Geometric interpretation of four-wave mixing

The nonlinear phenomenon of four-wave mixing (FWM) is investigated using a method, where, without the need of calculus, both phase and amplitudes of the mixing ﬁelds are visualized simultaneously, giving a complete overview of the FWM dynamics. This is done by introducing a set of Stokes-like coordinates of the electric ﬁelds, which reduce the FWM dynamics to a closed two-dimensional surface, similar to the Bloch sphere of quantum electrodynamics or the Pointcar´e sphere in polarization dynamics. The coordinates are chosen so as to use the gauge invariance symmetries of the FWM equations which also give the conservation of action ﬂux known as the Manley-Rowe relations. This reduces the dynamics of FWM to the one-dimensional intersection between the closed two-dimensional surface and the phase-plane given by the conserved Hamiltonian. The analysis is advantageous for visualizing phase-dependent FWM phenomena which are found in a large variety of nonlinear systems and even in various optical communication schemes.


I. INTRODUCTION
Four-wave mixing (FWM) is a third-order nonlinear process arising from the interaction of four waves. This phenomenon is found in many systems ranging from plasma physics [1] to optics [2][3][4]. In nonlinear optical systems FWM can be intuitively understood by quantum mechanically describing the electric fields. In such a description, two photons of frequencies ω 2 and ω 3 are annihilated while two photons of frequency ω 1 and ω 4 are created (or vice versa) while keeping energy and momentum conserved. These processes can be either nondegenerate, as sketched in Fig. 1, or degenerate (when ω 2 = ω 3 ).
The fiber optical parametric process of FWM is currently used for several optical communication schemes such as amplification [5], phase conjugation [6][7][8], noiseless frequency conversion [9], and pulse regeneration [10,11]. Furthermore, the process shows great prospects in the field of fiber-based quantum information since it offers the possibility of, e.g., single-photon frequency translation [12] and quantum-statepreserving pulse reshaping [13]. These schemes predominantly use two different setups known as phase conjugation (PC), where the two inner bands are stronger than the outer, and Bragg scattering (BS), where one of the inner bands and one of the outer bands are stronger than the two remaining, see Figs. 1(b) and 1(c), respectively. The two setups are governed by the same physics since the power flow from the inner bands to the outer or vice versa and the equations governing FWM and their analytic solutions are well known [2][3][4][14][15][16]. The difference between PC and BS though appear in the linearized theory, where they have considerably different solutions and, in quantum mechanical treatments, even different noise properties [9]. The linearized theory though results in incorrect predictions, e.g., of what physical parameters yield optimal power conversion [17,18]. Furthermore, standard linearized and nonlinear solution methods of FWM only give information about the amplitude and relative phase separately. Though in a recent analysis both phase and amplitude were found and visualized in the linearized regime using a coordinate transform to Stokes-like variables [19] following a routine which has previously been used to describe the evolution of polarization [20].
Here we show a simple method, which, without the use of calculus, allows calculating and visualizing simultaneously the field amplitudes and relative phase of the full nonlinear problem. The method gives the dynamics as a closed two-dimensional surface in a three -dimensional space, see Fig. 1(d). This is similar to the Poincaré sphere of polarization dynamics or the Bloch sphere of quantum electrodynamics. With the used method the amplitude and phase of the fields are found without having to evaluate any integrals or differential equations, but simply by plotting two implicit equations and observing their intersection. A similar approach has previously been used for three-wave mixing in quadratic nonlinear media [21]. This has been used for optimizing the efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings of second-order Nonlinear materials [22][23][24].
In the following we start by introducing the governing equations for FWM in Sec. II. Furthermore, Sec. II reviews the structure of the Hamiltonian and the use of the Poisson bracket formalism for deriving governing equations. Then in Sec. III we introduce the geometric representation of the FWM equations using Stokes vectors and the Hamiltonian in this new space are found. With this, the evolution in the reduced phase space is investigated in Sec. IV and the known results are recovered. In Sec. V the geometric representation is used to investigate the PC and BS configurations to visualize the difference between the two. Finally, the paper is concluded in Sec. VI.

