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 Ex and Ey 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 = [βx (ω 0)+βy (ω 0)]/2, where each β(ω 0) is a natural wavenumber (dispersion function), and to write the field components as

Ex(t,z)=Ax(t,z)exp[i(k0zω0t)]+c.c.,
Ey(t,z)=Ay(t,z)exp[i(k0zω0t)]+c.c.

The evolution of the wave amplitudes (polarization components) Ax and Ay is governed by the coherently-coupled NS equations

izAx=βx(it)Ax+γ(Ax2+2Ay23)Ax+γAy2Ax*3,
izAy=βy(it)Ay+γ(2Ax23+Ay2)Ay+γAx2Ay*3,

where βx and βy are modified dispersion functions and γ is the nonlinearity coefficient [17]. In the frequency domain each β(ω) = Σn=0=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 Ax = Bx exp(iδk 0 z) and Ay = By exp(-iδk 0 z), where δk 0 = [βx (ω 0) -βy (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 δn in the range 10-7-10-5 [22]. The wavenumber shift δk 0 = (ω 0 δn /c and the beat length 2π/δk 0 = λ 0/δn . 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 (δn ≫ 10-7) is governed by the incoherently-coupled NS equations

izBx=βx(it)Bx+γ(Bx2+2By23)Bx,
izBy=βy(it)By+γ(2Bx23By2)By,

where each β(ω) = Σn=1=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 βx(n) (ω 0) = βy(n) (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 (δn ~ 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

izAξ=β(it)Aξ+(8γ9)(Aξ2+Aη2)Aξ,
izAη=β(it)Aη+(8γ9)(Aξ2+Aη2)Aη,

where β(ω) = Σn=2=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 a1 and a2 denote the Stokes vectors of waves with frequencies ω 1 and ω 2, respectively, and let 〈〉 denotes an ensemble average. Then the polarization correlation

a1·a2=a1·a2inexp(δω2δτ23),

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〉 = [Aξ ,Aη ]T, one can rewrite Eqs. (7) and (8) in the compact form

izA=β(it)A+γ¯AAA,

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

A=exp(iω1t)A1+exp(iω2t)A2

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

DA1=iβ1A1+iγ[(P1+P2)A1+A2A1A2],
DA2=iβ2A2+iγ[A1A2A1+(P1+P2)A2],

where D = d/dz, j = 1 or 2, the wavenumber βj =β(ω j) and the power Pj = 〈AjAj 〉. 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

DAj=iHjAj,

where the (hermitian) operator

Hj=[βj+γ(Pj+3Pk2)]I+γak·σ2

and kj. In Eq. (15) the term γPj represents SPM, whereas the term γPk represents CPM. By using the identity a⃗·σ⃗|A〉 = P|A〉, one can rewrite operator (15) in the equivalent form

Hj=[βj+γ(Pj+3Pk2)]I+γat·σ2,

where the total Stokes vector at = a1 + a2.

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

DAjAk=iAj(HkHj)Aj
=i[βkβj+γ(PjPk)]AjAk.

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

DAjσAj=iAj[σ,Hj]Aj,

where [σ⃗,Hj ] denotes the commutator σHj - Hj σ⃗. By using the identities [σ⃗,I] = 0 and [σ⃗, a⃗·σ⃗] = 2ia⃗ × σ⃗, one finds that [σ⃗,Hj ] = -iγσ⃗ × at and, hence, that

Daj=aj×at.

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

Dat=0,
D(aj·ak)=0.

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 |〈Aj |Ak 〉|2=(PjPk + aj·ak)/2.

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

θj(z)=βjz+γ0z[Pj(z)+3Pk(z)]dz2.

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

DBj=iHtBj,

where the (common) operator Ht = γat · σ⃗/2. Since Ht is hermitian, there exists a (common) unitary operator Ut such that

Bj(z)=Ut(z)Bj(0).

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

Aj(z)=exp(iHtz+iθj)Aj(0).

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 kj and the two-vector sum in at 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

A=j=13exp(iωjt)Aj

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

DA1=iH1A1+A3A2A2,
DA2=iH2A2+(A2A3A1+A2A1A3).

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

Hj=[βj+γ(Pj+3ΣPk2)]I+γat·σ2,

where the summation involves the two indices kj and at = a1 + a2 + a3. 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

DA1A1=(A3A2A1A2A2A3A2A1),
DA2A2=2(A2A3A2A1A3A2A1A2).

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

D(P1P3)=0,
D(P1+P2+P3)=0.

