Incoherently coupled Gaussian soliton pairs in biased photorefractive crystal having both the linear and quadratic electro-optic effect

Abstract

Propagation characteristics of incoherently coupled Gaussian soliton pairs are studied in photorefractive media having both the linear and electro-optic effect. Applying paraxial ray approximation, a very large family of bright–bright Gaussian soliton pairs in photorefractive crystals having both the linear and quadratic electro-optic effect has been identified. These composite solitons can exist only when the carrier beams have the same polarization, wavelength and are mutually incoherent. It is shown that the incoherently coupled soliton pair cannot propagate as a stationary entity if its components have different spatial widths. A wide parameter space of spatial width of the soliton pair and its power has been found where it can exist as a stationary entity. The paraxial theory shows that the identified family of soliton pairs is stable. An interesting finding is that of bistable states in this system for the degenerate case of the power being same for both components of the soliton pair. A relevant example has been taken for illustration of our results where the crystal is PMN-0.33PT.

1 Introduction

Optical spatial solitons have been attracting much interest in recent times. In particular, optical spatial solitons in photorefractive media are really interesting due to their formation at low laser powers and the saturable nature of the non-linearity in these materials. The self-focussing induced due to the refractive index profile produces an index waveguide which balances the diffraction. This results in a robust, nonuniform beam, i.e., a spatial soliton formation.

Spatial solitons in photorefractive materials were first predicted theoretically in 1992 by Segev et al. and henceforth discovered experimentally by Duree et al [1, 2]. When a beam of light illuminates a photorefractive crystal, it causes generation of charge carriers. These charge carriers migrate under the action of an external electric field, transient pyroelectric field or a bulk photovoltaic field. This results in a space charge field being set up. This space charge field is responsible for inducing a refractive index change through the electro-optic effect.

Photorefractive solitons can be classified on the basis of the mechanism of migration of charge carriers which in turn affects the buildup the space charge field. Screening solitons [3], photovoltaic solitons [4], screening photovoltaic solitons [5] are the broad categories of the spatial solitons observed in photorefractive crystals. In addition, the pyroelectric effect with the photovoltaic field has been shown to support the self trapping in photorefractive media [6].

The coupling of two solitons pertains to a state in which an effective index waveguide is created which guides each component of the soliton pair. Switching off one of the soliton beams results in the collapse of the soliton pair. If the two beams are mutually incoherent, them an incoherently coupled soliton pair results. There have been many investigations on the incoherent coupling of photorefractive solitons [7,8,9,10,11,12]. As opposed to this, if the two soliton beams are mutually coherent with a constant phase difference between them, a coherently coupled pair results [13,14,15].

Photorefractive solitons were first predicted in conventional photorefractive media where the linear electro-optic effect contributed to the change in refractive index [10,11,12,13,14,15,16,17,18]. Segev et al. proved that photorefractive solitons can also be supported in the centrosymmetric photorefractive media where the quadratic electro-optic effect governs the change of refractive index [16]. Many new electro-optic materials have been recently discovered which have the electro-optic effect due to both the linear and quadratic electro-optic effects near the phase-transition temperature. There is an interesting interplay between the linear and quadratic electro-optic effect which leads to many interesting self trapping phenomena in such materials [17]. Some of these are ferroelectric KTaxNb1-xO3 (KTN) crystals [18], Pb(Zn1/3Nb2/3)O3-PbTiO3 (PZN-PT) single crystals [19], Pb(Mg1/3Nb2/3)O3](1 − x)-(PbTiO3)x (PMN-PT) [20] single crystals, among others. Photorefractive solitons have been predicted in such novel photorefractive media and their coupling has also been investigated recently [12, 17, 21,22,23,24].

In this paper, we predict a new and very large family of bright–bright incoherently coupled Gaussian soliton pairs in such biased photorefractive media having both the linear and quadratic electro-optic non-linearity. Also, we investigate and illustrate various important properties of such soliton pairs in the present work. We use the paraxial approximation to derive the dynamical equations of the spatial width of these Gaussian soliton pairs. We discuss the existence condition for the family of soliton pairs, various aspects and properties of the soliton pairs along with illustrating their dynamical evolution and stability properties.

