## Abstract

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

©2004 Optical Society of America

## 1. Introduction

In recent years there has been a resurgence of interest in the parametric amplification (PA) of optical signals [1, 2]. Because of recent improvements in highly-nonlinear fibers, it is now a routine matter to produce four-wave-mixing (FWM) gains higher than 40 dB [3, 4] over bandwidths 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 by Hansryd [5] and PA driven by two pump waves (nondegenerate FWM) was reviewed by McKinstrie [6, 7]. In this paper the latter process is studied in detail. (For completeness, the former process is discussed briefly.)

PA is driven most strongly when the pumps have parallel polarization vectors. However, the signal gain depends sensitively on the input signal polarization: It is maximal when the signal is polarized parallel to the pumps and is minimal when the signal is polarized perpendicular to the pumps [1, 2]. Because transmission fibers are not polarization maintaining, practical amplifiers must 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 signal polarization. However, real fibers are birefringent. In birefringent polarization-maintaining fibers (which we refer to as fibers with constant birefringence) the signal gain does depend on the signal polarization [8]. Inoue [9] analyzed PA in birefringent non-polarization-maintaining fibers (which we refer to as fibers with random birefringence). In such fibers the random reorientation of the birefringence axes washes out whatever signal-polarization dependence is associated with birefringence: If the input polarizations of the pumps are perpendicular, the signal gain does not depend on the input signal polarization. (It was for such fibers that the use of perpendicular pumps was first proposed.) Inoue’s predictions were verified by experiments with long (20 Km) dispersion-shifted fibers [10, 11]. Recent experiments were made with highly-nonlinear fibers [12, 13, 14].

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

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

## 2. Governing equations

Let *E*_{x}
and *E*_{y}
denote the electric-field components of a light wave, measured relative to the birefringence axes of a polarization-maintaining fiber. It is convenient to measure (angular) frequencies relative to a reference frequency *ω*
_{0} and wavenumbers relative to the associated reference wavenumber *k*
_{0} = [*β*_{x}
(*ω*
_{0})+*β*_{y}
(*ω*
_{0})]/2, where each *β*(*ω*
_{0}) is a natural wavenumber (dispersion function), and to write the field components as

The evolution of the wave amplitudes (polarization components) *A*_{x}
and *A*_{y}
is governed by the coherently-coupled NS equations

where *β*_{x}
and *β*_{y}
are modified dispersion functions and *γ* is the nonlinearity coefficient [17]. In the frequency domain each *β*(*ω*) = ${\mathrm{\Sigma}}_{n=0}^{\infty}$=*β*
^{(n)}(*ω*
_{0})*ω*^{n}
/*n*!-*k*
_{0} is a Taylor expansion of a dispersion function about the reference frequency. In the time domain the frequency difference *ω* is replaced by the time derivative *i∂*_{t}
.

According to Eqs. (3) and (4), the wave amplitudes are subject to (dispersive) wavenumber shifts of opposite sign. One can make the associated phase shifts explicit by defining *A*_{x}
= *B*_{x}
exp(*iδk*
_{0}
*z*) and *A*_{y}
= *B*_{y}
exp(-*iδk*
_{0}
*z*), where *δk*
_{0} = [*β*_{x}
(*ω*
_{0}) -*β*_{y}
(*ω*
_{0})]/2. By substituting these definitions in Eqs. (3) and (4), one finds that the coherent-coupling terms are multiplied by the phase factors exp(∓4*iδk*
_{0}
*z*). Typical fibers have differential indices of refraction *δ*_{n}
in the range 10^{-7}-10^{-5} [22]. The wavenumber shift *δk*
_{0} = (*ω*
_{0}
*δ*_{n}
/*c* and the beat length 2*π*/*δk*
_{0} = *λ*
_{0}/*δ*_{n}
. For typical fibers the beat lengths are in the range 0.15–15.0 m. These beat lengths are all much shorter than the parametric gain length, which is typically of order 100 m. Thus, in the context of PA, the coherent coupling terms oscillate rapidly and can be neglected: PA in a polarization-maintaining fiber with constant (high) birefringence (*δ*_{n}
≫ 10^{-7}) is governed by the incoherently-coupled NS equations

where each *β*(*ω*) = ${\mathrm{\Sigma}}_{n=1}^{\infty}$=*β*
^{(n)}(*ω*
_{0})*ω*^{n}
/*n*!represents the higher-order (convection and dispersion) terms in the Taylor expansion of a dispersion function [17]. One often makes the simplifying assumption that ${\beta}_{x}^{\left(n\right)}$ (*ω*
_{0}) = ${\beta}_{y}^{\left(n\right)}$ (*ω*
_{0}) for *n* ≥ 2. In a fiber with constant birefringence, the polarization components convect at different speeds, but experience similar dispersion. The cross-phase modulation (CPM) coefficient differs from the self-phase modulation (SPM) coefficient by a factor of 2/3 and the coherent-coupling terms are absent. The consequences of the constant-birefringence model were studied in [8].

Fibers with low birefringence (*δ*_{n}
~ 10^{-7}) have cross-sections that are almost circular. Because it is difficult to manufacture such cross-sections reproducibly, the orientation of the birefringence 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 to each section individually. Between sections one changes the orientation of the birefringence axes and the value of the birefringence parameter *δk*
_{0}. In a frame rotating with the birefringence axes of the sections, the first change causes the Stokes vector of a (virtual) reference wave (with frequency *ω*
_{0}) to rotate about the 3-axis (in Stokes space), whereas the second change causes it to rotate about the 1-axis. The combined effect of many such rotations moves the Stokes vector randomly over the entire Poincaré sphere and causes the associated Jones vector to change randomly. By averaging Eqs. (3) and (4) over the Poincaré sphere, one finds that PA in a non-polarization-maintaining fiber with variable (low) birefringence is governed by the incoherently-coupled NS equations

where *β*(*ω*) = ${\mathrm{\Sigma}}_{n=2}^{\infty}$=*β*
^{(n)}(*ω*
_{0})*ω*^{n}
/*n*! represents the higher-order (dispersion) terms in the Taylor expansion of the (common) dispersion function [18, 19, 20, 21]. The subscripts *ξ* and *η* denote polarization components measured relative to basis vectors that track the polarization of the reference wave (one basis vector remains parallel to the Jones vector of the reference wave and the other remains perpendicular to it.) In a fiber with random birefringence both polarization components convect at the same speed and experience the same dispersion. [There are no convection terms in Eqs. (7) and (8) because *t* represents the retarded time.] The SPM and CPM coefficients are the same and the coherent-coupling terms are absent.