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¯ωj 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 ωj are much smaller than the reference (carrier) frequency ω 0 and the concomitant approximation that the nonlinearity coefficient γ does not depend on ωj . Had we used the Maxwell equations to derive Eqs. (28) and (29) directly, the nonlinear terms would have included factors of 1 + ωj /ω 0 [27]. We would have eliminated these factors from Eqs. (31) and (32) by rewriting them in terms of the photon fluxes Fj = Pj /h¯(ω 0 + ωj ), and would have obtained the MRW equations [(33) and (34), with Pj replaced by Fj ]. By combining the MRW equations with the frequency-matching condition, we would have confirmed that the total power is conserved exactly. In typical experiments |ωj )/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

DA1σA1iA1[σ,Ht]A1=(A3A2A1σA2A2A3A2σA1),
DA2σA2iA2[σ,Ht]A2=(A2A3A2σA1A3A2A1σA2
+A2A1A2σA3A1A2A3σA2).

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

Dat=0.

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 ωj . 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

DAl=iHlAl,
DA3=iH3A3+A1A2A2,

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 at = a1 + a2), 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 |Aj 〉 = exp(iHtz + j ) |Bj 〉. Then

DBl=0,
DB3=B1B2B2exp[i(2θ2θ3θ1)].

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

P3(z)=Γ(z)P2A1A22,

where the spatial growth factor

Γ(z)=γ2sin2(kz)k2

and the FWM wavenumber (mismatch)

k=[β12β2+β3+γ(2P2P1)]2

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

P3(z)=Γ(z)P1P22(1+e1·e2)2,

where e1 and e2 are unit vectors parallel to a1 and a2, 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.

Tables Icon

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

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 (〈e1 · e2〉 = 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

DA1=iH1A1+A3A2A2,
DA2=iH2A2,
DA3=iH3A3+A1A2A2.

The pump is subject to a phase shift (which depends on P 2), but no polarization rotation (because at = a2 and |A 2〉 is an eigenvector of a1 · σ⃗). The signal and idler are subject to the same phase shift (which depends on P 2) and polarization rotation (which depends on a2). Let |Aj 〉 = exp(iH 2 z)|Bj 〉). Then

DB1=i(β1β2+γP2)B1+B3B2B2,
DB2=0.
DB3=i(β3β2+γP2)B3+B1B2B2.

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

D1B1=B3B2B2,
D3*B3=B2B1B2.

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

(D3*D1Iγ2P2B2B2)B1=0.

Let |E 〉 and |E 〉 be unit vectors parallel and perpendicular to |B 2〉, and (since wave 1 is the signal) let |B 1〉/P11/2 = S |E 〉 + S |E 〉. Then, by using this decomposition, one can rewrite Eq. (54) in the matrix form

[D3*D1γ2P2200D3*D1][SS]=0.

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

k±=(δk1δk3)2±[(δk1+δk3)24γ2P22]12,

which are the roots (eigenvalues) of the characteristic equation (k - δk 1)(k + δk 3) + γ 2 P22 = 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

κ=[γ2P22(δβ+γP2)2]12.

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(iγa2 · σz/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 kj , a contribution γP 2 comes from δkj , 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(iγa2·σz/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

A=j=14exp(iωjt)Aj

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

DA1=iH1A1+(A4A2A3+A4A3A2),
DA2=iH2A2+(A3A4A1+A3A1A4),

where H 1 and H 2 are defined by Eq. (30). For nondegenerate FWM the summation in Eq. (30) involves the three indices kj and at = a1 + a2 + a3 + a4. 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

DA1A1=(A4A2A1A3+A4A3A1A2
A2A4A3A1A3A4A2A1),
DA2A2=(A3A4A2A1+A3A1A2A4
A4A3A1A2A1A3A4A2).

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

D(P1P4)=0,
D(P2P3)=0,
D(P1+P2+P3+P4)=0,

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

DA1σA1γA1σ×atA1=(A4A2A1σA3+A4A3A1σA2)
A2A4A3σA1A3A4A2σA1),
DA2σA2γA2σ×atA2=(A3A4A2σA1+A3A1A2σA4)
A4A3A1σA2A1A3A4σA2).

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

D(A1σA1+A2σA2)=γ(A1σ×atA1+A1σ×atA1)
+(A4A2A1σA3A2A4A3σA1
+A3A1A2σA4A1A3A4σA2).

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

Dat=0.

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

DAl=iHlAl,
DA4=iH4A4+(A1A2A3+A1A3A2),

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 at = a1+ a2 + a3), 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 |Aj 〉 = exp(iHtz + j ) |Bj 〉. Then

DBl=0,
DB4=(B1B2B3+B1B3B2)exp[i(θ2+θ3θ4θ1)].

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

P4(z)=Γ(z)(P2A3A12+P3A1A22
+A2A1A1A3A3A2+A3A1A1A2A2A3),

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

k=[β1β2β3+β4+γ(P2+P3P1)]2.

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

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

where each el is a unit vector parallel to al.

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 (〈ej · ek) = 0 when kj).

