## Abstract

Exact formulas are obtained for the amplitudes of light waves involved in four-wave-mixing cascades near the zero-dispersion frequency of a fiber. The cascade that is initiated by two strong pump waves is phase insensitive, whereas the cascade that is initiated by two strong pump waves and a weak signal wave is phase sensitive. In both cascades, the number of waves that have significant power increases with distance.

©2006 Optical Society of America

## 1. Introduction

Parametric devices based on four-wave mixing (FWM) in a fiber can amplify, frequency convert, phase conjugate and sample optical signals [1, 2]. Degenerate FWM involves one strong pump wave, which interacts with weak signal and idler waves. Nondegenerate FWM involves two strong pumps, a weak signal and a weak idler. One can optimize the performance characteristics of these FWM processes, on which the aforementioned applications are based, by using a pump whose frequency (or two pumps whose average frequency) is near the zero-dispersion frequency (ZDF) of the fiber.

Not only can two primary pumps amplify a signal and generate an idler, they can also generate secondary pumps by pump–pump FWM. If the primary–pump frequencies differ significantly, pump–pump FWM is limited by dispersion. However, if the primary-pump frequencies are similar, multiple FWM processes can occur and many secondary pumps (harmonics) can be generated.

Harmonic generation (frequency cascading) should be studied quantitatively, for a variety of reasons. First, it limits the performance of phase-insensitive (PI) parametric devices designed to process many signal channels simultaneously [3]. Second, it might also limit the performance of a phase-sensitive (PS) amplifier, in which two pumps amplify a signal whose frequency is the average of the pump frequencies [4, 5, 6]. However, it produces combs of pumps (or signals), with well-defined frequency and phase differences, which have applications in multiple-channel optical communications [7] and optical metrology [8].

## 2. Frequency cascades

Wave propagation in a dispersive nonlinear medium (fiber) is governed by the nonlinear Schroedinger equation (NSE)

where A is the slowly-varying (light) wave amplitude, *∂*_{z}
is the spatial derivative *∂*/*∂*_{z}
,*τ* is the retarded time *t*-*β*
_{1}
_{z}
, where *β*
_{1} is the group slowness, *β*=${\sum}_{n=2}^{\infty}$
*β*_{n}
(*i∂*_{τ}
)
^{n}
=*n*! is the Taylor expansion of the dispersion function and *γ* is the (Kerr) nonlinearity coefficient. The group-slowness, dispersion and nonlinearity coefficients are evaluated at the carrier frequency.

FWM processes are driven by nonlinearities and modified by dispersion-induced wavenumber shifts (phase mismatches). Formulas for these shifts were derived in [3]. Current FWM experiments involve highly-nonlinear fibers (HNFs) with third- and fourth-order dispersion coefficients β_{3}≈0.03 ps^{3}/km and β_{4}≈-3×10^{-4} ps^{4}/km, respectively, nonlinearity coefficients *γ*≈10/km-W (all evaluated at the ZDFs), and lengths *l*≈1 km, and pumps with powers *P*≈0.3 W [1, 2]. In these experiments the nonlinearity parameter *γPl*≈3. Consider the interaction of two pumps, whose average frequency (measured relative to the ZDF) is 0, with a signal and an idler, whose frequencies are -6 and 6 Tr/s, respectively. For this interaction the half-difference frequency *ω*_{d}
=6 Tr/s and the dispersion (phase-mismatch) parameter β_{4}
${\omega}_{d}^{4}$*l*=24≈-0.02, which is much smaller than the nonlinearity parameter. Now consider the interaction of two pumps, whose average frequency ω_{a}≈3 Tr/s, with a signal and an idler, whose frequencies are 0 and 6 Tr/s, respectively. For this interaction ω
_{d}
=3 Tr/s and the dispersion parameter (β_{3}ω
_{a}
)${\mathrm{\omega}}_{d}^{2}$*l*=2≈0.4, which is also smaller than the nonlinearity parameter. (β_{3}ω
_{a}
is the second-order dispersion coefficient evaluated at the average pump frequency.) For 10-Gb/s communication systems the channel spacing is about 0.3 Tr/s. The preceding numerical examples suggest that the effects of dispersion on a comb of 20 channels (centered on the ZDF), are weak. These examples were conservative: HNFs exist with smaller third-order dispersion coefficients and larger nonlinearity coefficients, and pumps exist with higher powers.

