0. (103)

Let |E‖〉 and |E⊥〉 be unit vectors parallel and perpendicular to|B1〉, and (with the subscript

3 omitted for brevity) let|B3〉/P1/23 = B‖|E‖〉 + B⊥|E⊥〉 and (since wave 2 is the signal)

|B2〉/P1/22 = S‖|E‖〉+S⊥|E⊥〉. Then

[

D4D2 + γ2P1P3(1+3|B‖|2) γ2P1P3B‖B∗

⊥

γ2P1P3B∗‖B⊥ D4D2 + γ2P1P3|B‖|

2

][

S‖S⊥

]

= 0. (104)

It follows from Eq. (104) that the characteristic BS wavenumbers

k± = (δk2 +δk4)/2± [(δk2−δk4)2/4− γ2P1P3∆±]1/2, (105)

where∆± = [−(1+4|B‖|

2)± (1+8|B‖|2)1/2]/2 (106)

are the normalized eigenvalues of the coupling matrix. Because∆± ≤ 0, the BS process isintrinsically stable. Notice that the wavenumber mismatch(δk4−δk2)/2 is theP2 → 0 limit ofthe mismatch (75). It follows from Eqs. (104)–(106) that

∣

∣

∣

∣

S⊥S‖

∣

∣

∣

∣

2

±

=(1+8|B‖|

2)1/2± (1+2|B‖|2)

(1+8|B‖|2)1/2∓ (1+2|B‖|2)(107)

and, hence, (for normalized signal eigenvectors) that

|S‖|2± =

(1+8|B‖|2)1/2∓ (1+2|B‖|

2)

2(1+8|B‖|2)1/2. (108)

The wavenumber mismatch depends on the pump powers, but not the pump polarizations. Incontrast, the determinant of the coupling matrix and the signal-eigenvector polarizations dependon the pump polarizations, but not the pump powers.

By combining Eqs. (101) and (102), and retaining the idler vector rather than the signalvector, one finds that

[D2D4I + γ2(P1|B3〉〈B3|+ |〈B3|B1〉|2

+ 〈B1|B3〉|B1〉〈B3|+ 〈B3|B1〉|B3〉〈B1|)]|B2〉 = 0. (109)


Equation (109) differs slightly from Eq. (103). Let|B4〉/P1/24 = I‖|E‖〉+ I⊥|E⊥〉. Then

[

D2D4 +4γ2P1P3|B‖|2 2γ2P1P3B‖B∗

⊥

2γ2P1P3B∗‖B⊥ D2D4 + γ2P1P3

][

I‖I⊥

]

= 0. (110)

The characteristic wavenumbers associated with Eq. (110) are identical to those associated withEq. (104), as they must be. However, the idler-eigenvector polarizations are different. It followsfrom Eqs. (106) and (110) that

∣

∣

∣

∣

I⊥I‖

∣

∣

∣

∣

2

±

=(1+8|B‖|

2)1/2± (4|B‖|2−1)

(1+8|B‖|2)1/2∓ (4|B‖|2−1)(111)

and, hence, (for normalized idler eigenvectors) that

|I‖|2± =

(1+8|B‖|2)1/2∓ (4|B‖|

2−1)

2(1+8|B‖|2)1/2. (112)

Each signal-idler eigenmode evolves independently of the other, in the same way that the signaland idler evolve in scalar BS [6].

For reference, Eqs. (106), (108) and (112) can be rewritten,respectively, in the Stokes-vectorforms

∆± = [−(3+2~e1 ·~e3)± (5+4~e1 ·~e3)1/2]/2, (113)

(~e2 ·~e1)± = ∓ (2+~e1 ·~e3)/(5+4~e1 ·~e3)1/2. (114)

(~e4 ·~e1)± = ∓ (1+2~e1 ·~e3)/(5+4~e1 ·~e3)1/2. (115)

A complementary analysis of BS, which is based on the matrix version of Eqs. (101) and (102),is described in the Appendix.