In the linear regime, the rotating-frame Stokes vector of a monochromatic wave remains constant while the laboratory-frame Stokes vector rotates rapidly and randomly about the Poincaré sphere. According to Eqs. (7) and (8), every frequency component of a multichromatic wave behaves in the same way. This behavior allows the fiber to distinguish the input polarizations of the pumps and signal. In reality, polarization-mode dispersion (PMD) causes the relative orientation of the Stokes vectors of waves with different frequencies to change [22]. Let *a*⃗_{1} and *a*⃗_{2} denote the Stokes vectors of waves with frequencies *ω*
_{1} and *ω*
_{2}, respectively, and let 〈〉 denotes an ensemble average. Then the polarization correlation

where *δω* is the frequency difference *ω*
_{2} - *ω*
_{1} and 〈*δτ*
^{2}〉 is the mean-square differential group delay of the fiber section [23]. For typical parameters, the polarization-decorrelation length predicted by Eq. (9) is of order 1 Km. This prediction is consistent with the results of recent experiments with a 1 Km-long fiber [13], which exhibited a strong dependence on the input polarizations: Equations (7) and (8) constitute a reasonable model for these and similar experiments. However, for experiments with very-long fibers or pumps with very-large frequency differences, a different model is required.

By introducing the Jones vector ∣*A*〉 = [*A*_{ξ}
,*A*_{η}
]^{T}, one can rewrite Eqs. (7) and (8) in the compact form

where *γ*¯ = 8*γ*/9 and 〈∣〉 denotes an inner (dot) product. Henceforth, for simplicity of notation, the overbar on *γ* will be omitted and the factor of 8/9 will be implied. Equation (10) is the Manakov equation [16].

The Jones-vector and Stokes-vector (Poincaré-sphere) formalisms, and many mathematical identities that relate them, were described in detail by Gordon and Kogelnik [24]. We will use these notations and results throughout this paper, without further comment.

## 3. Waves with incommensurate frequencies

Consider the interaction of two waves with different frequencies (*ω*
_{1} and *ω*
_{2}). By substituting the ansatz

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

where *D* = *d*/*dz*, *j* = 1 or 2, the wavenumber *β*_{j}
=*β*(*ω*
_{j}) and the power *P*_{j}
= 〈*A*_{j}
∣*A*_{j}
〉. By using the identity |*A*〉〈*A*| = (*PI* + *a*⃗·*σ*⃗)/2, where *σ*⃗ is the Pauli spin operator and *a*⃗ = 〈*A*|*σ*|*A*〉 is the Stokes (polarization) vector, one finds that

where the (hermitian) operator

and *k*≠*j*. In Eq. (15) the term *γP*_{j}
represents SPM, whereas the term *γP*_{k}
represents CPM. By using the identity *a*⃗·*σ*⃗|*A*〉 = *P*|*A*〉, one can rewrite operator (15) in the equivalent form

where the total Stokes vector *a*⃗_{t} = *a*⃗_{1} + *a*⃗_{2}.

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

Although the vectors |*A*
_{1}〉 and |*A*
_{2}〉 evolve (in manners to be determined), the magnitude of their inner product 〈*A*
_{1}|*A*
_{2}〉 is constant, as are their powers *P*
_{1} and *P*
_{2}.

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

where [*σ*⃗,*H*_{j}
] denotes the commutator *σ*⃗*H*_{j}
- *H*_{j}
*σ*⃗. By using the identities [*σ*⃗,*I*] = 0 and [*σ*⃗, *a*⃗·*σ*⃗] = 2*ia*⃗ × *σ*⃗, one finds that [*σ*⃗,*H*_{j}
] = -*iγσ*⃗ × *a*⃗_{t} and, hence, that

Had we used operator (15) rather than operator (16), we would have obtained the standard equation *D*
*a*⃗_{j} = *a*⃗_{j} × *a*⃗_{k} [25, 26], which manifests the fact that (nonlinear) polarization rotation is produced by CPM, but not by SPM. It follows from Eq. (20) that

In Stokes space each individual Stokes vector rotates about the total Stokes vector, which is constant. As in other rigid vector rotations, the lengths and relative orientations of the vectors are constant. Equation (22) is consistent with Eq. (18) and the identity |〈*A*_{j}
|*A*_{k}
〉|^{2}=(*P*_{j}*P*_{k}
+ *a*⃗_{j}·*a*⃗_{k})/2.

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

(For waves with incommensurate frequencies, the powers are constants. However the transformation is also valid for variable powers.) Then

where the (common) operator *H*_{t}
= *γa*⃗_{t} · *σ*⃗/2. Since *H*_{t}
is hermitian, there exists a (common) unitary operator *U*_{t}
such that

Furthermore, since *H*_{t}
is a constant (matrix) operator with zero trace, *U*_{t}
= exp(*iH*_{t}*z*) 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

Equation (26) has the canonical form of a unitary transformation (a unimodular transformation combined with a phase shift). As predicted by Eq. (20), the two waves are subject to the same polarization rotation (which depends on the powers and polarizations). They are also subject to different 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 incommensurate frequencies (no linear combination of frequencies, with integer coefficients, equals zero): The one index *k* is replaced by a sum over the *n* - 1 indices *k* ≠ *j* and the two-vector sum in *a*⃗_{t} is replaced by an *n*-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 modify their phases and polarizations, it also allows them to exchange power.

