## Abstract

In this work we applied a Hamiltonian formalism to simplify the equations of non-degenerate nonlinear four-wave mixing to the one-degree-of-freedom Hamiltonian equations with a three-parameter Hamiltonian. Thereby, a problem of signal amplification in a phase-sensitive double-pumped parametric fiber amplifier with pump depletion was reduced to a geometrical study of the phase portraits of the one-degree-of-freedom Hamiltonian system. For a symmetric case of equal pump powers and equal signal and idler powers at the fiber input, it has been shown that the theoretical maximum gain occurs on the extremal trajectories. However, to reduce the nonlinear interaction of waves, we proposed to choose the separatrix as the optimal trajectory on the phase plane. Analytical expressions were found for the maximum amplification, as well as the length of optical fiber and the relative phase of interacting waves allowing this amplification. Using the proposed approach, we optimized of the phase-sensitive parametric amplifier. As a result, the optimal parameters of the phase-sensitive amplifier were found and the maximum possible signal amplification was realized in a broad range of signal wavelengths.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The resonant four-wave mixing is an effective mechanism for energy transfer between waves propagating in a nonlinear medium, for example, in an optical fiber. This nonlinear process is employed in fiber-optic parametric amplifiers where the signal wave (or waves) is amplified by energy transfer from pump waves [1–4]. If we neglect the losses in the optical fiber, the four-wave mixing in a nonlinear medium can be described by a Hamiltonian system of equations with four degrees of freedom. In addition to the Hamiltonian, these equations have three independent integrals, called the Manley-Rowe relations. This allows us to integrate the system of four-wave mixing equations and reduce the dynamics of nonlinear system to an one-degree-of-freedom Hamiltonian system [6–13].

In this paper we performed the canonical transformation of variables in the complete Hamiltonian system describing the interaction of four waves and we obtained Manley-Rowe relations as the integrals of motion corresponding to the cyclic variables. As a result, the initial system of four pairs of Hamiltonian equations was reduced to one pair of equations in which the Hamiltonian depends on three integrals of motion of the complete system. To study the dynamics of the one-degree-of-freedom Hamiltonian system for all possible values of the parameters, another canonical transformation of variables was done and a system with dimensionless variables was obtained with Hamiltonian depending only on three dimensionless parameters. Two dimensionless parameters are determined only by the initial data and one parameter depends on the initial data and the properties of the parametric amplifier.

The optimization problem for the phase-sensitive fiber optical parametric amplifier (PS-FOPA) consists in finding the parameters of the system leading to the maximum energy transfer from the pump waves to the signal and idler waves. The maximum theoretical signal (idler) amplification is determined by the condition of the complete pump power transfer to signal and idler. However, the maximum amplification can be achieved only if the corresponding trajectory of the one-degree-of-freedom Hamiltonian system exists. In turn, the question of existence of this trajectory requires a comprehensive study of the phase portrait of the system, which fully describes the system dynamics [7, 9, 11]. Such analysis of the phase portraits of the dynamical system was used in [14] to describe evolution of the waves, involved in one-pump FOPA, and solve some optimization problems arising in the design of the specific PS-FOPA scheme.

In this work we studied the phase portraits of nonlinear system in the symmetric case when two pump waves have equal powers at fiber input, as well as signal and idler waves. The maximum theoretical signal gain, required length of the optical fiber, and relative phase of the interacting waves at the beginning of the fiber were found from analytic expressions. The proposed approach was applied to the optimization of a phase-sensitive double-pumped parametric amplifier with signal wavelength varying in wide spectral range.

## 2. Basic equations and Hamiltonian formulation

Consider the four-wave resonance conditions for a two-pump FOPA: *ω*_{1} + *ω*_{2} = *ω*_{3} + *ω*_{4}, where frequencies with indexes 1 and 2 correspond to the pump waves, the frequency with index 3 corresponds to the signal wave, and the frequency with index 4 corresponds to the idler.

Four-wave mixing between continuous linearly co-polarized waves is described by the well-known four coupled differential equations for complex wave amplitudes *A _{l}*,

*l*= 1, 2, 3, 4:

*γ*is the Kerr nonlinearity coefficient,

*β*=

_{l}*β*(

*ω*) is the propagation constant of the

_{l}_{l}th wave [2].