First, consider the case in which the pumps are parallel (B‖ = 1 andB⊥ = 0). In this casethere is no pump-pump polarization rotation. The eigenvalues∆± = −1,−4. The+ root (−1)is associated with signal and idler eigenvectors that are perpendicular to the (common) pumpvector and, hence, do not rotate. The− root (−4) is associated with signal and idler eigenvectorsthat are parallel to the pump vector and, hence, do not rotate. Formulas for the signal and idlerpowers are stated in [6]. For short distances (before the idler power is comparable to the signalpower), the idler produced by two pumps and a weak signal should grow in the same way asan idler produced by two pumps and a strong signal [Eq. (76)].The results described in thisparagraph are consistent with the first and third rows of Table 2. (See Figures 3a, 3c and 6.)

1

2 3

4 ω||

⊥

12

34

ω||

⊥

Fig. 6. Eigenpolarizations of BS driven by parallel pumps.


Second, consider the case in which the pumps are perpendicular (B‖ = 0 andB⊥ = 1). In thiscase there is also no pump-pump polarization rotation. The eigenvalues∆± = 0,−1. The− root(−1) is associated with a signal eigenvector that is parallel to the lower-frequency pump vectorand an idler eigenvector that is parallel to the higher-frequency pump vector. The+ root (0) isassociated with a signal eigenvector that is perpendicularto the lower-frequency pump vector.In this configuration no idler is produced: Even for perpendicular pumps, BS does not exhibitthe polarization-independence required of a practical FWM process. The results described inthis paragraph are consistent with the second and fourth rows of Table 2. (See Figures 3b, 3dand 7.)

12

34 ω

||

⊥

ω

1

23 4

||

⊥Fig. 7. Eigenpolarizations of BS driven by perpendicular pumps. The dashed lines denotesidebands that propagate independently.

6. Summary

The parametric amplification (PA) of optical signals is madepossible by four-wave mixing(FWM). In low-birefringence fibers the birefringence axes and strength vary randomly withdistance. Wave propagation in such fibers is governed by the Manakov equation. In this paperthe Manakov equation was used to make a detailed study of degenerate and nondegenerateFWM. Inoue’s analysis of FWM in fibers with random birefringence [9] was extended by theinclusion of the effects of linear and nonlinear wavenumbermismatches [which are caused byself-phase modulation (SPM) and cross-phase modulation (CPM)], and nonlinear polarizationrotation (which is caused by CPM).

Equations were derived from the Manakov equation, which govern the initial (linear) andfinal (nonlinear) evolution of the pump, signal and idler waves. These FWM equations [(28)and (29) for the degenerate interaction, and (59) and (60) for the nondegenerate interaction]show that the photon flux and photon-weighted polarization flux are conserved exactly.

In the linear regime the growth of the signal and idler powersis independent of the pump-induced rotation of the signal and idler polarization vectors. Formulas were derived for theinitial quadratic growth of the idler power [(45) and (76)],and the subsequent exponentialgrowth of the signal and idler powers [(55)–(57) and (86)–(89)]. These formulas are valid forarbitrary pump powers and polarizations. Signal amplification and idler generation are madepossible by nonlinear coupling and are inhibited by linear and nonlinear wavenumber shifts.Although the former depends on the pump powers and polarizations, the latter depend only thepump powers.

At present, there is considerable interest in PA driven by two pumps (nondegenerate FWM).Let the frequenciesω1 < ω2 < ω3 < ω4. Then, if the signal and idler frequencies are locatedsymmetrically relative to the pump frequencies (for example, if waves 2 and 3 are the pumps,and waves 1 and 4 are the signal and idler), the process is called phase conjugation (PC) and ispotentially unstable (signal gain is possible). In contrast, if the signal and idler frequencies are


located asymmetrically (for example, if waves 1 and 3 are thepumps, and waves 2 and 4 aresignal and idler), the process is called Bragg scattering (BS) and is intrinsically stable (no sig-nal gain is possible). The polarization dependence of both processes was studied in detail (forthe linear regime). For (nondegenerate) PC in a fiber with constant (high) birefringence, themaximal growth rate is associated with parallel pumps and signal. Let this rate [2γ(P2P3)

