## Abstract

Phase-sensitive amplification (PSA) has the potential to improve significantly the performance of optical communication systems. PSA is known to occur in *χ*
^{(2)} devices, and in a fiber interferometer, which is an example of a *χ*
^{(3)} device. In this report some four-wave mixing processes are described, which produce PSA directly in fibers.

©2004 Optical Society of America

## 1. Introduction

Long-haul communication systems require optical amplifiers to compensate for fiber loss. Current systems use erbium-doped and 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, one could also use phase-sensitive amplifiers (PSAs) in communication systems. The potential advantages of PSAs include, but are not limited to, noise reduction [1], the reduction of noise-induced frequency [2] and phase [3] fluctuations, dispersion compensation [4], and the suppression of the modulational instability (MI) [5].

The standard scheme for PSA (and squeezed-state generation) is degenerate parametric amplification (PA) in a *χ*
^{(2)} medium [6]. PSA is also possible in a *χ*
^{(3)} medium: Some schemes are based on degenerate backward [7, 8] and near-forward [9] four-wave mixing (FWM) in a three-dimensional medium. In these schemes the idlers are generated within the medium. Phase sensitivity (and squeezing) is obtained by the use of a beam-splitter [7, 9] or a mirror [8] to combine the signal and idler. The amplification of vacuum fluctuations by nondegenerate FWM (MI) in a one-dimensional medium, such as a fiber, produces squeezed states [10, 11, 12]. However, no signal is involved in this process. The PSA of a signal does occur in a nonlinear fiber interferometer [13, 14]. However, in this scheme the signal gain only increases as a quadratic function of the fiber length. In this report some schemes for the PSA of signals are described, which are based solely on FWM processes in a fiber, and in which the signal gain increases as an exponential function of the fiber length.

## 2. Parametric amplification in a *χ*^{(2)} medium

PA (frequency down-conversion) in a *χ*
^{(2)} medium occurs when a pump wave interacts with signal and idler waves of lower frequency. If the signal and idler frequencies are identical, the process is said to be degenerate. (The backward and near-forward FWM schemes mentioned in Section 1 are even more degenerate, because the pump frequencies equal the common signal and idler frequency.) Degenerate PA is governed by the frequency-matching condition *ω*
_{2} = 2*ω*
_{1}, where *ω*
_{2} and *ω*
_{1} are the pump and signal frequencies, respectively, and the amplitude equations

where *β* = 2*β*
_{1} - *β*
_{2} is the linear wavenumber mismatch and *$\overline{\gamma}$* is the nonlinear coupling coefficient, which is proportional to *χ*
^{(2)}. One can choose the amplitude units in such a way that |*A*_{j}
|^{2} is proportional to the photon flux *P*_{j}
.

Suppose that wave 2 is a strong pump and wave 1 is a weak signal. Then, in the small-signal (undepleted-pump) approximation, *A*
_{2}(*z*) =*A*
_{2}(0). Let

Then the transformed signal amplitude obeys the (linearized) equation

where *δ* = *β*/2 and *γ*= 2*$\overline{\gamma}$
A*
_{2}. Equation (4) has the solution

where the transfer functions

and the growth rate *κ* = (|*γ*|^{2} - *δ*
^{2})^{1/2}. The transfer functions satisfy the auxiliary equation |*μ*|^{2} - |*ν*|^{2} = 1. The input-output relation described by Eqs. (5)–(7) is the defining property of a PSA. In quantum optics this relation is called a squeezing transformation [6]. To illustrate the effects of PSA, consider the simplest case, in which *δ* = 0, and suppose that *A*
_{2} is real (measure the signal phase relative to the pump phase). Then, according to Eqs. (5)–(7), the in-phase signal quadrature (*B*
_{1}+${\mathit{\text{iB}}}_{1}^{*}$)/2 is amplified by the factor exp(*κz*), whereas the out-of-phase quadrature (*B*
_{1} - ${\mathit{\text{iB}}}_{1}^{*}$)/2 is attenuated by the same factor [multiplied by the factor exp(-*κz*)]. This property is responsible for the phenomena described in Section 1.