In the absence of dispersion, the NSE has the simple solution

Because solution (2) only contains the effects of nonlinearity, it is convenient to let *P* be a reference power, *A*/*P*
^{1/2}→A and *γPz*→*z*, in which case the amplitude and distance variables are dimensionless, and *γ* is absent from the solution.

#### 2.1. Pump–pump cascade

Consider the two-frequency boundary (initial) condition

where *ϕ*=*ωτ*+*ϕ*
_{1}, which corresponds to two pumps (1 and -1) with frequency difference 2ω and initial phase difference 2ϕ_{1}. The input power has the time-average ${\rho}_{+}^{2}$+${\rho}_{-}^{2}$ and the contribution 2*ρ*+*ρ*- cos(2*ϕ*), which oscillates at the difference frequency. By using the identity [9]

where *J*_{m}
is the Bessel function of order *m*, *x* is the distance parameter 2*ρ*+*ρ*-*z* and θ=2ϕ, one can write solution (2) as the harmonic series

where

*n* is an odd integer and the common (n-independent) phase factor exp[*i*(${\rho}_{+}^{2}$+${\rho}_{-}^{2}$)*z*] was omitted for simplicity. (The harmonic *A*_{n}
has frequency -*nω*.) For continuous-wave inputs, *ρ*
_{±} and *ϕ*
_{1} are constant, whereas for pulsed inputs they vary with time (*τ*). Equations (5) and (6) are valid for arbitrary temporal variations. However, the harmonic series they describe is only useful if the input parameters vary slowly compared to the difference-frequency oscillation.

As distance increases, so also does the number of harmonics with significant power: Eq. (6) describes a FWM cascade. This cascade has several interesting properties, which follow from Eq. (6) and the identity [9]

First, observe that the harmonics do not depend on ϕ1: the pump–pump cascade is PI. Second,

which shows that the time-averaged power is constant, consistent with solution (2). Third,

If the initial power spectrum is symmetric about the average frequency (*ρ*
_{+}=*ρ*
_{-}), it remains symmetric [*A*_{n}
(*x*)=*A*-*n*(*x*)]. In contrast, if it is asymmetric initially it remains asymmetric. However, the degree of asymmetry decreases nonmonotonically with distance (as 1/*x*). Fourth, the product term

In the presence of dispersion, the input modes are constrained to interact with only one other mode (degenerate FWM) or two other modes (nondegenerate FWM), and the modal evolution is periodic [10, 11]. However, in the absence of dispersion, the transfer of power to other modes is unabated, and the modal evolution is aperiodic.

Formulas similar to (6) apply to the stimulated-Raman-scattering (SRS) cascades studied in [12, 13]. These SRS cascades are (conceptually) simple, because each mode is coupled to one higher-frequency (anti-Stokes) mode and one lower-frequency (Stokes) mode. In contrast, the FWM cascade is complicated, because each mode is coupled to every pair or triplet of other modes that satisfies the frequency-matching conditions for degenerate or nondegenerate FWM, respectively [14, 15]. It is remarkable that similar formulas apply to such different cascades.

