## Abstract

In a previous paper the generalized Schroedinger equation that governs wave propagation in a rapidly-spun fiber was derived. In this paper the aforementioned equation is used to study four-wave mixing (FWM). The properties of FWM associated with a rapidly-spun fiber are described, and contrasted to those associated with constantly-birefringent and randomly-birefringent fibers. FWM driven by perpendicular linearly-polarized pump waves, or counter-rotating circularly-polarized pump waves, provides polarization-independent signal amplification and phase-conjugation, whereas FWM driven by co-rotating circularly-polarized pump waves provides polarization-independent frequency conversion.

©2006 Optical Society of America

## 1. Introduction

Parametric devices (amplifiers, frequency convertors and phase conjugators) based on four-wave mixing (FWM) in highly-nonlinear fibers have many uses in optical communication systems [1, 2]. Because transmission fibers are not polarization maintaining, practical devices must produce output signal and idler waves whose powers do not depend on the polarizations of the input signals. It has been known for many years that the amplification produced by perpendicular linearly-polarized (LP) pump waves in a randomly-birefringent fiber (RBF) is signal-polarization independent [3, 4]. Two recent papers predicted that the frequency conversion produced by co-rotating circularly-polarized (CP) pumps in a rapidly-spun fiber (RSF) is also polarization independent [5, 6]. Although this prediction was validated approximately by experiments [5], the theoretical model on which it was based was oversimplified. A detailed analysis of nonlinear wave propagation in a RSF was made in [7]. Now that a firm theoretical foundation has been established, FWM in a RSF can be studied in detail.

Nonlinear wave propagation in a lossless RSF is governed by the generalized Schroedinger equation (GSE)

where *B*=[*B*_{x}*,B*_{y}
]t is the amplitude vector of the wave and *b* is the dispersion function of the fiber. In the frequency domain, *β*(ω)=∑
_{n}
≥*2β*_{n}
(*ω*_{0}
)*ω*^{n}*/n*!, where *ω*
_{0} is the carrier frequency of the wave and *ω* is the difference between the actual and carrier frequencies. To convert from the frequency domain to the time domain, one replaces *ω* by *i∂ _{t}*. The focus of this paper is FWM in a RSF, for which

*γ*

_{b}=2

*γ*/3 and

*γb*=

*γ*/3, where γ is the (Kerr) nonlinearity coefficient of the fiber [8]. In this case Eq. (1) is valid in a frame that rotates slowly, but deterministically, because of the residual effects of spinning [7]. Independent symbols for the nonlinearity coefficients were used to broaden the scope of the results: Wave propagation in a constantly-birefringent fiber (CBF) is governed by Eq. (1), supplemented by birefringence and walk-off terms, with

*γ*

_{a}=2γ/3 and

*γb*=

*γ*/3 [9]. In this case the GSE is valid in the laboratory frame. Wave propagation in a RBF is also governed by Eq. (1), with

*γ*

_{a}=8

*γ*/9 and

*γ*

_{b}=0 [10, 11]. In this case the GSE is valid in a frame that rotates randomly with the polarization axes of a reference wave. For media with nonresonant electronic responses

*γ*

_{b}

*/γ*

_{a}=1/2, like CBFs and RSFs, for media with electrostrictive responses

*γ*

_{b}

*/γ*

_{a}=0, like RBFs, and for media with molecular-orientational responses

*γ*

_{b}

*/γ*

_{a}=3 [12]. The descriptors for the aforementioned fibres are summarized in Table 1.

This paper is organized as follows: Nonlinear polarization rotation is studied in Section 2, and the configurations that allow the waves to retain their input polarizations are identified. Nonlinear phase modulation is studied in Section 3, for waves with the aforementioned configurations. The polarization properties of degenerate and nondegenerate FWM are studied in Sections 4 and 5, respectively, for arbitrary values of *γ*_{a} and *γ*_{b}. The properties of FWM in a RSF are compared to the known properties of FWM in a CBF or a RBF. Finally, in Section 6 the main results of this paper are summarized.

## 2. Nonlinear polarization rotation

In FWM, one or two pump waves and a signal wave combine to produce an idler wave. Studies of FWM are simplified by the assumption that the polarizations of the interacting waves are constant. Linear polarization rotation does not occur in a RSF, because its linear response is isotropic. However, nonlinear polarization rotation can occur, and must be studied.

The Jones-and Stokes-vector formalisms used to describe polarization evolution are reviewed in [13]. For any amplitude (Jones) vector *B*, which was defined after Eq. (1), the associated Stokes vector *S⃗*=(*S*
_{1},*S*
_{2},*S*
_{3}), where the Stokes components

The common notation *B⃗*=(*B*
_{1},*B*
_{2},*B*
_{3}) was not used to denote the Stokes vector and its components, because in Section 4 *B*
_{1}, *B*
_{2} and *B*
_{3} will be used to denote frequency components. It follows from definitions (2)–(4) that the wave power *P*=(${S}_{1}^{2}$+${S}_{2}^{2}$+${S}_{3}^{2}$)^{1/2}.