The system of Eqs. (1)–(4) is a Hamiltonian system for the four canonical couples $\left({A}_{l},{A}_{l}^{*}\right)$ [7, 8, 13]:

The Hamiltonian *H* does not depend on *z* and, consequently, is conserved. Besides that, there are 3 independent invariants (the Manley-Rowe relations) [13], thus, the system of equations (5) is integrated. These invariants can easily be found in new suitable variables, which will be introduced below.

Since we are considering the parametric amplifier, we are interested in the evolution of powers and phases, so it is convenient to express the complex amplitude as ${A}_{l}={P}_{l}^{1/2}\mathrm{exp}\left\{i{\theta}_{l}\right\}$, where *θ _{l}* is the phase and

*P*is the power [12].

_{l}This substitution is canonical, so the system of Eqs. (5) is Hamiltonian for real variables (*P _{l}*,

*θ*), [12]:

_{l}*θ*= −

*θ*

_{1}−

*θ*

_{2}+

*θ*

_{3}+

*θ*

_{4}is the relative phase of interacting waves.

The Hamiltonian *H* includes the phases *θ _{l}* in the form of their linear combination

*θ*, therefore it is convenient to consider

*θ*and any three phases as the new variables. We express the signal phase though the other phases

*θ*

_{3}=

*θ*+

*θ*

_{1}+

*θ*

_{2}−

*θ*

_{4}and introduce new pairs of canonical variables:

The old variables can be expressed through the new ones in the following way:

Substituting these values in the expression for the Hamiltonian (9), we obtain the Hamiltonian in the form:

*β*=

*β*

_{4}+

*β*

_{3}−

*β*

_{2}−

*β*

_{1}.

As can be seen from (12), the Hamiltonian does not depend on phases *θ*_{1}, *θ*_{2} and *θ*_{4}, so the variables *p*_{1}, *p*_{2}, *p*_{4} preserve at any propagation distance along the fiber, representing Manley-Rowe relations. It also follows from these relations, that the total power is invariant: *P _{T}* =

*P*

_{1}(

*z*) +

*P*

_{2}(

*z*) +

*P*

_{3}(

*z*) +

*P*

_{4}(

*z*) =

*p*

_{1}+

*p*

_{2}+

*p*

_{4}.

Now the dynamics of the system can be described by the two equations for the pair of variables (*p*_{3}*, θ*):

*p*

_{1},

*p*

_{2}and

*p*

_{4}are constants.

If we know a solution of the system (13), we will be able to solve the complete system (7)–(8) with the initial data:

Unknown powers of the waves *P*_{1}, *P*_{2} and *P*_{4} will be found from Eqs. (11), and phases of the waves *θ _{l}*(

*z*) – from Eq. (8) with known powers

*P*(

_{l}*z*) and relative phase

*θ*(

*z*) on the right-hand side.

The Hamiltonian (14) can also be symmetrized by introducing the signal power increment *x* = *P*_{3}(*z*) − *P*_{30}. Then the wave powers take the form:

After introducing the new variables (*x, θ*) the Hamiltonian (14) reads:

*P*

_{0}=

*P*

_{10}+

*P*

_{20}−

*P*

_{30}−

*P*

_{40}and

*P*are defined in (16).

_{l}The equations of motion can be presented in the normalized form:

*Z*=

*γz*is the normalized distance.

It is worth noting, that the equations obtained above are equivalent to the equations given in [2]: Eqs. (16) and (3.50), the Hamiltonian (17) and the conserved quantity *K*_{2} in (3.57), the equations of motion (18), (19) and (3.59), (3.53), respectively. In addition, in the phase-insensitive case, when *P*_{10} = *P*_{20}, *P*_{40} = 0 and *x* = *F*, the derived system of Eqs. (18)–(19) is equivalent to the system (8)–(9) from [12].

The resulting system (18)–(19) with initial data *x| _{Z}*

_{=0}= 0,

*θ|*

_{Z}_{=0}=

*θ*

_{0}describes all possible solutions of the original system (1)–(4).

#### 2.1. Space of parameters

Since the Hamiltonian (17) depends on five parameters: ∆*β*/*γ*, *P*_{10}, *P*_{20}, *P*_{30} and *P*_{40}, we should consider all possible phase portraits of an one-degree-of-freedom Hamiltonian system to study the complete dynamics of the nonlinear system.