The evolution of a pump–pump cascade is illustrated in Figs. 1 and 2, for the initial conditions *ρ*
_{+}=0.71 and *ρ*
_{-}=1.00. [A cascade with *ρ*
_{±}=1 is illustrated in Fig. 5.] In Fig. 1 the pump spectrum is displayed for two values of the distance parameter *x*=2*ρ*
_{+}
*ρ*
_{-}
*z*. The first spectrum is produced (mainly) by three distinct FWM processes. The degenerate FWM processes in which 2_{γ-1}→*γ*
_{-3}+*γ*
_{1} and 2_{γ1}→*γ*
_{-1}+*γ*
_{3} (where *γ*_{n}
represents a photon with frequency -*nω*) seed modes -3 and 3. Once these modes have been seeded, their growth is modified by the nondegenerate FWM process in which *γ*
_{-1}+*γ*
_{1}→*γ*
_{-3}+*γ*
_{3}. Henceforth, these processes will be denoted by (-3,-1,-1,1), (-1,1,1,3) and (-3,-1,-1,3), respectively. The second spectrum contains many modes of comparable power, whose origins are complicated. For example, mode 5 is seeded by the degenerate processes (1,3,3,5) and (-3,1,1,5), and the nondegenerate processes (-1,1,3,5) and (-3,-1,3,5). Once it has been seeded, its growth is modified by the nondegenerate processes (-5,-3,3,5) and (-5,-1,1,5). We val-idated the spectra displayed in Fig. 1 (and Figs. 3 and 5) by calculating Fourier transforms of solution (2) numerically.

In Fig. 2(*a*) the powers of modes -1, -3 and -5 are plotted as functions of the distance parameter *x*. Mode -1 is depleted rapidly, as it transfers power to the other modes. At *x*=3, modes -3 and -5 are more powerful then mode -1, as shown in Fig. 1(*b*). However, for longer distances the powers of modes -3 and -5 also decrease nonmonotonically, as they transfer power to higher-order modes. In Fig. 2(*b*) the power asymmetry ${\sum}_{n=1}^{\infty}$[|*A*
_{-n}(*x*)|^{2}-|*A*_{n}
(*x*)|^{2}] is plotted as a function of distance. It decreases to 0 in the oscillatory manner predicted by Eq. (9).

#### 2.2. Pump–signal cascade

Now consider the three-frequency initial condition

where ρ_{0} and ϕ_{0} are constants (or slowly-varying functions of time). This condition corresponds to two pumps of equal power and a signal whose frequency is the average of the pump frequencies (0). The input power has the time-average 2ρ^{2}+${\mathrm{\rho}}_{0}^{2}$, and the contributions 2ρ^{2} cos(2*ϕ*) and 4*ρ*
_{0}
*ρ* cos*ϕ*
_{0} cos*ϕ*, which oscillate at the frequencies 2*ω* and *ω*, respectively. By using identity (4) twice, one finds that solution (2) can be written as the harmonic series (5), where

*n* is an integer, *x*=2ρ^{2}
*z*, ε_{0}=2(*ρ*
_{0}=*ρ*)cosϕ_{0}, *J′*
_{l}
(*y*)=*dJ*_{l}
(*y*)/*dy* and the common phase factor exp[*i*(2ρ^{2}+${\mathrm{\rho}}_{0}^{2}$)*z*] was omitted for simplicity (*l* is an integer, not a length). Let A(*n*;*m*) denote the *m*th term on the right side of Eq. (12). Then *A*(*n*,-*m*)=*A*(-*n*,*m*), from which it follows that the three-input cascade is symmetric [*A*_{n}
(*x*)=*A*
_{-n}(*x*)]. Observe that the amplification of the 0-frequency signal, and the generation of even-harmonic idlers, depends on *ϕ*
_{0}: The signal–idler cascade is PS. The phase shift *ϕ*
_{0}→*ϕ*
_{0}+*π* causes cos*ϕ*
_{0}, sin*ϕ*
_{0} and ε_{0} to change sign. These changes have no effect on the amplitudes of the odd harmonics (pump modes), and change the signs, but not the magnitudes, of the even harmonics (signal and idler modes). Hence, the period of phase sensitivity is [0,π).

In the linear regime (ε_{0}
*x*≪1), the pumps do not depend on ρ_{0}, whereas the signal and idlers are all proportional to ρ_{0}. In this regime, the exact formula (12) reduces to the approximate formula

which demonstrates clearly the aforementioned phase sensitivity (of the even harmonics).