2 Mathematical formulation

We consider two optical beams propagating collinearly along the z-direction in a photorefractive crystal having both the linear and quadratic electro-optic effect. The optical c-axis of the photorefractive crystal is along the x-direction. The beam is polarized along the x-direction. We will consider diffraction of the beam to be along the same x-direction. The external electric field is applied along the x-direction and, therefore, a space charge field is setup in the photorefractive crystal. We express the optical field of the incident beam in terms of slowly varying envelopes ${\psi _1}$ and ${\psi _2}$, i.e., ${\vec {E}_1}=\hat {x}{\psi _1}(x,z)\exp ({\text{ikz}}),{\vec {E}_2}=\hat {x}{\psi _2}(x,z)\exp ({\text{ikz}})$ where $\hat {x}$ is the unit vector along the x-direction, $k={k_0}{n_e}$ where ${n_e}$ is the unperturbed index of refraction, and ${\lambda _0}$ is the free space wavelength. The optical beams satisfy the following envelope equations [10]:

$$\left( {i\frac{\partial }{{\partial z}}+\frac{1}{{2{k_0}{n_{\text{e}}}}}\frac{{{\partial ^2}}}{{\partial {x^2}}}+\frac{k}{{{n_{\text{e}}}}}\Delta n({E_{{\text{sc}}}})} \right){\psi _1}=0,$$

(1)

$$\left( {i\frac{\partial }{{\partial z}}+\frac{1}{{2{k_0}{n_{\text{e}}}}}\frac{{{\partial ^2}}}{{\partial {x^2}}}+\frac{k}{{{n_{\text{e}}}}}\Delta n\left( {{E_{{\text{sc}}}}} \right)} \right){\psi _2}=0,$$

(2)

where the change in the refractive index is due to both the linear and quadratic electro-optic effect,[12]

$$\Delta n\left( {{E_{{\text{sc}}}}} \right)= - \frac{1}{2}{k_0}n_{{\text{e}}}^{3}{r_{{\text{eff}}}}{E_{{\text{sc}}}} - \frac{1}{2}{k_0}n_{{\text{e}}}^{3}{g_{{\text{eff}}}}\epsilon _{0}^{2}{({\epsilon _r} - 1)^2}E_{{{\text{sc}}}}^{2},$$

(3)

where g_eff is the effective quadratic electro-optic coefficient, r_eff is the linear electro-optic coefficient, ${\epsilon _o}$and ${\epsilon _r}$ are the vacuum and relative dielectric constants, respectively.

The expression for the space charge field neglecting the effect of diffusion can be stated [3] as follows:

$${E_{{\text{sc}}}}=\frac{{{I_\infty }+{I_{\text{d}}}}}{{I+{I_{\text{d}}}}}{E_0}$$

(4)

where I = $I(x,z)={I_1}(x,z)+{I_2}(x,z)$ represents the total intensity of the two mutually incoherent optical beams and ${I_\infty }$ is the asymptotic value of I at $x \to \pm \infty .$ ${I_d}$ is the dark irradiance. ${E_0}=E(x \to \infty )$ is the space charge field at $x \to \infty .$ If the spatial extent of the soliton beams is much less than the x-width W of the photorefractive crystal, ${E_0} \approx V/W$, where V is the applied external voltage. We shall transform to dimensionless coordinates:

$\xi =\frac{z}{{kx_{0}^{2}}},s=\frac{x}{{{x_0}}},I=\frac{{{n_e}}}{{2{\eta _0}}}\left( {|{\psi _1}{|^2}+|{\psi _2}{|^2}} \right)$ and ${\psi _1}=\sqrt {\frac{{2{\eta _0}{I_{\text{d}}}}}{{{n_{\text{e}}}}}} {U_1},$ ${\psi _2}=\sqrt {\frac{{2{\eta _0}{I_{\text{d}}}}}{{{n_{\text{e}}}}}} {U_2}$ where ${x_0}$ is an arbitrary scale paramter and ${\eta _0}=\sqrt {\frac{{{\mu _0}}}{{{\epsilon _0}}}} .$

The soliton beam intensity has been scaled with respect to the dark irradiance I_d, i.e., ${I_1}=|{U_1}{|^2}{I_{\text{d}}}$ and ${I_2}=|{U_2}{|^2}{I_{\text{d}}}.$

