## Abstract

The four-sideband model of parametric instabilities driven by orthogonal pump waves in birefringent fibers is developed and validated by numerical simulations. A polynomial eigenvalue equation is derived and used to determine how the spatial growth rates and frequency bandwidths of various instabilities depend on the system parameters. The maximal growth rate is associated with a group-speed matched four-sideband process (coupled modulation instability), whereas broad-bandwidth gain is associated primarily with a two-sideband process (phase conjugation). This four-sideband model facilitates the design of parametric amplifiers driven by two pump waves with different frequencies and polarizations.

© 2003 Optical Society of America

## 1. Introduction

All fiber-optic communication systems need optical amplifiers to compensate for fiber loss. Current systems use Erbium-doped, Raman or hybrid fiber amplifiers. One could also use parametric amplifiers, which are based on optical mixing [1]. Not only can parametric amplifiers provide broad-bandwidth, low-noise amplification at arbitrary wavelengths, they can also conjugate the signals and convert their wavelengths. These features are important for wavelength-division-multiplexed transmission systems and wavelength-based routers.

Parametric amplification driven by two pump waves (1 and 2) involves (at least) four product waves (signal and idler sidebands) that are coupled by three distinct four-wave-mixing (FWM) processes. For example, suppose that the signal frequency *ω*
_{1-}=*ω*
_{1}-*ω*, where *ω* is the frequency difference (modulation frequency). Then the modulation interaction (MI) in which 2*ω*
_{1}=*ω*
_{1-}+*ω*
_{1+} produces an idler with frequency *ω*
_{1+}=*ω*
_{1}+*ω*, the Bragg-scattering (BS) process in which *ω*
_{1-}+*ω*
_{2}=*ω*
_{1}+*ω*
_{2-} produces an idler with frequency *ω*
_{2-}=*ω*
_{2} -*ω* and the phase-conjugation (PC) process in which *ω*
_{1}+*ω*
_{2}=*ω*
_{1-}+*ω*
_{2+} produces an idler with frequency *ω*
_{2+}=*ω*
_{2} +*ω*. Each of the idlers is coupled to the others by the appropriate FWM process (BS, MI or PC).

Parametric amplification driven by pump waves with parallel polarization vectors was studied theoretically [2, 3, 4, 5] and experimentally [6, 7, 8, 9]. Parallel signals (whose polarization vectors are aligned with the pump vectors) experience maximal gain, whereas orthogonal signals experience minimal gain (no gain in a high-birefringence fiber): In the parallel-pump configuration the signal gain depends sensitively on the signal polarization. Because transmission fibers are not polarization maintaining, practical amplifiers must operate on signals with arbitrary polarizations. One can minimize the polarization dependence of the gain by using orthogonal pumps [10, 11, 12]. However, if one does so one encounters the effects of birefringence, which were omitted from the aforementioned theoretical studies.

The propagation of light waves in birefringent polarization-maintaining fibers is governed by a pair of nonlinear Schroedinger (NS) equations (one equation for each polarization component of the electric field). In this paper the coupled NS equations [13] are used to derive four ordinary-differential equations (one for each sideband), which model parametric instabilities driven by orthogonal pumps. The four-sideband (FS) equations facilitate studies of how the existence and growth of various (two-, three- and foursideband) instabilities depend on the system parameters, and what system parameters allow the broad-bandwidth signal gain required for practical amplifiers. The analysis of this paper also unifies two sets of previous analyses, of instabilities driven by pumps with the same polarization, but different frequencies [14, 15, 16, 17], and instabilities driven by pumps with the same frequency, but orthogonal polarizations [18, 19, 20, 21].