Degenerate FWM involves three waves with frequencies that satisfy the matching condition 2*ω*
_{2} = *ω*
_{3} + *ω*
_{1}. By substituting the ansatz

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

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

where the summation involves the two indices *k* ≠ *j* and *a*⃗_{t} = *a*⃗_{1} + *a*⃗_{2} + *a*⃗_{3}. One can deduce the 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 is cited implicitly.

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

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

Equations (33) and (34) are exact consequences of Eqs. (28) and (29), which, in turn, are exact consequences of the Manakov equation. They imply that the total power is constant, and the difference between the signal and idler powers is constant. The first implication is true, but the second must be false because it contradicts the Manley–Rowe–Weiss (MRW) equations [28, 29], upon which the photon interpretation of FWM is based: For every pair of pump photons that is destroyed, one signal photon and one idler photon are created. Since the signal and idler frequencies are different, the photon energies *h*¯*ω*_{j}
are also different, so the signal and idler powers must increase at different rates. The Manakov equation is based on the envelope approximation that the relative frequencies *ω*_{j}
are much smaller than the reference (carrier) frequency *ω*
_{0} and the concomitant approximation that the nonlinearity coefficient *γ* does not depend on *ω*_{j}
. Had we used the Maxwell equations to derive Eqs. (28) and (29) directly, the nonlinear terms would have included factors of 1 + *ω*_{j}
/*ω*
_{0} [27]. We would have eliminated these factors from Eqs. (31) and (32) by rewriting them in terms of the photon fluxes *F*_{j}
= *P*_{j}
/*h*¯(*ω*
_{0} + *ω*_{j}
), and would have obtained the MRW equations [(33) and (34), with *P*_{j}
replaced by *F*_{j}
]. By combining the MRW equations with the frequency-matching condition, we would have confirmed that the total power is conserved exactly. In typical experiments |*ω*_{j}
)/*ω*
_{0}| ~ 10^{-2}. For such frequencies the envelope approximation is accurate and the distinction between power and photon flux is (quantitatively) unimportant.

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

$$\phantom{\rule{15em}{0ex}}+\u3008{A}_{2}\mid {A}_{1}\u3009\u3008{A}_{2}\mid \overrightarrow{\sigma}\mid {A}_{3}\u3009-\u3008{A}_{1}\mid {A}_{2}\u3009\u3008{A}_{3}\mid \overrightarrow{\sigma}\mid {A}_{2}\u3009).$$

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

Even in the presence of FWM the total polarization vector is conserved. Equation (37) is an exact consequence of the Manakov equation. However, as discussed after Eq. (34), the Manakov equation does not account for the dependence of *γ* on *ω*_{j}
. Had we retained this frequency dependence, 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 equations that are similar to Eqs. (35) and (36) for any hermitian operator *G*. (The FWM terms always sum 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) is generated in the fiber by FWM. The linearized analysis of this process is based on the approximation that the idler does not affect the input waves, in which case

where *l* = 1 or 2. The inputs are subject to different phase shifts (which depend on *P*
_{1} and *P*
_{2}) and the same polarization rotation (which depends on *a*⃗_{t} = *a*⃗_{1} + *a*⃗_{2}), as discussed in Section 3. The idler is subject to a phase shift (which depends on *P*
_{1} and *P*
_{2}) and the same polarization rotation as the inputs. In the notation of Section 3, let |*A*_{j}
〉 = exp(*iH*_{t}*z* + *iθ*_{j}
) |*B*_{j}
〉. Then

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

By integrating Eq. (41) one finds that

where the spatial growth factor

and the FWM wavenumber (mismatch)

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

where *e*⃗_{1} and *e*⃗_{2} are unit vectors parallel to *a*⃗_{1} and *a*⃗_{2}, respectively.

When *k* = 0 the idler power increases quadratically with distance. When *k* ≠ 0 the idler power is a periodic function of distance, with period and maxima that are inversely proportional to *k*. The nonlinear contribution to *k* 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 than that produced in a polarization-maintaining fiber by a factor of (9/8)^{2} ≈ 1.3 (1.0 dB). As the alignment between the input polarizations decreases, so also does the output idler power. In particular, if the input polarizations are perpendicular, no idler is produced.

The properties of degenerate FWM are summarized in Table 1. Each row pertains to a particular combination of input polarizations, which is illustrated in Figure 1. The first column describes 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 by averaging the right side of Eq. (45) over the Poincaré sphere (〈*e*⃗_{1} · *e*⃗_{2}〉 = 0).

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

The pump is subject to a phase shift (which depends on *P*
_{2}), but no polarization rotation (because *a*⃗_{t} = *a*⃗_{2} and |*A*
_{2}〉 is an eigenvector of *a*⃗_{1} · *σ*⃗). The signal and idler are subject to the same phase shift (which depends on *P*
_{2}) and polarization rotation (which depends on *a*⃗_{2}). Let |*A*_{j}
〉 = exp(*iH*
_{2}
*z*)|*B*_{j}
〉). Then

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

By defining the wavenumber shift *δk*
_{1} = *β*
_{1} - *β*
_{2} + *γP*
_{2} and the operator *D*
_{1}=*D* - *iδk*
_{1}, and making similar definitions for wave 3, one can rewrite Eqs. (49) and (51) in the compact form

By combining Eqs. (52) and (53), one finds that

Let |*E*
_{∥}〉 and |*E*
_{⊥}〉 be unit vectors parallel and perpendicular to |*B*
_{2}〉, and (since wave 1 is the signal) let |*B*
_{1}〉/${P}_{1}^{1/2}$ = *S*
_{∥}|*E*
_{∥}〉 + *S*
_{⊥}|*E*
_{⊥}〉. Then, by using this decomposition, one can rewrite Eq. (54) in the matrix form