The evolution of a pump–signal cascade is illustrated in Figs. 3 and 4, for the initial conditions ρ=1.0, ρ_{0}=0.1 and *ϕ*
_{0}=0.0, which correspond to a weak signal that is in-phase with strong pumps (whose average phase is 0). In Fig. 3 the power spectrum is displayed for two values of the distance parameter *x*=2ρ^{2}
*z*. The first spectrum contains the pump modes ±1 and ±3, whose origin was discussed in the context of Fig. 1. In the context of Fig. 3 the input pumps have equal powers, so the cascade is symmetric. The first spectrum also shows an amplified signal (mode 0) and generated idlers (±2), which will be referred to as signals for simplicity. The second spectrum consists of many pump modes, and many signal modes of comparable power. The generation of higher-order signals will be discussed quantitatively in Section 3.

In Fig. 4(*a*) the powers of signal modes 0, -2 and -4 are plotted as functions of the distance parameter *x*. At *x*=1.4 the first two modes have comparable power, and at *x*=2.6 all three modes have comparable power: The signal cascade develops rapidly, because it is driven by the pump cascade. At *x*=3 modes ±4 are the strongest signals, as shown in Fig. 3(*b*). None of these modes grows exponentially with distance. In Fig. 4(*b*) the total pump power and total signal power are plotted as functions of distance. For short distances the pumps transfer power to the signals, and at *x*=4.6 the pump and signal powers are equal. For longer distances the signals transfer power back to the pumps: Despite the fact that the power exchanges among the pumps and among the signals are not periodic, the power exchange between the pumps and signals is quasi-periodic.

The evolution of another pump–signal cascade is illustrated in Figs. 5 and 6, for the initial conditions *ρ*=1.0, *ρ*
_{0}=0.1 and *ϕ*
_{0}=*π*/2, which correspond to a weak signal that is out-of-phase with strong pumps. In Fig. 5 the power spectrum is displayed for two values of the distance parameter *x*. The main difference between Figs. 3 and 5 is the presence of a strong signal cascade in the former, and the absence of a strong signal cascade in the latter. The pump spectra in Fig. 5 are similar to those in Fig. 3, but the pump powers are slightly higher because no power is transferred to the signals.

In Fig. 6(*a*) the powers of signal modes 0, -2 and -4 are plotted as functions of the distance parameter *x*. Because it is out-of-phase with the pumps, mode 0 is attenuated. Modes -2 and -4 evolve in an oscillatory manner, with powers that never exceed the input power of mode 0. The generation of weak higher-order signals will be discussed quantitatively in Section 3. In Fig. 6(*b*) the total pump power and total signal power are plotted as functions of distance. Although a large amount of power is transferred among the pumps, and a small amount of power is transferred among the signals, no power is transferred between the pumps and signals.

Pump and signal spectra were also obtained for *ϕ*
_{0}=*π*/4 and *ϕ*
_{0}=3*π*/4. Both initial conditions lead to well-developed spectra, like those shown in Fig. 3(*b*), at *x*=3. For the first condition the average signal power was about 0.04, whereas for the second it was about 0.03. The signal evolution was also determined for *x*<10. In the first case the total pump (signal) power decreases (increases) monotonically, and the pump and signal powers are comparable near *x*=6.1. In the second case the total pump (signal) power increases (decreases) slightly for short distances (*x*<0.3), then decreases (increases) monotonically. The pump and signal powers are comparable near *x*=6.8. Figures 3–6 and the preceding results demonstrate clearly the phase sensitivity of the pump–signal cascades.

## 3. Perturbation analyses of the cascades

Solutions (6) and (12) are exact. However, they provide little insight into the development of, and interaction among, the constituent FWM processes. In this section, the initial developments of the cascades (*z*≪1) are analyzed perturbatively. The specific processes that produce the first few pump and signal harmonics, and the phase-sensitivity of the pump–signal interaction, are studied in detail for symmetric initial conditions (ρ_{±}=1).

#### 3.1. Pump–pump cascade

Equation (6) was obtained for a two-mode input (±1). For short distances, the neighboring harmonics are much weaker than the input modes, so one can use perturbation analysis to model their evolution approximately. By doing so, one finds that each input mode acquires phase shifts, of magnitudes 1
_{s}*z* and 2
_{c}*z*, due to self-phase modulation (SPM) and cross-phase modulation (CPM), respectively. The subscripts *s* and *c* label the mechanisms, but do not affect the values of the coefficients, which are 1 and 2, respectively.