Tables Icon

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

 

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 γAy2 Ax*/3 and γAx2 Ay*/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 - γA|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

DA1=iH1A1+(A4A2A3+A4A3A2),
DAl=iHlAl,
DA4=iH4A4+(A1A2A3+A1A3A2),

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 at = a2 + a3). 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 at). Let |A 1〉 = exp(iH 2 z)|B 1〉, |A l〉 = exp(iHlz)|B 1〉 and |A 4〉 = exp(iH 3 z)|B 4〉. Then

DB1=i(β1β2+γP2)B1+(B4B2B3+B4B3B2),
DBl=0,
DB4=i(β4β3+γP3)B4+(B1B2B3+B1B3B2).

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

D1B1=(B4B2B3+B4B3B2),
D4*B4=(B2B1B3+B3B1B2).

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

[D4*D1Iγ2(P2B3B3+P3B2B2
+B2B3B2B3+B3B2B3B2)]B1=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〉/P31/2 = B |E 〉 + B |E 〉 and (since wave 1 is the signal) |B 1〉/P11/2 = S |E 〉 + S |E 〉. Then

[D4*D1γ2P2P3(1+3B2)2γ2P2P3BB*2γ2P2P3B*BD4*D1γ2P2P3B2][SS]=0.

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

k±=(δk1δk4)2±[(δk1+δk4)24γ2P2P3Δ±]12,

where

Δ±=(1±B)2

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

κ±={γ2P2P3Δ±[δβ+γ(P2+P3)2]2}12.

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

SS±2=[(1B)(1+B)]±1

and, hence, (for normalized signal eigenvectors) that

S±2=(1±B)2.

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

Δ±=(3+e2·e3)2±[2(1+e2·e3)]12,
(e1·e2)±=±[(1+e2·e3)2]12,
(e4·e2)±=±[(1+e2·e3)2]12.

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 D4* 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

DAl=iHlAl,
DA2=iH2A2+(A3A4A1+A3A1A4),
DA4=iH4A4+(A1A2A3+A1A3A2),

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 at = a1 + a3). 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 at). Let |Al 〉 = exp(iHlz)|Bl 〉, |A 2〉 = exp(iH 1 z)|B 2〉, and |A 4〉 = exp(iH 3 z)|B 4〉. Then

DBl=0,
DB2=i(β2β1+γP1)B2+(B3B4B1+B3B1B4),
DB4=i(β4β3+γP3)B4+(B1B2B3+B1B3B2).

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

D2B2=(B3B4B1+B3B1B4),
D4B4=(B1B2B3+B1B3B2).

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

[D4D2I+γ2(P3B1B1+B1B32
+B3B1B3B1+B1B3B1B3)]B2=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〉/P31/2 = B |E 〉 + B |E 〉 and (since wave 2 is the signal) |B 2〉/P21/2 = S |E 〉 + S |E 〉. Then

[D4D2+γ2P1P3(1+3B2)γ2P1P3BB*γ2P1P3B*BD4D2+γ2P1P3(B2)][SS]=0.

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

k±=(δk2+δk4)2±[(δk2δk4)24γ2P1P3Δ±]12,

where

Δ±=[(1+4B2)±(1+8B2)12]2

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

SS±2=(1+8B2)12±(1+2B2)(1+8B2)12(1+2B2)

and, hence, (for normalized signal eigenvectors) that

S±2=(1+8B2)12(1+2B2)2(1+8B2)12.

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

[D2D4I+γ2(P1B3B3+B3B12
+B1B3B1B3+B3B1B3B1)]B2=0.

Equation (109) differs slightly from Eq. (103). Let |B 4〉 / P41/2 = I |E 〉 + I |E 〉. Then

[D2D4+4γ2P1P3B22γ2P1P3BB*2γ2P1P3B*BD2D4+γ2P1P3][II]=0.

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

II±2=(1+8B2)12±(4B21)(1+8B2)12(4B21)

and, hence, (for normalized idler eigenvectors) that

I±2=(1+8B2)12(4B21)2(1+8B2)12.

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

Δ±=[(3+2e1·e3)±(5+4e1·e3)12]2,
(e2·e1)±=(2+e1·e3)(5+4e1·e3)12·
(e4·e1)±=(1+2e1·e3)(5+4e1·e3)12.

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.)

 

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

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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± = 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.)

 

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

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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( -z + 2)|C 1〉, |B 2〉 = exp( 2)|C 2〉, |B 3〉 = exp( 3)|C 3〉 and |B 4〉 = exp(- -z + 3)|C 4〉. Then the pump vectors |C 2〉 = P21/2 and |C 3〉 = P3 1/2 , and the signal and idler vectors are governed by the equations

DC1=(C4C2C3+C4C3C2),
D+C4=(C2C1C3+C3C1C2),

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

[D02iγPBiγPB0DiγPB02iγPBiγPBD+0iγPB00D+][SSI*I*]=0,

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* and I* must be the conjugate of the relation between S and S .