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

which are the roots (eigenvalues) of the characteristic equation (*k* - *δk*
_{1})(*k* + *δk*
_{3}) + *γ*
^{2}
${P}_{2}^{2}$ = 0. Because the nonlinear coupling term in Eq. (56) is negative, the MI is potentially unstable [Im(*k*) ≠ 0]. By defining the linear wavenumber mismatch *δβ* = (*β*
_{1} - 2*β*
_{2} + *β*
_{3}) /2, one obtains the growth-rate formula

Notice that the wavenumber mismatch *δβ* + *γP*
_{2} is the *P*
_{1} → 0 limit of the mismatch (44). As the signal and idler propagate, their powers increase exponentially with distance. Formulas for the (parallel) signal and idler powers are stated in [6]. When the mismatch is optimal (*δβ* = - *γP*
_{2}) the maximal growth rate *κ* = *γP*
_{2}. The signal and idler wavenumbers are *k*
_{1} = *β*
_{1} - *δβ* + *γP*
_{2} and *k*
_{3} = *β*
_{3} - *δβ* + *γP*
_{2}, respectively. [For each wavenumber, one contribution *γP*
_{2}/2 comes from the diagonal term in *H*
_{2} and the other contribution *γP*
_{2}/2 comes from the identity exp(*iγa*⃗_{2} · *σ*⃗*z*/2)|*E*
_{∥}〉 = exp(*iγP*
_{2}
*z*/2|*E*
_{∥}〉.] These results are the standard results for scalar FWM driven by one pump [2], with the nonlinearity coefficient reduced by a factor of 9/8. The associated reduction in growth rate was observed in numerical simulations of noise amplification by MI in fibers with random birefringence [32]. For short distances (before the idler power is comparable to the signal power), the idler produced by a pump and a weak signal should grow in the same way as an idler produced by a pump and a strong signal [Eq. (42)]. The results described in this paragraph are consistent with the first row of Table 1. (See Figures 1*a* and 2*a*.)

The perpendicular part of Eq. (55) represents stable signal and idler propagation with the wavenumbers *k*
_{1} = *β*
_{1} + *γP*
_{2} and *k*
_{3} = *β*
_{3} + *γP*
_{2}, respectively. [For each *k*_{j}
, a contribution *γP*
_{2} comes from *δk*_{j}
, a contribution *γP*
_{2}/2 comes from the diagonal term in *H*
_{2} and the opposite contribution - *γP*
_{2}/2 comes from the identity exp(*iγa*⃗_{2}·*σ*⃗*z*/2)|*E*
_{⊥}) = exp(- *iγP*
_{2}
*z*/2)|*E*
_{⊥}〉.] The pump 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 this paragraph are consistent with the second row of Table 1. (See Figures 1b and 2b.)

In neither case do the signal and idler vectors rotate, because they are both parallel or perpendicular to the pump vector. However, if *S*
_{∥} and *S*
_{⊥} are both nonzero, polarization rotation occurs (in addition to the rotation associated with *H*
_{2}) because the polarization components grow at different rates.

The effects of PMD on degenerate FWM, in a 2 Km-long fiber with a PMD coefficient of 0.05 ps/(Km)^{1/2}, were studied recently [33]. Most of the other fiber and pump parameters were typical 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 by 80 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 fibers with lengths of order 1 Km.

## 5. Nondegenerate four-wave mixing

Nondegenerate FWM involves four waves with frequencies that satisfy the matching condition *ω*
_{2} + *ω*
_{3} = *ω*
_{4} + *ω*
_{1}. By substituting the ansatz

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

where *H*
_{1} and *H*
_{2} are defined by Eq. (30). For nondegenerate FWM the summation in Eq. (30) involves the three indices *k* ≠ *j* and *a*⃗_{t} = *a*⃗_{1} + *a*⃗_{2} + *a*⃗_{3} + *a*⃗_{4}. One can deduce the equations for waves 3 and 4 from the equations for waves 2 and 1 by interchanging the subscripts 2 and 3, and the 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

$$\phantom{\rule{4.5em}{0ex}}-\u3008{A}_{2}\mid {A}_{4}\u3009\u3008{A}_{3}\mid {A}_{1}\u3009-\u3008{A}_{3}\mid {A}_{4}\u3009\u3008{A}_{2}\mid {A}_{1}\u3009),$$

$$\phantom{\rule{4.5em}{0ex}}-\u3008{A}_{4}\mid {A}_{3}\u3009\u3008{A}_{1}\mid {A}_{2}\u3009-\u3008{A}_{1}\mid {A}_{3}\u3009\u3008{A}_{4}\mid {A}_{2}\u3009).$$

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

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

$$\phantom{\rule{15em}{0ex}}-\u3008{A}_{2}\mid {A}_{4}\u3009\u3008{A}_{3}\mid \overrightarrow{\sigma}\mid {A}_{1}\u3009-\u3008{A}_{3}\mid {A}_{4}\u3009\u3008{A}_{2}\mid \overrightarrow{\sigma}\mid {A}_{1}\u3009),$$

$$\phantom{\rule{15em}{0ex}}-\u3008{A}_{4}\mid {A}_{3}\u3009\u3008{A}_{1}\mid \overrightarrow{\sigma}\mid {A}_{2}\u3009-\u3008{A}_{1}\mid {A}_{3}\u3009\u3008{A}_{4}\mid \overrightarrow{\sigma}\mid {A}_{2}\u3009).$$