First, consider self-polarization rotation (SPR), which involves one wave (with arbitrary frequency). It follows from the aforementioned definitions and Eq. (1) that

where *D=d/dz* and *S⃗*_{3} is the vector formed by *S*
_{3} and the associated unit vector in Stokes space. Equation (5) shows that SPR is produced by the *γ*_{b}
term in the GSE, but not the *γ*_{a}
term [14]. SPR does not occur if *S*
_{3}=0 or *S*
_{1}=0 and *S*
_{2}=0. The first condition implies that the wave is LP, in an arbitrary (transverse) direction, whereas the second implies that it is CP. Equation (5) is equivalent to the polarization equation of [15], for which *γb*=*γ*/3.

Second, consider cross-polarization rotation (CPR), which involves more than one wave. By substituting the ansatz

in Eq. (1) and collecting terms of like frequency, one finds that

where *S⃗*_{j}
is the Stokes vector of wave *j, S*_{j}*k* is the *k*-component of the Stokes vector of wave *j*, and *S⃗*_{jk}
is the vector formed by *S*_{jk}
and the associated unit vector. The equation for *S⃗*_{2} is similar. Equation (7) shows that CPR is produced by both the *γ*_{a} and *γ*_{b} terms in the GSE. CPR does not occur if *S*
_{13}=0, *S*
_{23}=0 and *S⃗*_{1}×*S⃗*_{2}=0, which corresponds to LP waves whose Jones vectors are parallel or perpendicular, or *S*
_{11}=0, *S*
_{12}=0, *S*
_{21}=0 and *S*
_{22}=0, which corresponds to CP waves whose Jones vectors are parallel or perpendicular (which co-or counter-rotate). The first term on the right side of Eq. (7) is equivalent to the CPR term in [16], for which *γa*=8*γ*/9 and *γb*=0. It is easy to extend these results to arbitrary numbers of waves with incommensurate frequencies, which experience CPR, but not FWM.

In comparison, SPR does not occur in a CBF if the wave is LP along either birefringence axis, and CPR does not occur if both waves are LP along the birefringence axes. SPR does not occur in a RBF, regardless of how the wave is polarized, and CPR does not occur for parallel or perpendicular LP waves, co-or counter-rotating CP waves, or any other aligned or orthogonal waves.

## 3. Nonlinear phase modulation

The analysis of Section 2 showed that polarization rotation does not occur if the input waves are CP, or LP in parallel or perpendicular directions. In Sections 3–5 these polarization conditions will be assumed. For degenerate FWM the frequency-and wavenumber-matching conditions are 2*ω*_{2} = *ω*_{3} + *ω*_{1} and 2*k*
_{2}=*k*
_{3}+*k*
_{1}, respectively, whereas for nondegenerate FWM the matching conditions are *ω*_{2}+*ω*
_{3}=*ω*
_{4}+*ω*
_{1} and *k*
_{2}+*k*
_{3}=*k*
_{4}+*k*
_{1}. Not only do the wavenumber conditions depend on the linear wavenumbers *k*_{j}
=*β*(*ω*_{j}
), they also depend on the nonlinear wavenumber shifts associated with self-phase modulation (SPM) and cross-phase modulation (CPM).

First, consider SPM, which involves one wave. Let *B*=*eP*
^{1/2}, where *B, e* and *P* are the Jones vector, unit polarization vector and power of the wave, respectively. It follows from these definitions that *P*=*B†B*. For a LP wave *e* is [1,0]
^{t}
or [0,1]
^{t}
, whereas for a CP wave it is [1, *i*]
^{t}
/2^{1/2} or [1,-*i*]
^{t}
/2^{1/2}. One can multiply these unit vectors by arbitrary phase factors without changing their lengths. By substituting ansaetze of this form in Eq. (1), one obtains expressions for the nonlinear wavenumber shifts associated with SPM. The results are listed in Table 2. The first entry means that for a LP wave the SPM contribution to *k* is (*γ*_{a}
+*γ*_{b}
)*P*, which depends on both *γ*_{a}
and *γ*_{b}
. In contrast, for a CP wave the SPM contribution is *γ _{a}P*, which does not depend on

*γb*.

Second, consider CPM, which involves more than one wave. Let *B*_{j}
=*e*_{j}*P*^{1/2}*j*, where *B, e* and *P* were defined in the preceding paragraph and *j*=1 or 2. Then, by substituting ansaetze of the form (6) in Eq. (1) and collecting terms of like frequency, one obtains expressions for the nonlinear wavenumber shifts associated with CPM. The results are listed in Table 3. The first entry means that, if waves 1 and 2 are both *x*-polarized, the CPM contribution to *k*
_{1} is 2(*γ*_{a} + *γ*_{b})*P*_{2}
. If the waves are LP, the *γ*_{b}
term in the GSE contributes to the CPM of waves that are parallel, but not perpendicular, whereas if the waves are CP, it contributes to the CPM of waves that are counter-rotating, but not co-rotating.

