1. Introduction

The nonlinear propagation of coherent optical fields has been widely explored in the framework of nonlinear optics [1, 2, 3, 4], while the linear propagation of incoherent fields has been widely studied in the framework of statistical optics [5, 6, 7, 8, 9, 10, 11]. However, it is worthnoting that these two fundamental fields of optics have been mostly developed independently, so that a complete and satisfactory understanding of incoherent nonlinear optics is still lacking.

Incoherent nonlinear optics received a growing recent interest since the first experimental demonstration of incoherent solitons in photorefractive crystals [12, 13, 14]. The incoherent soliton consists of a phenomenon of self-trapping of partially incoherent light in a noninstantaneous response nonlinear medium [15, 16, 17, 18, 19, 20]. The notable simplicity of experiments performed in photorefractive media allowed for a fruitful investigation of the dynamics of incoherent nonlinear waves [2, 14]. Let us note that incoherent solitons [21, 22, 23, 24, 25, 26, 27] and remarkable dynamical features inherent to incoherent nonlinear waves have also been recently investigated in instantaneous response nonlinear media [28, 29, 30, 31, 32, 33, 34, 35].

Here we analyze the evolution of the polarization properties of a partially coherent optical field that propagates in a nonlinear (Kerr) medium in the presence of a phase-mismatch between the orthogonal polarization components. We show that an initially unpolarized field exhibits, as a rule, an irreversible process of repolarization, characterized by a monotonous increase of its degree of polarization during the nonlinear propagation. As a remarkable result, the unpolarized beam is shown to irreversibly evolve towards a linearly polarized state, without loss of energy or intensity.

The repolarization process is characterized by the spontaneous emergence of a strong mutual correlation between the circular polarization components of the optical field. This result is in apparent contradiction with previous studies of polarization changes induced by nonlinear propagation. It is indeed well established that the propagation of an optical field in a (birefringent or isotropic) Kerr medium is characterized by a depolarization process [37, 38, 39, 40, 41, 42], in which the random polarization states of the field exhibit a diffusion process over the Poincaré sphere [41]. However, in these previous works the influence of dispersive effects, such as diffraction or chromatic dispersion, have been systematically ignored in the analysis of the polarization properties of the nonlinear field. In the following, we shall see that diffraction (or chromatic dispersion) plays a central role in the process of repolarization.

The depolarization process reported in Refs.[37, 38, 39, 40, 41, 42] is accompanied by a process of entropy production, a feature which is consistent with the intuitive idea that a loss of correlation in the polarization components constitutes a loss of information in the optical field. Note that this aspect has been the subject of a detailed investigation in the context of light scattering in random media [9]. In contrast with these previous works, we show here that an increase of “disorder” in the optical field requires the spontaneous emergence of a strong correlation in the polarization components of the optical field, i.e., an increase of its degree of polarization. This repolarization process is shown to result from the natural tendency of the optical field to reach a thermodynamic equilibrium state that realizes the maximum of nonequilibrium entropy.

We analyze the process of thermal wave relaxation to equilibrium on the basis of the kinetic wave theory [43, 44, 45, 46, 47, 48, 49, 50, 51]. This theory has been recently applied to optical waves in various circumstances [30, 31, 32, 35] (for a simple introduction to the thermalization of optical waves, see, e.g., Ref. [33]). The kinetic theory relies on a natural asymptotic closure induced by the dispersive properties of the waves, which leads to a kinetic description of the wave interaction that is formally based on irreversible kinetic equations. Such irreversible behavior is expressed by the H-theorem of entropy growth [46, 47, 48], in analogy with the Boltzmann’s H-theorem relevant for gas kinetics [52]. It results that, in spite of the formal reversibility of the equation that governs optical wave propagation, the kinetic equation describes an irreversible evolution of the field to thermodynamic equilibrium [33, 51]. In this respect, we show that the process of repolarization results from the natural irreversible evolution of the optical field towards thermal equilibrium. The theory then reveals that it is thermodynamically advantageous for the optical field to evolve towards a highly polarized state, because this permits the optical field to reach the “most disordered state”, i.e., the state of maximum nonequilibrium entropy.