Real highly-nonlinear fibers, with which parametric amplifiers are constructed, are not polarization maintaining. For short fibers the polarization misalignment is small and the results of this paper are applicable: In a recent experiment with a 0.3-Km fiber [22], polarization axes were identified and the measured characteristics of PC were consistent with the predictions of the polarization-maintaining theory [23]. To model parametric instabilities in long fibers accurately one must include the effects of polarization-mode dispersion [24]. Such a task is beyond the scope of this paper. Current parametricamplification experiments involve fibers with lengths in the range 0.3 Km–3.0 Km: Typical fibers are long enough to exhibit some effects of polarization-mode dispersion, but not long enough to decorrelate completely the input and output polarizations. For these fibers it is reasonable to expect that the measured characteristics of the instabilities will be intermediate between those of instabilities driven by parallel pump waves, which were analyzed previously [4], and instabilities driven by orthogonal pump waves, which are analyzed in this paper.

The objectives of this paper are the development and validation of the FS model of instabilities driven by orthogonal pumps in birefringent polarization-maintaining fibers, and the application of this model to the search for conditions under which broad-bandwidth amplification is possible. Specific amplifier designs will be discussed in detail elsewhere.

## 2. Four-sideband equations

Let *E*_{x}
(*t*, *z*) and *E*_{y}
(*t*, *z*) be the electric-field components of a light wave in a fiber. It is often convenient to measure (angular) frequencies relative to a reference frequency *ω*
_{0} and wavenumbers relative to the associated reference wavenumbers, and write the field components as

A fiber is said to be highly birefringent if its beat length is much shorter than its physical length. In such a fiber the wave amplitudes *A*_{x}
and *A*_{y}
satisfy the incoherently-coupled NS equations

where each *β*(*ω*)=${\sum}_{n=1}^{\infty}$
${\beta}_{0}^{\left(n\right)}$
*ω*^{n}
/*n*! represents the higher-order terms in the Taylor expansion of the dispersion function about *ω*
_{0}, *γ* is the self-nonlinearity coefficient and *∊*=2/3 is the ratio of the cross- and self-nonlinearity coefficients [13].

Equations (3) and (4) have the (two-frequency) equilibrium solution

where *P*
_{1} and *P*
_{2} are the pump powers, *ω*
_{1} and *ω*
_{2} are the pump frequencies (measured relative to *ω*
_{0}), and the pump phases

Equations (7) and (8) describe two pumps with nonlinear wavenumber shifts imposed by self-phase modulation (SPM) and cross-phase modulation (CPM). In a highly-birefringent fiber there is no (parasitic) pump-pump FWM when the polarization vectors of the pumps are aligned with the principal axes of the fiber.

One can simplify the analysis of two-pump processes by making the substitution

in which case one encounters dispersion terms of the form

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$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=\sum _{m=0}^{\infty}\sum _{n=m}^{\infty}\frac{{\beta}_{0}^{\left(n\right)}{\omega}_{p}^{n-m}}{\left(n-m\right)!}\frac{{\left(i{\partial}_{t}\right)}^{m}{B}_{p}}{m!}$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=\sum _{m=0}^{\infty}\left[\sum _{l=0}^{\infty}\frac{{\beta}_{0}^{\left(m+l\right)}{\omega}_{p}^{l}}{l!}\right]\frac{{\left(i{\partial}_{t}\right)}^{m}{B}_{p}}{m!},$$

where *p* is the pump index (1 or 2) and *l*=*m*-*n*. In the first line of Eq. (11) the null term ${\beta}_{0}^{\left(0\right)}$ [*B*_{p}
exp(-*iω*_{p}*t*)] was added to simplify the algebra that follows. In the last line of Eq. (11) the term in square brackets is ${\beta}_{p}^{\left(m\right)}$
, the *m*th derivative of the dispersion function evaluated at *ω*_{p}
. By using this result, one finds that

where *β*
_{1}(*ω*) represents the higher-order (*m*≥1) terms in the Taylor expansion of *β*_{x}
about *ω*
_{1} and *β*
_{2}(*ω*) represents the expansion of *β*_{y}
about *ω*
_{2}.