In comparison, for one LP wave in a CBF the SPMcoefficient is *γ*, whereas for two LP waves

the CPM coefficients are 2*γ* (parallel) and 2*γ*/3 (perpendicular). The ratio of the perpendicular and parallel coefficients is 1/3. The CP results do not apply to a CBF. For one wave with arbitrary polarization in a RBF the SPM coefficient is 8*γ*/9, whereas for two waves the CPM coefficients are 16*γ*/9 (aligned) and 8*γ*/9 (orthogonal). The ratio of the CPMcoefficients is 1/2.

## 4. Degenerate four-wave mixing

As stated in Section 3, degenerate FWM involves three waves whose frequencies satisfy the matching condition 2ω_{2}=ω_{3}+ω_{1}. By substituting the ansatz

in Eq. (1) and collecting terms of like frequency, one finds that

where the wavenumbers *k*
_{1}–*k*
_{3} include both linear and nonlinear contributions, as discussed in Section 3.

#### 4.1. Idler generation

Consider the initial evolution of degenerate FWM. In the first case of interest, two strong input 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 inputs, in which case *B*_{j}*(z)=C*_{j}
exp(*ik*_{j}*z*), where*C*_{j}
is a constant vector and *j*=1 or 2. Once the input polarizations have been specified, subject to the constraints of Section 2, the SPM and CPM contributions to *k*_{j}
can be determined. Let *B*
_{3}(*z*)=*C*
_{3}(*z*)exp(*ik3z*), where *k*
_{3} is to be determined. Then it follows from Eq. (11) that

where *δ*=(*k*
_{1}-2*k*
_{2}+*k*
_{3})/2 is the wavenumber mismatch. Equation (12) has the solution

where *e*_{3}
and c are the polarization and magnitude of the coupling vector [*γ _{a}*(

*e*

^{†}

_{1}

*e*

_{2})

*e*

_{2}+

*γ*(${e}_{2}^{t}$

_{b}*e*

_{2})

*e*

^{*}

_{1}](

*P*

_{1}${P}_{2}^{2}$ )

^{1/2}, respectively. Once

*e*

_{1}and

*e*

_{2}have been specified,

*e*

_{3},

*χ*and the CPM contributions to

*k*

_{3}and

*d*can be determined. For idler generation

*C*

_{3}(0)=0, in which case it follows from Eq. (13) that

The output idler power does not depend on the phase of *e*
_{3} (or the phases of *e*
_{1} and *e*
_{2}).

The properties of degenerate FWM are summarized in Table 4. Each column pertains to a particular combination of input polarizations, which is illustrated in Fig. 1 or 2. The first row describes the output idler polarization (apart from a phase factor that is of secondary importance), the second row describes the magnitude of the coupling vector and the third row describes the nonlinear contribution to the wavenumber mismatch. For parallel LP inputs the *γ*_{a}
and *γ*_{b}
terms in the GSE both contribute to FWM, whereas for perpendicular LP inputs only the *γb* term contributes. In contrast, for co-rotating CP inputs only the γa term contributes. For counter-rotating CP inputs (Fig. 2
*b*) the coupling coefficient is zero, so the idler vector and mismatch coefficient are not defined. According to the results of Section 2, nonlinear polarization rotation does not occur for any of the aforementioned configurations.

The idler vector is *e*
_{3}, the coupling coefficient c should be multiplied by (*P*_{1}${P}_{2}^{2}$
)^{1/2} and the mismatch coefficient d should be multiplied by (*2P*_{2}*-P*_{1}
)/2. †For the second configuration, d=(*γa*+3*γb*)*P*1/2-*γb*
*P*
_{2}. These results are consequences of Eq. (12).

In comparison, for degenerate FWM in a CBF the coupling coefficients are *γ* (parallel LP inputs) and *γ*/3 (perpendicular LP inputs) [17, 18]. The maximal idler powers generated by these configurations (when *δ*=0) can be normalized to 1 and 1/9, respectively. Because of birefringence, the mismatch coefficients associated with these configurations cannot be zero simultaneously. The CP results do not apply to a CBF. For degenerate FWM in a RBF the LP and CP results are identical. The coupling coefficients are 8*γ*/9 (aligned inputs) and 0 (orthogonal inputs), so the normalized idler powers generated by these configurations are 1 and 0, respectively [19, 20].

#### 4.2. Modulation interaction

In the second case of interest, a weak signal wave (1) and a strong pump wave (2) are launched into the fiber and a weak idler wave (3) is generated. This process is called the modulation interaction (MI). The linearized analysis of MI is based on the assumption that the signal and idler do not affect the pump, in which case *B2(z)*=*C*_{2}
exp(*ik*_{2}*z*), where *C*_{2}
is a constant vector. However, the idler power quickly becomes comparable to the signal power, so the effects of the idler on the signal cannot be neglected. By making the substitutions

in Eqs. (9) and (11), one obtains the MI equations

where the wavenumber mismatch d was defined after Eq. (12).