Making use of the dimensionless coordinates, the space charge field can be written as

$${E_{sc}}=\frac{{1+\rho }}{{1+|{U_1}{|^2}+|{U_2}{|^2}}}{E_0}$$

(5)

and the evolution equations become

$$i\frac{{\partial {U_1}}}{{\partial \xi }}+\frac{1}{2}\frac{{{\partial ^2}{U_1}}}{{\partial {s^2}}} - {\beta _1}\frac{{1+\rho }}{{1+|{U_1}{|^2}+|{U_2}{|^2}}}{U_1} - {\beta _2}{\left( {\frac{{1+\rho }}{{1+|{U_1}{|^2}+|{U_2}{|^2}}}} \right)^2}{U_1}=0,$$

(6)

$$i\frac{{\partial {U_2}}}{{\partial \xi }}+\frac{1}{2}\frac{{{\partial ^2}{U_2}}}{{\partial {s^2}}} - {\beta _1}\frac{{1+\rho }}{{1+|{U_1}{|^2}+|{U_2}{|^2}}}{U_2} - {\beta _2}{\left( {\frac{{1+\rho }}{{1+|{U_1}{|^2}+|{U_2}{|^2}}}} \right)^2}{U_2}=0,$$

(7)

where

${\beta _1}=\frac{{{{({k_0}{x_0})}^2}n_{{\text{e}}}^{4}{r_{{\text{eff}}}}}}{2}{E_0},$ ${\beta _2}=\frac{{{{({k_0}{x_0})}^2}n_{e}^{4}{g_{{\text{eff}}}}\epsilon _{0}^{2}{{({\epsilon _{\text{r}}} - 1)}^2}}}{2}E_{0}^{2}$ alongwith $\rho ={I_\infty }/{I_d}.$

The third and fourth terms in Eqs. (6) and (7) signify the non-linear contribution to the refractive index of the photorefractive crystal. Equations (6) and (7) cannot be solved exactly to obtain an analytic solution. Hence, these are usually solved numerically. But there are several methods to approximately solve these equations. Segev’s method [4], Akhmanov’s paraxial method [25], Anderson’s variational method [26] and Vlasov’s moment method [27] can be used. We shall obtain physically acceptable soliton states in the present work making use of the paraxial approximation alongwith a variational solution. We shall take $\rho =0$ in our investigation as appropriate for the case of bright solitons. Following [8, 9], the slowly varying beam envelope can be expressed as follows:

$${U_1}(\xi ,s)={A_1}(\xi ,s){e^{ - i{\nu _1}(s,\xi )}},$$

(8)

$${U_2}(\xi ,s)={A_2}(\xi ,s){e^{ - i{\nu _2}(s,\xi )}},$$

(9)

where ${A_1}(\xi ,s)$ and ${A_2}(\xi ,s)$ are purely real quantities and $\nu (\xi ,s)$ represents the phase. Substituting these two ansatz in (6) and (7) gives

$$\frac{{\partial {A_1}}}{{\partial \xi }} - \frac{{\partial {A_1}}}{{\partial s}}\frac{{\partial {\nu _1}}}{{\partial s}} - \frac{1}{2}{A_1}\frac{{{\partial ^2}{\nu _1}}}{{\partial {s^2}}}=0,$$

(10)

$$\frac{{\partial {A_2}}}{{\partial \xi }} - \frac{{\partial {A_2}}}{{\partial s}}\frac{{\partial {\nu _2}}}{{\partial s}} - \frac{1}{2}{A_2}\frac{{{\partial ^2}{A_2}}}{{\partial {s^2}}}=0,$$

(11)

$${A_1}\frac{{\partial {\nu _1}}}{{\partial \xi }}+\frac{1}{2}\frac{{{\partial ^2}{A_1}}}{{\partial {s^2}}} - \frac{1}{2}{A_1}{\left( {\frac{{\partial {\nu _1}}}{{\partial s}}} \right)^2} - {\beta _1}{\Phi _1}(\xi ,s){A_1} - {\beta _2}{\Phi _2}(\xi ,s){A_1}=0,$$

(12)

$${A_2}\frac{{\partial {\nu _2}}}{{\partial \xi }}+\frac{1}{2}\frac{{{\partial ^2}{A_2}}}{{\partial {s^2}}} - \frac{1}{2}{A_2}{\left( {\frac{{\partial {\nu _2}}}{{\partial s}}} \right)^2} - {\beta _1}{\Phi _1}(\xi ,s){A_2} - {\beta _2}{\Phi _2}(\xi ,s){A_2}=0,$$

(13)

where

$${\Phi _1}(\xi ,s)=\frac{1}{{1+|{A_1}{|^2}+|{A_2}{|^2}}},$$

(14)

$${\Phi _2}(\xi ,s)={\left( {\frac{1}{{1+|{A_1}{|^2}+|{A_2}{|^2}}}} \right)^2}.$$

(15)

In Eqs. (12) and (13), ${\Phi _1}(\xi ,s)$ and ${\Phi _2}(\xi ,s)$ account for the non-linearity due to space charge field induced refractive index change due to the linear and quadratic electro-optic effect, respectively.