In equilibrium ${B}_{1}^{\left(0\right)}$=${P}_{1}^{1/2}$, and ${B}_{2}^{\left(0\right)}$=${P}_{2}^{1/2}$. By linearizing Eqs. (12) and (13) about this (simple) equilibrium one finds that

Before this point, no assumptions were made about the frequency spectra of the amplitude perturbations ${B}_{1}^{\left(1\right)}$ and ${B}_{2}^{\left(1\right)}$. Now suppose that the equilibrium is perturbed by an *x*-polarized signal sideband with (absolute) frequency *ω*
_{1-}=*ω*
_{1}-*ω*. Because of the the way in which *B*
_{1} was defined [Eq. (9)], this sideband is represented by a component of ${B}_{1}^{\left(1\right)}$ with (relative) frequency -*ω*. It follows from Eqs. (14) and (15) that the signal sideband produces an *x*-polarized idler sideband with (relative) frequency *ω*, and *y*-polarized idler sidebands with frequencies -*ω* and *ω*, as shown in Fig. 1. Each

of these idlers is coupled to the others. No new sidebands (at different frequencies) are produced. (This statement is not valid for parallel pumps.) One can manifest this FS interaction by writing

By substituting ansaetze (16) and (17) in Eqs. (14) and (15), and collecting terms of like frequency, one obtains the FS equations

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where the (dispersive) wavenumber mismatches

Equations (18)–(23) differ from Eqs. (8)–(13) of [4] in two ways: First, the ratio of the self- and cross-nonlinearity coefficients is *∊*, which does not equal 2, and second, the wavenumber mismatches depend on the dispersion functions *β*_{x}
and *β*_{y}
, which are not equal.

If one makes the *a priori* assumption that *x*-polarized sidebands with (absolute) frequencies *ω*
_{1}±=*ω*
_{1}±*ω* interact with *y*-polarized sidebands with frequencies *ω*
_{2}±=*ω*
_{2}±*ω* [23], one can write the sideband amplitudes in the form

By substituting ansaetze (24) and (25) in Eqs. (3) and (4), linearizing the equations that result and collecting terms of like frequency, one obtains Eqs. (18)–(21), in which

These formulas are equivalent to formulas (22) and (23).

FS equations similar to Eqs. (18)–(21), but without the higher-order dispersion terms, were derived independently by Seve *et al.* [25], and Forest and Wright [26]. The former authors made a detailed study of instabilities driven by orthogonal pump waves, both of whose frequencies are in the normal-dispersion regime, whereas the latter made a detailed study of the conditions required for instability. As the results of Section 3 show, high gain occurs when both pump frequencies are in the anomalous-dispersion regime and broad-bandwidth gain occurs when one pump frequency is in the anomalous-dispersion regime and the other is in the normal-dispersion regime. Furthermore, for the broad-bandwidth gain required of practical parametric amplifiers, the effects of higherorder dispersion are important [27, 28]: Our results are complementary to those of the aforementioned authors.

## 3. Results

The solutions of Eqs. (18)–(21) can be written as linear combinations of the functions exp(*ik*_{j}*z*), where each wavenumber *k*_{j}
is a root of the characteristic equation

$$\times \left[{\left(k-\delta {\beta}_{2o}\right)}^{2}-\delta {\beta}_{2e}\left(2\gamma {P}_{2}+\delta {\beta}_{2e}\right)\right]$$

$$-{\left(2\gamma \u220a\right)}^{2}\delta {\beta}_{1e}\delta {\beta}_{2e}{P}_{1}{P}_{2}=0.$$

In Eq. (28) *δβ*
_{1o}(*δβ*
_{1e}) and *δβ*
_{2o} (*δβ*
_{2e}) denote the odd (even) terms in the Taylor expansions of *βx*(*ω*
_{1}+*ω*) and *β*_{y}
(*ω*
_{2}+*ω*) about the frequencies *ω*
_{1} and *ω*
_{2}, respectively. Physically, *dδβ*
_{1o}/*dω* and *dδβ*
_{2o}/*dω* are the average slownesses (inverse group speeds) of the sidebands of pumps 1 and 2, respectively, which are comparable to the pump slownesses ${\beta}_{1}^{\left(1\right)}$ and ${\beta}_{2}^{\left(1\right)}$.