As explained in Section 2, one can simplify the analysis of MI by assuming that the pump is LP or CP. For example, if the pump is x-polarized the vector equations (17) and (18) can be rewritten as the scalar equations

These equations show that *C*
_{1x} is coupled to *C**_{3x} and *C*
_{1y} is coupled to *C**_{3y}. In the first case the signal and idler are parallel to the pump, whereas in the second case they are perpendicular to the pump: In neither case does polarization rotation occur. Each two-mode interaction is independent of the other, and is governed by the canonical equations

For the first interaction the coupling coefficient *k*=(*γ*_{a}
+*γ*_{b}
)${C}_{2x}^{2}$
. It follows from Tables 2 and 3 that the nonlinear contribution to the mismatch coefficient *δ*=(*γ*_{a}
+*γ*_{b}
)*P*_{2x}
. In contrast, for the second interaction *k*=${\gamma \mathit{\text{bC}}}_{2x}^{2}$
and *δ*=-*γbP*_{2x}
. The solutions of Eqs. (23) and (24) can be written in the input–output form

where the transfer functions

and the MI wavenumber *k*=(*δ*
^{2}-|*k*|^{2})^{1/2}. The transfer functions satisfy the auxiliary equation |*µ*|^{2}-|*ν*|^{2}=1. For idler generation *C**_{3}(0)=0, in which case it follows from Eqs. (25)–(28) that

The output signal and idler (sideband) powers do not depend on the pump phase. For short distances (*kz*≪1) or low pump powers (|*k*|≪|d |), Eq. (30) reduces to Eq. (14). If |*k*|>|d |, the MI is unstable, and the sideband powers grow exponentially with distance. The maximal gain exponent (*δ*=0) is |*k*|. The analysis of MI driven by a CP pump is similar.

The properties of MI are summarized in Table 5. Each column pertains to a particular combination of pump and sideband polarizations, which is illustrated in Fig. 3 or 4. The first row describes the coupling coefficient (apart from a phase factor that is of secondary importance) and the second row describes the nonlinear contribution to the mismatch coefficient. For LP sidebands that are parallel to the pump the *γ*_{a}
and *γ*_{b}
terms in the GSE both contribute to MI, whereas for LP sidebands that are perpendicular to the pump only the *γ*_{b}
term contributes. In contrast, for CP sidebands that co-rotate with the pump only the *γ*_{a}
term contributes. For CP sidebands that counter-rotate to the pump (Fig. 4
*b*) the coupling coefficient is zero. Although the mismatch coefficient is defined, the sidebands are independent of each other, so what matters is their (common) pump-CPM coefficient, which is listed in Table 3. The cases described herein (two strong inputs and one strong pump) are different realizations of the same degenerate-FWM process, so there are many similarities between the properties summarized in Tables 4 and 5, and illustrated in the associated figures.

ForMI, the coupling coefficient *k* and the mismatch coefficient *δ* should both be multiplied by *P*_{2}
. These results are consequences of Eqs. (17) and (18).

MI driven by a LP pump (two counter-rotating CP pumps with the same frequency) in a RSF was studied numerically and experimentally in [21]. The results of this paper are of considerable scientific interest, but are not directly relevant to current research on parametric devices [1, 2], because the pump frequency was far into the normal-dispersion regime, rather than near the zero-dispersion frequency.

MI in an idealized isotropic fiber (a mythical polarization-maintaining fiber with zero birefringence) was studied in [22] and MI in a RBF was studied in [20]. The results listed in Table 5 reduce to the results of these papers in the appropriate limits (*γa*=2*γ*/3 and *γ*_{b}
=*γ*/3 for the former fiber, and *γ*_{a}
=8*γ*/9 and *γb*=0 for the latter).

A highly-birefringent fiber (HBF) is a CBF whose birefringence beat-length is much shorter than the FWM lengths associated with typical pump powers. Wave propagation in a HBF is governed by a modified GSE, in which the *γ*_{a}
term and the incoherently-coupled part of the *γ*_{b}
term in Eq. (1) are retained, and the coherently-coupled part of the *γ*_{b} term is omitted [9]. In this GSE, *γ*_{s}
=*γ* is the SPM coefficient, which describes the effect of each polarization component on itself, and *γc*=2*γ*/3 is the CPM coefficient, which describes the effect of each component on the other. MI in a HBF was studied in [23, 24]. Despite the differences between the relevant GSEs, one can deduce the properties of MI driven by a LP pump in a HBF from those of MI driven by a CP pump in a RSF, by replacing + and − (in the tables) with || and ⊥, and *γ _{a}* and

*γ*+ 2

_{a}*γ*

_{b}with

*γs*and

*γc*, respectively.