II. FOUR-WAVE EQUATIONS AND THEIR SYMMETRIES
In this section the governing equations of FWM are stated and the use of the Poisson bracket formalism to derive the equations of motion along with the Hamiltonian is briefly reviewed. In a normalized form, the well-known governing equations for the nondegenerate FWM process in nonlinear optical fibers are [3,4] These are obtained by a slowly varying amplitude approximation of Maxwell's equations for the resonant interaction of four light waves propagating in a centrosymmetric material. The functions q j are the normalized slowly varying complex field envelopes and the parameters j = β(ω j ) γ K P are the normalized wave numbers [physical wave number β(ω j )] of the j = 1,2,3,4 fields while ξ = γ K P z is the normalized propagation length (physical propagation length z). Both j 's and ξ are normalized with respect to the nonlinear propagation length γ K P , where γ K is the nonlinear Kerr coefficient, for simplicity assumed to be frequency independent, and P is the total action flux. Furthermore, the field envelopes are normalized with respect to P such that The equations have the canonical Hamiltonian structure |q n | 4 +2(q * 1 q 2 q 3 q * 4 + q 1 q * 2 q * 3 q 4 ). (4) In Eq. (4) the first sum corresponds to group velocity dispersion while the second sum concerns the nonlinear phase modulation and the last terms give rise to energy conversion. One can further define the complex Poisson bracket such that the equations of motion can be written in a form similar to that of quantum mechanics The complex-valued Poisson bracket formalism is equivalent to the usual real-valued position-momentum formalism described in Ref. [25], but with the use of q j and q * j as the choice of canonical variables. Notice that q j and q * j are independent variables in such complex formalism.
The four-wave equations, Eqs. (1), have certain differential symmetries, namely that they are invariant under the gauge transforms From Noether's theorem we know that differential symmetries result in conserved quantities [25] and thus the four-gauge transforms result in the constants of motion also known as the Manley-Rowe relations (MRR)s [26,27]. The physical significance of the MRRs can easily be understood by a quantum mechanical interpretation as follows.
The first two of the relations signify that in the inner and outer bands photons are created or annihilated in pairs. The third and fourth relations signify that creating the photon pairs in the inner or outer bands annihilates photon pairs in the outer or inner bands. The MRRs are seen to be an overfull set and only three of them are independent. Finally, we notice that the Hamiltonian, Eq. (4), is also a constant of motion. We end this section by sketching the standard procedure for calculating the FWM dynamics. The Hamiltonian, and thus the governing equations for the fields, is seen only to depend on the where the phase is defined by q j (ξ ) = |q j (ξ )|e iφ j (ξ ) . Because the Hamiltonian only depends on the phase difference, the four complex, corresponding to eight real, coupled nonlinear differential equations governing FWM can be reduced to four equations 043805-2 governing the real amplitudes |q j | and one governing equation for the phase difference φ. Then using three of the conserved quantities, Eqs. (8), these can be reduced to a single potential equation for either of the amplitudes, which is solvable using elliptic integrals. Subsequently all the other amplitudes can be found using the MRRs and the phase difference can be calculated using the conservation of the Hamiltoian [15,16]. In the following the problem is solved in another way without using integrals. This gives a geometric interpretation of all the amplitudes and the phase difference simultaneously.

