## Abstract

Phase-sensitive amplification (PSA), which is produced by degenerate four-wave mixing (FWM) in a randomly-birefringent fiber, has the potential to improve the performance of optical communication systems. Scalar FWM, which is driven by parallel pumps, is impaired by the generation of pump–pump and pump–signal harmonics, which limit the level, and modify the phase sensitivity, of the signal gain. In contrast, vector FWM, which is driven by perpendicular pumps, is not impaired by the generation of harmonics. Vector FWM produces PSA with the classical properties of a one-mode squeezing transformation.

©2007 Optical Society of America

## 1. Introduction

Long-haul communication systems require optical amplifiers to compensate for fiber loss. Current systems use erbium-doped or Raman fiber amplifiers. These amplifiers are examples of phase-insensitive amplifiers (PIAs), which produce signal gain that is independent of the signal phase. In principle, phase-sensitive amplifiers (PSAs) could also be used. The potential advantages of PSAs include, but are not limited to, noise reduction [1], the reduction of noise- and collision-induced phase [2] and frequency [3] fluctuations, and dispersion compensation [4].

Previous papers [5, 6, 7] showed that degenerate four-wave mixing (FWM) in a
randomlybirefringent fiber (RBF) produces phase-sensitive amplification (PSA),
provided that the signal frequency (ω_{0}) is the average of the
pump frequencies (ω_{-1} and ω_{1}).
Degenerate scalar and vector FWM are illustrated in Figs. 1(a) and 1(b), respectively. In the former process (inverse modulation
interaction)
γ_{-1}+γ_{1}→2γ_{0},
where γ_{j} represents a photon with frequency
ω_{j}. In the latter process (degenerate phase conjugation)
γ_{-1}+γ_{1}→γ∥
+γ_{˔}, where the subscript 0 was omitted for
simplicity.

If one assumes that each interaction involves only the aforementioned pumps and signal, then each interaction produces PSA with the classical properties of a one-mode squeezing transformation [8]. FWM processes are driven by pump- and signal-induced nonlinearities and limited by dispersion-induced wavenumber shifts. If the pump frequencies differ significantly, strong dispersion prevents other (secondary) FWM processes from occurring and the preceding assumption is valid. However, it is difficult to phase lock pumps with dissimilar frequencies, which are usually produced by two separate lasers. In contrast, it is easy to phase lock pumps with similar frequencies, which can be produced by one laser and a phase modulator. However, if the pump frequencies are similar, dispersion is too weak to counter nonlinearity and secondary FWM processes occur.

A previous paper on scalar FWM [9] showed that, if the pump frequencies are comparable to the zero-dispersion frequency (ZDF) of the fiber, a cascade of product waves (harmonics) is produced. In the low-pump-power regime that is characteristic of photon-generation experiments [10], weak harmonics are produced, which do not affect the primary process significantly. However, in the high-pump-power regime that is characteristic of signal-amplification experiments, strong harmonics are produced, which limit the level, and modify the phase sensitivity, of the signal gain. In this paper vector FWM is studied in detail. If the pumps are orthogonal, and the input-signal power is split evenly between the pump polarizations, vector FWM does not produce pump-pump harmonics, but does produce PSA with the aforementioned properties.

This paper is organized as follows: In Section 2 the coupled-mode equations (CMEs), which model wave propagation in a dispersionless RBF, are stated and solved for arbitrary input conditions. In Sections 3 and 4 these solutions are used to study pump–pump and pump–signal FWM cascades, respectively. Simulations that quantify the effects of dispersion are described in Section 5. Finally, in Section 6 the main results of this paper are summarized.

## 2. Wave propagation in a fiber

Wave propagation in a RBF is modeled by the coupled Schroedinger equations (CSEs)

where *z* is distance, ∂_{z} = ∂/*∂ _{z}*,

*X*and

*Y*are the amplitude (polarization) components of the wave and β is the dispersion function of the fiber. In the frequency domain β(ω)=∑

_{n≳2}β

_{n}(ω

_{c})ω

^{n}/

*n*!, where ω

_{c}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∂*

_{τ}, where τ =

*t*-β

_{1}

*z*is the retarded time and β

_{1}(ω

_{c}) is the group slowness. The nonlinearity coefficient

*γ*=8γ

*/9, where γ*

_{K}_{K}is the Kerr coefficient. Equations (1) and (2) are valid in a frame that rotates randomly with the polarization axes of a reference wave. They omit the effects of polarization-mode dispersion [11], which are weak for the narrow spectral bandwidths and short fiber lengths of current experiments.