1/2]be normalized to 2. Then the growth rate associated with parallel pumps and a perpendicularsignal is 0. The growth rate associated with perpendicular pumps has a maximum of 2/3≈ 0.67and depends sensitively on the signal polarization [8]. ForPC in a fiber with random birefrin-gence, the maximal growth rate of 2×8/9≈ 1.8 is associated with parallel pumps and signal.The growth rate associated with parallel pumps and a perpendicular signal is 0. The growthrate associated with perpendicular pumps is 8/9 ≈ 0.89 and does not depend on the signalpolarization: PC driven by perpendicular pumps in a fiber with random birefringence exhibitsthe signal-polarization insensitivity required of a practical amplification process. Although theBS process is stable, it does transfer power from the signal to a frequency-converted idler.However, this power transfer depends sensitively on the signal polarization, for arbitrary pumppolarizations: BS does not exhibit the signal-polarization insensitivity required of a practicalfrequency-conversion process. Detailed measurements arebeing made to determine how wellthe Manakov equation models the behavior of PC and BS in Km-long highly-nonlinear fibers.

The saturation of scalar FWM (which is caused by pump depletion and nonlinear detuning),was discussed in detail by McKinstrie [31]. The saturation of vector FWM will be discussedelsewhere.

Acknowledgments

We thank a reviewer for bringing to our attention [26], [32] and [33].

7. Appendix: Eigenvectors for nondegenerate four-wave mixing

In Section 5 the ratiosS⊥/S‖ and I⊥/I‖ associated with the PC and BS eigenvectors weredetermined. In this appendix the complete PC and BS eigenvectors are determined.

First, consider PC, which is governed by Eqs. (80)–(82). Letφ2 and φ3 denote the ini-tial phases of the pumps, and letδ± denote the wavenumber mismatches(δk1 ± δk4)/2.Furthermore, let|B1〉 = exp(iδ−z + iφ2)|C1〉, |B2〉 = exp(iφ2)|C2〉, |B3〉 = exp(iφ3)|C3〉 and

|B4〉 = exp(−iδ−z + iφ3)|C4〉. Then the pump vectors|C2〉 = P1/22 and |C3〉 = P1/2

3 , and thesignal and idler vectors are governed by the equations

D−|C1〉 = iγ(〈C4|C2〉|C3〉+ 〈C4|C3〉|C2〉), (116)

D+〈C4| = −iγ(〈C2|C1〉〈C3|+ 〈C3|C1〉〈C2|), (117)

whereD± = D± iδ+. If one decomposes each vector into components that are parallel andperpendicular to pump 2, as described after Eq. (85), one finds that

D− 0 −2iγPB‖ −iγPB⊥

0 D− −iγPB⊥ 02iγPB‖ iγPB⊥ D+ 0iγPB⊥ 0 0 D+

S‖S⊥I∗‖I∗⊥

= 0, (118)

whereP = (P2P3)1/2, andB‖ and B⊥ are real. One can obtain the idler equations from the

signal equations by interchangingS andI, and taking the complex conjugate of the equationsthat result. It follows from this observation thatI‖ can only differ fromS‖ by a phase factor, andthe relation betweenI∗‖ andI∗⊥ must be the conjugate of the relation betweenS‖ andS⊥.


Let κ denote a PC eigenvalue (growth rate) and let∆ = (κ2 +δ 2+)/γ2P2. Then the PC eigen-

valuesκ± = ±(γ2P2∆±−δ 2

+)1/2, (119)

where the associated eigenvalues∆± are the solutions (roots) of the characteristic equation

∆2−2(1+B2‖)∆+B4

⊥ = 0. (120)

It follows from Eq. (120) that∆± = (1±B‖)

2, (121)

in agreement with Eq. (88). Suppose thatS‖ = 1. Then it follows from Eqs. (118)–(121) that

(S⊥)± =±B⊥

1±B‖, (122)

(I∗‖ )± =∓i(κ±− iδ+)

γP(1±B‖), (123)

(I∗⊥)± =−iγPB⊥

κ± + iδ+. (124)

Equation (122) is consistent with Eq. (90). It follows from Eqs. (121) and (123) that|I‖|2± = 1,

and it follows from Eqs. (121), (123) and (124) thatI∗⊥/I∗‖ = ±B⊥/(1±B‖). These results areconsistent with the predictions made after Eq. (118) and theresults of Section 4.