Let κ denote a PC eigenvalue (growth rate) and let Δ = (κ 2 + δ+2)/γ 2 P 2. Then the PC eigenvalues

κ±=±(γ2P2Δ±δ+2)12,

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

Δ22(1+B2)Δ+B4=0.

It follows from Eq. (120) that

Δ±=(1±B)2,

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

(S)±=±B1±B,
(I*)±=i(κ±iδ+)γP(1+B),
(I*)±=iγPBκ±+iδ+.

Equation (122) is consistent with Eq. (90). It follows from Eqs. (121) and (123) that |I |±2 = 1, and it follows from Eqs. (121), (123) and (124) that I*/I* = ±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* and S is coupled to I*. 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( 1)|C 1〉, |B 2〉 = exp( +z + 1)|C 2〉, |B 3〉 = exp( 3)|C 3〉 and |B 4〉 = exp( +z + 3)|C 4〉. Then the pump vectors |C 1〉 = P11/2 and |C 3〉 = P31/2 , and the signal and idler vectors are governed by the equations

DC2=(C3C4C1+C3C1C4),
D+C4=(C1C2C3+C1C3C2),

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

[D02iγPBiγPB0D0iγPB2iγPB0D+0iγPBiγPB0D+][SSII]=0.

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

k±=±(γ2P2Δ±+δ2)12,

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

Δ2(1+4B2)Δ+4B4=0.

It follows from Eq. (129) that

Δ±=[1+4B2(1+8B2)12]2.

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

(S)±=BBΔ±B2,
(I)±=2γPBk±+δ,
(I)±=(k±δ)BγP(Δ±B2).

By using Eq. (129), written in the form (Δ - B2)(Δ - 1 - 3B2) = B2 (1 - B2), one can show that Eq. (131) is consistent with Eq. (107). It follows from Eqs. (129), (132) and (133) that (I /I )± = 2B B ± - 1). By using Eq. (129), written in the form (Δ - 1)(Δ - 4B2) = 4B2(1 - B2), 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].

References and links

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4. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002). [CrossRef]  

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

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

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

8. C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003). [CrossRef]  

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

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

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

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

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

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

15. S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

16. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

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

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

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

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

21. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996). [CrossRef]  

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

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

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

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

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

27. R. W. Boyd, Nonlinear Optics (Academic, San Diego,1992), Sections 1.4, 2.3 and 6.5.

28. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956). [CrossRef]  

29. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

30. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993). [CrossRef]  

31. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993) and references therein. [CrossRef]  

32. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998). [CrossRef]  

33. Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.

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

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

36. G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt. Soc. Am. B 9, 1047–1061 (1992). [CrossRef]  

References

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  • |

  1. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
    [Crossref]
  2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
    [Crossref]
  3. J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
    [Crossref]
  4. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
    [Crossref]
  5. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002) and references therein.
    [Crossref]
  6. C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 and 956 (2002) and references therein.
    [Crossref]
  7. C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
    [Crossref] [PubMed]
  8. C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
    [Crossref]
  9. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
    [Crossref]
  10. R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
    [Crossref]
  11. K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).
    [Crossref]
  12. K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
    [Crossref]
  13. S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
    [Crossref]
  14. T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photon. Technol. Lett. 15, 1573–1575 (2003).
    [Crossref]
  15. S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.
  16. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  17. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
    [Crossref]
  18. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
    [Crossref] [PubMed]
  19. S. G. Evanglides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.,  10, 28–35 (1992).
    [Crossref]
  20. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
    [Crossref]
  21. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996).
    [Crossref]
  22. H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego,2002), pp. 725𠄿861.
  23. M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
    [Crossref]
  24. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000).
    [Crossref] [PubMed]
  25. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton-soliton collisions,” Opt. Lett. 20, 2060–2062 (1995).
    [Crossref] [PubMed]
  26. D. Wang and C. R. Menyuk, “Reduced model of the evolution of the polarization states in wavelength-division-multiplexed channels,” Opt. Lett. 23, 1677–1679 (1998).
    [Crossref]
  27. R. W. Boyd, Nonlinear Optics (Academic, San Diego,1992), Sections 1.4, 2.3 and 6.5.
  28. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
    [Crossref]
  29. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
  30. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
    [Crossref]
  31. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993) and references therein.
    [Crossref]
  32. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
    [Crossref]
  33. Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.
  34. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
    [Crossref]
  35. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Widely tunable wavelength conversion using nondegenerate fiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communications, Rimini, Italy, 21–25 September 2003, paper We3.7.3.
  36. G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt. Soc. Am. B 9, 1047–1061 (1992).
    [Crossref]

2003 (4)

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[Crossref]

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

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

2002 (4)

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

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

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

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

2001 (1)

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[Crossref]

2000 (2)

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

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

1998 (2)

1996 (2)

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

T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996).
[Crossref]

1995 (1)

1994 (2)

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

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

1993 (3)

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[Crossref]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[Crossref]

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993) and references therein.
[Crossref]

1992 (3)

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

G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt. Soc. Am. B 9, 1047–1061 (1992).
[Crossref]

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

1991 (1)

1987 (1)

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

1982 (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[Crossref]

1978 (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1957 (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

1956 (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Agrawal, G. P.