As stated in the Introduction, and discussed quantitatively in [9], if the frequencies of the interacting waves are comparable to the ZDF of fiber, the effects of dispersion are much weaker than those of nonlinearity and can be neglected. In this limit (β = 0), the CSEs reduce to the coupled-mode equations (CMEs)

which model the effects of self-phase modulation (SPM) and cross-phase modulation (CPM). The notation and language of this paper are based on the assumption that the basis vectors for the wave amplitude are linearly polarized. However, Eqs. (3) and (4) are valid for any pair of orthogonal vectors, including counter-rotating circularly-polarized vectors. Each polarization component depends implicitly on the retarded time.

The CMEs have the simple solutions

Because solutions (5) and (6) contain only the effects of nonlinearity, it is
convenient to let *P* be a reference power,
*X*/*P*
^{1/2}→*X*,*Y*/*P*
^{1/2}→*Y*
and γ*P _{z}*→

*z*, in which case the amplitude and distance variables are dimensionless, and γ is absent from the solutions.

## 3. Pump-pump cascade

Consider the two-frequency boundary (initial) conditions

where ϕ_{+}=-ωτ+ϕ_{1}(0)
and ϕ_{-1}=ωτ+ϕ_{-1}(0).
These conditions correspond to two pumps (±1) with frequencies
±ω, which are inclined at the angles θ ±
relative to the *x*-axis. (The average of the pump frequencies equals
the carrier frequency.) The input power has the time average
ρ^{2}
_{+}+ρ^{2}
_{-} and
the contribution
2ρ_{+}ρ_{-}cos(θ_{+}-θ_{-})cos(ϕ_{+}-ϕ_{-}
), which oscillates at the difference frequency 2ω. The input power
depends on the phase difference
ϕ_{d}=[ϕ_{1}(0)-ϕ_{-}1(0)]/2,
but does not depend on the phase average
ϕ_{a}=[ϕ_{1}(0)+ϕ_{-1}(0)]/2.
By measuring phase relative to the reference phase ϕ_{a}, and
time relative to the reference time ϕ_{d}/ω, one can
rewrite conditions (7) and (8) in the simpler forms

where ϕ=ωτ.

By using the identity
exp(*i*ζcosΨ)=∑_{m}
*i ^{m}*

*J*(ζ )exp(-

_{m}*imΨ*) [12], where

*m*is an integer,

*J*is the Bessel function of order

_{m}*m*, ζ is the distance parameter 2ρ+ρ-cos(θ

_{+}-θ

_{-})

*z*and Ψ is the phase parameter 2ϕ, one can write solutions (5) and (6) as the series

*X*(τ,

*z*)=∑

_{n}

*X*

_{n}(

*z*)exp(-

*in*ϕ)and

*Y*(τ,

*z*)= ∑

_{n}

*Y*(

_{n}*z*)exp(-

*in*ϕ), respectively. The frequency components (harmonics)

where *n* is an odd integer and the (common) phase factor
exp[*i*(ρ^{2}
_{+}+ρ^{2}
_{-})*z*]
was omitted for simplicity. For continuous-wave inputs
ρ_{±} are constant (as are ϕ* _{a}* and ϕ

*), whereas for pulsed inputs they vary slowly with time (as do ϕ*

_{d}_{a}and ϕ

_{d}). As distance increases, so also does the number of harmonics (modes) with significant power: Eqs. (11) and (12) describe a vector FWM cascade. Notice that the mode powers ∣

*X*∣

_{n}^{2}and ∣

*Y*∣

_{n}^{2}do not depend on the input phases: This pump–pump cascade is phase insensitive (PI). Two other properties follow from Eqs. (11) and (12), and the identity ∑

_{m}*J*

^{2}

*(ζ)=1 [12]. First,*

_{m}which shows that the time-averaged power in the x-component is constant, consistent with solution (5). Second,

which shows that the asymmetry in the *x*-component of the power
spectrum decreases nonmonotonically with distance (as 1/ζ). Similar
results apply to the *y*-component.