The first problem is to determine the phase space of the system. The phase *θ* is obviously 2*π*-periodic and there are no restrictions on the initial phase *θ*_{0}. The variable *x* is more complicated. Formally, it is necessary to check the non-negativity of the radicand in the Hamiltonian (17). However, as long as the powers of the waves *P _{l}* and initial powers

*P*

_{l}_{0}are nonnegative,

*x*should lay in the intervals:

Without loss of generality, we can assume that *P*_{10} ≥ *P*_{20} and *P*_{30} ≥ *P*_{40}, so *x* ∈ [−*P*_{40}*, P*_{20}]. The condition of the radicand non-negativity in the Hamiltonian (17) is naturally satisfied for this interval as a consequence of the positivity of the wave powers *P _{l}*.

For the initial data (0*, θ*_{0}) the Hamiltonian (17) reads:

This expression describes two fundamentally different cases [1]. If *P*_{40} = 0, then the initial value of the Hamiltonian does not depend on the initial phase *θ*_{0}. This phase-insensitive case was considered in [12]. If *P*_{40} *>* 0, then the initial value of the Hamiltonian depends on the phase *θ*_{0} and we observe phase-sensitive amplification, which is the main focus of this work.

#### 2.2. Maximum gain

In case of phase-insensitive amplification, it is easy to find a necessary condition for achieving maximum gain [12]. Since the Hamiltonian (20) equals to zero at fiber input, the existence of a phase trajectory corresponding to the maximum signal gain *x* = *P*_{20} (*θ*_{1} is arbitrary) requires equality to zero of the Hamiltonian:

Equation (22) implies that for small signal power and equal powers of the pumps at fiber input, maximum gain occurs when wavevector mismatch approaches zero in a phase-insensitive amplifier.

Equation (21) is also correct for the phase-sensitive amplification, therefore, a necessary condition for the existence of a trajectory corresponding to the maximum gain can be presented as follows:

*θ*

_{0}. The coefficient of maximum amplification in both cases is:

## 3. Phase portraits

As we intend to analyze the behavior of the trajectories of the Hamiltonian system (18), (19) for all sets of parameters, it is more convenient to introduce a new dimensionless variable:

where*p*lies in the range between 0 and 0.5. The Hamiltonian takes the form:

*Q*=

_{l}*P*/2

_{l}*R*can be expressed using Eqs. (16) as follows:

*δ*

_{1}= (

*P*

_{10}−

*P*

_{20})/(2

*R*) ≥ 0,

*δ*

_{2}= (

*P*

_{30}−

*P*

_{40})/(2

*R*) ≥ 0.

The equations of motion take the form:

*l*= 2

*RZ*is the normalized distance.

Therefore, the Hamiltonian now depends on three dimensionless parameters: *k*, *δ*_{1} and *δ*_{2}. Zero value of *k* corresponds to non-dispersive case. Zero initial value of *p* corresponds to the phase-insensitive case. If the initial value of *p* does not equal to zero, it becomes the fourth parameter in the phase-sensitive case.

In order to fully describe a phase portrait of the dynamical system (28)–(29), it is necessary to find all stationary points and special trajectories.

#### 3.1. Stationary points

Let us find the stationary points (*p, θ*) of the dynamical system (28)–(29) that describe the steady state co-propagation of four nonlinearly interacting waves in such systems. There are four conditions when the right-hand side of Eq. (28) becomes zero:

Substituting *θ _{A}* and

*θ*on the right-hand side of Eq. (29), we find equations for the conjugate variable

_{B}*p*at the stationary points:

Asymptotes of this function at the points *p* = 0 and *p* = 1/2 are equal to:

Since the continuous function *f*_{±} takes opposite signs at the end points of an interval [0, 1/2], at least one solution of the equation *f*_{±}(*p*) = 0 exists. We denote such solutions *p _{A}* and

*p*.

_{B}After substituting *p _{C}* = 0 and

*p*= 1/2 in Eq. (29), the right-hand side of the equation tends to infinity. Thus, the system dynamics at these points are not defined. Note, that in contrast to the Kepler problem, the Hamiltonian (26) is defined at these points. We use this fact to express cos

_{D}*θ*from Eq. (26) in terms of the Hamiltonian. As a result, the right-hand side of Eq. (29) can be expressed only in terms of

*p*:

*H*

_{0}is a constant defining the level of the Hamiltonian.