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

$$\phantom{\rule{15em}{0ex}}+\mathit{i\gamma}(\u3008{A}_{4}\mid {A}_{2}\u3009\u3008{A}_{1}\mid \overrightarrow{\sigma}\mid {A}_{3}\u3009-\u3008{A}_{2}\mid {A}_{4}\u3009\u3008{A}_{3}\mid \overrightarrow{\sigma}\mid {A}_{1}\u3009\text{}$$

$$\phantom{\rule{15em}{0ex}}+\u3008{A}_{3}\mid {A}_{1}\u3009\u3008{A}_{2}\mid \overrightarrow{\sigma}\mid {A}_{4}\u3009-\u3008{A}_{1}\mid {A}_{3}\u3009\u3008{A}_{4}\mid \overrightarrow{\sigma}\mid {A}_{2}\u3009).$$

By combining Eq. (68) with the associated equation for waves 3 and 4, one finds that

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 fiber by FWM. The linearized analysis of this process is based on the approximation that the idler does not affect the input waves, in which case

where *l* = 1, 2 or 3. The inputs are subject to different phase shifts (which depend on *P*
_{1}, *P*
_{2} and *P*
_{3}) and the same polarization rotation (which depends on *a*⃗_{t} = *a*⃗_{1}+ *a*⃗_{2} + *a*⃗_{3}), as discussed in Section 3. The idler is subject to a phase shift (which depends on *P*
_{1}, *P*
_{2} and *P*
_{3}) and the same polarization rotation as the inputs. In the notation of Section 3, let |*A*_{j}
〉 = exp(*iH*_{t}*z* + *iθ*_{j}
) |*B*_{j}
〉. Then

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 or transformed) vectors of inputs 2 and 3: The initial evolution of nondegenerate FWM does not depend on the (rigid) polarization rotation.

By integrating Eq. (73) one finds that

$$\phantom{\rule{2.4em}{0ex}}+\u3008{A}_{2}\mid {A}_{1}\u3009\u3008{A}_{1}\mid {A}_{3}\u3009\u3008{A}_{3}\mid {A}_{2}\u3009+\u3008{A}_{3}\mid {A}_{1}\u3009\u3008{A}_{1}\mid {A}_{2}\u3009\u3008{A}_{2}\mid {A}_{3}\u3009),$$

where the spatial growth factor Γ was defined in Eq. (43) and the FWM wavenumber (mismatch)

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

where each *e*⃗_{l} is a unit vector parallel to *a*⃗_{l}.

When *k* = 0 the idler power increases quadratically with distance. When *k* ≠ 0 the idler power is a periodic function of distance, with period and maxima that are inversely proportional to *k*. The nonlinear contribution to *k* 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, the idler power produced by parallel inputs in a fiber with random birefringence is lower than that produced 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 a particular combination of input polarizations, which is illustrated in Figure 3. The first column describes the output idler polarization and the second column describes the FWM power efficiency (normalized to a maximal efficiency of 1). For the polarization combinations described in the second and third rows, in which the inputs 2 and 3 are perpendicular, the idler power produced in a fiber with random birefringence is lower than that produced by parallel inputs 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 Poincaré sphere (〈*e*⃗_{j} · *e*⃗_{k}) = 0 when *k*≠*j*).

For the special case in which *k* = 0, the results listed in Table 2 were obtained by Inoue [9], who analyzed FWM in a system of concatenated, birefringent fiber segments. By applying the Jones-vector formalism to the Manakov equation (10), one simplifies the FWM analysis significantly. Inoue neglected the coherent-coupling terms ${\gamma A}_{y}^{2}$
${A}_{x}^{*}$/3 and ${\gamma A}_{x}^{2}$
${A}_{y}^{*}$/3 *a priori*. These terms were retained in the derivation of the Manakov equation. When averaged over the Poincaré sphere, they contributed to the term - *γ*〈*A*|*A*〉 |*A*〉/9, which reduced the nonlinear coefficient in the Manakov equation from *γ* to *γ*¯= 8*γ*/9. (They also eliminated the nonlinear polarization rotation associated with SPM.) Inoue used his results for nondegenerate FWM to infer the properties of degenerate FWM (|*A*
_{3}〉 = |*A*
_{2}〉). By doing so, one reproduces the results of 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 pump waves (2 and 3) are launched into the fiber and a weak idler wave (4) is generated in the fiber by FWM. This process is often called phase conjugation (PC). Since the idler power quickly becomes comparable to the signal power, the effects of the idler on the signal cannot be neglected, in which case

where *l* = 2 or 3. The pumps are subject to different phase shifts (which depend on *P*
_{2} and *P*
_{3}) and the same polarization rotation (which depends on *a*⃗_{t} = *a*⃗_{2} + *a*⃗_{3}). The signal and idler are subject to the same phase shift (which depends on *P*
_{2} and *P*
_{3}) and polarization rotation (which depends on *a*⃗_{t}). Let |*A*
_{1}〉 = exp(*iH*
_{2}
*z*)|*B*
_{1}〉, |*A*
_{l}〉 = exp(*iH*_{l}*z*)|*B*
_{1}〉 and |*A*
_{4}〉 = exp(*iH*
_{3}
*z*)|*B*
_{4}〉. Then

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

By defining the wavenumber shift *δk*
_{1} = *β*
_{1} - *β*
_{2} + *γP*
_{2} and the operator *D*
_{1}=*D* - *iδk*
_{1}, and making similar definitions for wave 4, one can rewrite Eqs. (80) and (82) in the compact form

By combining Eqs. (83) and (84), one finds that

$$+\u3008{B}_{2}\mid {B}_{3}\u3009\mid {B}_{2}\u3009\u3008{B}_{3}\mid +\u3008{B}_{3}\mid {B}_{2}\u3009\mid {B}_{3}\u3009\u3008{B}_{2}\mid )]\u3008{B}_{1}\mid =0.$$