The two-sideband interactions (MI, BS and PC) associated with Eqs. (18)–(21) also have characteristic wavenumbers (roots of characteristic equations). By using the relations described after Eq. (23), one can deduce the required formulas for orthogonal-pump interactions from the existing formulas for parallel-pump interactions [4]. The MI wavenumber

where the linear wavenumber mismatch *δβ*=(*δβ*
_{1-}+*δβ*
_{1+})/2 (and the second wavenumber is -*k*). The MI of pump 1 does not depend on birefringence or the odd-order dispersion coefficients (evaluated at *ω*
_{1}). Its (spatial) growth rate Im(*k*) is maximal when the linear and nonlinear mismatches cancel (*δβ*=-*γ*
*P*
_{1}). Similar results apply to the MI of pump 2. The BS wavenumber

where the mismatch *δβ*=(*δβ*
_{2-}-*δβ*
_{1-})/2. BS depends on birefringence, all orders of dispersion and both pump powers. It is intrinsically stable. Similar results apply to the BS process that involves the 1+ and 2+ sidebands. The PC wavenumber

where the mismatch *δβ*=(*δβ*
_{1-}+*δβ*
_{2+})/2. PC depends on birefringence and dispersion. Like MI, its growth rate is maximal when the linear and nonlinear mismatches cancel [*δβ*=-*γ*(*P*
_{1}+*P*
_{2})/2]. Unlike MI, the linear mismatch depends on dispersion coefficients evaluated at both pump frequencies [even-order coefficients evaluated at the average pump frequency (*ω*
_{1}+*ω*
_{2})/2] and the nonlinear mismatch depends on both pump powers. Similar results apply to the PC process that involves the 1+ and 2- sidebands.

For infinitessimal modulation frequencies each *k*_{j}
≈0. Suppose that the 1- sideband is the signal. Then a short calculation shows that the (normalized) sideband powers

The signal and idler powers grow quadratically with distance, rather than exponentially.

For finite frequencies the gain is exponential (when it exists). Typically, the side-bands are not all driven resonantly (at their natural wavenumbers) over a broad range of frequencies. MI exists for low modulation frequencies. When the sideband pairs have similar group speeds they interact strongly: a complete FS interaction occurs. Conversely, when the sideband pairs have dissimilar group speeds they interact weakly: The MIs of the pumps are almost independent. PC exists for high modulation frequencies (if the PC wavenumber-matching conditions are satisfied). These conclusions follow from Eqs. (28)–(31).

Characteristic equations of the form (28) arose in studies of instabilities driven by pumps with the same polarization, but different frequencies [14, 15, 16, 17], and instabilities driven by pumps with the same frequency, but orthogonal polarizations [18, 19, 20, 21]. In the former studies group-speed mismatch was caused by dispersion, whereas in the latter it was caused by birefringence. In this paper

where *s*=*δn*/*c* is the differential slowness and *δn* is the differential index of refraction. [If *s*>0 the *x*-axis is called (colloquially) the slow axis.] The birefringent and dispersive contributions to the group speeds are independent [25]. For certain pump frequencies the birefringent and dispersive contributions to ${\beta}_{1}^{\left(1\right)}$-${\beta}_{2}^{\left(1\right)}$ cancel, and the FS interaction is group-speed matched, as shown in Fig. 2.

Differential indices vary from 10^{-7} (transmission fibers) to 10^{-5} (polarization maintaining fibers). To illustrate the use of the FS equations we chose the intermediate value *δn*=10^{-6}, for which *s*=3 ps/Km. We also chose the (representative) fiber parameters ${\beta}_{0}^{\left(3\right)}$=0.1 ps^{3}/Km, ${\beta}_{0}^{\left(4\right)}$=3×10^{-4} ps^{4}/Km, *γ*=10/Km-W and *l*=0.3 Km, and the common pump power *P*=1 W. The dispersion coefficients represent values associated with the zero-dispersion frequency *ω*
_{0}.