If the Hamiltonian (26) becomes zero (*H*_{0} = 0), the function *g*(*H*_{0}*, p*) has a finite limit $\frac{1}{2}\left(k+{\delta}_{1}-{\delta}_{2}+1\right)$ for *p* → +0. The Hamiltonian can be factorized:

For ${H}_{0}=\frac{1}{2}\left(k+{\delta}_{1}-{\delta}_{2}\right)$, the function *g*(*H*_{0}*, p*) has the finite limit $\frac{1}{2}\left(k+{\delta}_{1}-{\delta}_{2}-1\right)$ if (1/2 – p) = *q* → +0. In this case the isoline of the Hamiltonian also splits into two parts:

## 4. Symmetrical case

In this section we consider the most common and practically important symmetrical case, when the pump powers are equal at fiber input, as well as the signal and idler powers. It corresponds to *δ*_{1} = *δ*_{2} = 0 and leads to a substantial simplification of the Hamiltonian (26) and the corresponding equations of motion:

#### 4.1. Stationary points of the dynamical system

In the symmetrical case, the stationary points can be easily found analytically. From Eq. (35) we find four conditions under which the right-hand side of the equation becomes zero:

Substituting these values on the right-hand side of Eq. (36), we obtain equations governing the existence of stationary points:

Therefore, four families of stationary points exist:

The inner stationary points *A* and *B* lie inside the band *p* ∈ [0, 1/2 when *k* ∈ [−3, 3] and *k* ∈ [−1, 1] correspondingly. The boundary point *C* exists when *k* ∈ [−3, 1] and the boundary point *D* exists when *k* ∈ [−1, 3]. Stationary points lying on the axis *p* = 0 correspond to a system without input signal/idler power and points with coordinate *p* = 0.5 correspond to a system without pump power.

#### 4.2. Analysis of the phase portraits

Now let us consider phase portraits corresponding to the Hamiltonian (34) for different values of the parameter *k*. A type of the phase portrait is determined by the existence of inner and boundary stationary points. Thus, analyzing phase trajectories in the vicinity of stationary points, we can obtain complete information concerning the dynamics of the nonlinear system. Note, that the Hamiltonian (34) is invariant up to a constant −*k*/2 after the transformation of variables *k* → −*k*, *p* → 1/2 − *p*. Therefore, the phase portraits for *k* and −*k* can be obtained from each other by reflection about the axis *p* = 1/4. This is a consequence of the invariance of the system with respect to the permutation of the signal and the pump.

The main types of phase portraits that describe dynamics of the system under consideration are shown in Fig. 1. When *k* ∈ [−1, 1] all stationary points *A*, *B*, *C*, and *D* exist and are shown in Fig. 1(b), when *k* ∈ [−3, −1] two stationary points *A* and *C* exist, when *k* ∈ [1, 3] two stationary points *B* and *D* exist, and finally, when |*k| >* 3 no stationary points exists. The signal amplification will be the smallest in a system with less variation of *p* in the curves of the phase portraits without stationary points in Fig. 1(d). Exact expressions governing signal amplification in the whole range of the parameter *k* values are given in appendix 7.

Next we find a trajectory, which starts at some sufficiently small power level 0 < *p*_{0} *<<* 1 and reaches a maximum attainable power for a given value of parameter *k*. The analysis of phase portraits shows that there are four types of trajectories connecting points on the phase plane, corresponding to the minimum and maximum signal powers. We will henceforth refer to these trajectories as “extremal”. The first type includes trajectories starting at point (*p*_{0}, 0) and reaching point (*p*, 0), where *p > p*_{0} is the maximum value of *p* along such trajectories. The second type includes trajectories starting at (*p*_{0}, 0) and reaching (*p, π*). Trajectories of the third type are those which start at (*p*_{0}*, π*) and reach (*p, π*), and trajectories of the fourth type start at (*p*_{0}*, π*) and end at (*p*, 0). For all the other possible trajectories the maximum attainable power *p* lies between the extrema of the corresponding extremal trajectories.

As an example, we considered the phase portrait presented in Fig. 2(b). The phase trajectory starting at the point (*p*_{0}, 0) reaches a maximum at the point (*p*_{1}, 0), which lies on the line *θ* = 0. Moving to the right along the line *p* = *p*_{0}, we can reach even larger values of *p* also lying on the line *θ* = 0. Finally, we reach the point (*p*_{0}*, π*), which determines the trajectory on which the maximum value $p={p}_{A}+{\left({p}_{A}^{2}-\frac{1}{6}H\left({p}_{0},\pi \right)\right)}^{1/2}$ is reached for all initial relative phases.

One could also choose among the extremal trajectories the trajectory corresponding to the fastest dimensionless signal power growth for any *k*. All extremal trajectories are tangent to the line *p* = *p*_{0}, so the derivative of *p* in (35) equals to zero at the beginning of the fiber. To achieve the maximum value of the derivative, leading to the fastest growth of *p*, the initial phase at the fiber input should be properly selected.