Let |*E*
_{∥}〉 and |*E*
_{⊥}〉 be unit vectors parallel and perpendicular to |*B*
_{2}〉, and (with the subscript 3 omitted for brevity) let |*B*
_{3}〉/${P}_{3}^{1/2}$ = *B*
_{∥}|*E*
_{∥}〉 + *B*
_{⊥}|*E*
_{∥}〉 and (since wave 1 is the signal)
|*B*
_{1}〉/${P}_{1}^{1/2}$ = *S*
_{∥}|*E*
_{∥}〉 + *S*
_{⊥}|*E*
_{∥}〉. Then

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

where

are the normalized eigenvalues of the coupling matrix [associated with the projection operators in Eq. (85)]. Each eigenvalue (subscript + or -) is associated with a second-order (differential or polynomial) equation, to which both signs (non-subscript + and -) on the right side of Eq. (87) apply. Because Δ_{±} ≥ 0, the PC process is potentially unstable. By defining the linear wavenumber mismatch *δβ* = *β*
_{1} - *β*
_{2} - *β*
_{3} + *β*
_{4})/2, one obtains from Eq. (87) the growth-rate formula

Notice that the wavenumber mismatch *δβ* + *γ*(*P*
_{2} + *P*
_{3})/2 is the *P*
_{1} → 0 limit of the mismatch (75). It follows from Eqs. (86)–(88) that

and, hence, (for normalized signal eigenvectors) that

The wavenumber mismatch depends on the pump powers, but not the pump polarizations. In contrast, the determinant of the coupling matrix and the signal-eigenvector polarizations depend 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 factors exp[±*i*(*δk*
_{4} - *δk*
_{2})*z*/2], which do not alter |*I*
_{∥}| and |*I*
_{⊥}|.) Thus, Eqs. (90) and (91) also apply to the idler. Each signal-idler eigenmode evolves independently of the other, in the same way that the 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

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

First, consider the case in which the pumps are parallel (*B*
_{∥} = 1 and *B*
_{⊥} = 0). In this case there is no pump-pump polarization rotation. The eigenvalues Δ_{±} = 4, 0. The + root (4) is associated with signal and idler eigenvectors that are parallel to the (common) pump vector and, hence, do not rotate. The signal and idler wavenumbers are *k*
_{1} = *β*
_{1} - *δβ* + 3*γ*(*P*
_{2} + *P*
_{3})/2 and *k*
_{4} = *β*
_{4} - *δβ* + 3*γ*(*P*
_{2} + *P*
_{3})/2, respectively. When the mismatch is optimal [*δβ* = -*γ*(*P*
_{2} + *P*
_{3})/2] the maximal growth rate *κ*= 2*γ*(*P*
_{2}
*P*
_{3})^{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 coefficient modified by a factor of 8/9. The - root (0) is associated with signal and idler eigenvectors that are perpendicular to the pump vector and, hence, do not rotate. They propagate independently and stably, with wavenumbers *β*
_{1} + 3*γ*(*P*
_{2} + *P*
_{3})/2 and *k*
_{4} = *β*
_{4} + 3*γ*(*P*
_{2} + *P*
_{3})/2, respectively. For short distances (before the idler power is comparable to the signal power), the idler produced by two pumps and a weak signal should grow in the same way as an idler produced by two pumps and a strong signal [Eq. (76)]. The results described in this paragraph are consistent with the first and fourth rows of Table 2. (See Figures 3*a*, 3*d* and 4.)

Second, consider the case in which the pumps are perpendicular (*B*
_{∥} = 0 and *B*
_{⊥} = 1). In this case there is also no pump-pump polarization rotation. The eigenvalues Δ_{±} = 1, 1 and the propagation matrix in Eq. (86) is diagonal, with both elements equal to ${D}_{4}^{*}$
*D*
_{1} - *γ*
^{2}
*P*
_{2}
*P*
_{3}. Because of this degeneracy, the signal gain does not depend on the signal polarization. It should be re-emphasized that the term perpendicular is used in the sense of Jones vectors: The preceding results apply equally to pumps with perpendicular linear polarizations, circular polarizations with opposite helicity or any other orthogonal polarizations. For some input signal polarizations the (original) signal and idler vectors will rotate, whereas for others they will not. The special cases in which the signal is parallel or perpendicular to the pumps are illustrated in Figure 5. When the mismatch is optimal [*δβ* = -*γ*(*P*
_{2} + *P*
_{3})/2] the maximal growth rate *κ* = *γ*(*P*
_{2}
*P*
_{3})^{1/2} . For the special case in which the waves are linearly polarized, these results are similar to the standard perpendicular-polarization results [8]. (To be precise, the SPM coefficient, which determines the nonlinear wavenumber mismatch, changes from 1 to 8/9 and the CPM coefficient, which determines the coupling strength, changes from 2/3 to 8/9.) The results described in this paragraph are consistent with the second and third rows of Table 2. (See Figures 3b, 3c and 5.)

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

In the third case a weak signal wave (2) and two powerful pump waves (1 and 3) are launched into the fiber and a weak idler wave (4) is generated in the fiber by FWM [34, 35]. We call this process Bragg scattering (BS). (The terminology comes from a spatial analog of this process [36], in which a wave with parallel wavevector **k**, that is incident on a grating with wavevector -2**k**, is reflected as a wave with parallel wavevector -**k**.) Since the idler power quickly becomes comparable to the signal power, the effects of the idler on the signal cannot be neglected, in which case

where *l* = 1 or 3. The pumps are subject to different phase shifts (which depend on *P*
_{1} and *P*
_{3}) and the same polarization rotation (which depends on *a*⃗_{t} = *a*⃗_{1} + *a*⃗_{3}). The signal and idler are subject to the same phase shift (which depends on *P*
_{1} and *P*
_{3}) and polarization rotation (which depends on *a*⃗_{t}). Let |*A*_{l}
〉 = exp(*iH*_{l}*z*)|*B*_{l}
〉, |*A*
_{2}〉 = exp(*iH*
_{1}
*z*)|*B*
_{2}〉, and |*A*
_{4}〉 = exp(*iH*
_{3}
*z*)|*B*
_{4}〉. Then