## 3. Four-wave mixing in a *χ*^{(3)} medium

Unfortunately, the two-wave mixing process described in Section 2 does not occur in a *χ*
^{(3)} medium. However, it is possible to produce, in a *χ*
^{(3)} medium, an idler that is a non-frequency-shifted image of the signal. The bidirectional scheme [15] involves forward and backward pumps, with frequencies *ω*
_{3} and *ω*
_{1}, and a forward signal with frequency *ω*
_{2} = (*ω*
_{3} + *ω*
_{1})/2. The interaction of these waves produces a backward idler with frequency *ω*
_{2}. In the unidirectional scheme described in this report, all the waves propagate in the forward direction. The scalar version of this degenerate FWM process (inverse MI) is illustrated in Fig. 1. It is governed by the frequency-matching condition *ω*
_{3} + *ω*
_{1} = 2*ω*
_{2} and the amplitude equations

where *β* = 2*β*
_{2} - *β*
_{3} - *β*
_{1} is the linear wavenumber mismatch and *$\overline{\gamma}$* is the nonlinear coupling coefficient, which is proportional to *χ*
^{(3)} [16]. In this scalar process the signal and idler are identical. The self-phase modulation (SPM) and cross-phase modulation (CPM) coefficients are not exactly equal to *$\overline{\gamma}$* and 2*$\overline{\gamma}$*, respectively. However, the deviations from the stated values are qualitatively unimportant and, for typical frequencies, are quantitatively insignificant.

Suppose that waves 3 and 1 are strong pumps and wave 2 is a weak signal. Then, in the small-signal approximation, the pump photon-fluxes are constant:

In a *χ*
^{(3)} medium the pumps are subject to SPM and CPM. Let *A*
_{3} (0) = *B*
_{3}, *A*
_{1}(0) = *B*
_{1} and

Then the transformed signal amplitude obeys the (linearized) equation

where *δ* = *β*/2 + *$\overline{\gamma}$*(*P*
_{3}+*P*
_{1})/2 and *γ* = 2*$\overline{\gamma}$
B*
_{3}
*B*
_{1}. Equation (14) has the same form as Eq. (4), so the input-output relation is described by Eqs. (5)–(7): Degenerate scalar FWM provides PSA in a fiber.

Nondegenerate FWM in a *χ*
^{(3)} medium involves pumps with frequencies *ω*
_{4} and *ω*
_{1}, a nominal signal with frequency *ω*
_{2} and a nominal idler with frequency *ω*
_{3}. If the pumps are co-polarized (scalar FWM), so are the signal and idler. If the pumps are cross-polarized (vector FWM), the signal is aligned with one pump and the idler is aligned with the other. Nondegenerate vector FWM is governed by the frequency-matching condition *ω*
_{4} + *ω*
_{1} = *ω*
_{2} + *ω*
_{3} and the amplitude equations

where *β* = *β*
_{2} + *β*
_{3} - *β*
_{4} - *β*
_{1} is the linear wavenumber mismatch, *$\overline{\gamma}$* is the nonlinear coupling coefficient for co-polarized waves and *ε* is the ratio of the coupling coefficients for cross-polarized and co-polarized waves. For (polarization-maintaining) fibers with constant dispersion *ε* = 2/3 [17] and for (non-polarization-maintaining) fibers with random dispersion *ε* = 1 [18]. Equations (15)–(18) also govern the degenerate FWM process in which *ω*
_{3} = *ω*
_{2}. This process is the vector version of the aforementioned unidirectional scheme. It is illustrated in Fig. 2.

Suppose that waves 4 and 1 are strong pumps and waves 2 and 3 are the polarization components of a weak signal. Then, in the small-signal approximation, the pump photon-fluxes are constant:

As in the scalar process described above, the pumps are subject to SPM and CPM. Let *A*
_{4}(0) = *B*
_{4}, *A*
_{1}(0) = *B*
_{1},

Then the transformed signal components obey the (linearized) equations

where *δ* = *β*/2 + *$\overline{\gamma}$*(*P*
_{4} + *P*
_{1})/2 and *γ*= *$\overline{\gamma}$
εB*
_{4}
*B*
_{1}. It follows from Eqs. (23) and (24) that

where *μ* and *ν* were defined in Eqs. (6) and (7). In quantum optics the input-output relation defined by Eqs. (25) and (26) is called a two-mode squeezing transformation [6]. If there is no input idler [*B*
_{3}(0) = 0], the output idler is proportional to the complex conjugate of the input signal. Consequently, this FWM process is called phase conjugation (PC). If the input signal is split evenly between the two polarizations [|*B*
_{3}(0)| = |*B*
_{2}(0)|], Eq. (25) reduces to Eq. (5): Degenerate vector FWM produces PSA in a fiber.

In nondegenerate vector FWM the parallel component of the signal interacts with the perpendicular component of the idler, and *vice versa*. Because both interactions amplify the signal component and generate the idler component in the same way, the total output powers of the signal and idler are proportional to the total input power of the signal: The signal and idler gains produced by nondegenerate vector FWM do not depend on the signal polarization [18]. In contrast, Eqs. (25) and (26) imply that the signal gain produced by degenerate vector FWM depends on the signal polarization.