Besides its fundamental interest, this repolarization process may find applications as a universal polarizer performing repolarization of unpolarized incoherent light with almost 100% efficiency, in contrast with standard polarizers that unavoidably waste 50% of unpolarized light [5, 6, 7, 8]. Let us remark that universal polarizers of coherent light have been recently proposed by the groups of Boyd [53] and Millot [54, 55]. The repolarization process reported here has a fundamental different physical origin, since it relies on the natural thermalization of the optical field. It is thus inherently associated to the stochastic nature of the optical field, a distinguished feature that enables the repolarization of incoherent spatio-temporal fields.

2. Resonantly coupled NLS equations

As a starting point, let us consider a partially coherent optical field that propagates close to the z axis of a lossless Kerr medium. The electric field vector E thus lies in the perpendicular plane r=(x,y), and the evolution of its orthogonal polarization components is known to be governed by the resonantly coupled NonLinear Schrödinger (NLS) equations [1, 2, 4]

where the components E=(E+,E-) respectively refer to the right- and left-handed circularly polarized components of the optical field. The parameter δ=Δk/2 represents the wave-vector mismatch between the orthogonal polarization components. Such a phase-mismatch may originate in a non-collinear configuration of the interaction of the beams [1, 2, 4, 56]. It may also originate in natural crystal birefringence, in which case Δk=k_y-k_x with k_x,y=n_x,y ω₀/c, n_x,y referring to the refraction indexes along the principal axes (x,y) of the crystal (c being the speed of light in vacuum and ω0 the optical frequency) [1, 2, 4, 56]. The laplacian operator∇²⊥=∂ ²/∂x ²+∂ ²/∂y ² models optical diffraction in the paraxial approximation, where α=1/2k. The nonlinear Kerr effect is responsible for a self-phase modulation of each polarization component, as well as a cross-phase modulation between them, where Γ=2γ/3, γ being the nonlinear Kerr coefficient. Let us note that, in substance, the role of the nonlinearity is to drive the random optical field to thermal equilibrium. In this respect, the repolarization process takes place regardless of the ratio between the cross- and self-phase modulation coefficients, i.e., the factor 2 in Eqs.(1-2).

It is important to remark that Eqs.(1-2) also hold for the description of the spatio-temporal evolution of the optical field, which is governed by spatial diffraction and temporal chromatic dispersion [1, 2]. Indeed, the substitution

is known to transform Eqs.(1-2) into the equations that govern the spatio-temporal evolutions of two orthogonal (TE and TM) modes in a planar wave-guide geometry, where β represents the dispersion coefficient [2]. In this wave-guided configuration, the two beams are only allowed to diffract along one (y-)direction and the parameter δ accounts for the birefringence induced by the wave-guide geometry [2, 57, 58, 59]. One may then remark that, in the anomalous dispersion regime (β < 0) the spatio-temporal equations for E± turn out to be formally analogous to Eqs.(1-2). The present work then reveals that, as a result of a spatio-temporal thermalization process [32], the incoherent optical field could be repolarized in both the spatial and temporal domains, i.e., every ‘portion’ of the unpolarized beam irreversibly evolves towards a linearly polarized state.

According to this discussion, the repolarization effect may be expected to also occur in the purely temporal case, a feature that would be relevant for the study of the propagation of a partially polarized wave in a low-birefringent optical fiber system [11]. However, it is now well-recognized that, contrary to the two-dimensional case, the thermalization process occurs very slowly in the one-dimensional case. Our preliminary numerical study confirms this fact and reveals that the repolarization effect does not occur efficiently in the one-dimensional geometry.

Let us finally note that the influence of diffraction (or chromatic dispersion) was neglected [i.e., α=0 in Eqs.(1,2)] in the previous works [37, 38, 39, 40, 41, 42], where the nonlinear polarization properties of optical fields were studied. This assumption is not justified when one deals with partially incoherent fields that exhibit rapid spatiotemporal fluctuations. Let us note that, under the assumption α=0, Eqs.(1,2) recover ordinary differential equations, a feature which considerably simplifies the analysis of optical wave propagation. In the following we shall see that diffraction or dispersion effects play a central role in the process of spontaneous repolarization.