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

By defining the wavenumber shift *δk*
_{2} = *β*
_{2} - *β*
_{1} + *γP*
_{1} and the operator *D*
_{2} = *D* - *iδk*
_{2}, and making similar definitions for wave 4, one can rewrite Eqs. (99) and (100) in the compact form

By combining Eqs. (101) and (102), one finds that

$$+\u3008{B}_{3}\mid {B}_{1}\u3009\mid {B}_{3}\u3009\u3008{B}_{1}\mid +\u3008{B}_{1}\mid {B}_{3}\u3009\mid {B}_{1}\u3009\u3008{B}_{3}\mid )]\u3008{B}_{2}\mid =0.$$

Let |*E*
_{∥}〉 and |*E*
_{⊥}〉 be unit vectors parallel and perpendicular to |*B*
_{1}〉, and (with the subscript 3 omitted for brevity) let |*B*
_{3}〉/${P}_{3}^{1/2}$ = *B*
_{∥}|*E*
_{∥}〉 + *B*
_{⊥}|*E*
_{⊥}〉 and (since wave 2 is the signal) |*B*
_{2}〉/${P}_{2}^{1/2}$ = *S*
_{∥} |*E*
_{∥}〉 + *S*
_{⊥}|*E*
_{⊥}〉. Then

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

where

are the normalized eigenvalues of the coupling matrix. Because Δ_{±} ≤ 0, the BS process is intrinsically stable. Notice that the wavenumber mismatch (*δk*
_{4} - *δk*
_{2})/2 is the *P*
_{2} → 0 limit of the mismatch (75). It follows from Eqs. (104)–(106) that

and, hence, (for normalized signal eigenvectors) that

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

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

$$+\u3008{B}_{1}\mid {B}_{3}\u3009\mid {B}_{1}\u3009\u3008{B}_{3}\mid +\u3008{B}_{3}\mid {B}_{1}\u3009\mid {B}_{3}\u3009\u3008{B}_{1}\mid )]\mid {B}_{2}\u3009=0.$$

Equation (109) differs slightly from Eq. (103). Let |*B*
_{4}〉 / ${P}_{4}^{1/2}$ = *I*
_{∥}|*E*
_{∥}〉 + *I*
_{⊥}|*E*
_{⊥}〉. Then

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

and, hence, (for normalized idler eigenvectors) that

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

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

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 and *B*
_{⊥} = 0). In this case there is no pump-pump polarization rotation. The eigenvalues Δ_{±} = - 1, -4. The + root (-1) is associated with signal and idler eigenvectors that are perpendicular to the (common) pump vector and, hence, do not rotate. The - root (-4) is associated with signal and idler eigenvectors that are parallel to the pump vector and, hence, do not rotate. Formulas for the signal and idler powers are stated in [6]. For short distances (before the idler power is comparable to the signal power), the idler produced by two pumps and a weak signal should grow in the same way as an idler produced by two pumps and a strong signal [Eq. (76)]. The results described in this paragraph are consistent with the first and third rows of Table 2. (See Figures 3a, 3c and 6.)

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

## 6. Summary

The parametric amplification (PA) of optical signals is made possible by four-wave mixing (FWM). In low-birefringence fibers the birefringence axes and strength vary randomly with distance. Wave propagation in such fibers is governed by the Manakov equation. In this paper the Manakov equation was used to make a detailed study of degenerate and nondegenerate FWM. Inoue’s analysis of FWM in fibers with random birefringence [9] was extended by the inclusion of the effects of linear and nonlinear wavenumber mismatches [which are caused by self-phase modulation (SPM) and cross-phase modulation (CPM)], and nonlinear polarization rotation (which is caused by CPM).

Equations were derived from the Manakov equation, which govern the initial (linear) and final (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 powers is independent of the pump-induced rotation of the signal and idler polarization vectors. Formulas were derived for the initial quadratic growth of the idler power [(45) and (76)], and the subsequent exponential growth of the signal and idler powers [(55)–(57) and (86)–(89)]. These formulas are valid for arbitrary pump powers and polarizations. Signal amplification and idler generation are made possible 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 the pump powers.

At present, there is considerable interest in PA driven by two pumps (nondegenerate FWM). Let the frequencies *ω*
_{1} < *ω*
_{2} < *ω*
_{3} < *ω*
_{4}. Then, if the signal and idler frequencies are located symmetrically 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 is potentially unstable (signal gain is possible). In contrast, if the signal and idler frequencies are located asymmetrically (for example, if waves 1 and 3 are the pumps, and waves 2 and 4 are signal and idler), the process is called Bragg scattering (BS) and is intrinsically stable (no signal gain is possible). The polarization dependence of both processes was studied in detail (for the linear regime). For (nondegenerate) PC in a fiber with constant (high) birefringence, the maximal growth rate is associated with parallel pumps and signal. Let this rate [2γ(*P*
_{2}
*P*
_{3})^{1/2}] be normalized to 2. Then the growth rate associated with parallel pumps and a perpendicular signal is 0. The growth rate associated with perpendicular pumps has a maximum of 2/3 ≈ 0.67 and depends sensitively on the signal polarization [8]. For PC in a fiber with random birefringence, 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 growth rate associated with perpendicular pumps is 8/9 ≈ 0.89 and does not depend on the signal polarization: PC driven by perpendicular pumps in a fiber with random birefringence exhibits the signal-polarization insensitivity required of a practical amplification process. Although the BS 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 pump polarizations: BS does not exhibit the signal-polarization insensitivity required of a practical frequency-conversion process. Detailed measurements are being made to determine how well the 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 discussed elsewhere.