## 5. Nondegenerate four-wave mixing

As stated in Section 3, nondegenerate FWM involves four waves whose frequencies satisfy the matching condition ω_{2}+ω_{3}=ω_{4}+ω_{1}. By substituting the ansatz

in Eq. (1) and collecting terms of like frequency, one finds that

where the wavenumbers *k*
_{1}–*k*
_{4} include both linear and nonlinear contributions, as discussed in Section 3.

#### 5.1. Idler generation

Consider the initial evolution of nondegenerate FWM. In the first case of interest, three strong input waves (1–3) 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 inputs, in which case *B*_{j}*(z)*=*C*_{j}
exp(*ik*_{j}*z*), where *C*_{j}
is a constant vector and *j*=1–3. Once the input polarizations have been specified, subject to the constraints of Section 2, the SPM and CPM contributions to *k*_{j}
can be determined. Let *B*
_{4}(*z*)=*C*
_{4}(*z*)exp(*ik4z*), where *k*
_{4} is to be determined. Then it follows from Eq. (35) that

where *δ*=(*k*
_{1}-*k*
_{2}-*k*
_{3}+*k*
_{4})/2 is the wavenumber mismatch. Equation (36) has the solution

where *e*
_{4} and *χ* are the polarization and magnitude of the coupling vector {*γ*_{a}
[(*e†*_{1}*e2*)*e*
_{3}+*e*†
_{1}*e*_{3}
)*e*
_{2}]+2γ*b*(${e}_{2}^{t}$*e3*)*e**1}(*P*1*P*2*P*3)^{1/2}, respectively. Once *e*_{1}*–e*_{3}
have been specified, *e*
_{4}, c and the CPM contributions to *k*
_{4} and d can be determined. For idler generation *C*
_{4}(0)=0, in which case it follows from Eq. (37) that

The output idler power does not depend on the phase of *e*
_{4} (or the phases of *e*_{1}*–e*_{3}
).

The properties of nondegenerate FWM are summarized in Tables 6 and 7. Each column pertains to a particular combination of input polarizations, which is illustrated in Fig. 5 or 6. The first row describes the output idler polarization (apart from a phase factor that is of secondary importance), the second row describes the magnitude of the coupling vector and the third row describes the nonlinear contribution to the wavenumber mismatch. For parallel LP inputs the *γ*_{b}
and *γb* terms in the GSE both contribute to FWM, whereas for perpendicular LP inputs only one term contributes. In contrast, for co-rotating CP inputs only the *γ*_{a}
term contributes, whereas for counter-rotating CP inputs neither term contributes or both terms contribute. For the configuration in which neither term contributes (Fig. 6
*b*) the coupling coefficient is zero, so the idler vector and mismatch coefficient are not defined. According to the results of Section 2, nonlinear polarization rotation does not occur for any of the aforementioned configurations.

The idler vector is *e*
_{4}, the coupling coefficient *χ* should be multiplied by (*P*_{1}*P*_{2}*P*_{3}
)^{1/2} and the mismatch coefficient d should be multiplied by (*P*_{2}
+*P*_{3}
-*P*_{1}
)/2. These results are consequences of Eq. (36).

In comparison, for nondegenerate FWM in a CBF the coupling coefficients are 2*γ* (first LP configuration) and 2*γ*/3 (other LP configurations) [17, 18]. The maximal idler powers generated by these configurations (when *δ*=0) can be normalized to 1, 1/9, 1/9 and 1/9, respectively. The mismatch coefficients associated with these configurations cannot be zero simultaneously, because of birefringence, and the fact that the nonlinear contribution to the mismatch associated with the second configuration differs from the contributions associated with the other configurations. The CP results do not apply to a CBF. For nondegenerate FWM in a RBF the LP and CP results are identical. The coupling coefficients are 16*γ*/9 (first configuration), 0 (second configuration) and 8*γ*/9 (third and fourth configurations), so the normalized idler powers generated by these configurations are 1, 0, 1/4 and 1/4, respectively [19, 20].

The idler vector is *e*
_{4}, the coupling coefficient c should be multiplied by (*P*_{1}*P*_{2}*P*_{3}
)^{1/2} and the mismatch coefficient d should be multiplied by (*P*_{2}
+*P*_{3}
-*P*_{1}
)/2. These results are consequences of Eq. (36).

#### 5.2. Phase conjugation

In the second case of interest, a weak signal wave (1) and two strong pump waves (2 and 3) are launched into the fiber and a weak idler wave (4) is generated. This process is called phase conjugation (PC). The linearized analysis of PC is based on the assumption that the signal and idler do not affect the pumps, in which case *B*_{j}
(*z*)=*C*_{j}
exp(*ikjz*), where *C*_{j}
is a constant vector and *j*=2 or 3. However, the idler power quickly becomes comparable to the signal power, so the effects of the idler on the signal cannot be neglected. By making the substitutions

in Eqs. (32) and (35), one obtains the PC equations

where the wavenumber mismatch *δ* was defined after Eq. (36).