3. Repolarization effect

Physical insight into the repolarization effect may be obtained by means of the numerical integration of Eqs.(1-2). We consider an initial state of the random optical field, E(rr, z=0), that we assume to be of zero mean (〈E+〉=〈E-〉=0), and characterized by a homogenous statistics. This latter assumption refers to the physical limit in which the correlation length, λ _c, of the random field is much smaller than the size of the optical beam (the so-called “quasi-homogenous statistics”) [5, 6, 7]. Let us underline that the orthogonal polarization components are assumed to be fully decorrelated at the input of the medium, 〈E+(r, z=0)E*_(r, z=0)〉=0, where <. >denotes an average over the realizations. The degree of polarization of the optical field is defined from the 2×2 coherence matrix J [5, 6, 7, 8],

where ‘Det’ and ‘Tr’ denote the determinant and trace operations. The quantityP is bounded by 0 and 1, which respectively correspond to unpolarized and fully polarized fields. In the basis of circular polarization, the elements of the coherence matrix read J_ij(z)=〈E_i(r, z)E*j (r, z)〉 (j=+,-) [60].

Figure 1 illustrates a typical evolution of the degree of polarizationP of a partially coherent optical field that propagates in the Kerr medium. As remarkably illustrated in Fig. 1(a), the initially unpolarized field [𝒫(z=0) ≃ 0] evolves towards a highly polarized state (𝒫^eq ≃ 0.95), without loss of energy or intensity. Let us note that the degree of mutual correlation between the polarization components E+ and E- follows exactly the same evolution as 𝒫(z). This is merely due to the fact that that the averaged intensity of each polarization component is almost preserved during the propagation [〈|E+|²〉(z)=〈|E_|²〉(z)], which thus entails that 𝒫 ∝|〈E+E*_〉| [5, 61].

Let us remark that, in principle, the experiment aimed at observing this spontaneous repolarization effect is conceptually simple. Since our model neglects the temporal dynamics of light fluctuations, the numerical simulations of Fig. 1 model the propagation of a stationary speckle beam that exhibits random polarization states over its transverse cross section. Such a stationary speckle beam may be generated in practice by reflecting a laser beam on a static random screen [5, 6]. The speckle beam is then launched in the Kerr material, and the repolarization effect may be expected to be observed in a single passage through the material. In particular, the degree of polarization 𝒫(z) [given in Eq.(4)] may be measured at different propagation lengths z by performing an average over different realizations of the stationary speckle beams. For a material sufficiently long, the increase of the degree of polarization 𝒫(z) is expected to saturate up to its equilibrium value 𝒫^eq, as illustrated in Fig. 1(a). Note that, since we are considering a two-dimensional propagation problem in a Kerr material, it is important to consider a defocusing nonlinearity so as to avoid the self-filamentation and the collapse of the speckle beam. Note however that this problem may be circumvented by using a saturable nonlinearity [1, 2, 4]. Indeed, the numerical simulations reveal that the thermalization process occurs efficiently even in the presence of a focusing saturable nonlinearity.

The numerical simulation reported in Fig. 1 would correspond to an experimental configuration in which an unpolarized field of intensity 𝓘=2.5×10 ¹³W/m² and spatial correlation length λ_c ≃ 6µm, is injected in a nonlinear medium (e.g., semiconductor crystal) characterized by an index of refraction of 3.5, a birefringence of |n_x-n_y| ≃ 10^-3 and a nonlinear coefficient of Γ=1.67×10^-10m/W. With these parameters the intensity of the optical field is smaller than the critical intensity of polarization instability, $I\widetilde{-}\frac{2}{3}{I}_c$, where 𝓘_c=2δ/Γ [3]. However, let us underline that the repolarization effect has been shown to occur under rather general conditions. More specifically, the key parameters that govern wave propagation may be determined from the three relevant characteristic lengths, namely, the nonlinear length L_nl=1/Γ𝓘, the birefringence length L_δ=2π/Δk=π/δ and the diffraction length L_α=λ²_c/α. The numerical simulations reveal that the repolarization effect may occur efficiently for a relatively wide range of characteristic lengths, i.e., typically $\frac{1}{10}<L_i⁄L_j<10,L_{i,j}$ referring to the three lengths Lnl, Lδ, Lα. In particular, we note that the repolarization effect also occurs for an intensity of the optical field greater than the critical intensity of polarization instability, 𝓘 > 𝓘_c.

Let us finally mention that the repolarization effect may also be studied in another experimental configuration, namely the planar wave-guide geometry discussed at the end of Section 2. In this configuration the speckle beam exhibits both spatial and temporal polarization fluctuations and the repolarization effect is expected to occur in both the spatial and temporal domains (see Section 2). A notable advantage of this configuration relies on the fact that the birefringence is induced by the wave-guide geometry and may thus be controlled in the process of wave-guide fabrication [2, 57, 58, 59].