## 7. Appendix: Eigenvectors for nondegenerate four-wave mixing

In Section 5 the ratios *S*
_{⊥}/*S*
_{∥} and *I*
_{⊥}/*I*
_{∥} associated with the PC and BS eigenvectors were determined. In this appendix the complete PC and BS eigenvectors are determined.

First, consider PC, which is governed by Eqs. (80)–(82). Let *ϕ*
_{2} and *ϕ*
_{3} denote the initial phases of the pumps, and let *δ*
_{±} denote the wavenumber mismatches (*δk*
_{1} ± *δk*
_{4})/2. Furthermore, let |*B*
_{1}〉 = exp(*iδ*
_{-z} + *iϕ*
_{2})|*C*
_{1}〉, |*B*
_{2}〉 = exp(*iϕ*
_{2})|*C*
_{2}〉, |*B*
_{3}〉 = exp(*iϕ*
_{3})|*C*
_{3}〉 and |*B*
_{4}〉 = exp(-*iδ*
_{-z} + *iϕ*
_{3})|*C*
_{4}〉. Then the pump vectors |*C*
_{2}〉 = ${P}_{2}^{1/2}$ and |*C*
_{3}〉 = P_{3}
^{1/2} , and the signal and idler vectors are governed by the equations

where *D*
_{±} = *D*
_{±}
*iδ*
_{+}. If one decomposes each vector into components that are parallel and perpendicular to pump 2, as described after Eq. (85), one finds that

where *P* = (*P*
_{2}
*P*
_{3})^{1/2}, and *B*
_{∥} and *B*
_{⊥} are real. One can obtain the idler equations from the signal equations by interchanging *S* and *I*, and taking the complex conjugate of the equations that result. It follows from this observation that *I*
_{∥} can only differ from *S*
_{∥} by a phase factor, and the relation between ${I}_{\parallel}^{*}$ and ${I}_{\perp}^{*}$ must be the conjugate of the relation between *S*
_{∥} and *S*
_{⊥}.

Let *κ* denote a PC eigenvalue (growth rate) and let Δ = (*κ*
^{2} + ${\delta}_{+}^{2}$)/*γ*
^{2}
*P*
^{2}. Then the PC eigenvalues

where the associated eigenvalues Δ_{±} are the solutions (roots) of the characteristic equation

It follows from Eq. (120) that

in agreement with Eq. (88). Suppose that *S*
_{∥} = 1. Then it follows from Eqs. (118)–(121) that

Equation (122) is consistent with Eq. (90). It follows from Eqs. (121) and (123) that |*I*
_{∥}${|}_{\pm}^{2}$ = 1, and it follows from Eqs. (121), (123) and (124) that ${I}_{\perp}^{*}$/${I}_{\parallel}^{*}$ = ±*B*
_{⊥}/(1 ± *B*
_{∥}). These results are consistent with the predictions made after Eq. (118) and the results of Section 4.

For the special case in which *B*
_{∥} = 0, it follows from Eq. (118) that *S*
_{∥} is coupled to ${I}_{\perp}^{*}$ and *S*
_{⊥} is coupled to ${I}_{\parallel}^{*}$. Because *S*
_{∥} and *S*
_{⊥} are uncoupled, the natural choice of eigenvectors for the signal 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 are not unique. In the natural basis each signal eigenvector is perpendicular to the associated idler eigenvector. However, a signal vector that represents a superposition of eigenstates is not perpendicular to the associated idler vector.

Second, consider BS, which is governed by Eqs. (98)–(100). Let *ϕ*
_{1} and *ϕ*
_{3} denote the initial phases of the pumps, and let *δ*
_{±} denote the wavenumber mismatches (*δk*
_{2} ± *δk*
_{4})/2. Furthermore, let |*B*
_{1}〉 = exp(*iϕ*
_{1})|*C*
_{1}〉, |*B*
_{2}〉 = exp(*iδ*
_{+z} + *iϕ*
_{1})|*C*
_{2}〉, |*B*
_{3}〉 = exp(*iϕ*
_{3})|*C*
_{3}〉 and |*B*
_{4}〉 = exp(*iδ*
_{+z} + *iϕ*
_{3})|*C*
_{4}〉. Then the pump vectors |*C*
_{1}〉 = ${P}_{1}^{1/2}$ and |*C*
_{3}〉 = ${P}_{3}^{1/2}$ , and the signal and idler vectors are governed by the equations

where *D*
_{±} = *D*±*iδ*
_{-}. If one decomposes each vector into components that are parallel and perpendicular to pump 2, as described after Eq. (103), one finds that

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 Δ = (*k*
^{2} - *δ*
^{2})/*γ*
^{2}
*P*
^{2}. Then the BS eigenvalues

where the associated eigenvalues Δ_{±} are the roots of the characteristic equation

It follows from Eq. (129) that

Although the sign of Δ in this appendix is the opposite of the sign in Section 4, the ∓ on the right side of Eq. (130) ensures that the root identification is the same. [See Eq. (106).] Suppose that *S*
_{∥} = 1. Then it follows from Eqs. (127)–(130) that

By using Eq. (129), written in the form (Δ - ${B}_{\parallel}^{2}$)(Δ - 1 - 3${B}_{\parallel}^{2}$) = ${B}_{\parallel}^{2}$ (1 - ${B}_{\parallel}^{2}$), one can show that Eq. (131) is consistent with Eq. (107). It follows from Eqs. (129), (132) and (133) that (*I*
_{⊥}/*I*
_{∥})_{±} = 2*B*
_{∥}
*B*
_{⊥}(Δ_{±} - 1). By using Eq. (129), written in the form (Δ - 1)(Δ - 4${B}_{\parallel}^{2}$) = 4${B}_{\parallel}^{2}$(1 - ${B}_{\parallel}^{2}$), one can show that this result is consistent with Eq. (111).

## Acknowledgments

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

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