Next we propose how to choose the optimal initial phase different from *θ*_{0} = *π*/2, which is optimal for a system without pump depletion [2]. Consider the phase derivative with respect to the power:

It can be seen that for *θ* = 0 and *θ* = *π* the phase variation is infinitely large. If sin *θ* ≠ 0, then for the small *p <<* 1 the derivative will still be large. To prevent rapid growth of the phase one should put:

It corresponds to value of the phase *θ _{C}* at the singular point

*C*= (0

*, θ*). Therefore, we suggest choosing the starting point (

_{C}*p*

_{0}

*, θ*) of the power flow for nonzero initial power

_{C}*p*

_{0}. This point lies in the vicinity of stationary point

*C*, which is the beginning of the corresponding separatrix. The proposed trajectory will also be close to the separatrix due to the continuous dependence of the trajectories on the initial data.

Important to note, that weak dependence of the phase on the power in (38) can be interpreted as a weak nonlinearity, since signal power does not depend on phase in linear case.

#### 4.3. Finding the maximum gain

Contour lines of the Hamiltonian (34) at *H* = 0 and *H* = *k*/2 split into two curves:

Both curves families coincide if *k* = 0:

*p*= 0,

*p*= 1/2 and one vertical line

*θ*= arccos(−1/2) = 2

_{s}*π*/3.

If *k* ≠ 0 four families of curves exist:

In symmetrical case, the initial signal and idler wave powers are the same and equal to *P*_{30} = *P*_{40}. In this case an initial value of the dimensionless variable *p* equals to *p*_{0} = *P*_{40}/(2*R*).

If *k* = 0 separatrix is a vertical line, and moving along this line we can approach *p* = 1/2 but it requires an infinite fiber span length. Therefore, the maximum theoretical gain *G _{max}* is:

This expression for the gain has been previously obtained in (24). Here we also defined the phase *θ _{s}* = 2

*π*/3, at which such gain is possible. For any other values of the initial phase the gain will be smaller.

Analysis of the function *p* = *f* (*θ*) shows, that if *k* ∈ [−3, 0) and *θ* = 0 the maximum of the separatrix is *p*(*k*) = (3 + *k*)/6; if *k* ∈ (0, 1) and *θ* = *π* the maximum of the separatrix is p = (1 – *k*)/2. It means, that the attainable along the separatrix maximum gain equals to:

*p*

_{0}and reaches the maximum value

*p*(

*k*). The typical shapes of the separatrixes for

*k*= −0.5 and

*k*= 0.5 are shown in Fig. 3(a) by black lines.

Analysis of the function *p* = *f* (*θ*) − 1/2 shows, that if *k* ∈ [−1, 0) and *θ* = *π* the minimum of the separatrix is *p*(*k*) = −*k*/2; if *k* ∈ (0, 3) and *θ* = 0 the minimum of the separatrix is *p* = k/6. In this case the attainable along the separatrix maximum gain equals to:

*p*

_{0}, which should be larger than the separatrix minimum

*p*(

*k*). The typical shapes of the separatrixes for

*k*= −0.5 and

*k*= 0.5 are shown in Fig. 3(a) by green lines.

Figure 3(b) shows function *p*(*k*), proportional to the maximum attainable signal gain, in cases when the evolution of dynamical system occurs along the separatrixes of two different types considered above. The black line corresponds to the separatrix of the first type (motion along the curve *p* = *f* (*θ*)) and the green line corresponds to the separatrix of the second type (motion along the curve *p* = *f* (*θ*) − 1/2). To initiate the system evolution along the certain separatrix, the initial power *p*_{0} should be within the regions filled with color. As the initial signal (idler) power *p*_{0} is usually much smaller than the pump power, the separatrix of the first type can used for any sufficiently small *p*_{0}.

## 5. Analytical solution

An analytical solution of the system of Eqs. (18)–(19) has been previously obtained in the elliptic functions and studied in many papers [2, 7, 12]. The solution was expressed in terms of the roots of a fourth-degree polynomial of *x*. In the case of zero idler *P*_{40} = 0 at fiber input and zero Hamiltonian (17), the solution was derived using the Cardano formula, inasmuch as the fourth-degree polynomial was the product of the variable *x* and the cubic equation. In the paper [12], the roots of the cubic equation were found on the assumption that the initial signal power *P _{s}* is small in comparison with the pump power, and an approximate solution of the system of equations was derived in terms of hyperbolic functions.