III. REDUCTION TO FOUR-WAVE SURFACES
In this section the dynamics of the four-wave equations are reduced to dynamics on a closed surface in a three-dimensional space. A simple example of the method is the projection of polarization onto the surface of a sphere. In that case the reduction of dimension is performed by using the Stokes parameters, a set of invariant coordinates with respect to the phase symmetry of the polarization, that project the dynamics onto the Poincaré sphere, see, e.g., Ref. [20]. In our case of FWM, a set of Stokes-like parameters which are invariant under the same gauge transforms, Eqs. (7), must be chosen. Such a set of real parameters X, Y , and Z are with κ = (p 1 − p 2 − p 3 + p 4 ) −1 and p n being some constants that can be chosen freely only restrained by p 1 − p 2 − p 3 + p 4 = 0 corresponding to Z not being a combination of the conserved quantities, Eqs. (8). The choices of the p n 's focus the analysis to specific energy conversions in the FWM dynamics.
Three examples of such choices are as follows.
(1) By choosing p j = 1, while p n =j = 0, Z yield the intensity of the j th field.
(2) Choosing p 1 = p 4 = 1 and p 2 = p 3 = 0, then Z corresponds to the intensity of the side bands such that Z is maximal when most intensity is in the side bands and minimal when it is in the inner bands. Such a choice would be useful for analyzing optical amplification.
(3) Finally, by choosing p 1 = −p 2 = 1 and p 3 = p 4 = 0 the value of Z corresponds to the difference between one of the inner bands and one of the side bands. This describe frequency conversion. The new coordinate Z is connected to either of the field envelopes through |q n | 2 = Z − Z n for n = 1 and 4 and |q n | 2 = Z n − Z for n = 2 and 3, where Since the MRRs along with energy conservation are an overcomplete set one can write the Z n 's in terms of only two of the K n 's, e.g., K 1 and K 2 . By taking the square modulus of X + iY and expressing the |q n | 2 's in terms of the Z coordinate one obtain the expression By defining we immediately acquire the implicit equation, = 0, from Eq. (11), which yields a surface in the (X,Y,Z) space. The FWM dynamics is thus restricted to the dynamics on the surface given by = 0. This surface is determined strictly from the definitions of (X,Y,Z) along with the MRRs and not the physical parameters of the system such as group velocity dispersion and nonlinearity and neither the phases of the fields. The surface describes the relative phase of the fields through the X and Y coordinates and the chosen field intensities through Z, specified by the choice of the p n constants.
To plot the implicit equation = 0 it is necessary to parametrize the coordinates. One way to do this is to write the coordinates as (X,Y,Z) = (R cos(θ ),R sin(θ ),Z) then noticing that, from Eq. (12), = 0 implies Thus, since R is positive, we can write the parametrization as ⎛ with the two parameters θ ∈ [0; 2π ] and Z ∈ [Z min ; Z max ]. Notice that the surface = 0 could equally well be described by a line in the (R,Z) coordinates having only the single parameter Z. The advantage of using a three-dimensional space is the ability to visualize the relative phase of the field envelopes which is given by θ . The minimum and maximum of Z depend on the roots Z n of the polynomial n (Z − Z n ) and if choosing κ > 0 they are given by Z min = max(0,K 1 ) + Z 1 and Z max = −max(0,K 2 ) + Z 2 . Independent of the choice of p n 's the four-wave surface has three typical shapes shown in Figs. 2(a) to 2(c). The citrus-or rugby-shaped surface shown in Fig. 2(a) occur when K 1 = K 2 = 0, which can be seen by noting that K 1 = K 2 = 0 implies K 3 = K 4 giving (for κ > 0) Z 1 = Z 4 = Z min and Z 2 = Z 3 = Z max such that the top and bottom are double roots of n (Z − Z n ). A schematic illustration of such a configuration is shown in Fig. 2(d). Similarly, the teardrop shown in Fig. 2(b) occurs when either K 1 = 0 K 2 = 0, as schematically illustrated in Fig. 2(e) for K 2 = 0. Finally, the deformed sphere shown in Fig. 2(c) occurs when K 1 = 0 and K 2 = 0, see Fig. 2(f). The singular points occur when there is a possibility of a full conversion of energy, either to the inner bands or the side bands which is known only to be possible when either K 1 = 0 and/or K 2 = 0 and the singularity is a sign that it, in theory, takes infinite propagation to achieve [15].
Using that the Hamiltonian is also a constant of motion, the trajectory for a specific configuration of physical parameters can be found as the intersection between the four-wave surface and the Hamiltonian. The Hamiltonian depends on 043805-3 the dispersion and the phase of the fields, the wave-number mismatch = 1 − 2 − 3 + 4 , the phase mismatch = + n Z n , and an unimportant constant which we remove by rescaling. Thereby the Hamiltonian in terms of the (X,Y,Z) coordinates is given by which is noted to be independent of Y , linear in X, and quadratic in Z. Since the Hamiltonian is a constant of motion one has the restriction H = H 0 with H 0 being the Hamiltonian for some specific values of (X,Y,Z), e.g., the initial values. The FWM dynamics is thus also confined by the Hamiltonian being a constant of motion and thus the full dynamics for a given set of initial fields and wave-number mismatch is found by the intersection between the Hamiltonian plane, H = H 0 and the FWM surface = 0. Similar to the FWM surface, we need to parametrize the line of intersection to plot it. Again we do this by choosing (X,Y,Z) = (R cos(θ ),R sin(θ ),Z), but while only R was restricted by the coordinate Z when dealing with the FWM surface, now also θ is restricted by the conservation of the Hamiltonian H = H 0 , through where X 0 and Z 0 are the initial values of X and Z. The observant reader may notice the similarities between the method described here and using the the conservation of the Hamiltonian to plot the FWM dynamics in a phase plane as described in Refs. [16,19]. While the phase plane is a simple way to predict the FWM dynamics from the knowledge of the Hamiltonian it is limited to show the prediction for a specific set of physical parameters (here ) at a time. The method presented here, on the contrary, is able to show the predictions of multiple sets of physical parameters simultaneously.