The last two terms control the diffraction effects and hence lead to a self-trapped soliton propagation.

3 Results and discussion

We shall look for self-similar spatial soliton solutions for which the electromagnetic field energy is confined to the center of the beam. It is important to note that Eqs. (6) and (7) are modified coupled nonlinear Schrodinger equations which are non-integrable in nature. Gaussian solutions are the most widely used approximate solutions to solve non-integrable systems for solitons [26, 28,29,30,31,32,33]. The Gaussian solution does not deviate too much from the numerically computed solutions. The Gaussian solution gives good analytical results comparable to those of the actual solution found from pure numerical simulations [8]. Hence, it can be used for approximate analytical treatment of this system and extract useful results. The term “quasi-soliton” is best suited when such an approximation is taken and henceforth the term “soliton” in this article may be understood to be a “quasi-soliton” [21]. Hence, we assume quasi-soliton solutions for (12) and (13) and following Ref. [8, 9], we shall take the following ansatz in our investigation:

$${A_1}(\xi ,s)=\frac{{\sqrt {{P_1}} }}{{\sqrt {{f_1}(\xi )} }}\exp \left[ {\frac{{ - {s^2}}}{{2r_{1}^{2}f_{1}^{2}(\xi )}}} \right],$$

(16)

$${\nu _1}(\xi ,s)=\frac{{{s^2}}}{2}{\eta _1}(\xi )+{\phi _1}(\xi ),$$

(17)

$${\eta _1}(\xi )= - \frac{1}{{{f_1}(\xi )}}\frac{{d{f_1}(\xi )}}{{d\xi }},$$

(18)

$${A_2}(\xi ,s)=\frac{{\sqrt {{P_2}} }}{{\sqrt {{f_2}(\xi )} }}\exp \left[ {\frac{{ - {s^2}}}{{2r_{2}^{2}f_{2}^{2}(\xi )}}} \right],$$

(19)

$${\nu _2}(\xi ,s)=\frac{{{s^2}}}{2}{\eta _2}(\xi )+{\phi _2}(\xi ),$$

(20)

$${\eta _2}(\xi )= - \frac{1}{{{f_2}(\xi )}}\frac{{d{f_2}(\xi )}}{{d\xi }},$$

(21)

where P₁ and P₂ are the peak powers of these solitons, r₁ and r₂ are positive constants, $f(\xi )$ is the variable beam width parameter such that the product ${r_1}{f_1}(\xi ),{r_2}{f_2}(\xi )$ gives the spatial widths of the respective solitons. r₁, ${f_1}(\xi ),$ ${f_2}(\xi ),$ ${\eta _1}(\xi ),{\eta _2}(\xi )$ are the variational parameters which we are interested with the solution in (16) being the variational solution. Following a similar procedure to solve equations(12) − (13) as done in Ref [8, 9], we expand ${\Phi _1},{\Phi _2}$ in series and under first-order approximation,

$${\Phi _1} \cong \frac{1}{{(1+({P_1}/{f_1})+({P_2}/{f_2}))}}+{s^2}\left( {\frac{{{P_1}}}{{r_{1}^{2}f_{1}^{3}}}+\frac{{{P_2}}}{{r_{2}^{2}f_{2}^{3}}}} \right)\frac{1}{{{{\left( {1+({P_1}/{f_1})+({P_2}/{f_2})} \right)}^2}}},$$

(22)

$${\Phi _2} \cong \frac{1}{{{{(1+({P_1}/{f_1})+({P_2}/{f_2}))}^2}}}+2{s^2}\left( {\frac{{{P_1}}}{{r_{1}^{2}f_{1}^{3}}}+\frac{{{P_2}}}{{r_{2}^{2}f_{2}^{3}}}} \right)\frac{1}{{{{\left( {1+({P_1}/{f_1})+({P_2}/{f_2})} \right)}^3}}}.$$