As explained in Section 2, one can simplify the analysis of PC by assuming that the pumps are LP or CP. For example, if the pumps are both *x*-polarized the vector equations (41) and (42) can be rewritten as the scalar equations

These equations show that *C*_{1x}
is coupled to *C**_{4x} and *C*
_{1y} is coupled to *C**_{4y}. In the first case the signal and idler are parallel to the pumps, whereas in the second case they are perpendicular to the pumps: In neither case does polarization rotation occur. Each two-mode interaction is independent of the other, and is governed by the canonical equations

which are equivalent to Eqs. (23) and (24). For the first interaction the coupling coefficient *k*=2(*γa*+*γb*)*C*_{2x}*C*_{3x}
. It follows from Tables 2 and 3 that the nonlinear contribution to the mismatch coefficient *γ*=(*γa*+*γb*)(*P*_{2x}
+*P*_{3x}
)/2. In contrast, for the second interaction *k*=2γ*bC*_{2x}*C*_{3x}
and *δ*=-(*γ*_{a}
+3*γ*_{b}
)(*P*_{2x}
+*P*_{3x}
)/2. The solutions of Eqs. (47) and (48) are described by equations that are similar to Eqs. (25)–(28), where the PC wavenumber *k*=(*δ*
^{2}-|*k*|^{2})^{1/2}. For idler generation *C**_{4}(0)=0, in which case it follows from the aforementioned equations that

Once again, the output sideband powers do not depend on the pump phases. For short distances (*kz*≪1) or low pump powers (|*k*|≪|ε|), Eq. (50) is consistent with Eq. (38). If |*k*|>|d |, the PC process is unstable, and the sideband powers grow exponentially with distance. The maximal gain exponent (*δ*=0) is |*k*|. The analyses of PC driven by perpendicular LP pumps, or co-or counter-rotating CP pumps, are similar.

The properties of PC are summarized in Tables 8 and 9. Each column pertains to a particular combination of pump and sideband polarizations, which is illustrated in Figs. 7–10. The first row describes the coupling coefficient (apart from a phase factor that is of secondary importance) and the second row describes the nonlinear contribution to the mismatch coefficient. For LP sidebands that are parallel to themselves and the pumps, and CP sidebands that are driven by counter-rotating pumps, the *γ*_{a}
and *γ*_{b}
terms in the GSE both contribute to PC, whereas for the other configurations the *γ*_{a}
and *γ*_{b}
terms contribute separately, or not at all. For CP side-bands that counter-rotate to the pumps (Fig. 9
*b*) the coupling coefficient is zero. Although the mismatch coefficient is defined, the sidebands are independent of each other, so what matters is their (common) pump-CPM coefficient, which is listed in Table 3. The cases described herein (three strong inputs and two strong pumps) are different realizations of the same nondegenerate-FWM process, so the properties summarized in Tables 8 and 9 are similar to those summarized in Tables 6 and 7.

For PC waves 2 and 3 are the pumps, the coupling coefficient *k* should be multiplied by (*P*_{2}*P*_{3}
)^{1/2} and the mismatch coefficient d should be multiplied by (*P*_{2}
+*P*_{3}
)/2, whereas for BS waves 1 and 3 are the pumps, k should be multiplied by (*P*
_{1}
*P*
_{3})1/2 and d should be multiplied by (*P*
_{3}-*P*
_{1})/2. These results are consequences of Eqs. (41) and (42), and Eqs. (53) and (54), respectively.

As explained in the introduction, practical parametric devices must produce output sidebands whose powers do not depend on the polarization of the input signal. Tables 8 and 9 show that the configurations with perpendicular LP pumps and counter-rotating CP pumps both produce polarization-independent outputs, for arbitrary values of *γ*_{a}
and *γ*_{b}
. For the former configuration the gain coefficient is *γ*_{a}
, whereas for the latter it is *γ*_{a}
+2*γ*_{b}
.

PC driven by counter-rotating CP pumps in a RSF was studied numerically and experimentally in [21], for pump frequencies that were far into the normal dispersion regime. PC in an idealized isotropic fiber (a mythical polarization-maintaining fiber with zero birefringence) was studied in [22, 25] and PC in a RBF was studied in [20]. The results listed in Tables 8 and 9 reduce to the results of these papers in the appropriate limits (*γ*_{a}
=2*γ*/3 and *γ*_{b}
=*γ*/3 for the former fiber, and *γ*_{a}
=8*γ*/9 and *γ*_{b}
=0 for the latter). In particular, for LP pumps in an idealized fiber the coupling (gain) coefficients are 2*γ* (first configuration of Table 8) and 2*γ*/3 (other three configurations), so the gain ratios are 1, 1/3, 1/3, and 1/3. For CP pumps the gain coefficients are 4*γ*/3 (first configuration of Table 9), 0 (second configuration) and 4*γ*/3 (third and fourth configurations), so the gain ratios are 1, 0, 1 and 1. Notice that the gain coefficient for counter-rotating CP pumps is twice as large as the coefficient for perpendicular LP pumps [22, 25]. In contrast, for LP and CP pumps in a RBF the gain coefficients are 16*γ*/9 (first configuration of Table 8 or 8), 0 (second configuration) and 8*γ*/9 (third and fourth configurations), so the gain ratios are 1, 0, 1/2 and 1/2.