The evolution of the pump–pump cascade is illustrated in Fig. 2, for the initial conditions
ρ_{±}=1 and θ_{-}=0. In each
row the polarization components of the pump spectrum (pump spectra) are displayed
for two distances, *z*=0 and *z*=2. The first row
corresponds to pumps that are parallel initially (θ_{+}=0), the
second row corresponds to oblique pumps (θ_{+}=π/4)
and the third row corresponds to perpendicular pumps
(θ_{+}=π/2). Figures 2(a) and 2(b) show that, if the pumps are *x*-polarized
initially, they remain *x*-polarized: The CMEs do not allow a
transfer of power from one component to the other. If the pumps are oblique a FWM
cascade also occurs. However, the cascade in Fig. 2(d) is less developed than the cascade in Fig. 2(b),because the oscillations in total power are weaker
(ζ is smaller). If the pumps are perpendicular the total power is
constant, so no cascade occurs, as shown in Fig. 2(*f*). It was shown in [13, 14] that perpendicular pumps in RBFs do not produce the lowest
harmonics (modes -3 and 3). The preceding analysis shows that they produce no
harmonics whatsoever.We validated the spectra displayed in Fig. 2 (and Figs. 3, 6 and 7) by calculating the Fourier transforms of solutions (5) and
(6) numerically.

## 4. Pump-signal cascades

Now consider the three-frequency initial conditions

where ρ, ρ_{0}, θ_{0} and
ϕ_{0} are constants (or slowly-varying functions of time).
These conditions correspond to two pumps of equal power and a signal whose frequency
is the average of the pump frequencies (0). The pumps are perpendicular and the
signal is polarized at the angle θ_{0} to pump -1. For these
conditions, the input power has the time average
2ρ^{2}+ρ^{2}
_{0} and the
contribution 2ρρ_{e} cos(ϕ
-ϕ_{e}), which oscillates at the difference frequency
ω. The (effective) amplitude and phase parameters

Notice that ρ_{e}, which determines the strength of the power
oscillations, depends on ϕ_{0}, unless θ_{0}=0
or π/2. By using the aforementioned identity, one can write solutions (5)
and (6) as the series *X*(τ, *z*)=∑_{n}*X _{n}*(

*z*)exp[-

*in*(ϕ-ϕ

_{e})] and

*Y*(τ,

*z*)=∑

_{n}*Y*(

_{n}*z*)exp[-

*in*(ϕ -ϕ

_{e})], respectively.The harmonics

where *n* is an integer,
ζ=2ρρ_{e}
*z* and the
(common) phase factor
exp[*i*(2ρ^{2}+ρ^{2}
_{0})]
was omitted.Equations (19) and (20) describe a pump–signal cascade, and are valid for arbitrary
signal polarizations and signal phases. In this paper the polarization dependence of
the cascade is studied for the case in which ϕ_{0}=0, and the
phase dependence of the cascade is studied for the case in which
θ_{0}=π/4.

First, suppose that ϕ_{0}=0. Then
ρ_{e}=ρ_{0}[1+sin(2θ_{0})]^{1/2}
and ϕ_{e}=0 or π. These parameter specifications are
equivalent to
ρ_{e}=ρ_{0}(cosθ_{0}+sinθ_{0})
and ϕ_{e}=0. Equations (19) and (20) reduce to the simpler equations

where ζ was defined after Eq. (20). Notice that the harmonics have *x* and
*y* components for arbitrary θ_{0}: This
pump-signal cascade is polarization diverse.