(23)

Substituting Eqs. (16)-(23) in (12) and (13), and equating the coefficients of s² on both sides of the resulting equations, we get,

$$\frac{{{d^2}{f_1}}}{{d{\xi ^2}}}=\frac{1}{{r_{1}^{4}f_{1}^{3}}} - 2{\beta _1}\left( {\frac{{{P_1}}}{{r_{1}^{2}f_{1}^{2}}}+\frac{{{P_2}{f_1}}}{{r_{2}^{2}f_{2}^{3}}}} \right)\frac{1}{{{{\left( {1+\frac{{{P_1}}}{{{f_1}}}+\frac{{{P_2}}}{{{f_2}}}} \right)}^2}}} - 4{\beta _2}\left( {\frac{{{P_1}}}{{r_{1}^{2}f_{1}^{2}}}+\frac{{{P_2}{f_1}}}{{r_{2}^{2}f_{2}^{3}}}} \right)\frac{1}{{{{\left( {1+\frac{{{P_1}}}{{{f_1}}}+\frac{{{P_2}}}{{{f_2}}}} \right)}^3}}},$$

(24)

$$\frac{{{d^2}{f_2}}}{{d{\xi ^2}}}=\frac{1}{{r_{2}^{4}f_{2}^{3}}} - 2{\beta _1}\left( {\frac{{{P_1}{f_2}}}{{r_{1}^{2}f_{1}^{3}}}+\frac{{{P_2}}}{{r_{2}^{2}f_{2}^{2}}}} \right)\frac{1}{{{{\left( {1+\frac{{{P_1}}}{{{f_1}}}+\frac{{{P_2}}}{{{f_2}}}} \right)}^2}}} - 4{\beta _2}\left( {\frac{{{P_1}{f_2}}}{{r_{1}^{2}f_{1}^{3}}}+\frac{{{P_2}}}{{r_{2}^{2}f_{2}^{2}}}} \right)\frac{1}{{{{\left( {1+\frac{{{P_1}}}{{{f_1}}}+\frac{{{P_2}}}{{{f_2}}}} \right)}^3}}}.$$

(25)

Eqn (24) and (25) can be used to illustrate the evolution of spatial width of the two solitons with distance of propagation. For stable bright soliton pairs which do not diverge, we must look for the points of equilibrium of the above differential equations by applying the condition $\frac{{{d^2}{f_1}}}{{d{\xi ^2}}}=\frac{{{d^2}{f_2}}}{{d{\xi ^2}}}=0.$ Hence, we can obtain the following relations:

$$\frac{1}{{r_{1}^{4}}} - 2{\beta _1}\left( {\frac{{{P_1}}}{{r_{1}^{2}}}+\frac{{{P_2}}}{{r_{2}^{2}}}} \right)\frac{1}{{{{\left( {1+{P_1}+{P_2}} \right)}^2}}} - 4{\beta _2}\left( {\frac{{{P_1}}}{{r_{1}^{2}}}+\frac{{{P_2}}}{{r_{2}^{2}}}} \right)\frac{1}{{{{\left( {1+{P_1}+{P_2}} \right)}^3}}}=0,$$

(26)

$$\frac{1}{{r_{2}^{4}}} - 2{\beta _1}\left( {\frac{{{P_1}}}{{r_{1}^{2}}}+\frac{{{P_2}}}{{r_{2}^{2}}}} \right)\frac{1}{{{{\left( {1+{P_1}+{P_2}} \right)}^2}}} - 4{\beta _2}\left( {\frac{{{P_1}}}{{r_{1}^{2}}}+\frac{{{P_2}}}{{r_{2}^{2}}}} \right)\frac{1}{{{{\left( {1+{P_1}+{P_2}} \right)}^3}}}=0.$$

(27)

To find stationary incoherently coupled composite bright–bright soliton pair, we need to look for the common solutions of (26) and (27). These can be found out by equating the LHS of both equations which gives us the condition r₁ = r₂. This implies that each component of the soliton pair has equal spatial width. Hence, stable propagation of a soliton pair is only possible if the spatial width of each component is the same. Setting r₁ = r₂ = r in (26) and (27), we get a relationship between the peak power of these solitons and their spatial widths,

$$\frac{1}{{{r^2}}}=\frac{{2{\beta _1}({P_1}+{P_2})}}{{{{\left( {1+{P_1}+{P_2}} \right)}^2}}}+\frac{{4{\beta _2}({P_1}+{P_2})}}{{{{\left( {1+{P_1}+{P_2}} \right)}^3}}}.$$

