Separate spatial Holographic-Hamiltonian soliton pairs and solitons interaction in an unbiased series photorefractive crystal circuit

Xin Cai; Jinsong Liu; Shenglie Wang

doi:10.1364/OE.17.002287

1. Introduction

Photorefractive (PR) solitons have become a convenient playground in which to study soliton phenomena[1–27], since they are observed at low light powers and exhibit robust trapping in both transverse dimensions. To date, a quasi-steady-state PR soliton has been predicted[1] and found experimentally[2], three different kinds of steady-state PR solitons (screening solitons[3–6], open- and closed-circuit photovoltaic solitons[7–10] and screening-photovoltaic solitons[11, 12]) which are supported by the self-phase-modulation self-focusing mechanism have been predicted and found experimentally. Cohen et al. have predicted a new kind of spatial solitons which are supported by the holographic focusing mechanism, namely holographic solitons for the first time[13]. Thereafter, Liu has proposed holographic solitons supported by dissipative system, namely dissipative holographic soliton[14], Friedler et al. have shown that the giant Kerr nonlinearity in the regime of electromagnetically induced transparency in vapor could lead to the formation of holographic vector solitons[15], Freedman et al. have discussed the relation between grating-mediated wave guiding and holographic solitons[16], and Cohen et al. have shown that holographic solitons could be supported by periodically poled photovoltaic photorefractives[17]. Combining the holographic focusing mechanism with the self-phase-modulation self-focusing mechanism, holographic photovoltaic solitons, holographic screening solitons and holographic screening-photovoltaic solitons have been predicted [18] and the first one has been found experimentally[19]. All these investigations on the PR soliton, soliton pair and soliton interaction were concerned with only one piece of PR crystal[20–27]. Several years ago, Liu et al. presented a new kind of soliton pair, namely separate soliton pair, which could form in a serial crystal circuit consisting of two PR crystals connected by wire leads in a chain with or without a voltage source unit under certain conditions. The numerical results showed that the two solitons in a soliton pair can affect each other by the light-induced current and their coupling can affect their spatial profiles, dynamical evolutions, stabilities and self-deflection[28, 29]. However, all studies on separate soliton pair have been only considered in two crystals that are illuminated by two laser beams respectively so far. What will happen if two laser beams, signal beam and pump beam, incident upon one of the crystals in a serial PR crystal circuit and one laser beam incident upon the other respectively? Can a dissipative holographic soliton and a Hamiltonian soliton form individually within each crystal in the circuit? If it can, do the two solitons, formed separate in the two crystals, interact or affect each other?

In this paper, we investigate steady-state PR solitons formed in a series PR crystal circuit in which at least one being PV-PR (photovoltaic-photorefractive). By use of the well-known transport model of photorefractive effect, successfully used to develop the theories of a Hamiltonian soliton[4] and a holographic soliton[13–18], we predicate that one crystal can support a Hamiltonian soliton and the other crystal can support a holographic soliton in the case that the spatial extent of the optical wave is much less than the width of the crystal. We name the two solitons, formed separately in the two crystals, a separate holographic-Hamiltonian soliton pair. For an example, we consider that a non-photovoltaic PR crystal that can support a holographic soliton and a PV-PR crystal that can support a Hamiltonian soliton are connected as a series circuit. As a result, there are two types of the separate soliton pair: dark-dark and bright-dark. Because the two crystals are connected electronically in such a circuit, the PV-PR crystal acts as a source to bias the other crystal and the light-induced current can flow from it to the other when it is illuminated, and as a result, the current will vary with the intensities of incident laser beams. Therefore, for an unbiased PR crystal circuit, changing the intensity of the laser beam incident upon one crystal, not only will the soliton formed in that crystal change, but the soliton formed in the other crystal will also change. The interaction is collisionless. However, the effect of the input intensity of pump beam on the light-induced current is too weak to affect the other soliton. That is, the Hamiltonian screening-photovoltaic soliton can affect the holographic screening soliton by light-induced current whereas the reverse is not true, i.e., the interaction is unilateral.

This paper is organized as follows. In Sec. 2, the theoretical model is built upon the well-known transport model of photorefractive effect ignoring the diffusion effects. In Sec. 3, the coupling effect between the two solitons in a separate soliton pair on the intensity profiles is investigated numerically. Finally, we draw some conclusions in Sec. 4.

2. Theoretical model

We consider the series PR crystal circuit, in which two crystals, denote by P and P̂, at least one being PV-PR are connected electronically in a chain by electrode leads as shown in Fig. 1. For each crystal, electrodes are made on the two surfaces with their normal parallel to c-axis of the crystal. The two optical beams I and Î are made to propagate in the two crystals along z and ẑ axes and are permitted to diffract only along the x and x̂ directions respectively, and I_p, an optical beam with a uniform spatial distribution in both transverse dimensions as the pump beam, is also made to propagate in the crystal along the z axis and makes a small angle θ in the crystal with the signal beam I. Moreover, let us assume that the two incident optical beams I and Î are both linearly polarized along the x and x̂ directions, respectively. The pump beam I_p is also linearly polarized light and its polarized direction makes an angle φ with the x axis. In the limit in which the optical wave has a spatial extent ∆x much less than the x width of the crystal (the x-direction being taken parallel to the c-axis), we predict that each crystal can support a spatial bright (dark) soliton.