IV. REDUCED PHASE-SPACE EVOLUTION
In this section we derive the dynamic equations for the new coordinates. In the new coordinates the Poisson brackets are given by Using these and the properties of the Poisson bracket the dynamical equations for the (X,Y,Z) coordinates are  In this form the coupling between X and Y , i.e., the evolution of the phase, consist of a contribution equivalent to a harmonic oscillator with frequency − 4Z. This shows that the evolution of the phase is given by both a contribution from linear dispersion and from nonlinear dispersion −4Z. Furthermore, Z and Y are coupled nonlinearly through ∂ /∂Z equivalent to a nonlinear oscillator. The exact evolution can be calculated using the standard potential approach, see the Appendix A. Let us analyze the linearized equations by assuming that two of the bands are much stronger than the two other. We choose the p n parameters such that Z corresponds to the weak bands, i.e., the signal and idler bands, and thus ZX, ZY , and Z n with n > 1 are considered small. This gives the linearized equations where, for brevity, we have defined the constants which depend on the choice of the p n 's and the MRRs. We now compare the PC and BS configurations. As described in the Introduction for the PC configuration the inner bands are of comparable strength and likewise are the outer bands. In the BS configuration, one of the inner bands and one of the outer bands are of comparable strength and so are the remaining two. With equal weights on the weak bands the PC configuration give, to first order in the weak bands, that A PC ≈ 0 and B PC ≈ −4P p P q ≈ −1, where P p and P q are the powers of the two strong pumps. Similarly, the BS configuration gives A BS ≈ 0 and B BS ≈ 4P p P q ≈ 1. There is thus a subtle difference between the linearized evolution of PC and BS in that BS can be written in the formẆ = −M × W , where W = (X/ P p P x ,Y / P p P x ,Z) and M = (4 P p P x ,0, ). The linearized evolution in Stokes space for BS thus corresponds to motion on a spherical surface as is also found in Ref. [19]. On the other hand, the linearized evolution for PC is in general not bounded. This difference can be understood by noting that the BS configuration will be an extreme case of the type sketched in Figs. 2(c) and 2(f) while PC can be of any of the types sketched in Fig. 2 and thus may experience full energy transfer such that the assumption of strong pumps may only be valid initially.

V. EXAMPLES
In the following we use the method to describe two well known setups. First in Sec. V A a parametric amplifier, which corresponds to the PC configuration, and then in Sec. V B frequency conversion which corresponds to the BS configuration.

A. Phase conjugation
Let us analyze a setup where the two inner bands are of comparable strength and similar are the two outer bands, e.g. q 2 and q 3 are the pumps and q 1 and q 4 are known as the signal and idler. We choose p 1 = p 4 = 1 such that Z = Z min signifies that all energy is in the inner bands, q 2 and q 3 , while Z = Z max signifies that all energy is in the outer bands q 1 and q 4 . We will consider a case with equal pumps K 2 = 0, but unequal sidebands such that K 1 = 0.2( = 0). The values of K 1 and K 2 along with the p n 's fully determine the shape of the four-wave surface. The considered scheme describes a two-pump fiber optical parametric amplifier with the possibility of full energy transfer. We investigate the two cases: (i) the case where one of the sidebands (i.e., the idler) is initially zero giving phase insensitive amplification and (ii) the case where none of the fields are initially zero, thereby resulting in phase sensitive amplification.

Phase insensitive amplification
We consider the case where the initial amplitudes are |q 1 (0)| = 1/5, |q 2 (0)| = |q 3 (0)| = 2/5, and |q 4 (0)| = 0 such that K 1 = 0.2 and K 2 = 0. The relative phase is unimportant in this case. To determine the evolution on the four-wave surface we only need to calculate the intersection between the Hamiltonian and the four-wave surface. We investigate two cases (1) zero wave-number mismatch = 0, which is what is found to be optimal in linear theory [28] and (2) optimal wave-number mismatch for full energy transfer [14]. For zero wave-number mismatch, = 0, the Hamiltonian, given by Eq. (15), is shown as the blue parabolic curved plane in Fig. 3. As described in Sec. III the FWM dynamics is then restricted to the intersection between the Hamiltonian and the four-wave surface and is shown as the green solid line. As seen, the zero wave-number mismatch does not reach the top of the teardrop, i.e., it does not yield the full energy transfer as is well known from standard nonlinear analysis [3,4,15,16].

043805-5
We now turn to the case of optimal energy transfer. The wave-number mismatch of optimal energy transfer is found by setting H 0 = H opt and isolating , where the index 0 denotes the fields at ξ = 0 and the index opt denotes the optimal energy transfer. For optimal energy transfer at least one of the fields is fully depleted such that X opt = 0 and furthermore Z opt = Z max or Z opt = Z min depending on the choice of p n 's. The resulting wave-number mismatch is thus found from Eq. (15) to be where, expressed in K 1 and K 2 , one has n Z n = 1 + 2κK 1 (p 1 − p 4 ) + 2κK 2 (p 2 − p 3 ). Notice that the optimal wave-number mismatch depends on X 0 and thus if all the fields have initial nonzero amplitude then opt depends on the initial phase difference of the fields. Such a setup is known as phase sensitive amplification and is discussed in a moment. In the case considered here p 1 = p 4 = 1, p 2 = p 3 = 0, and K 2 = 0 we get κ = 1 2 , X 0 = 0, Z 0 = K 1 2 , Z opt = 1 2 , and n Z n = 1 such that opt = K 1 . This gives the route for full energy transfer shown in Fig. 3 as the black dashed line.