To find a general solution in terms of elliptic functions, where all the coefficients are expressed explicitly, we use the system of Eqs. (28)–(29). An equation governing *p* evolution takes the form:

If *H* = 0, the right-hand side of Eq. (45) can be expressed as a product of *p* and the cubic polynomial. Thus, a trajectory on the phase plane exists, for which the roots of the equation can be found in explicit form.

Next we continue to study the symmetrical case *δ*_{1} = *δ*_{2} = 0. In this case, the right-hand side of Eq. (45) can be expressed as a product of two quadratic polynomials:

*H*

_{0}=

*H*(

*p*

_{0},

*θ*

_{0}) is the value of the Hamiltonian at the initial point (

*p*

_{0},

*θ*

_{0}).

The roots of the quadratic polynomials are equal to:

For the Hamiltonian *H* one can write out two representations:

It follows from the representations (49), that for a fixed *p* ∈ [0, 1/2] the value of the Hamiltonian lies in the interval *H*(*p*, *θ*) ∈ [*H*(*p*, *π*), *H* (*p*, 0)]. In addition, by taking into account that *H _{A}* is a maximum for

*H*(

*p*, 0) and

*H*is a minimum for

_{B}*H*(

*p*,

*π*), the radical expressions in Eqs. (47)–(48) are always nonnegative:

*H*≥

_{A}*H*

_{0}and

*H*

_{0}≥

*H*. Therefore, all roots (47)–(48) are real at any

_{B}*θ*

_{0}

*p*

_{0}∈ [0, 1/2].

Let us consider the most practically important case *k* ∈ (−3, 0). If the starting point of the power flow (*p*_{0}, *θ*_{0}) lies inside the area confined by the separatrix on the phase plane, the corresponding phase trajectory will lie inside this area. In this case, *H*_{0} ∈ (0, *H _{A}*) and the roots of Eq. (46) are arranged as follows:

Equation (51) defines the fiber length *l*(*p*) if $\left({p}_{0},{\theta}_{0}\right)=\left({p}_{A}^{-},0\right)$. For an arbitrary initial condition (*p*_{0}, *θ*_{0}) the fiber length can be found as *l*(*p*) − *l*(*p*_{0}), where the dimensionless signal power varies from *p*_{0} to *p*.

If the starting point of the power flow (*p*_{0}, *θ*_{0}) lies outside the area confined by the separatrix on the phase plane, the points ${p}_{A}^{-}$ and ${p}_{B}^{-}$ are interplaced in (50) and, correspondingly, in Eqs. (51)–(52).

For relatively small *p*_{0}, the value of Hamiltonian *H*_{0} is also small, leading to ${p}_{A}^{+}>>{p}_{A}^{-}$, $\left|{p}_{B}^{-}\right|$ and *m*^{2} ≈ 1. This observation can be used to obtain the approximate formulas in terms of hyperbolic functions [12].

For the separatrix the initial value of the Hamiltonian *H*_{0} is equal to zero, so the solution of Eq. (46) can expressed in terms of the logarithmic function (see formula 380.111 in [17]):

Equation (54) also implies a well-known fact that an infinite distance is required to reach the stationary point *C* moving along the separatrix. For this reason the limits of integration along the separatrix in Eq. (53) were chosen from a point with maximum power (2*p _{A}*, 0) to a point with some power

*p*. Equation (53) is one of the key equations, which provides definition for the amplifier length required to reach the maximum signal gain at nonzero parameter

*k*.

## 6. Optimization of phase-sensitive parametric amplifier

In this section we perform the optimization of phase-sensitive parametric amplifiers using a geometric approach based on the phase portrait of dynamic system. Consider pump, signal and idler evolution on highly-nonlinear fiber with the parameters presented in [15]: zero dispersion wavelength *λ _{zdwl}* = 1552.78 nm, third-order dispersion coefficient

*β*

_{30}= 0.073 ps

^{3}/km, fourth-order dispersion coefficient

*β*

_{40}= −1.7·10

^{−4}ps

^{4}/km, coefficient of nonlinearity

*γ*= 7.5 (W km)

^{−1}. Powers of the both pump sources equal to

*P*

_{1}=

*P*

_{2}= 2.1 W, signal and idler powers

*P*

_{3}=

*P*

_{4}= 0.01 mW, and wavelengths of the pumps

*λ*

_{1}= 1495.9 nm and

*λ*

_{2}= 1611.9 nm.