(28)

The Eq. (28) can be said to be an existence equation of the bright incoherently coupled composite soliton pair. Since r cannot be negative, LHS will be positive. Hence, ${\beta _1}$ and ${\beta _2}$ should be such that RHS be positive and satisfy the Eq. (28). For a given photorefractive crystal, i.e., for a given value of ${\beta _1}$ and ${\beta _2}$, the Eq. (28) gives us the minimum spatial width of the propagating soliton pair. The spatial width of the propagating solitons will be minimum when

$${P_1}+{P_2}=\frac{{ - 2{\beta _2} \pm \sqrt {\beta _{1}^{2}+2{\beta _1}{\beta _2}+4\beta _{2}^{2}} }}{{{\beta _1}}}.$$

(29)

Also, Eq. (28) clearly tells us that for a given value of ${P_1}+{P_2},$ the solitons’ width depends inversely upon both the external bias field (second term on RHS) and the square root of the external bias field(first term on RHS). The magnitude of the linear and quadratic electro-optic coefficient decides which term will be dominant. This behavior is different from the previously studied cases in non-centrosymmetric photorefractive media [8] and centrosymmetric photorefractive media [9] as now the refractive index change is governed by both the linear (Pockels’ effect) and quadratic electro-optic effect(dc Kerr effect) simultaneously.

To illustrate our results, we shall consider the PMN-0.33PT crystal. The parameters are shown clearly in Table 1.

Table 1 Parameters taken in our theoretical investigation for PMN-0.33PT crystal [17, 23, 34, 35]

The existence curves of these bright–bright soliton pairs in photorefractive media having both the linear and quadratic effect in both, the high power regime and the low power regime have been plotted in Figs. 1 and 2. Every point on any curve of these figures can be said to represent a stably travelling incoherently coupled soliton pair with a specific peak power and spatial width. Hence, these figures capture the existence conditions for a large family of bright incoherently coupled soliton pairs of varying width and peak power in photorefractive media having both the linear and quadratic electro-optic effect. We can conclude from Figs. 1 and 2 that the peak power of one soliton component can be a fraction of the peak power of the other soliton component of the incoherently coupled soliton pair. This has an important practical application in that an optical beam of appropriate spatial width can be made to propagate stably as a self-trapped soliton by making use of another co-propagating strong optical beam.

Fig. 1

Existence curve for bright–bright soliton pairs in the high power regime

Fig. 2

Existence curve for bright–bright soliton pairs in the low-power regime

In Figs. 3 and 4, we plot the dependence of the soliton width with peak power of one component assuming the power of the other component to be constant in the high-power and low-power regime, respectively. From the figures, we conclude that with a fixed value of P₂, the other component can exist for different values of P₁ for both the low- and high-power regime.

Fig. 3

Spatial width of incoherently coupled bright–bright soliton pairs in the high-power regime as a function of peak power P₁ when P₂ is constant

Fig. 4

Spatial width of incoherently coupled bright–bright soliton pairs in the low-power regime as a function of peak power P₁ when P₂ is constant

In view of the above, it is now logical to consider the case where the power of both components of the soliton pair is same, i.e., P₁ = P₂ = P. Substituting in (28), we get

$$\frac{1}{{{r^2}}}=\frac{{4{\beta _1}P}}{{{{(1+2P)}^2}}}+\frac{{8{\beta _2}P}}{{{{(1+2P)}^3}}}.$$

(30)

The spatial width of the propagating solitons for this case will be minimum when

$$P=\frac{{ - {\beta _2} \pm \tfrac{1}{2}\sqrt {\beta _{1}^{2}+2{\beta _1}{\beta _2}+4\beta _{2}^{2}} }}{{{\beta _1}}}.$$

(31)

Again considering the PMN-0.33PT crystal and using the parameter values as specified in Table 1, we plot the dependence of soliton width with the degenerate peak power P in Fig. 5. It is very clear from the figure that bistable states are present in this case, i.e., two different soliton pairs exist with the same spatial width but different peak power.