The spatial growth rate obtained by solving Eq. (28) numerically is plotted as a function of the modulation frequency *ω* and the pump frequency *ω*
_{2} in Fig. 3, for *ω*
_{1}=-5 THz. (In this paper all frequencies are angular frequencies: THz is an abbreviation for Trad/s.) The ridges that are parallel to the *ω*
_{2}-axis are associated with MIs [Eq. (29)]. For most values of *ω*
_{2}, the group speeds of the pumps are dissimilar and the MIs of the pumps are independent. Because *ω*
_{1} is constant, the ridge associated with pump 1 has constant width. Because *ω*
_{2} varies, the ridge associated with pump 2 has variable width and does not exist for *ω*
_{2}>0 (normal dispersion region). The ridges that extend to large values of *ω* are associated with PC processes [Eq. (31)]. To demonstrate this fact, the PC gain loci are plotted in Fig. 4. Negative modulation frequencies correspond to cases in which the 1- sideband is the signal, whereas positive modulation frequencies correspond to cases in which the 1+ sideband is the signal. Because Eq. (28) is an even function of the modulation frequency, both PC processes manifest themselves in Fig. 3. To connect Fig. 4 to Fig. 3 the reader should project the mirror image of the 1-PC process (-*ω*→*ω*) onto the right half-plane: The 1- PC process is responsible for the ridges that cover the points (15,-20) and (20, 6), whereas the 1+ PC process is responsible for the ridges that cover (20, 0) and (20, 20). When *ω*
_{2}≈-9.2 THz group-speed matching occurs and all four sidebands interact strongly. In this case Eq. (28) is biquadratic and can be solved analytically for arbitrary *β*
_{1e} and *β*
_{2e}. If one ignores the difference between the pump frequencies, one finds that

where *δβ*
_{o} (*δβ*_{e}
) denote the odd (even) terms in the Taylor expansion of the (common) dispersion function. The maximal FS growth rate *κ*≈*γ*(1+*∊*)*P* is higher than the maximal growth rates associated with MI (*κ*=*γP*) and PC (*κ*=*γ∊P*). For the stated parameters, the value of the FS growth rate predicted by Eq. (38) agrees fairly well with the value obtained by solving Eq. (28) numerically (16.7 and 16.0, respectively). Group-speed matching also occurs when *ω*
_{2}≈9.2 THz. In this case the presence of a modulationally stable pump (2) reduces slightly the growth rates associated with the unstable pump (1). The matching conditions for PC are not satisfied.

The signal-gain curves obtained by solving the sideband equations numerically for *ω*
_{1}=-5.0 THz and *ω*
_{2}=-9.2 THz are displayed in Fig. 5(a). The solid, dot-dashed and dashed curves represent the signal gains (ratios of the output and input signal powers) associated with the FS process [Eqs. (18)–(21)], the PC processes [Eqs. (18) and (21) for the external sidebands, and Eqs. (19) and (20) for the internal sidebands] and the MIs [Eqs. (18) and (19) for pump 1, and Eqs. (20) and (21) for pump 2], respectively. Clearly, the FS gain-level is higher and the FS gain-bandwidth is broader that the levels and bandwidths of the constituent processes (PC and MI). The signal gain and idler gains (ratios of the output idler powers and the input signal power) obtained by solving the FS equations numerically are displayed in Fig. 5(b). The gains of the idlers (1+, 2- and 2+) are comparable to that of the signal (1-), which justifies our assertion that all four sidebands interact strongly.