Phase sensitive amplification
Before we move on to the BS configuration let us take a look at the phase sensitive amplification scheme, where none of the fields are initially zero. Specifically we consider the initial amplitudes |q 1 (0)| = 5/20, |q 2 (0)| = |q 3 (0)| = √ 7/20, and |q 4 (0)| = √ 1/20 such that K 1 = 0.2 and K 2 = 0 thereby giving the same four-wave surface as the above phase insensitive amplification. In this case the relative phase is important as is shown in Fig. 4. We choose the wave-number mismatch to be optimal for a relative phase of θ = 0 shown as the black dashed line. By varying the relative phase from 0 to 7/20, and |q 4 (0)| = √ 1/20, but with the relative phases of θ = 0 (dashed black line), θ = π/3 (solid yellow line), θ = π/2 (solid green line), θ = 2π/3 (solid red line), and θ = π (solid blue line). The wave-number mismatch is chosen for optimal energy transfer for θ = 0. The p n parameters are p 1 = p 4 = 1 and p 2 = p 3 = 0 as in Fig. 3.
π the Hamiltonian plane is moved towards negative values of X. This can be understood by noting that Z does not change with the phase and thus the change of phase only change X 0 in the Hamiltonian plane being given by H = H 0 and thus results in a translation in X. The solid yellow, green, red, and blue lines, respectively, correspondto θ = π/3, θ = π/2, θ = 2π/3, and θ = π . It is clear that θ = π yields an amplification which is far from the optimal θ = 0.

B. Bragg scattering
Let us now take a look at a setup where one of the sidebands is comparable in strength with one of the inner bands and similar the remaining bands are of similar strength. This configuration is known as BS and in such a setup both K 1 and K 2 are nonzero. In this case it is impossible to convert all the energy from the pumps to the signal and idler. On the other hand it is always possible to fully convert the energy of the signal onto the idler. We consider a case with fairly strong pumps, specifically the initial amplitudes are |q 1 (0)| = |q 3 (0)| = √ 2/100 and |q 2 (0)| = |q 4 (0)| = √ 48/100. Furthermore, we choose p 1 = −p 3 = 1 while p 2 = p 4 = 0 such that Z is maximal when the total energy of bands 1 and 3 is in band 1 and minimal when it is in band 3. The four-wave surface is shown in Fig. 5 where we have rescaled the X and Y axis with the |q 2 (0)q 4 (0)| = (P 2 P 4 ) 1 2 showing how the BS dynamics for strong pumps is restricted to a sphere in the rescaled Stokes space as described in Sec. IV.
For a given we now find the initial relative phase that give a chosen optimal energy conversion. This is done by setting H 0 = H opt and isolating the corresponding relative phase θ opt 0 , while noting that Z is phase independent. For optimal conversion X opt = 0 and we get from Eq.
With this we can for a given physical parameter , find the optimal relative phase for a preferred frequency conversion. In case we wish to convert all the energy from the bands 1 and 3 into the band 1 we insert Z opt = Z max (dashed black line) while if we wish to convert it into band 3 we insert Z opt = Z min (solid black line).

VI. CONCLUSION
We have shown a method to visualize the amplitude and phase of FWM processes simultaneously. By defining a set of Stokes-like parameters which are invariant under the same gauge transforms as the traditional FWM equations of motion, the FWM dynamics is reduced to the evolution on a closed surface in the three-dimensional space spanned by the Stokes-like parameters. This four-wave surface is determined solely by the Manley-Rowe relations and is independent of the physical parameters of the nonlinear system. For a given set of physical parameters, i.e., dispersion and Kerr nonlinearity, and initial conditions, i.e., initial wave intensities and phases, the wave dynamics is further restricted by the conservation of the Hamiltonian. The specific evolution of the four-wave dynamics is thus found by the intersection of a parabolic plane spanned by the Hamiltonian and the four-wave surface determined by the Manley-Rowe relations. The method is illustrated in three examples used in optical communications schemes; phase insensitive amplification, phase sensitive amplification, and frequency conversion.