4. Interpretation of the repolarization process

This repolarization effect is surprising because the propagation of the optical field in the material exhibits the essential properties of being conservative (lossless) and reversible [Eqs.(1-2) are invariant under the transformation z→-z, E→E*]. Note that this contrasts with the function of standard polarizers, which are inherently dissipative and irreversible devices. In spite of the reversible nature of optical wave propagation, the repolarization effect is characterized by a gain of statistical ‘information’ (correlation) between the polarization components of the optical field. More precisely, according to the traditional definition of the entropy of polarization [7, 9], W=-Tr[Jlog(J)], such a reversible process of repolarization would be characterized by an intriguing process of entropy reduction.

In order to clarify this intriguing result, let us gain some insight into the mechanism underlying the repolarization effect. It is important to underline that the conservative and reversible properties of optical wave propagation are reflected by the conservation of two important quantities in Eqs.(1-2): the intensity $I=\frac{1}{A}{\int }_A\mid \mathbf{E}{\mid }^{2}d\mathbf{r}$ (A representing the area of integration), and the total energy (Hamiltonian) ℋ=𝓔+𝓔_nl, which has a linear (𝓔) and a nonlinear (E_nl) contribution [2].

A key signature of the repolarization process comes from the analysis of the linear energy 𝓔=𝓔_kin+𝓔_res, which exhibits a ‘kinetic’ contribution that originates in optical diffraction

and a resonant phase-sensitive contribution that originates in the phase-mismatch term

 

Fig. 1. Spontaneous polarization of an unpolarized field during its propagation in a loss-less Kerr medium in the presence of a phase-mismatch between the orthogonal polarization components (δ=3.2×10³m^-1). Numerical simulations of Eqs.(1-2) showing the evolution during the propagation of the degree of polarization 𝒫 (a), the kinetic and resonant energies (b), and the nonequilibrium entropy S(c). The process of entropy production is saturated once the equilibrium state is reached, as described by the H-theorem of entropy growth (z is in units of the nonlinear length L_nl=1/Γ𝓘=0.24mm).

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Let us now remark that the natural tendency of a conservative and reversible (nonintegrable Hamiltonian) system is to increase its disorder by evolving towards an equilibrium state [47, 52]. In complete analogy, we may expect that the natural tendency of the optical field is also to increase its disorder. Such a “disordered state” of the optical wave would be characterized by the presence of rapid fluctuations of the field amplitude E(rr). A natural measure of the amount of fluctuations is provided by the kinetic energy 𝓔_kin, since it measures the gradient of the spatial variations of the vector field E(rr) [see Eq.(5)]. Accordingly, the contribution of the kinetic energy is expected to increase during the propagation, as illustrated in Fig. 1(b). However, the total energy must remain constant, so that an increase in the kinetic contribution E_kin should be accompanied by a reduction of the resonant contribution 𝓔_res. This is confirmed by the numerical simulations [see Fig. 1(b)], which reveal, in particular, that the nonlinear contribution 𝓔_nl keeps an almost constant value during the propagation. Let us now remark that, as indicated by the expression of the resonant contribution 𝓔_res [Eq.(6)], a variation of E_res naturally entails the emergence of a correlation between the polarization components |〈E+E*_〉|, i.e., an increase of the degree of polarization 𝒫. This merely explains why it is advantageous for the optical field to evolve towards a highly polarized state in order to reach the “most disordered state”. In other terms, an increase of the amount of disorder in the optical field requires the emergence of a strong correlation between the polarization components. Note that a similar mechanism of self-organization induced by wave-thermalization may be encountered in the spontaneous formation of coherent structures in turbulent environments [30, 34, 49, 51, 62, 63, 64].

This simple interpretation reveals the central role of the phase-mismatch in the repolarization process, a peculiar feature which is confirmed by the fact that the repolarization effect does not take place for δ=0. Along the same way, let us emphasize the essential role of diffraction in the repolarization effect. Indeed, it becomes apparent from the previous discussion that the spontaneous emergence of polarization is fundamentally associated to the increase of kinetic energy, which means that the repolarization process cannot take place if the influence of diffraction is neglected [α=0 in Eqs.(1,2)]. This merely explains why the repolarization process was not identified in previous studies of nonlinear polarization evolution of random fields, since in these works the influence of diffraction (or dispersion) was ignored [37, 38, 39, 40, 41, 42].