PC in a HBF was studied in [23, 24]. Despite the differences between the relevant GSEs, one can deduce the properties of PC driven by LP pumps in a HBF from those of PC driven by CP pumps in a RSF, by replacing + and - (in the tables) with || and ⊥, and *γ*_{a} and *γ*_{a}+2*γ*_{b}
with *γ*_{s}
and gc, respectively.

For PC waves 2 and 3 are the pumps, the coupling coefficient *k* should be multiplied by (*P*_{2}*P*_{3}
)^{1/2} and the mismatch coefficient d should be multiplied by (*P*_{2}
+*P*_{3}
)/2, whereas for BS waves 1 and 3 are the pumps, *k* should be multiplied by (*P*_{1}*P*_{3}
)^{1/2} and d should be multiplied by (*P*_{3}*-P*_{1}
)/2. These results are consequences of Eqs. (41) and (42), and Eqs. (53) and (54), respectively.

#### 5.3. Bragg scattering

In the third case of interest, a weak signal wave (2) and two strong pump waves (1 and 3) are launched into the fiber and a weak idler wave (4) is generated. This process is called Bragg scattering (BS). The linearized analysis of BS is based on the assumption that the signal and idler do not affect the pumps, in which case *B*_{j}*(z)*=*C*_{j}
exp(*ik*_{j}*z*), where *C*_{j}
is a constant vector and *j*=1 or 3. By making the substitutions

in Eqs. (33) and (35), one obtains the BS equations

where the wavenumber mismatch d was defined after Eq. (36).

One can simplify the analysis of BS by assuming that the pumps are LP or CP. For example, if the pumps are co-rotating CP waves the vector equations (53) and (54) can be rewritten as the scalar equations

These equations show that *C*
_{2+} is coupled to *C*
_{4+} and *C*
_{2−} is coupled to *C*_{4−}. In the first case the signal and idler are parallel to the pumps, whereas in the second case they are perpendicular to the pumps: In neither case does polarization rotation occur. Each two-mode interaction is independent of the other, and is governed by the canonical equations

which are similar, but not equivalent, to Eqs. (47) and (48). For the first interaction the coupling coefficient *k*=2*γ*_{a}
*C*
_{1}+*C**_{3}+. It follows from Tables 2 and 3 that the nonlinear contribution to the mismatch coefficient *δ*=*γ*_{a}
(*P*
_{3}+-*P*
_{1}+)/2. In contrast, for the second interaction *k*=(*γ*_{a}
+2*γb*)*C*
_{1}+*C**_{3}+ and *δ*=*γ*_{a}
(*P*
_{3}+-*P*
_{1}+)/2. The solutions of Eqs. (59) and (60) can be written in the input–output form

where the transfer functions

and the BS wavenumber *k*=(*δ*
^{2}+|*k*|^{2})^{1/2}. The transfer functions satisfy the auxiliary equation |*µ*|^{2}+|*ν*|2=1. For idler generation *C**_{4}(0)=0, in which case it follows from Eqs. (61)–(64) that

Once again, the output sideband powers do not depend on the pump phases. For short distances (*kz*≪1) or low pump powers (|*k*|≪|d |), Eq. (66) is consistent with Eqs. (38) and (50). Under these conditions BS and PC develop in the same way. However, because *k* is always real, the BS process is always stable: The signal and idler exchange power periodically, and the maximal idler power cannot exceed the input signal power. The power-transfer efficiency |*k*|^{2}/*k*
^{2} attains its maximal value 1 when *δ*=0. The analyses of BS driven by counter-rotating CP pumps, or parallel or perpendicular LP pumps, are similar.

The properties of BS are also summarized in Tables 8 and 9. Each column pertains to a particular combination of pump and sideband polarizations, which is illustrated in Figs. 11–14. The first row describes the coupling coefficient (apart from a phase factor that is of secondary importance) and the second row describes the nonlinear contribution to the mismatch coefficient. For LP sidebands that are parallel to themselves and the pumps, and a CP signal (2) that counter-rotates to the higher-frequency pump (3), the *γ*_{a}
and *γ*_{b}
terms in the GSE both contribute to BS, whereas for the other configurations the *γ*_{a}
and *γ*_{b}
terms contribute separately, or not at all. For counter-rotating CP pumps, and a signal (2) that co-rotates with the higher-frequency pump (3), the coupling coefficient is zero (Fig. 14
*a*). Although the mismatch coefficient is defined, the sidebands are independent of each other, so what matters are their pump-CPM coefficients, which are listed in Table 3.