The evolution of the pump–signal cascade is illustrated in Fig. 3, for the initial conditions ρ=1,
ρ_{0}=0.1 and ϕ_{0}=0. In each row power
spectra are displayed for two distances, *z*=0 and 4. The first row
corresponds to a signal that is parallel to pump -1 initially
(θ_{0}=0), the second row corresponds to a signal that is
polarized at 45° to the pumps (θ_{0}=π/4)
and the third row corresponds to a signal that is parallel to pump 1
(θ_{0}=π/2). Power is normalized to
10^{-4}, so the input pump powers
*P*
_{±1}=1 correspond to 40 dB and the input
signal power *P*
_{0}=0.01 corresponds to 20 dB. Figures 3(a) and 3(b) show that an *x*-polarized signal
produces an *x*-polarized pump–signal cascade. This cascade is
similar to the cascade illustrated in Figs. 2(a) and 2(b), but is less developed, because the signal in the latter
cascade is weaker than pump 1 in the former. Notice that a
*y*-polarized cascade develops sympathetically, even though the
ycomponent of the input power was not modulated. For *z*=4, only a
slight asymmetry between the *x*- and *y*-components
of this vector cascade exists (which is clearly visible in modes ±2).
Similar remarks apply to Figs. 3(e) and 3(*f*), which illustrate the vector cascade
produced by a *y*-polarized signal. The strongest vector cascade
occurs when θ_{0}=π/4, in which case the input signal
power is split evenly between the pump polarizations, and the cascade that develops
is symmetric. This case corresponds to the strongest modulation of the input power
(ρ_{e}=2^{1/2}ρ_{0}).

Second, suppose that θ_{0}=π/4. Then
ρ_{e}=2ρ̂_{0}∣cosϕ_{0}∣,
where ρ̂_{0}=ρ_{0}/2^{1/2},
and ϕ_{e}=0 or π. These parameter specifications are
equivalent to ρ_{e}=2ρ̂_{0}
cosϕ_{0} and ϕ_{e}=0. Equations (19) and (20) reduce to the simpler equations

where ζ was defined after Eq. (20). Notice that the harmonics depend on
ϕ_{0}: This pump–signal cascade is phase sensitive (PS). It
follows from Eq. (23) that

The formula for Y_{0} is identical. In the linear regime (ζ≪1),
the output signal is proportional to the input signal. (Because
ρ_{0}/ρ≪1, it is possible that
ζ≪1 and
ρ^{2}
*z*~1 simultaneously.)In this
regime,

Although Eq. (26) is only part of an approximate solution of the CMEs, it is
the exact solution of the FWM equations for the standard PS process [5, 6, 7], which involves only modes -1, 0 and 1 [Fig. 1(b)]. It follows from Eq. (26), and its counterpart for Y_{0}, that the signal
power

It follows from Eq. (27) that the signal gain
P_{0}(*z*)/ρ^{2}
_{0} attains
its extremal values when
2ϕ_{0}=tan^{-1}(1/ρ^{2}
*z*).
Let μ=1+*i*ρ^{2}
*z* and
*ν*=*i*ρ^{2}
*z*.
Then the first-quadrant value of 2ϕ_{0} corresponds to the
maximal gain (∣μ∣
+∣ν∣)^{2}, whereas the third-quadrant value
corresponds to the minimal gain
(∣μ∣-∣ν∣)^{2}. It
also follows from Eq. (23) that, in the linear regime,

For long distances (ρ^{2}
*z*>1),
*X*
_{-2}≈*X*
_{0},
unless ϕ_{0} ≈π/2 or 3π/2.
Similar results apply to *Y*
_{2}.The dependence of the signal
power (gain) on phase and distance is illustrated in Fig. 4,for the initial conditions ρ=1 and
ρ_{0}=0.01 (which correspond to pump powers of 1 W and a
signal power of 0.1 mW). The contour spacing is 2.5 dB, so the white regions
correspond to positive gains higher than 18.75 dB, whereas the black region
corresponds to negative gain lower than -18.75 dB. For short distances the gain is
maximal when the input phase ϕ_{0} ≈ π/4,
whereas it is minimal when ϕ_{0} ≈ 3π/4. In
contrast, for long distances the maximal- and minimal-gain conditions are
ϕ_{0} ≈ 0 and ϕ_{0}
≈ π/2, respectively. This behavior distinguishes FWM in a
fiber from difference-frequency generation in a crystal, for which the maximal- and
minimal-gain conditions are ϕ_{0}=0 and
ϕ_{0}=π/2, respectively, for all distances [8].