5. Thermodynamic equilibrium spectra

The previous reasoning indicates that the repolarization effect finds its origin in the natural thermalization of the optical field towards an equilibrium state. The thermalization of a nonlinear wave may be considered as an irreversible process of diffusion in phase-space, which is due to the nonintegrable character of Eqs.(1-2) [47, 48]. It results that, despite the formal reversibility of Eqs.(1-2), the optical field E exhibits an irreversible evolution to thermal equilibrium during its propagation [33]. This is in complete analogy with kinetic gas theory: in spite of the formal reversibility of the newtonian equations underlying the evolution of a classical gas, it is well-known that the gas system exhibits an irreversible evolution to thermodynamic equilibrium [52].

The essential properties underlying the thermalization of a nonlinear wave may be described by the kinetic wave theory. The kinetic wave theory has been applied to optical waves in various situations [30, 31, 32, 33, 51]. However, to our knowledge it is the first time that resonantly coupled NLS equations are analyzed in the framework of the kinetic wave theory. In this respect, we remark that the kinetic equations of the resonantly coupled NLS equations cannot be derived in the basis of circular polarization, because of the existence of a strong correlation between the components E+ and E_, i.e., 〈E+E*_〉≠0. One has thus to find an appropriate transformation in which the linear evolutions of the polarization components are independent (i.e., the linear Hamiltonian 𝓔 is ‘diagonalized’). It turns out that the required basis is simply given by the rectilinear polarization basis (x,y). The rectilinear polarization components (E_x,E_y) are no longer correlated and, as we shall see, the repolarization effect manifests itself by means of a transfer of intensity from one polarization component to the other.

The basis of rectilinear polarization, E=(E_x,E_y), is defined by the following variables E_x=(E₊+E_)/√2,E_y=i(E_-E+)/√2). In this basis Eqs.(1-2) are known to take the following form [1, 2, 3, 4],

It becomes apparent from these equations that the linear evolutions of the fields E_x and E_y are no longer coupled. In particular, contrary to the circular polarization basis, the fields E_x,y exhibit two distinct dispersion relations,

which result to be splitted by the phase-mismatch parameter δ [see Fig. 2(a)].

The kinetic wave equations that govern the evolution of the spatial spectra of the fields may be derived by following the standard procedure outlined in Ref. [48]. One obtains two coupled kinetic equations that govern the coupled evolutions of the spatial spectra of the optical field E, n_x(k, z)=〈|Ẽ_x(k, z)|²〉 and n_y(k, z)=〈|Ẽ_y(k, z)|²〉, where Ẽ_x,y(k, z) is the Fourier’s transform of E_x,y(r, z). We refer the reader to the Appendix for details regarding the kinetic wave equations.

It is important to underline that the kinetic equations [Eqs.(17,18) in the Appendix] exhibit properties similar to the celebrated Boltzmann’s kinetic equation that governs the nonequilibrium evolution of a classical gas [52]. In particular, the kinetic equations (17,18) conserve the total intensity (‘number of quasi-particles’) of the optical field, 𝓘=𝓘_x+𝓘_y[𝓘=∫[n_x(k, z)+n_y(k, z)]dk] and the (linear) energy 𝓔=𝓔_x+𝓔_y[𝓔x,y=∫K_x,y n_x,y(k, z)dk]. The irreversible character of the kinetic equation is expressed by the H-theorem of entropy growth d_z S ≥ 0, where the nonequilibrium entropy reads 𝓢=𝓢_x+𝓢_y, 𝓢_x,y=∫log[n_x,y(k, z)]dk. It is because this entropy satisfies a H-theorem that it turns out to be the relevant entropy for the description of the repolarization process of the optical field [see Fig. 1(c)]. Indeed, contrary to the traditional entropy of polarizationW [7, 9], the nonequilibrium entropy 𝓢 increases during the propagation of the field, and the process of entropy growth saturates once the equilibrium state is reached, d_z 𝓢 ≃ 0 [see Fig. 1(c)].