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

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[Crossref]

Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.

Andrekson, P. A.

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

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[Crossref]

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

Benedetto, S.

Bergano, N. S.

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

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego,1992), Sections 1.4, 2.3 and 6.5.

Brar, K.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Brentel, J.

Cao, X. D.

Carena, A.

Centanni, J. C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Chen, H. H.

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

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

Curri, V.

Evanglides, S. G.

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

Gaudino, R.

Goh, C. S.

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

Gordon, J. P.

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

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

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

Hansryd, J.

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

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[Crossref]

Headley, C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Hedekvist, P. O.

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

Heismann, F.

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Inoue, K.

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

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

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

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Jopson, R.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Jopson, R. M.

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

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[Crossref]

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego,2002), pp. 725𠄿861.

Jorgensen, C.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Jorgensen, C. G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Karlsson, M.

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Kazovsky, L. G.

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

Kikuchi, K.

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

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

Kogelnik, H.

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

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego,2002), pp. 725𠄿861.

Lakoba, T. I.

Li, J.

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

Li, J. S.

Lin, Q.

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

Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.

Luther, G. G.

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Manley, J. M.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Marhic, M. E.

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

McKinstrie, C. J.

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

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

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

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993) and references therein.
[Crossref]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[Crossref]

G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt. Soc. Am. B 9, 1047–1061 (1992).
[Crossref]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Menyuk, C. R.

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

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

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

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

Mollenauer, L. F.

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

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

Nelson, L. E.

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego,2002), pp. 725𠄿861.

Poggiolini, P.

Radic, S.

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

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[Crossref]

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

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Raybon, G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Rowe, H. E.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Set, S. Y.

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

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[Crossref]

Tanemura, T.

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

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

Tench, R. E.

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[Crossref]

Uesaka, K.

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

Wai, P. K. A.

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

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

Wang, D.

Weiss, M. T.

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

Westlund, M.

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

Wong, K. K. Y.

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

Xie, C.

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

Yu, M.

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[Crossref]

Electron. Lett. (2)

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[Crossref]

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

IEEE J. Quantum Electron. (3)

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

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[Crossref]

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

IEEE J. Sel. Top. Quantum Electron. (2)

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

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

IEEE Photon. Tech-nol. Lett. (1)

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (4)

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[Crossref]

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

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

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

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

J. Lightwave Technol. (5)

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

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[Crossref]

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

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

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

J. Opt. Soc. Am. B (4)

Opt. Express. (1)

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Rev. E (1)

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[Crossref]

Proc. IRE (2)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

Proc. Nat. Acad. Sci. (1)

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

Sov. Phys. JETP (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Other (5)

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

R. W. Boyd, Nonlinear Optics (Academic, San Diego,1992), Sections 1.4, 2.3 and 6.5.

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego,2002), pp. 725𠄿861.

Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.

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

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Figures (7)

Fig. 1.
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.
Fig. 2.
Fig. 2. Eigenpolarizations of MI. The dashed lines denote sidebands that propagate independently.
Fig. 3.
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.
Fig. 4.
Fig. 4. Eigenpolarizations of PC driven by parallel pumps. The dashed lines denote sidebands that propagate independently.
Fig. 5.
Fig. 5. Eigenpolarizations of PC driven by perpendicular pumps.
Fig. 6.
Fig. 6. Eigenpolarizations of BS driven by parallel pumps.
Fig. 7.
Fig. 7. Eigenpolarizations of BS driven by perpendicular pumps. The dashed lines denote sidebands that propagate independently.

Tables (2)

Tables Icon

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

Tables Icon

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

Equations (144)

Equations on this page are rendered with MathJax. Learn more.