The dependence of the signal gain on phase and distance is also illustrated in Fig. 5, for the initial conditions ρ=1 and
ρ_{0}=0.01. In Fig. 5(a) the gain is plotted as a function of input phase,
for the case in which *z*=5. When ϕ_{0}=0, the
gain is 1+4(ρ^{2}
*z*)^{2}: The signal
power increases quadratically with distance. In contrast, when
ϕ_{0}=π/2 the gain is 1: The signal is not amplified
and no idlers are produced. The maximal gain
{[1+(ρ^{2}
*z*)^{2}]^{1/2}+ρ
^{2}
*z*}^{2}
≈4(ρ^{2}
*z*)^{2}
corresponds to ϕ_{0}=0.099, whereas the minimal gain
{[1+(ρ^{2}
*z*)^{2}]^{1/2}-ρ^{2}
*z*}^{2}≈
1/4(ρ^{2}
*z*)^{2} corresponds to
ϕ_{0}1=1.67. In Fig. 5(b) the gain is plotted as a function of distance, for
the extremal cases in which ϕ_{0}=0.099 and 1.67. The power of
the amplified signal (normalized to 10^{-4}) increases from 1 to 102 (20.1
dB), whereas the power of the attenuated signal decreases from 1 to
9.8×10^{-3} (-20.1 dB). These results show that vector FWM
provides (at least) 20 dB of PS amplification or attenuation. Neither process is
impeded by the generation of idlers. The evolution of the pump–signal cascade is
illustrated in Fig. 6, for the initial conditions ρ=1,
ρ_{0}=0.01 and ϕ_{0}=0.099 (maximal gain).
Spectra are displayed for *z*=0 and *z*=5.

Power is normalized to 10^{-6}, so the input pump powers
*P*
_{±1}=1 correspond to 60 dB and the input
signal power *P*
_{0}=10^{-4} corresponds to 20 dB.
(Each component of the signal has a power of 17 dB.) The output signal power is
about 40 dB (37 dB in each component). In addition to a strong output signal, FWM
produces two strong idler modes. Mode -2, which is *x*-polarized, is
generated by the FWM processes in which
2γ_{-1}→γ_{-2}+γ_{0}
and
γ_{-1}+γ_{0}→γ_{-2}+γ_{1},
and enhanced by the process in which
γ_{-1}+γ_{1}→γ_{-2}+γ_{2}.
Similar processes produce mode 2, which is y-polarized. The scalar pump–signal
cascade produces many idlers, whose powers are comparable to the signal power, and
whose presence limits the signal gain [9]. In contrast, the vector pump–signal cascade produces only
two idlers, whose presence does not affect the signal gain significantly.

Two other spectra are displayed in Fig. 7, for the initial conditions ρ =1 and
ρ_{0}=0.01, which were illustrated in Fig. 6(a), and the distance *z*=5. Figure 7(a) corresponds to ϕ_{0}=1.57, for
which no cascade occurs: The mode powers are constants. Similar behavior is
exhibited by the standard PS process [5, 6, 7] and the scalar cascade [9]. Figure 7(b) corresponds to ϕ_{0}=1.67
(minimal gain). The signal is attenuated, consistent with Fig. 5(b). In the standard
PS process the signal power is transferred to the pumps (±1). However, in
the vector cascade the signal power is transferred to the neighboring idlers
(±2). This behavior is consistent with Eqs. (26) and (28): For long distances, the condition
2ϕ_{0}=tan^{-1}(1/ρ^{2}
_{z})
is equivalent to the conditions sinϕ_{0} ≈ 1 and
cosϕ_{0} ≈
-1/2ρ^{2}
*z*, which imply that
*X*
_{0}≈-ρ̂_{0}/2ρ^{2}
*z*
and *X*
_{-2} ≈
-*i*ρ̂_{0}.

## 5. Numerical simulations of vector four-wave mixing

Current experiments involve highly-nonlinear fibers with dispersion coefficients
β _{3} ≈ 0.03 ps^{3}/Km and
β_{4} ≈ -3×10^{-4}
ps^{4}/Km, nonlinearity coefficients γ ≈ 10/Km-W (all
evaluated at the ZDFs), and pumps with powers *P* ≈ 0.3 W.
The mode frequencies depend on the intended application. We chose (real) frequency
spacings of 50 and 200 GHz (0.4 and 1.6 nm), which are the channel spacings for 10-
and 40-Gb/s systems, respectively.