As in standard statistical mechanics, the thermodynamic equilibrium state may be determined from the postulate of maximum entropy [52]. The equilibrium spectra n^eqx,y(k) that realize the maximum of 𝓢[n_x,n_y], subject to the constraints of conservation of 𝓔 and 𝓘, may readily be calculated by introducing the corresponding Lagrange’s multipliers, 1/T and -µ/T, yielding

where T denotes the temperature and µ the chemical potential, by analogy with thermodynamics [30, 31, 32, 33]. The temperature of the field represents the notion of thermal equilibrium between the orthogonal polarization components. Let us remark that the chemical potential of the two polarization components are identical. This results from the fact that the kinetic Eqs.(17,18) (and the resonantly coupled NLS equations) do not conserve the intensity of each polarization component, but only the total intensity 𝓘=𝓘_x+𝓘_y [65].

The thermalization of the optical field then refers to an irreversible relaxation process towards the “most disordered state”, as illustrated by the saturation of the process of entropy growth described by the H-theorem [see Fig. 1(c)]. As in conventional thermodynamics [52], the values of the macroscopic quantities (T,µ) are stationary and predictable using statistical mechanics. Indeed, the unknown parameters (T,µ) can be calculated from the two conserved quantities (𝓘,𝓔): For a given nonequilibrium initial condition of the optical field characterized by (I,E), Eqs.(11,12) determine the (asymptotic) equilibrium spectra of the field [30, 31, 32, 33].

Let us remark that, although the nonlinear coefficient γ appears explicitly in the kinetic equations [see Eqs.(17,18) in the Appendix], it does not enter the equilibrium distributions (11,12). Actually, the collision terms of the kinetic equations are proportional to γ², which means that the nonlinear coefficient γ accelerates the nonequilibrium dynamics of the field: The larger the parameter γ, the faster the field reaches the equilibrium distribution (11,12). This feature is clearly visible in the numerical simulations of the NLS equations. We remark that one encounters the same physical picture in kinetic gas theory: The Boltzmann’s equation that governs the nonequilibrium evolution of a classical gas depends on the nonlinear interaction coefficient (the so-called scattering cross-section), however the Maxwell’s equilibrium distribution is known to be independent of such nonlinear coefficient [52]. It becomes apparent from this discussion that the particular form of the considered nonlinearity is not essential for the repolarization process to occur. In particular, the ratio between the self- and cross-phase-modulation terms may be varied without affecting the equilibrium spectra given in Eqs.(11,12).

 

Fig. 2. (a) Interpretation of the repolarization process as a thermalization of intensity fluctuations from the x-polarized (‘excited’) state towards the y-polarized (‘ground’) state. (b) Degree of polarization at equilibrium 𝒫^eq vs the energy 𝓔, for a fixed value of the field intensity 𝓘. The continuous line refers to the theory [Eqs.(13,14)], the points (◇) to the numerical simulations. The parameters are the same as in Fig. 1 (the energy 𝓔 is varied by varying the coherence of the field: the coherence is degraded as the energy 𝓔 increases).

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6. Degree of polarization at equilibrium

In the basis of rectilinear polarization the dispersion relations K_x,y(k) of the polarization components E_x,y result to be splitted by the phase-mismatch parameter δ, i.e., K_x,y(k)=αk ²±δ[see Fig. 2(a)]. According to the equilibrium spectra (11,12), the optical field then exhibits an unbalanced intensity distribution between the E_x and E_y components. More precisely, assuming δ > 0, one remarks that, as the parameter δ increases, the intensity 𝓘_y(z)=∫n_y(k, z)dk polarized along y increases to the detriment of 𝓘_x. In a loose sense this means that, “it is energetically advantageous for the field to populate the lower-energy state”. This physical picture is schematically illustrated in Fig. 2(a). It suggests that the nonequilibrium dynamics of the optical field may be interpreted in analogy with a two-level system, in which the separation of the two-level energies is provided by the phase-mismatch parameter δ [see Fig. 2(a)].

Let us illustrate this analogy with a two-level system by considering the concrete example of an optical field that is initially polarized along the ‘excited’ x-state, 𝒫(z=0) ≃ 1. During its propagation through the Kerr medium, the four-wave mixing process is responsible for the generation of the orthogonal y-polarized component E _y, which leads to the expected effect of depolarization of the field [37, 38, 39, 40, 41, 42]. As a result of the thermalization process, the population of the y-polarized ‘ground’ state will grow during the field propagation. At some propagation distance z₀, the two levels will be equally populated and the field results to be unpolarized [𝒫(z₀=0) ≃ 0]. The thermalization process continues for larger propagation lengths, which eventually leads to an unbalanced intensity distribution that entails the repolarization of the field along the y-polarized ‘ground’ state.