The value of parameter *k* for varying signal wavelength is shown in Fig. 2(a). It can be seen from Fig. 2(a), that *k* ≤ 0 for the amplifier scheme under consideration. As mentioned above, we will not consider a case where *k* < −3 due to the smallness of the signal amplification. For a case *k >* −3 consider three characteristic points: *k* = −0.5 (*λ*_{3} = 1483 nm), *k* = −0.96 (*λ*_{3} = 1520 nm), and *k* = −2 (*λ*_{3} = 1473 nm), depicted in Fig. 2(a) with black dots and numbers “1”, “2”, “3”. The corresponding phase portraits are shown in Figs. 2(b)–2(d). If *k* = −0.5 and *k* = −0.96 all four types of stationary points and three types of phase trajectories exist. Trajectories around stationary points A and B are closed curves. These trajectories describe periodic intracavity dynamics of relative phase and pump, signal and idler powers, shown in Fig. 4. Separatrices are depicted by black dots. If signal power at the fiber input is small in comparison with the pump power, which is satisfied for most practical applications, the maximum signal (idler) gain will be achieved on the phase trajectory close to the separatrix surrounding the stationary point A, which was demonstrated in the previous section,

#### 6.1. Optimization of PS-FOPS using geometrical approach

The main optimization problem, which will be considered in this section, is a determination of maximum achievable signal (idler) gain, and HNLF length and relative phase allowing the realization of this gain at known powers of waves at fiber input.

A schematic depiction of system evolution along the phase trajectories at a given signal gain *G* = *p*(*L*)/*p*(0) is presented in Fig. 2(b). If input and output signal powers are known, coordinates *p*(0) = *P*_{3}(0)/*P _{T}* and

*p*(

*L*) =

*P*

_{3}(

*L*)/

*P*are also known. The points of intersection between the phase trajectories and horizontal lines

_{T}*p*=

*p*(0) and

*p*=

*p*(

*L*) determine the initial and final points of energy transfer along the trajectories. Each trajectory of interest is uniquely defined by relative phase

*θ*(0) at the fiber input. In our case, energy transfer takes place along the separatrix, so the line

*p*(

*L*) is given by equation

*p*(

*L*) = 2

*p*and touches the maximum of the separatrix at point (0, 2

_{A}*p*).

_{A}Using the analytical estimation (43) for *G _{max}*, we build a dependence of the maximum possible gain in dual-pumped phase-sensitive parametric amplifier on the signal wavelength (see Fig. 5). Fig. 5 also shows the value of the relative phase parameter

*θ*(0) and fiber length, at which the maximum gain with a given initial power

*P*

_{3}=

*P*

_{4}= 0.01 mW is achieved. The amplifier length is found from the expression (53), the phase parameter value coincides with the value of the phase at stationary point C:

*θ*= arccos(−(

_{C}*k*+ 1)/2). Black horizontal line indicates the gain (

*G*= 53.2 dB) that would be obtained when depleting completely the pumps. Red circles depict signal amplification, found from the solution of the system of Eqs. (35)–(36) at the optimum values of the relative phase and the fiber length. As can be seen from the figure, the real amplification of the signal in the phase-sensitive parametric amplifier coincides with the theoretical estimation for the maximum gain

*G*, since the signal power at the fiber input is much smaller than the pump power. To give a more realistic evaluation of the theoretical estimation and obtain signal amplification with the proposed optimal parameters in the presence of various four-wave mixing components, including degenerate four-wave mixing between pump-signal and pump-idler, we use the nonlinear Schrödinger equation (NLSE) (dashed line in Fig. 5). Numerical simulations reveal the existence of two gain dips centered on the pumps wavelengths, restricting the range over which the maximum gain predicted by theoretical estimation is applicable. In that case the predicted gain values in the spectral regions around each pump are not exact, as it is necessary to take into account two additional waves, which are symmetric of the signal and the idler with respect to the pumps adjacent to them. This explains the discrepancies between numerical simulations and theoretical predictions. The gain spectrum generally exhibits narrow peaks, dips or rapid variations on either side of the pumps, which are subject for further analysis in the frame of six-wave model (see [18] and citation therein).