For general values of *γ*_{a}
and *γ*_{b}
, no configuration produces outputs whose powers are independent of the input signal-polarization. However, for a RBF *γ*_{a}
=2*γ*_{b}
: The configurations shown in Fig. 12 have the same coupling coefficient, but different mismatch coefficients, whereas the configurations shown in Fig. 13 have the same coupling and mismatch coefficients. BS driven by co-rotating CP pumps produces polarization-independent output [5, 6].

BS in a RBF was studied in [20]. The results listed in Tables 8 and 9 reduce to the results of this paper in the appropriate limit. BS in a HBF was studied in [24]. The same substitutions that relate PC in a RSF to PC in a HBF also relate BS in these fibers.

## 6. Summary

In this paper the properties of four-wave mixing (FWM) in a rapidly-spun fiber (RSF) were studied. In a RSF the residual effects of birefringence and spinning cause the polarization vector of a wave to rotate slowly. However, the rotation rate does not depend on the wave frequency, so the polarization vectors of waves participating in FWMall rotate at the same rate: In the rotating frame, linear polarization rotation is absent, and the other linear and nonlinear properties of the fiber (dispersion and the Kerr effect) are isotropic.

The Kerr nonlinearity causes the polarization vector of an elliptically-polarized wave to rotate at a rate that is proportional to the wave power. However, if the wave is linearly-polarized (LP) or circularly-polarized (CP) self-polarization rotation does not occur. Likewise, two or more co-propagating waves experience cross-polarization rotation unless they are parallel or perpendicular LP waves, or co-or counter-rotating CP waves.

Detailed studies were made of FWM processes that involve waves with the aforementioned aligned and orthogonal polarization vectors. Because the transmission fibers used in optical communication systems are not polarization maintaining, practical devices based on FWM (such as parametric amplifiers, frequency convertors and phase-conjugators) must produce output sidebands (signals and idlers) whose powers do not depend on the polarizations of the input signals. To fulfill this requirement, the nonlinear-coupling and wavenumber-mismatch coefficients of the underlying FWM processes must both be polarization independent.

The modulation interaction (MI) involves three waves whose frequencies satisfy the matching condition 2*ω*
_{2}=ω_{3}+ω_{1}. In this process, wave 2 is the pump, and waves 1 and 3 are the signal and idler, respectively. The properties of MI are summarized in Table 5, and illustrated in Figs. 3 and 4. MI does not produce polarization-independent outputs: If the pump is LP, the signal is always coupled to an idler, in which case the signal is amplified and the idler is generated. However, the coupling and mismatch coefficients both depend on the signal polarization. In contrast, if the pump is CP, only a co-rotating signal is coupled to an idler. A counter-rotating signal is not amplified and no idler is generated.

Phase conjugation (PC) involves four waves whose frequencies satisfy the matching condition ω_{2}+ω_{3}=ω_{4}+ω_{1}. In this process waves 2 and 3 are the pumps, and waves 1 and 4 are the signal and idler, respectively. The properties of PC are summarized in Tables 8 and 9, and illustrated in Figs. 7–10. If the pumps are aligned (parallel LP or co-rotating CP), the coupling and mismatch coefficients are both polarization dependent. For the former configuration the coupling coefficient is always nonzero, so the signal is always amplified. For the latter configuration only a co-rotating signal is amplified. In contrast, if the pumps are orthogonal (perpendicular LP or counter-rotating CP), the coupling and mismatch coefficients are both polarization independent. The coupling coefficient for the latter configuration is twice as large as the coefficient for the former.

Bragg scattering (BS) also involves four waves whose frequencies satisfy the matching condition ω_{2}+ω_{3}=ω_{4}+ω_{1}. However, in this process waves 1 and 3 are the pumps, and waves 2 and 4 are the signal and idler, respectively. In contrast to MI and PC, which are unstable processes, BS is a stable process: The signal cannot be amplified, but power can be transferred from the signal to the idler. The properties of BS are also summarized in Tables 8 and 9, and illustrated in Figs. 11–14. If the pumps are aligned (parallel LP or co-rotating CP), a power transfer always occurs. For the former configuration the mismatch coefficient is polarization independent, but the coupling coefficient is polarization dependent. For the latter configuration the coupling and mismatch coefficients are both polarization independent. In contrast, if the pumps are orthogonal (perpendicular LP or counter-rotating CP), the power transfer is always polarization dependent. For the former configuration the coupling coefficient is polarization independent, but the mismatch coefficient is polarization dependent. For the latter configuration only a signal that co-rotates with the lower-frequency pump transfers power to an idler.

In conclusion, RSFs are suitable for use in practical parametric devices: Not only do they allow perpendicular LP and counter-rotating CP pumps to provide polarization-independent PC, which they can do in randomly-birefringent fibers (RBFs), they also allow co-rotating CP pumps to provide polarization-independent BS, which they cannot do in RBFs.

## Acknowledgment

We thank a reviewer for bringing to our attention [25].

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