For the special case in whichB‖ = 0, it follows from Eq. (118) thatS‖ is coupled toI∗⊥ andS⊥is coupled toI∗‖ . BecauseS‖ andS⊥ are uncoupled, the natural choice of eigenvectors for the sig-

nal subspace is[1,0]T and[0,1]T . (These eigenvectors were illustrated in Figure 5.) However,Eq. (122) implies that the signal eigenvectors are[1,1]T and[1,−1]T . Despite their differences,these results are not inconsistent: The eigenvectors associated with degenerate eigenvalues arenot unique. In the natural basis each signal eigenvector is perpendicular to the associated idlereigenvector. However, a signal vector that represents a superposition of eigenstates is not per-pendicular to the associated idler vector.

Second, consider BS, which is governed by Eqs. (98)–(100). Let φ1 and φ3 denote theinitial phases of the pumps, and letδ± denote the wavenumber mismatches(δk2 ± δk4)/2.Furthermore, let|B1〉 = exp(iφ1)|C1〉, |B2〉 = exp(iδ+z + iφ1)|C2〉, |B3〉 = exp(iφ3)|C3〉 and

|B4〉= exp(iδ+z+ iφ3)|C4〉. Then the pump vectors|C1〉= P1/21 and|C3〉= P1/2

3 , and the signaland idler vectors are governed by the equations

D−|C2〉 = iγ(〈C3|C4〉|C1〉+ 〈C3|C1〉|C4〉), (125)

D+|C4〉 = iγ(〈C1|C2〉|C3〉+ 〈C1|C3〉|C2〉), (126)

whereD± = D± iδ−. If one decomposes each vector into components that are parallel andperpendicular to pump 2, as described after Eq. (103), one finds that

D− 0 −2iγPB‖ −iγPB⊥

0 D− 0 −iγPB‖

−2iγPB‖ 0 D+ 0−iγPB⊥ −iγPB‖ 0 D+

S‖S⊥I‖I⊥

= 0. (127)

For BS there is no simple relation between the signal and idler eigenvectors in Jones space.Let k denote a BS eigenvalue (wavenumber) and let∆ = (k2−δ 2

−)/γ2P2. Then the BS eigen-values

k± = ±(γ2P2∆± +δ 2−)1/2, (128)


where the associated eigenvalues∆± are the roots of the characteristic equation

∆2− (1+4B2‖)∆+4B4

‖ = 0. (129)

It follows from Eq. (129) that

∆± = [1+4B2‖∓ (1+8B2

‖)1/2]/2. (130)

Although the sign of∆ in this appendix is the opposite of the sign in Section 4, the∓ on theright side of Eq. (130) ensures that the root identification is the same. [See Eq. (106).] SupposethatS‖ = 1. Then it follows from Eqs. (127)–(130) that

(S⊥)± =B‖B⊥

∆±−B2‖

, (131)

(I‖)± =2γPB‖

k± +δ−, (132)

(I⊥)± =(k±−δ−)B⊥

γP(∆±−B2‖)

. (133)

By using Eq. (129), written in the form(∆−B2‖)(∆− 1− 3B2

‖) = B2‖(1−B2

‖), one can showthat Eq. (131) is consistent with Eq. (107). It follows from Eqs. (129), (132) and (133) that(I⊥/I‖)± = 2B‖B⊥/(∆± − 1). By using Eq. (129), written in the form(∆ − 1)(∆ − 4B2

‖) =

4B2‖(1−B2

‖), one can show that this result is consistent with Eq. (111).


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Four-wave mixing in fibers with random birefringence