_{max}## 7. Conclusion

We have performed a theoretical analysis of phase-sensitive double-pumped fiber optical parametric amplification with pump depletion:

- The problem with one-degree-of-freedom Hamiltonian depending on 5-parameters and 1-parametric initial condition was reduced to the problem with a dimensionless Hamiltonian depending on 3 parameters and 2-parametric initial condition.
- We proved existence of stationary points and separatrices in general case and found the phase trajectory allowing the maximal theoretical gain.
- Theoretical estimation for the maximum signal gain was found in the symmetrical case of equal pump powers and equal signal and idler input powers, as well as the required length of the optical fiber and the relative phase of the interactive waves at the beginning of the fiber at which this gain is realized. We suggested the separatrix as the optimal trajectory for PS-FOPA. We found the analytic solution of the one-degree-of-freedom Hamiltonian system using the elliptic functions with exact explicit coefficients.
- We performed optimization of the specific phase-sensitive parametric amplifier using the analysis of the phase portrait of the corresponding dynamical system and the derived analytical expressions.

It should be mentioned that the theory developed in this paper is not valid in the narrow spectral regions around each pump, as four-wave model breaks down when the signal is very close to one of the pumps. In that case it is generally necessary to take into account two additional waves, which are symmetric of the signal and the idler with respect to the pumps adjacent to them. A more detailed investigation of this effect using six-wave model can be the subject of future research.

## Appendix: Finding the maximum signal gain along the extremal trajectories

In this section we find the maximum signal gain in the case when adynamical system evolves along the extremal trajectories corresponding to different values of parameter *k*. We assume that the initial signal (idler) power *p*_{0} is small 0 < *p*_{0} << 1. There are four types of the extremal trajectories:

- The dynamical system evolves from a point on the phase plane (
*p*_{0}, 0) to the point (*p*, 0), where*p*_{0}<*p*. Such trajectories are described by equation:*H*(*p*_{0}, 0) =*H*(*p*, 0). The equation has two solutions: and the second solution*p*= 2*p*−_{A}*p*_{0}exists, if the following condition is satisfied:*p*_{0}< (*k*+ 3)/6 −*p*_{0}≤ 1/2. - The dynamical system evolves from a point on the phase plane (
*p*_{0}, 0) to the point (*p*,*π*), where*p*_{0}<*p*. Such trajectories are described by equation:*H*(*p*_{0}, 0) =*H*(*p*,*π*). The equation has two solutions:$${p}_{B}^{\pm}=\frac{1-k}{4}\pm {\left[{\left(\frac{1-k}{4}\right)}^{2}+\frac{1}{2}{p}_{0}\left(k+3-6{p}_{0}\right)\right]}^{1/2}.$$Only one solution ${p}_{B}^{+}$ satisfies the conditions ${p}_{0}<{p}_{B}^{+}\le 1/2$. The first inequality yields*p*_{0}(*p*_{0}− 1/2) < 0, which is true for all*p*_{0}. The second inequality yields (*k*/6 −*p*_{0})(1/2 −*p*_{0}) > 0, which is true for 6*p*_{0}≤*k*. - The dynamical system evolves from a point on the phase plane (
*p*_{0}, 0) to the point (*p*,*π*), where*p*_{0}<*p*. Such trajectories are described by equation:*H*(*p*_{0}, 0) =*H*(p,*π*). The equation has two solutions:$${p}_{A}^{\pm}=\frac{k+3}{12}\pm {\left[{\left(\frac{k+3}{12}\right)}^{2}-\frac{1}{6}{p}_{0}\left(k-1+2{p}_{0}\right)\right]}^{1/2}.$$Only one solution ${p}_{B}^{+}$ satisfies the conditions ${p}_{0}<{p}_{A}^{+}\le 1/2$. The first inequality yields*p*_{0}(*p*_{0}– 1/2) < 0, which is true for all*p*0. The second inequality yields (*k*/6 −*p*_{0})(1/2 −*p*_{0}) > 0, which is true for 6*p*_{0}≤*k*. - The dynamical system evolves from a point on the phase plane (
*p*_{0},*π*) to the point (*p*,*π*), where*p*_{0}<*p*. Such trajectories are described by equation:*H*(*p*_{0},*π*) =*H*(*p*,*π*). The equation has two solutions: The second solution*p*= 2*p*−_{B}*p*_{0}exists, if the following condition is satisfied: −2*p*_{0}≤*k*< 1 − 4*p*_{0}.

The maximum powers attainable along the extremal trajectories are given in Table 1. To find the maximum theoretical gain for a given *k* ≠ 0, the maximum power *p* should be found. For *k* = 0, the maximum dimensionless power is equal to *p* = 1/2 and reached on the vertical separatrix.

## Funding

Russian Science Foundation (17-72-30006).

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