The power spectra of orthogonal pumps interacting with broad-bandwidth noise are displayed in Fig. 6 for *ω*
_{1}=-5.0 THz and *ω*
_{2}=-9.2 THz. These spectra were obtained by solving Eqs. (3) and (4) numerically, with noise included in the spectrum of the *x*-component at *z*=0. An idealized model of noise was used, in which the modulus of the noise field was the same for all frequency bins, but the phase varied randomly from bin to bin. The bin width was 6.2 GHz (angular frequency) and the input noise power was 10^{-8} W (-50 dBmW) per bin. The simulation results are shown in Fig. 6. The locations and heights of the primary peaks are consistent with the predictions of the FS model. (The peaks in Fig. 6 are 6-dB higher than those in Fig. 5 because of constructive interference between noise-driven sidebands with modulation frequencies *ω* and -*ω*.) The presence of the secondary peaks is a higher-order nonlinear effect driven by the pumps and primary sidebands. To check this assertion the simulation was repeated with lower pump power (*P*=0.5 W). The heights of the primary peaks were reduced and the secondary peaks were absent.

For a practical parametric amplifier, the gain curve should be as wide and flat as possible. It is clear from Figs. 3(b) and 4 that in this context, the optimal pump frequency *ω*
_{2}≈6 THz. The signal-gain curves obtained by solving the sideband equations numerically for *ω*
_{1}=-5.0 THz and *ω*
_{2}=5.9 THz are displayed in Fig. 7(a). (The line styles were defined in the discussion of Fig. 5.) For low modulation frequencies the FS gain is produced (mainly) by the MI of pump 1, whereas for high modulation frequencies the FS gain is produced (mainly) by PC. The signal and idler gains obtained by solving the FS equations numerically are displayed in Fig. 7(b). For low frequencies the gain of the 1+ idler is comparable to that of the 1- signal, as predicted by the MI equations. Notice that the nonlinear coupling provided by pump 2 allows the 2+ sideband to experience (sympathetic) gain.For high frequencies only the gain of the 2+ idler is comparable to the signal gain, which justifies our assertion that broad-bandwidth gain is produced by PC. The 2- idler does not experience significant gain for any frequency.

The power spectra of orthogonal pumps interacting with broad-bandwidth *x*-polarized noise are displayed in Fig. 8 for *ω*
_{1}=-5.0 THz and *ω*
_{2}=5.9 THz. There is good agreement between the predictions of the FS model (which were illustrated in Fig. 7) and the simulation results.

The spatial growth rate obtained by solving Eq. (28) numerically is plotted as a function of the modulation frequency *ω* and the pump frequency *ω*
_{2} in Fig. 9, for *ω*
_{1}=5 THz. Because *ω*
_{1}>0 (normal dispersion region) pump 1 is modulationally stable and the ridge that was prominent in Fig. 3 is absent from Fig. 9. The ridge that covers (2,-20) and broadens as *ω*
_{2} increases is associated with the MI of pump 2. The remaining ridges are associated with PC. To demonstrate this fact, the PC gain loci are plotted in Fig. 10 (which is related to Fig. 9 in the same way that Fig. 4 is related to Fig. 3). The 1- PC process is responsible for the ridges that cover (20,-18) and (20,-2), whereas the 1+PC process is responsible for the ridges that cover (20,-7) and (12, 20). For a broad range of modulation frequencies, the growth rates associated with the ridge that covers (20,-2) are higher that the PC and MI growth rates. Enhanced growth occurs because the PC gain locus intersects the MI gain locus of pump 2 (low-anomalous dispersion region). When *ω*
_{2}≈9.2 THz group-speed matching occurs. However, no instability occurs because both pumps are modulationally stable and the CPM coefficient *∊*<1 [Eq. (38) with *δβ*_{e}
>0]. Cases in which both pump frequencies are in the normal-dispersion regime were discussed in detail by Seve *et al.* [25]. Group-speed matching also occurs when *ω*
_{2}≈-9.2 THz. In this case the presence of a modulationally stable pump (1) reduces slightly the growth rates associated with the unstable pump (2). The matching conditions for PC are satisfied for a narrow range of modulation frequencies.