E x ( t , z ) = A x ( t , z ) exp [ i ( k 0 z ω 0 t ) ] + c . c . ,
E y ( t , z ) = A y ( t , z ) exp [ i ( k 0 z ω 0 t ) ] + c . c .
i z A x = β x ( i t ) A x + γ ( A x 2 + 2 A y 2 3 ) A x + γ A y 2 A x * 3 ,
i z A y = β y ( i t ) A y + γ ( 2 A x 2 3 + A y 2 ) A y + γ A x 2 A y * 3 ,
i z B x = β x ( i t ) B x + γ ( B x 2 + 2 B y 2 3 ) B x ,
i z B y = β y ( i t ) B y + γ ( 2 B x 2 3 B y 2 ) B y ,
i z A ξ = β ( i t ) A ξ + ( 8 γ 9 ) ( A ξ 2 + A η 2 ) A ξ ,
i z A η = β ( i t ) A η + ( 8 γ 9 ) ( A ξ 2 + A η 2 ) A η ,
a 1 · a 2 = a 1 · a 2 in exp ( δ ω 2 δ τ 2 3 ) ,
i z A = β ( i t ) A + γ ¯ A A A ,
A = exp ( i ω 1 t ) A 1 + exp ( i ω 2 t ) A 2
D A 1 = i β 1 A 1 + i γ [ ( P 1 + P 2 ) A 1 + A 2 A 1 A 2 ] ,
D A 2 = i β 2 A 2 + i γ [ A 1 A 2 A 1 + ( P 1 + P 2 ) A 2 ] ,
D A j = i H j A j ,
H j = [ β j + γ ( P j + 3 P k 2 ) ] I + γ a k · σ 2
H j = [ β j + γ ( P j + 3 P k 2 ) ] I + γ a t · σ 2 ,
D A j A k = i A j ( H k H j ) A j
= i [ β k β j + γ ( P j P k ) ] A j A k .
D A j σ A j = i A j [ σ , H j ] A j ,
D a j = a j × a t .
D a t = 0 ,
D ( a j · a k ) = 0 .
θ j ( z ) = β j z + γ 0 z [ P j ( z ) + 3 P k ( z ) ] dz 2 .
D B j = iH t B j ,
B j ( z ) = U t ( z ) B j ( 0 ) .
A j ( z ) = exp ( i H t z + i θ j ) A j ( 0 ) .
A = j = 1 3 exp ( i ω j t ) A j
D A 1 = i H 1 A 1 + A 3 A 2 A 2 ,
D A 2 = i H 2 A 2 + ( A 2 A 3 A 1 + A 2 A 1 A 3 ) .
H j = [ β j + γ ( P j + 3 Σ P k 2 ) ] I + γ a t · σ 2 ,
D A 1 A 1 = ( A 3 A 2 A 1 A 2 A 2 A 3 A 2 A 1 ) ,
D A 2 A 2 = 2 ( A 2 A 3 A 2 A 1 A 3 A 2 A 1 A 2 ) .
D ( P 1 P 3 ) = 0 ,
D ( P 1 + P 2 + P 3 ) = 0 .
D A 1 σ A 1 i A 1 [ σ , H t ] A 1 = ( A 3 A 2 A 1 σ A 2 A 2 A 3 A 2 σ A 1 ) ,
D A 2 σ A 2 i A 2 [ σ , H t ] A 2 = ( A 2 A 3 A 2 σ A 1 A 3 A 2 A 1 σ A 2
+ A 2 A 1 A 2 σ A 3 A 1 A 2 A 3 σ A 2 ) .
D a t = 0 .
D A l = iH l A l ,
D A 3 = iH 3 A 3 + A 1 A 2 A 2 ,
D B l = 0 ,
D B 3 = B 1 B 2 B 2 exp [ i ( 2 θ 2 θ 3 θ 1 ) ] .
P 3 ( z ) = Γ ( z ) P 2 A 1 A 2 2 ,
Γ ( z ) = γ 2 sin 2 ( kz ) k 2
k = [ β 1 2 β 2 + β 3 + γ ( 2 P 2 P 1 ) ] 2
P 3 ( z ) = Γ ( z ) P 1 P 2 2 ( 1 + e 1 · e 2 ) 2 ,
D A 1 = iH 1 A 1 + A 3 A 2 A 2 ,
D A 2 = iH 2 A 2 ,
D A 3 = iH 3 A 3 + A 1 A 2 A 2 .
D B 1 = i ( β 1 β 2 + γP 2 ) B 1 + B 3 B 2 B 2 ,
D B 2 = 0 .
D B 3 = i ( β 3 β 2 + γP 2 ) B 3 + B 1 B 2 B 2 .
