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)ωⁿ /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(∓4iδk 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 2π/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)ωⁿ /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)ωⁿ /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 a⃗₁ and a⃗₂ 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¯ = 8γ/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 + a⃗·σ⃗)/2, where σ⃗ is the Pauli spin operator and a⃗ = 〈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 a⃗·σ⃗|A〉 = P|A〉, one can rewrite operator (15) in the equivalent form

where the total Stokes vector a⃗_t = a⃗₁ + a⃗₂.

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 [σ⃗,H_j ] denotes the commutator σ⃗H_j - H_j σ⃗. By using the identities [σ⃗,I] = 0 and [σ⃗, a⃗·σ⃗] = 2ia⃗ × σ⃗, one finds that [σ⃗,H_j ] = -iγσ⃗ × a⃗_t and, hence, that

Had we used operator (15) rather than operator (16), we would have obtained the standard equation D a⃗_j = a⃗_j × a⃗_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_jP_k + a⃗_j·a⃗_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 = γa⃗_t · σ⃗/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_tz) 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 a⃗_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 2ω 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 a⃗_t = a⃗₁ + a⃗₂ + a⃗₃. 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

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 a⃗_t = a⃗₁ + a⃗₂), 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_tz + 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 e⃗₁ and e⃗₂ are unit vectors parallel to a⃗₁ and a⃗₂, 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.

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

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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 (〈e⃗₁ · e⃗₂〉 = 0).

 

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 that no idler is produced. Figures 1a and 1b correspond to rows 1 and 2 of Table 1, respectively.

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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 a⃗_t = a⃗₂ and |A ₂〉 is an eigenvector of a⃗₁ · σ⃗). The signal and idler are subject to the same phase shift (which depends on P ₂) and polarization rotation (which depends on a⃗₂). 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 - 2β 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(iγa⃗₂ · σ⃗z/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 1a and 2a.)

 

Fig. 2. Eigenpolarizations of MI. The dashed lines denote sidebands that propagate independently.

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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(iγa⃗₂·σ⃗z/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 a⃗_t = a⃗₁ + a⃗₂ + a⃗₃ + a⃗₄. 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

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

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

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

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 a⃗_t = a⃗₁+ a⃗₂ + a⃗₃), 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_tz + 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

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 e⃗_l is a unit vector parallel to a⃗_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 (〈e⃗_j · e⃗_k) = 0 when k≠j).

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

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Fig. 3. Polarization diagrams for nondegenerate FWM driven by three input waves. The symbols ∥ 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, respectively.

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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¯= 8γ/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 a⃗_t = a⃗₂ + a⃗₃). 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 a⃗_t). Let |A ₁〉 = exp(iH ₂ z)|B ₁〉, |A _l〉 = exp(iH_lz)|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

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 + 3γ(P ₂ + P ₃)/2 and k ₄ = β ₄ - δβ + 3γ(P ₂ + P ₃)/2, respectively. When the mismatch is optimal [δβ = -γ(P ₂ + P ₃)/2] the maximal growth rate κ= 2γ(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 + 3γ(P ₂ + P ₃)/2 and k ₄ = β ₄ + 3γ(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 3a, 3d and 4.)

 

Fig. 4. Eigenpolarizations of PC driven by parallel pumps. The dashed lines denote sidebands that propagate independently.

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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.)

 

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

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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 -2k, 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 a⃗_t = a⃗₁ + a⃗₃). 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 a⃗_t). Let |A_l 〉 = exp(iH_lz)|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

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