To check the analysis of Sections 3 and 4, we solved the CSEs numerically, using the
standard split-step method, for the initial conditions of Figs. 2, 3, 6 and 7. In the absence of dispersion (β_{3}=0
and β_{4}=0), the numerical results agree with the analytical
predictions (to the limit of numerical accuracy). In the presence of dispersion
(with the aforementioned characteristics), there are some quantitative differences,
but no qualitative differences, between the numerical results and the analytical
predictions.

The evolution of the pump–pump cascade is illustrated in Fig. 8, for the initial conditions
ρ_{±}=1 and θ_{-}=0, and the
distance *z*=2. The total mode power is plotted as a function of mode
number. Figure 8(a) corresponds to θ_{+}=0, which
was illustrated in Fig. 2(a). For a frequency spacing of 50 GHz, the difference
between the numerical results and the analytical predictions does not exceed 0.8%.
For a spacing of 200 GHz, moderate differences are visible. Dispersion breaks the
symmetry between the positive- and negative-mode powers. However, it is not strong
enough to suppress the cascade driven by parallel pumps. Figure 8(*b*) corresponds to the initial
condition θ_{+}=π/2, which was illustrated in Fig. 2(e). For both spacings, the numerical results agree
with the analytical predictions: Perpendicular pumps do not produce a cascade.

The evolution of the pump–signal cascade is illustrated in Fig. 9, for the initial conditions ρ = 1 and
ρ_{0}=0.01, which were illustrated in Fig. 6(a), and the distance *z*=5. The total
mode power is plotted as a function of mode number. Figure 9(a) corresponds to ϕ_{0} =0.099
(maximal gain in the zero-dispersion limit). For a frequency spacing of 50 GHz, the
difference between the numerical results and the analytical predictions does not
exceed 0.5%. For a spacing of 200 GHz, small differences in the powers of modes -3
and 3 are visible. However, for both spacings the signal is amplified by 20 dB. Figure 9(b) corresponds to ϕ_{0} = 1.67
(minimal gain in the zero-dispersion limit). For both spacings, the numerical
results agree with the analytical predictions: The signal is attenuated by 20 dB.

As the frequency spacing increases, so also do the differences between the numerical results and the analytical predictions. However, it remains true that perpendicular pumps do not produce a cascade, and a signal polarized at 45° to such pumps experiences significant PS amplification or attenuation. The extremal-gain conditions depend on the frequency spacing.

## 6. Summary

In this paper studies were made of the frequency cascades initiated by two strong pump waves (-1 and 1), and two strong pump waves and a weak signal wave (0), whose frequency is the average of the pump frequencies. These cascades are produced by vector four-wave mixing (FWM) in a randomly-birefringent fiber (RBF).

Wave propagation in a RBF is governed by coupled Schroedinger equations (CSEs).However, if the frequencies of the interacting waves are comparable to the zero-dispersion frequency (ZDF) of the fiber, the effects of dispersion are much weaker than those of nonlinearity and can be neglected. In this limit, the CSEs reduce to the coupled-mode equations (3) and (4), which were solved exactly.

The pump–pump cascade [Eqs. (11) and (12)] is phase insensitive. Parallel pumps produce a strong cascade with many harmonics. However, as the pump misalignment increases, the number and strength of the harmonics decrease. Perpendicular pumps do not produce a cascade.

The absence of a pump–pump cascade modifies the properties of the associated pump–signal cascade [Eqs. (17) and (18)], which is phase sensitive. If the signal is polarized at 45 ° to the pumps, only two strong harmonics (idlers) are produced (-2 and 2). These idlers are produced by the pumps and signal, but do not affect the signal adversely: For parameters that are typical of current experiments, the signal can be amplified or attenuated by more than 20 dB, depending on its input phase.

The analytical predictions of this paper were validated by numerical simulations based on the CSEs, which included the effects of dispersion.

In conclusion, vector FWM near the ZDF of a RBF produces phase-sensitive amplification with the classical properties of a one-mode squeezing transformation. This result is important, because it is easier to phase-lock pumps with similar frequencies (produced by one laser and a phase modulator) than pumps with dissimilar frequencies (produced by two separate lasers).

## Acknowledgments

The research of SR andMR was supported by the National Science Foundation, under contracts ECS-0406379 and PHY-0456974, respectively.

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