## 4. Cascaded four-wave mixing processes in a *χ*^{(3)} medium

The degenerate FWM processes described in Section 3 provide PSA because the frequency degeneracies allow the signal amplitudes to interact with their complex-conjugates [Eq. (5)]. The nondegenerate PC process [Eqs. (25) and (26)] also provides PSA if the idler amplitude is nonzero, and has a definite phase relative to the signal amplitude. Unfortunately, such an idler is usually absent. However, this absence can be filled by the prior use of another FWM process, called Bragg scattering (BS), to generate an idler that is a frequency-shifted, but non-conjugated, image of the signal. Once the idler has been generated by BS, PC can be used to provide PSA. The scalar and vector versions of these cascaded FWM processes are illustrated in Figs. 3 and 4, respectively.

Consider the BS process. Like the nondegenerate FWM process described in Section 4, BS is governed by the frequency-matching conditions *ω*
_{2} + *ω*
_{3} = *ω*
_{4} + *ω*
_{1} and the amplitude equations (15)–(18). Unlike the aforementioned FWM process, waves 1 and 3 are the pumps (rather than 1 and 4), wave 2 is the signal and wave 4 is the idler (rather than 3). Because power flows from the signal to the idler, the photon flux of the output idler cannot exceed the photon flux of the input signal: BS is intrinsically stable (does not provide gain).

Suppose that waves 1 and 3 are strong pumps, wave 2 is a weak signal and wave 4 is a weak idler. Then, in the small-signal approximation, the pump photon-fluxes are constant:

As in the PC process described above, the pumps are subject to SPM and CPM. Let *A*
_{1}(0) = *B*
_{1}, *A*
_{3}(0) = *B*
_{3},

Then the signal and idler amplitudes obey the (linearized) equations

where *δ* = *β*/2 + *$\overline{\gamma}$*(*P*
_{1} - *P*
_{3})/2 and *γ*=*$\overline{\gamma}$
εB*
_{1}
${B}_{3}^{*}$. It follows from Eqs. (35) and (36) that

where the transfer functions

and the wavenumber *k* = (|*γ*|^{2} + *δ*
^{2})^{1/2}. The transfer functions satisfy the auxiliary equation |*$\overline{\mu}$*|^{2}+|*$\overline{\nu}$*|^{2} = 1. In quantum optics the input-output relation defined by Eqs. (33)–(36) is called a beam-splitter transformation [6]. In the current context *B*
_{4}(0) = 0. For the ideal case in which *δ* = 0, *k* = *π*/2 and *γ* is real, *B*
_{2}(*z*′) = *B*
_{2}(0)/2^{1/2} and *B*
_{4}(*z*′) = *iB*
_{2}(0)/2^{1/2}: The output idler is a phase-shifted, but non-conjugated image of the signal. (The phase shift is required by photon-flux conservation.) Because the signal and idler frequencies are distinct, their relative phase can be modified between the BS and PC processes. (Because *k*
_{4} ≠ *k*
_{2}, propagation effects this change naturally.)

Now consider the PC process in which waves 5 and 1 are pumps, wave 2 is a signal and wave 4 is an idler. This process is governed by the frequency-matching conditions *ω*
_{5} + *ω*
_{1} = *ω*
_{2} + *ω*
_{4} and the amplitude equations (15)–(18), with the subscripts 3 and 4 replaced by 4 and 5, respectively. The effects of SPM and CPM on the pumps are described by Eqs. (19) and (20). The signal and idler amplitudes obey the (linearized) equations

where *δ* = *β*/2 + *$\overline{\gamma}$*(*P*
_{5}+*P*
_{1})/2, *γ* = *$\overline{\gamma}$
εB*
_{5}
*B*
_{1} and *β* = *β*
_{2} + *β*
_{4} - *β*
_{5} - *β*
_{1}. It follows from Eqs. (37) and (38) that

where *μ* and *ν* were defined in Eqs. (6) and (7). Because *B*
_{2}(*z*′) is proportional to *B*
_{2}(0) and ${B}_{4}^{*}$(*z*′) is proportional to ${B}_{2}^{*}$(0), cascaded BS and PC provide PSA in a fiber. By controlling the phase of pump 5 one controls the relative phase of the pumps, signal and idler and, hence, the orientation of the squeezing axis. For the ideal case described after Eq. (36), in which |*B*
_{4}(*z*′)| = |*B*
_{2}(*z*′)|, Eq. (39) reduces to Eq. (5). It should be noted that the pump frequencies are not unique: Other combinations of pump frequencies provide PSA by cascaded BS and PC. Furthermore, BS can also be used to combine a signal with the frequency-shifted, and conjugated, idler produced by prior PC.

## 5. Summary

This report described schemes to produce PSA in a fiber. The one-stage schemes are based on degenerate FWM, whereas the two-stage schemes are based on cascaded nondegenerate BS and PC, in either order. Although some scientific and practical issues remain to be addressed, the existence and study of these schemes can only advance the quest for PSA in optical communication systems.

## Acknowledgments

We thank M. G. Raymer for his constructive comments on the manuscipt.

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