Fig. 5

Spatial width of incoherently coupled bright–bright soliton pairs as a function of peak power, given that P₁ = P₂ = P

Finally, we will study the propagation of these incoherently coupled solitons. For that, we first choose a value of r and then select a point arbitrarily on the soliton existence curve corresponding to that particular value of r. Then, we solve Eqs. (24) and (25) numerically for f₁ and f₂ taking initial conditions ${f_1}(0)={f_2}(0)=1$ and $\frac{{d{f_1}}}{{d\xi }}=\frac{{d{f_2}}}{{d\xi }}=0$ at $\xi =0.$ For illustrating the above, let us take a point on the curve r = 1 in Fig. 1, from which we calculate P₁ = 10.00 and P₂ = 19.04. Solving (24) and (25) numerically, we plot the variation of f₁ and f₂ with the normalized distance $\xi$ as shown in Figs. 6 and 7, which shows that f₁ and f₂ remain constant during propagation in photorefractive crystals having both the linear and quadratic electro-optic effect. Hence, we can infer that the soliton pair is stable. We show this fact graphically in Figs. 8 and 9 by plotting each soliton component’s dynamic evolution using (16) and (19). Finally, we show the dynamic evolution of the bright–bright incoherently coupled soliton pair in Fig. 10.

Fig. 6

Evolution of the beam width of the first soliton component of the bright–bright incoherently coupled soliton pair with parameter P₁ = 10.00 and r = 1

Fig. 7

Evolution of the beam width of the second soliton component of the bright–bright incoherently coupled soliton pair with parameter P₂ = 19.04 and r = 1

Fig. 8

Dynamical evolution with normalized distance $\xi$ of the first soliton component of the bright–bright incoherently coupled soliton pair. P₁ = 10.00 and r = 1

Fig. 9

Dynamical evolution with normalized distance $\xi$ of the second soliton component of the bright–bright incoherently coupled soliton pair. P₂ = 19.04 and r = 1

Fig. 10

Dynamical evolution with normalized distance $\xi$ of the bright–bright incoherently coupled soliton pair. P₁ = 10.00 and P₂ = 19.04 and r = 1

4 Conclusions

We have presented the theoretical formulation for collinearly travelling incoherently coupled Gaussian soliton pairs in photorefractive materials having both the linear and quadratic electro-optic effect. These can be established provided that the two beams have the same polarization, wavelength and are mutually incoherent. We have found the existence condition for an infinitely large family of Gaussian soliton pairs in such photorefractive media and identified the relevant parameter space. Our results show that each component of the soliton pair should have same spatial width in order for the soliton pair to propagate stably. The individual components of the soliton pair can have different peak powers. We have shown the dependence of the spatial width of the soliton pair on the peak power of any one component given the peak power of the other component. Also, bistable states can be clearly seen in the case when power of both components is same. We express the condition for the minimum spatial width of the stably propagating solitons in terms of the total power of the soliton components. Finally, we examine the stability of these soliton pairs qualitatively and illustrate the dynamical evolution of these soliton pairs graphically.

It must be emphasized how our work differs considerably from previous research. While Ref [8] discusses the incoherently coupled Gaussian soliton pairs in photorefractive photovoltaic media, Ref [9] studies the incoherently coupled Gaussian soliton pairs in centrosymmetric photorefractive media. We study the Gaussian soliton pairs in novel photorefractive media having both the linear and quadratic electro-optic effect. Hence our ensuing results have a completely different theoretical foundation as compared to Ref [8, 9].

Also, in Ref [12], the authors have predicted and investigated incoherently coupled solitons in photorefractive media with both the linear and quadratic electro-optic effect. Our work differs from [12] in two main aspects. First, we follow the approach of Ref [8]. in using a Gaussian ansatz for the soliton beam solution and hence our theory pertains to a new type of family of Gaussian soliton pairs or “quasi-soliton” pairs. Second, and the most important distinguishing factor is that we have determined an analytical formula for existence of soliton that allows an infinite choice of solitons’ power. Each of these corresponds to unequivocal formation of stable bright–bright incoherently coupled soliton pairs. We have also found that one component of the incoherently coupled soliton pair can have an infinite number of different peak powers for a particular peak power of the other component, i.e., any of the two components of the incoherently coupled soliton pair can have power which is only a fraction of the other. This is not found in the result of Hao et al [12].