One can facilitate the analysis of the cascade by defining *A*_{n}
=*B*_{n}
exp[*i*(1
_{s}
+2
_{c}
)*z*], in which case the transformed amplitudes *B*
_{±}
_{1}≈1. Hence, the initial growth of mode 3 is governed by the equation

where 1
_{c}
, 1
_{d}
and 2
_{n}
denote the coefficients associated with CPM, the degenerate FWM process (-1,1,1,3) and the nondegenerate FWM process (-3,-1,1,3), respectively (*n* denotes a process, not a mode number). For the stated initial conditions, *B*
_{-3}=*B*
_{3}. Equation (14) can be solved iteratively, in powers of *z*. By doing so, one finds that

This result shows that the growth of mode 3 is seeded by degenerate FWM and enhanced by nondegenerate FWM, as stated in Section 2.1.

Equation (15) implies that |*B*
_{3}|^{2}≈*z*
^{2}. Because solution (6) conserves power, the input modes must be depleted concomitantly. The initial evolution of mode 1 is governed by the equation

where 1
_{d}
and 2
_{n}
denote the coefficients associated with the degenerate process (-1,1,1,3) and the nondegenerate process (-3,-1,1,3), respectively. For short distances, the degenerate term in Eq. (16) is larger than the nondegenerate term, which can be neglected. By solving what remains of Eq. (16) iteratively, one finds that

It follows from Eqs. (15) and (17) that |*B*
_{1}|^{2}+|*B*
_{3}|^{2}≈1 and *B*
_{1}
*B*
_{-1}+*B*
_{3}
*B*
_{-3}≈1-2*z*
^{2}: Power is conserved and the product term is not constant, as stated in Section 2.1.

By making the approximation *z*≪1 in Eq. (6), one finds that

where the phase factor exp(-*iz*) was added so that the modified-*A* and *B* amplitudes omit the same phase factor [exp(*i*3*z*)]. Equations (18) and (19) validate Eqs. (17) and (15), respectively.

#### 3.2. Pump–signal cascade

The analysis of Section 2.2 showed that, in the linear regime, the pumps are unaffected by the presence of the signals. Hence, the results of Section 3.1 also apply to the pump–signal cascade. What remains to be studied is the initial evolution of the signals. By making the approximation *z*≪1 in Eq. (12), one finds that

where *c*
_{0}=cos*ϕ*
_{0}, *s*
_{0}=sin*ϕ*
_{0}, and the common factor ρ0 was omitted for simplicity.

The standard PS process involves two pumps (±1), which amplify a signal (0), whose frequency is the average of the pump frequencies [4]. This degenerate FWM process, which is also called the inverse modulation interaction (MI), is governed by the equation

where 1
_{c}
and 2
_{m}
denote the coefficients associated with CPM and MI, respectively (*m* denotes a process, not a mode number). By solving Eq. (22) iteratively, one finds that

$$+\left({2}_{m}^{2}-{1}_{c}^{2}\right)\left({c}_{0}+i{s}_{0}\right){z}^{2}\u20442.$$

Equation (23) shows that the signal growth is driven by MI and modified by CPM. It agrees with Eq. (20) to order *z*, but not to order *z*
^{2}. The signal growth is modified by the depletion of pumps ±1 and the presence of pumps ±3, which drive the inverse MI (-3,0,0,3). However, the results of Section 3.1 show that these processes do not affect the second-order contribution to the signal. Instead, signal 0 is modified by its interactions with the signal harmonics ±2. These interactions will be discussed quantitatively after the growth of the harmonics has been analyzed.