D 1 B 1 = B 3 B 2 B 2 ,
D 3 * B 3 = B 2 B 1 B 2 .
( D 3 * D 1 I γ 2 P 2 B 2 B 2 ) B 1 = 0 .
[ D 3 * D 1 γ 2 P 2 2 0 0 D 3 * D 1 ] [ S S ] = 0 .
k ± = ( δ k 1 δ k 3 ) 2 ± [ ( δ k 1 + δ k 3 ) 2 4 γ 2 P 2 2 ] 1 2 ,
κ = [ γ 2 P 2 2 ( δβ + γ P 2 ) 2 ] 1 2 .
A = j = 1 4 exp ( i ω j t ) A j
D A 1 = iH 1 A 1 + ( A 4 A 2 A 3 + A 4 A 3 A 2 ) ,
D A 2 = iH 2 A 2 + ( A 3 A 4 A 1 + A 3 A 1 A 4 ) ,
D A 1 A 1 = ( A 4 A 2 A 1 A 3 + A 4 A 3 A 1 A 2
A 2 A 4 A 3 A 1 A 3 A 4 A 2 A 1 ) ,
D A 2 A 2 = ( A 3 A 4 A 2 A 1 + A 3 A 1 A 2 A 4
A 4 A 3 A 1 A 2 A 1 A 3 A 4 A 2 ) .
D ( P 1 P 4 ) = 0 ,
D ( P 2 P 3 ) = 0 ,
D ( P 1 + P 2 + P 3 + P 4 ) = 0 ,
D A 1 σ A 1 γ A 1 σ × a t A 1 = ( A 4 A 2 A 1 σ A 3 + A 4 A 3 A 1 σ A 2 )
A 2 A 4 A 3 σ A 1 A 3 A 4 A 2 σ A 1 ) ,
D A 2 σ A 2 γ A 2 σ × a t A 2 = ( A 3 A 4 A 2 σ A 1 + A 3 A 1 A 2 σ A 4 )
A 4 A 3 A 1 σ A 2 A 1 A 3 A 4 σ A 2 ) .
D ( A 1 σ A 1 + A 2 σ A 2 ) = γ ( A 1 σ × a t A 1 + A 1 σ × a t A 1 )
+ ( A 4 A 2 A 1 σ A 3 A 2 A 4 A 3 σ A 1
+ A 3 A 1 A 2 σ A 4 A 1 A 3 A 4 σ A 2 ) .
D a t = 0 .
D A l = iH l A l ,
D A 4 = iH 4 A 4 + ( A 1 A 2 A 3 + A 1 A 3 A 2 ) ,
D B l = 0 ,
D B 4 = ( B 1 B 2 B 3 + B 1 B 3 B 2 ) exp [ i ( θ 2 + θ 3 θ 4 θ 1 ) ] .
P 4 ( z ) = Γ ( z ) ( P 2 A 3 A 1 2 + P 3 A 1 A 2 2
+ A 2 A 1 A 1 A 3 A 3 A 2 + A 3 A 1 A 1 A 2 A 2 A 3 ) ,
k = [ β 1 β 2 β 3 + β 4 + γ ( P 2 + P 3 P 1 ) ] 2 .
P 4 ( z ) = Γ ( z ) P 1 P 2 P 3 ( 3 + 2 e 1 · e 2 + e 2 · e 3 + 2 e 3 · e 1 ) 2 ,
D A 1 = iH 1 A 1 + ( A 4 A 2 A 3 + A 4 A 3 A 2 ) ,
D A l = iH l A l ,
D A 4 = iH 4 A 4 + ( A 1 A 2 A 3 + A 1 A 3 A 2 ) ,
D B 1 = i ( β 1 β 2 + γP 2 ) B 1 + ( B 4 B 2 B 3 + B 4 B 3 B 2 ) ,
D B l = 0 ,
D B 4 = i ( β 4 β 3 + γP 3 ) B 4 + ( B 1 B 2 B 3 + B 1 B 3 B 2 ) .
D 1 B 1 = ( B 4 B 2 B 3 + B 4 B 3 B 2 ) ,
D 4 * B 4 = ( B 2 B 1 B 3 + B 3 B 1 B 2 ) .
[ D 4 * D 1 I γ 2 ( P 2 B 3 B 3 + P 3 B 2 B 2
+ B 2 B 3 B 2 B 3 + B 3 B 2 B 3 B 2 ) ] B 1 = 0 .
[ D 4 * D 1 γ 2 P 2 P 3 ( 1 + 3 B 2 ) 2 γ 2 P 2 P 3 B B * 2 γ 2 P 2 P 3 B * B D 4 * D 1 γ 2 P 2 P 3 B 2 ] [ S S ] = 0 .
k ± = ( δ k 1 δ k 4 ) 2 ± [ ( δ k 1 + δ k 4 ) 2 4 γ 2 P 2 P 3 Δ ± ] 1 2 ,
Δ ± = ( 1 ± B ) 2
κ ± = { γ 2 P 2 P 3 Δ ± [ δβ + γ ( P 2 + P 3 ) 2 ] 2 } 1 2 .
S S ± 2 = [ ( 1 B ) ( 1 + B ) ] ±1
S ± 2 = ( 1 ± B ) 2 .
Δ ± = ( 3 + e 2 · e 3 ) 2 ± [ 2 ( 1 + e 2 · e 3 ) ] 1 2 ,