According to this physical picture, the repolarization effect may be regarded as a thermalization process, which is characterized by an irreversible transfer of intensity from the ‘excited’ state, i.e. the x-polarized state, towards the ‘ground’ state, i.e. the y-polarized state (see Fig. 2a). As a result, the unpolarized field irreversibly evolves towards a state of rectilinear polarization, i.e., the y-polarized state for δ > 0, or the x-polarized state for δ < 0.

The degree of polarization of the optical field may be calculated explicitly from simple algebraic manipulations of the expressions of the intensity 𝓘 and the energy 𝓕 at equilibrium. One readily obtains the following set of coupled equations for the energy and the degree of polarization at equilibrium,

where

Note that, in order to compare the theory with the results of the numerical simulations, we substituted the continuous integrals with discrete sums in Eqs.(13,14), a feature that was discussed in details in Refs. [30, 33]. Accordingly, ∑_k denotes a sum over the discretized spatial frequency space k=(k_x,k_y).

Equations (13,14) should be interpreted as follows. Consider an initial incoherent state of the optical field which is unpolarized [P(z=0) ≃ 0] and characterized by some intensity 𝓘 and energy 𝓔. As a result of the nonlinear propagation, the field irreversibly relaxes to an equilibrium state whose degree of polarization is given by Eqs.(13,14). More precisely, the parametric plot of Eqs.(13,14) with respect to the variable µ provides the degree of polarization at equilibrium 𝒫^eq as a function of the two conserved quantities (𝓘, 𝓔). Such a polarization curve 𝒫^eq(𝓔) is represented in Fig. 2(b) for a fixed value of the field intensity 𝓘. Each numerical point (◇) has been obtained by performing an average of 𝒫(z) over 3000 L_nl once the equilibrium state was reached, i.e., once the entropy has reached a constant value (d _z𝓢 ≃ 0). Let us underline the quantitative agreement between the theory and the numerical simulations without adjustable parameters.

To interpret the results of Fig. 2(b), let us recall that since the initial field is unpolarized, the energy has essentially a kinetic contribution at z=0, 𝓔 ≃ 𝓔_kin(z=0), i.e. the resonant contribution is negligible 𝓔_res(z=0) ≃ 0 [see Fig. 1(b)]. As discussed above in Section 4, the kinetic energy 𝓔_kin provides a measure of the amount of incoherence in the optical field: the higher the kinetic energy, the lower the coherence of the field. Figure 2(b) thus reveals that, for a given intensity of the optical field 𝓘, the repolarization effect becomes less efficient as the coherence of the field is degraded. Referring back to the two-level analogy discussed above through Fig. 2(a), this result has a rather simple physical interpretation. Indeed, one should consider that the kinetic energy at equilibrium 𝓔_kin plays a role analogous to the temperature T, a feature discussed in details in Ref.[36]. It results that, by increasing the temperature, a transfer of quasi-particles occurs from the y-polarized ‘ground’ state towards the x-polarized ‘excited’ state. This property is consistent with the intuitive idea that the difference of populations in the two levels, |𝓘_y-𝓘_x|, decreases as the temperature is increased. This merely explains why the degree of polarization at equilibrium (𝒫^eq) decreases as the energy of the field increases, as illustrated in Fig. 2(b). Let us also remark that a polarization curve similar to that of Fig. 2(b) is obtained for an intensity of the field greater than the critical intensity of polarization instability, 𝓘 > 𝓘_c.

Let us finally note that, for small values of the energy 𝓔, the polarization components undergo a wave-condensation process [30, 33, 34, 49, 51, 62]. To our knowledge, the analysis of wave condensation in resonantly coupled NLS equations has not been studied previously. This actually refers to a rather difficult problem because, due to the presence of the phase-mismatch in Eqs.(1-2), the intensity (‘number of particles’) of each component is no longer a conserved quantity. The study of wave condensation in the framework of Eqs.(1-2) goes beyond the scope of the present work and is addressed to further research.

 

Fig. 3. Schematic illustration of the repolarization process on the Poincaré sphere representation: (a) An unpolarized field evolves towards a highly polarized state. Because of the reversibility of wave propagation, the phase conjugation (p.c.) of the repolarized field will regain its original unpolarized state at the entry of the medium [dashed arrow in (a)]. Such a depolarization process only refers to a transient evolution: The field would again reach a polarized state in a medium twice as long (b), in complete analogy with the celebrated Joule’s experiment of free expansion of a gas (c).