Suppose that pumps ±3 can be neglected, because they are weaker than pumps ±1. Then the initial growth of signal 2 is governed by the equation

where ∝ means includes, and 1
_{m}
, 2
_{b}
and 2
_{p}
denote the coefficients associated with the MI process (0,1,1,2), the Bragg scattering (BS) process (-1,0,1,2) and the phase conjugation (PC) process (-2,-1,1,2), respectively. BS and PC are both nondegenerate FWM processes. However, in the context of signal generation a distinction between them is made, because the former amplifies one signal at the expense of another (*γ*
_{0}+*γ*
_{1}→*γ*
_{2}+*γ*-_{1}), whereas the latter amplifies two signals simultaneously (*γ*
_{-1}+*γ*
_{1}→*γ*
_{2}+*γ*
_{-2}). For the stated initial conditions, *B*
_{-2}=*B*
_{2}. By solving Eq. (24) iteratively, one finds that

Equation (25) shows that signal 2 is generated by MI and BS. It agrees with Eq. (21) to order *z*, but not to order *z*
^{2}, because the second-order contributions canceled. Hence, the effects of pumps ±3 cannot be neglected. The contributions of these pumps to the growth of signal 2 are described by the equation

where *i*1
_{d}*z*≈*B*
_{3}, 2
_{p}
denotes the coefficient associated with the PC process (-1,0,2,3), and the first and second 2
_{b}
coefficients are associated with the BS processes (0,1,2,3) and (-3,-1,0,2), respectively. By solving Eq. (26) approximately, one finds that

Taken together, Eqs. (25) and (27) are equivalent to Eq. (21). It is sobering to realize that six distinct FWM processes contribute to signal 2 (which is only the first generated signal).

The contributions of signals ±2 to the growth of signal 0 is described by the equation

where the first and second 1
_{m}
coefficients are associated with the MI processes (0,1,1,2) and (-2,-1,-1,0), respectively, and the first and second 2
_{b}
coefficients are associated with the BS processes (-1,0,1,2) and (-2,-1,0,1), respectively. It follows from Eqs. (25) and (28), and the symmetry property *B*
_{-2}=*B*
_{2}, that

Taken together, Eqs. (23) and (29) are equivalent to Eq. (20). These equations show that the initial evolution of signal 0 is dominated by the standard process, but its subsequent evolution is modified by the presence of other harmonics: The phase sensitivity of the pump–signal cascade is similar, but not identical, to the phase sensitivity of the standard process (which is reviewed in the Appendix).

In addition to the standard PS process (inverse MI), in which two pumps (1 and -1) amplify a one-mode signal (0), there is an alternative PS process (MI), in which one pump (0) amplifies a two-mode signal (1 and -1) [10, 16]. The methods described herein can also be used to analyze the alternative process. Because it does not involve a second pump, there is no pump–pump cascade. However, the signal pair (-1,1) seeds the idler pairs (-*n*,*n*), where *n*≥2. If one were to amplify the second signal (-2;2) simultaneously, the second-harmonic idler seeded by the first signal would be superimposed on the second signal: Further work is required to quantify the effects of signal-signal crosstalk in multiple-channel implementations of the alternative process. Two-pump processes with two-mode signals are also PS [11].

## 4. Summary

In this report detailed studies were made of the frequency cascades initiated by two strong pump waves, and two strong pump waves and a weak signal wave, whose frequency is the average of the pump frequencies. These cascades are produced by four-wave mixing (FWM) in a fiber, and involve light waves whose frequencies are comparable to the zero-dispersion frequency of the fiber.

The aforementioned studies were based on the nonlinear Schroedinger equation, which can be solved exactly in the absence of dispersion. The pump–pump cascade [Eq. (6)] is phase insensitive. If the spectrum is symmetric initially, it remains symmetric. If the spectrum is asymmetric initially, it remains asymmetric, but the degree of asymmetry decreases with distance. In contrast, the pump–signal cascade [Eq. (12)] is phase sensitive. If the signal is in-phase with the pumps initially, it is amplified and many idlers are generated. If the signal is out-of-phase with the pumps initially, it exchanges power with the idlers, but does not receive power from the pumps. The phase sensitivity of the pump–signal cascade is similar, but not identical, to the phase sensitivity of the standard FWM process, which involves only the input waves (two pumps and one signal). In both cascades the wave evolution is aperiodic, which distinguishes them from their constituent processes (degenerate and nondegenerate FWM).