( e 1 · e 2 ) ± = ± [ ( 1 + e 2 · e 3 ) 2 ] 1 2 ,
( e 4 · e 2 ) ± = ± [ ( 1 + e 2 · e 3 ) 2 ] 1 2 .
D A l = iH l A l ,
D A 2 = iH 2 A 2 + ( A 3 A 4 A 1 + A 3 A 1 A 4 ) ,
D A 4 = iH 4 A 4 + ( A 1 A 2 A 3 + A 1 A 3 A 2 ) ,
D B l = 0 ,
D B 2 = i ( β 2 β 1 + γP 1 ) B 2 + ( B 3 B 4 B 1 + B 3 B 1 B 4 ) ,
D B 4 = i ( β 4 β 3 + γP 3 ) B 4 + ( B 1 B 2 B 3 + B 1 B 3 B 2 ) .
D 2 B 2 = ( B 3 B 4 B 1 + B 3 B 1 B 4 ) ,
D 4 B 4 = ( B 1 B 2 B 3 + B 1 B 3 B 2 ) .
[ D 4 D 2 I + γ 2 ( P 3 B 1 B 1 + B 1 B 3 2
+ B 3 B 1 B 3 B 1 + B 1 B 3 B 1 B 3 ) ] B 2 = 0 .
[ D 4 D 2 + γ 2 P 1 P 3 ( 1 + 3 B 2 ) γ 2 P 1 P 3 B B * γ 2 P 1 P 3 B * B D 4 D 2 + γ 2 P 1 P 3 ( B 2 ) ] [ S S ] = 0 .
k ± = ( δ k 2 + δ k 4 ) 2 ± [ ( δ k 2 δ k 4 ) 2 4 γ 2 P 1 P 3 Δ ± ] 1 2 ,
Δ ± = [ ( 1 + 4 B 2 ) ± ( 1 + 8 B 2 ) 1 2 ] 2
S S ± 2 = ( 1 + 8 B 2 ) 1 2 ± ( 1 + 2 B 2 ) ( 1 + 8 B 2 ) 1 2 ( 1 + 2 B 2 )
S ± 2 = ( 1 + 8 B 2 ) 1 2 ( 1 + 2 B 2 ) 2 ( 1 + 8 B 2 ) 1 2 .
[ D 2 D 4 I + γ 2 ( P 1 B 3 B 3 + B 3 B 1 2
+ B 1 B 3 B 1 B 3 + B 3 B 1 B 3 B 1 ) ] B 2 = 0 .
[ D 2 D 4 + 4 γ 2 P 1 P 3 B 2 2 γ 2 P 1 P 3 B B * 2 γ 2 P 1 P 3 B * B D 2 D 4 + γ 2 P 1 P 3 ] [ I I ] = 0 .
I I ± 2 = ( 1 + 8 B 2 ) 1 2 ± ( 4 B 2 1 ) ( 1 + 8 B 2 ) 1 2 ( 4 B 2 1 )
I ± 2 = ( 1 + 8 B 2 ) 1 2 ( 4 B 2 1 ) 2 ( 1 + 8 B 2 ) 1 2 .
Δ ± = [ ( 3 + 2 e 1 · e 3 ) ± ( 5 + 4 e 1 · e 3 ) 1 2 ] 2 ,
( e 2 · e 1 ) ± = ( 2 + e 1 · e 3 ) ( 5 + 4 e 1 · e 3 ) 1 2 ·
( e 4 · e 1 ) ± = ( 1 + 2 e 1 · e 3 ) ( 5 + 4 e 1 · e 3 ) 1 2 .
D C 1 = ( C 4 C 2 C 3 + C 4 C 3 C 2 ) ,
D + C 4 = ( C 2 C 1 C 3 + C 3 C 1 C 2 ) ,
[ D 0 2 iγP B iγP B 0 D iγP B 0 2 iγP B iγP B D + 0 iγP B 0 0 D + ] [ S S I * I * ] = 0 ,
κ ± = ± ( γ 2 P 2 Δ ± δ + 2 ) 1 2 ,
Δ 2 2 ( 1 + B 2 ) Δ + B 4 = 0 .
Δ ± = ( 1 ± B ) 2 ,
( S ) ± = ± B 1 ± B ,
( I * ) ± = i ( κ ± i δ + ) γP ( 1 + B ) ,
( I * ) ± = iγP B κ ± + i δ + .
D C 2 = ( C 3 C 4 C 1 + C 3 C 1 C 4 ) ,
D + C 4 = ( C 1 C 2 C 3 + C 1 C 3 C 2 ) ,
[ D 0 2 iγP B iγP B 0 D 0 iγP B 2 iγP B 0 D + 0 iγP B iγP B 0 D + ] [ S S I I ] = 0 .
k ± = ± ( γ 2 P 2 Δ ± + δ 2 ) 1 2 ,
Δ 2 ( 1 + 4 B 2 ) Δ + 4 B 4 = 0 .
Δ ± = [ 1 + 4 B 2 ( 1 + 8 B 2 ) 1 2 ] 2 .
( S ) ± = B B Δ ± B 2 ,
( I ) ± = 2 γP B k ± + δ ,
( I ) ± = ( k ± δ ) B γP ( Δ ± B 2 ) .

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