The signal-gain curves obtained by solving the sideband equations numerically for *ω*
_{1}=5.0 THz and *ω*
_{2}=-2.4 THz are displayed in Fig. 11(a). (The line styles were defined in the discussion of Fig. 5.) Consistent with the predictions of Eq. (28), the signal gain is significantly higher than the MI and PC gains for a broad range of modulation frequencies. The signal and idler gains obtained by solving the FS equations numerically are displayed in Fig. 11(b). For low frequencies the gain of the 2- idler is comparable to that of the 2+ signal and the nonlinear coupling provided by pump 1 allows the 1- sideband to experience (cooperative) gain. For high frequencies only the gain of the 1- idler is comparable to the signal gain, so the broad-bandwidth gain is produced by PC.

The power spectra of orthogonal pumps interacting with broad-bandwidth *y*-polarized noise are displayed in Fig. 12 for *ω*
_{1}=5.0 THz and *ω*
_{2}=-2.4 THz. Once again there is good agreement between the predictions of the FS model (which were illustrated in Fig. 11) and the simulation results.

In the preceding discussion it was stated that the growth rates of the MI, PC and FS instabilities are maximal when the associated wavenumber-matching conditions are satisfied. This statement was based on the linearized Eqs. (18)–(21), in which the coupling and mismatch terms depend on the initial pump powers, but not the signal or idler powers. It is the analog for waves of the statement that the growth rate of a driven linear oscillator is maximal when the oscillator is driven resonantly (the driving frequency equals its natural frequency). However, a typical oscillator is nonlinear (its natural frequency depends on its amplitude). As the amplitude of a nonlinear oscillation grows, the self-consistent frequency changes and the oscillator is detuned from its driver [29]. As a wave interaction proceeds, power is transferred from the pump wave(s) to the product waves [amplified signal and idler(s)]. Not only does pump depletion change the coupling strengths, it also changes the self-consistent wavenumber shifts (because SPM and CPM depend on the current pump and product powers). Nonlinear detuning prevents a complete transfer of power from the pump to the product waves, unless a nonzero mismatch is imposed initially to compensate for the (additional) nonlinear mismatch that accumulates [30, 31, 32]. This effect is important in parametric amplifiers used to produce short powerful pulses. Current 10-Gb/s communication systems are designed to transmit 128 signals with (time-averaged) powers of order 1 mW, so the total signal power is of order 0.1 W. Parametric amplifiers used in communication systems should be operated in the weakly-depleted regime to minimize the impairments associated with depletion-induced gain (and signal-power) fluctuations. This constraint requires the pump powers to be of order 1 W. For such pump powers the (additional) nonlinear wavenumber shifts would be measurable, but would not detune the amplification processes significantly [because most of the required signal amplification would occur before the signal power was high enough to alter (slightly) the nonlinear wavenumber shifts].

## 4. Summary

In this paper the four-sideband (FS) model of parametric instabilities driven by orthogonal pump waves was developed from the incoherently-coupled nonlinear-Schroedinger (NS) equations that govern light-wave propagation in highly-birefringent fibers. A polynomial eigenvalue equation was derived and used to determine (efficiently) how the spatial growth rates and frequency bandwidths of various (two-, three- and four-sideband) instabilities depend on the system parameters. For parameters of interest the FS equations were solved numerically to determine which sidebands participate in each instability. The maximal growth rate is associated with a group-speed matched four-sideband process (coupled modulation instability), whereas broad-bandwidth gain is associated primarily with a two-sideband process (phase conjugation). The predictions of the FS model were validated by comparing them to the results of numerical simulations based on the NS equations. This model facilitates the design of parametric amplifiers driven by two pump waves with different frequencies and polarizations. For the parameters of this paper no configuration was found that is better than the standard configuration, in which one pump frequency is in the anomalous dispersion region, the other is in the normal dispersion region and their average is close to the zero-dispersion frequency.

## Acknowledgments

We thank a reviewer and M. G. Forest for bringing to our attention [25] and [26], respectively.

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