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7. Discussion on the reversible property of the polarization process

Let us analyze in more details the physics underlying the reversible nature of the repolarization process. First of all we note that such a reversible property could be demonstrated in practice by means of a phase-conjugation experiment. For this purpose, consider that the initial unpolarized beam has reached a polarized state owing to the nonlinear propagation in a Kerr medium of length L. This is illustrated in Fig. 3(a) by making use of the Poincaré sphere representation [3, 5, 9]. Then suppose that the phase-conjugate of the repolarized field is created by some process (e.g., the degenerate four-wave mixing [1]) and is propagated backward in the material from z=L to z=0. As a result of the reversible nature of wave propagation, such a reflected beam will regain its original unpolarized state at z=0, as schematically illustrated with a dashed arrow in Fig. 3(a). In other terms the reflected beam exhibits a depolarization process, since it evolves from the initially polarized state at z=L to the unpolarized state at z=0. This seems to indicate that there exists some particular cases in which the expected effect of beam repolarization “does not work”. However, let us underline the key point that such a depolarization process only constitutes a transient evolution of the field. Indeed, let us assume that the phase-conjugated field is propagated backward in a material twice as long. Then the propagation length would become sufficient to permit the optical field to reach its equilibrium state, i.e., the polarized state [Fig. 3(b)].

To appreciate this aspect, it is instructive to consider the analogy with the celebrated Joule’s experiment of free expansion of a gas. As schematically illustrated in Fig. 3(c), a gas initially confined in the left compartment freely expands through a small hole, so that the two compartments are equally full at equilibrium. As it was pointed out in the Loschmidt’s paradox, if the velocities of all the particles were reversed at time t=T, then at time t=2T they will (in principle) regain the left compartment [52]. However, such apparent reduction of entropy only refers to a transient, since at later times the equilibrium state where the gas fill both compartments would be reached again. In complete analogy with the Joule’s experiment, a depolarization process of the optical field only refers to a transient evolution of the field. In other terms, the depolarization of the optical field is extremely improbable and could only occur under artificial initial conditions, while the generic behavior of the field is to increase its disorder by evolving towards a highly polarized state.

8. Conclusion

In summary we have identified a phenomenon of spontaneous polarization of partially incoherent unpolarized light. This effect may occurs thanks to a wave vector mismatch between the orthogonal polarization components of the field. According to the traditional definition of the entropy of polarization [7, 9], the repolarization effect would be characterized by a reduction of entropy. However, a kinetic approach of the problem revealed that the relevant physical entropy is provided by the nonequilibrium entropy 𝓢. In this way, the repolarization effect is characterized by a process of entropy production (d _z 𝓢 ≥ 0), which eventually saturates once the equilibrium state is reached (𝓢 ≃ constant).

The kinetic theory then reveals that the repolarization process finds its origin in the natural thermalization of the optical field towards an equilibrium state. Accordingly, it is thermodynamically advantageous for the optical field to evolve towards a highly polarized state, because this allows the field to reach the “most disordered state”. The kinetic approach allowed us to derive a polarization curve of the field at equilibrium, which has been found in quantitative agreement with the numerical simulations without adjustable parameters. In the basis of circular polarization, the repolarization effect manifests itself by means of the spontaneous emergence of a mutual correlation between the polarization components. In the rectilinear polarization basis, it is characterized by an irreversible transfer of intensity from the ‘excited-polarized state’ towards the ‘ground-polarized state’. In this way, the rectilinear polarization plays the role of a kind of “statistical attractor” for the unpolarized field. The repolarization effect may occur for both low or high pump powers, i.e., for an intensity smaller or greater than the critical intensity of polarization instability.

It is interesting to note that the wave vector mismatch responsible for the existence of the repolarization process appears naturally in the framework of quadratic (χ ⁽²⁾) nonlinearities. The present work may thus be extended to quadratic nonlinear materials [66, 67], in which the equilibrium distribution for the resonant fields are known to be of the same form as Eqs.(11,12) [32, 46]. Besides optics, the present work may also be relevant for multicomponent Bose-Einstein condensates, in which the resonant coherent coupling between the components is introduced by means of a Josephson’s coupling [68].