In this report, the properties of solutions (6) and (12) were described for continuous-wave inputs. However, the solutions are also valid for pulsed inputs, in which *ρ*
_{∓}, *ϕ*
_{1}, *ρ*
_{0} and *ϕ*
_{0} are slowly-varying functions of time.

## Appendix: Standard process

In this appendix the phase sensitivity of the standard FWM process (-1,0,0,1) is reviewed. The initial evolution of signal 0 is governed by the equation

where δ=[2β(ω_{0})-β(ω_{1})-β(ω_{-1})]/2+γ(|*B*
_{1}|^{2}+|*B*
_{-1}|^{2})/2 is the wavenumber mismatch. Let *P* be a reference power. Then, by making the substitutions *B*
_{0}=*P*
^{1/2}→*B*
_{0}, *B*
_{±1}/*P*
^{1/2}→*ρ* exp(±*iϕ*
_{1}), *δ*/(*γP*)→*δ* and *γPz*→*z*, one can rewrite (30) in the dimensionless form

which is consistent with Eq. (22).

One can write the solution of Eq. (31) in the input–output form

where the transfer functions

and the (spatial) growth rate *κ*=(4*ρ*
^{4}-*δ*
^{2})^{1/2}. The transfer functions satisfy the auxiliary equation |*µ*|^{2}-|*ν*|^{2}=1. It follows from the initial condition *B*
_{0}(0)=*ρ*
_{0} exp(*iϕ*
_{0}), which is consistent with condition (11), and Eqs. (32)–(34), that

where the subscripts *r* and *i* denote real and imaginary parts, respectively. Equation (35) shows that the signal gain depends on the input phase *ϕ*
_{0}, and the period of phase sensitivity is [0,*π*). The gain attains its extremal values when

The first-quadrant value of 2*ϕ*_{m}
corresponds to the maximal gain (|*µ*|+*ν*_{i}
)^{2}, whereas the third-quadrant value corresponds to the minimal gain (|*µ*|-*νi*)^{2}. It follows from the auxiliary equation that (|*µ*|-*νi*)^{2}=1/(|*µ*|+*νi*)^{2}: The minimal gain is the reciprocal of the maximal gain.

In the presence of dispersion, one can tune the pump frequencies in such a way that the dispersive contribution to the wavenumber mismatch cancels the nonlinear contribution. In this common case, δ=0, *µ*_{i}
=0 and ∇=2*ρ*
^{2}. The gain attains its maximal value (*µ*_{r}
+*ν*_{i}
)^{2}=exp(4*ρ*
^{2}
*z*) when ϕ_{0}=*π*/4, and its minimal value exp(-4*ρ*
^{2}
*z*) when ϕ_{0}=3*π*/4. The values of *ϕ*_{m}
do not depend on distance.

In the absence of dispersion, such tuning is impossible. In this case, which is relevant to this report, δ=ρ^{2}, ∇=3^{1/2}ρ^{2} and the values of *ϕ*_{m}
depend on distance. For short distances Eqs. (32)–(34) imply that

where *c*
_{0}=cos*ϕ*
_{0} and *s*
_{0}=sin*ϕ*
_{0}. Equation (37) is consistent with Eq. (23). The gain attains its (approximate) maximal value 1+4(*ρ*
^{2}
*z*)+8(*ρ*
^{2}
*z*)^{2} when *ϕ*
_{0}=*π*/4, and its minimal value 1-4(*ρ*
^{2}
*z*)+8(*ρ*
^{2}
*z*)^{2} when *ϕ*
_{0}=3*π*/4, as in the common case. However, for long distances the gain attains its maximal value [(1+${\nu}_{i}^{2}$
)^{1/2}+*ν*_{i}
]^{2}≈16sinh^{2}(3^{1/2}
*ρ*
^{2}
*z*)/3 when *ϕ*
_{0}=*π*/6, and its minimal value 3/[16sinh^{2}(3^{1/2}
*ρ*
^{2}
*z*)] when *ϕ*
_{0}=2*π*/3.

## Acknowledgment

We thank L. Schenato for his constructive comments on the manuscript.

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