Fig. 1. Illustration of the series PR crystal circuit consisting of two PR crystals in which at least one crystal is PV-PR. (a) The two crystal’s c-axes are oriented in a right-handed screw sense denoted by ⇈. (b) One crystal’s c-axis is oriented in a right-handed screw sense but the other crystal’s c-axis is oriented in a left-handed screw sense, denoted by ⇅. P and P̂ denote the two crystals. C and Ĉ denote the two c-axes. I_p denotes the incident pump beam. I and Î denote the incident bright or dark solitonlike one-dimensional laser beams.

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Considering first the crystal P, let ϕ denote the slowly varying envelope of the electric-field component of the signal beam. In the slowly varying approximation, ϕ satisfies the following paraxial wave equation[18]:

\frac{1}{2 k} \frac{\partial^{2} ϕ}{\partial x^{2}} + i \frac{\partial ϕ}{\partial z} - \frac{k_{0}}{2} n_{e}^{3} r_{33} E_{sc} ϕ + (Γ_{0} - \frac{i}{2} Γ) \frac{I_{p}}{I_{p} + I} ϕ - i \frac{ϑ_{0}}{2} ϕ = 0

where k = n_e k ₀, k ₀ =2π /λ ₀, λ ₀ is the free-space wavelength of the lightwave employed, n_e is the unperturbed extraordinary refractive index, ϑ ₀ is the absorption coefficient of the crystal, r ₃₃ is the electro-optical coefficient, Γ and Γ₀ are the intensity and phase coupling coefficients respectively of the two-wave mixing, I(x,z) = (n_e/2η ₀)|ϕ(x,z)|² , η ₀ =(μ ₀/ℰ ₀)^1/2 and I_p = (n_e/2η ₀)|ϕ_p|² . The expression of E_sc can be found from the standard set of rate and continuity equations and Gauss’s law, which describe the photorefractive effect in a medium in which the photovoltaic current is nonzero and the electrons are the sole charge carriers. In the steady state, the one-dimensional equations are [7, 28]

γ_{R} n N_{D}^{+} = s_{i} (I + I_{b}) (N_{D} - N_{D}^{+}),

\frac{\partial E_{sc}}{\partial} = \frac{e}{ε_{0} ε_{r}} (N_{D}^{+} - N_{A} - n),

J = eμn E_{sc} + k_{B} Tμ \frac{\partial n}{\partial x} + k_{p} s_{i} (N_{D} - N_{D}^{+}) I,

\frac{\partial J}{\partial x} = 0 or J = const .

Here N_D ⁺ and N_D are the ionized donor density and donor density respectively, N_A is the acceptor density, n is the electron density, s_i is the photoexcitation cross section, γ_R is the carrier recombination rate, μ and e are the electron mobility and the charge respectively, k_p is the photovoltaic constant, k_B is Boltzmann’s constant, T is the absolute temperature, ℰ ₀ is the free space permittivity, ℰ_r is the relative static dielectric constant, J is the light-induced current density in the crystal, and I_b, is the intensity of the background light. For the system considered in this paper, we have I_b = I + I_d , where I_d is the dark irradiance. We assume that I_d ≪ I_p , thus we have I_b ≈ I_p . Moreover, we have ignored any z spatial dependence by assuming that the variables involved vary much more rapidly in the x-direction in Eqs. (2)–(5). We can take a similar way to that in [4] and [12] to greatly simplify Eqs. (2)–(5) by keeping in mind that the following inequalities hold true in typical PR or PV-PR media: N _D ⁺ ≪ n , N_D ≪ n and N_A ≪ n as well as |(ℰ ₀ ℰ_r / eN_A)(∂E_sc / ∂x)| ≪ 1 if I(x,z) varies slowly with respect to x . In this case, Eqs. (2) and (3) yield the following results:

N_{D}^{+} \approx N_{A}

n = s_{i} (N_{D} - N_{A}) (I + I_{p}) / (γ_{R} N_{A}) .

At this point, let us also assume that the power density I = I(x, z) attains asymptotically a constant value at x → ±∞ ,i.e., I(x → ±∞, z) = I _∞ . In these regions of constant illumination, Eqs. (2)–(5) require that E_sc is independent of x, i.e., E_sc (x → ±∞, z) = E ₀. Therefore, from Eq. (7) the electron density n in the regions (x → ±∞), denoted by n _∞, can be subsequently determined and is given by

n_{\infty} = s_{i} (N_{D} - N_{A}) (I_{\infty} + I_{p}) / (γ_{R} N_{A}) .

On the other hand, from Eq. (4), the current density J in the regions (x → ±∞) can be given by

J_{\infty} = eμ n_{\infty} E_{0} + k_{p} s_{i} (N_{D} - N_{A}) I_{\infty},

where J _∞ = J(x → +∞,z). Substituting Eq. (8) into Eq. (9), we have

J_{\infty} = eμ n_{\infty} [E_{0} + E_{p} I_{\infty} / (I_{\infty} + I_{p})] .

where E_p = k_pγ_RN_A / (eμ) is the photovoltaic field constant of the crystal P.

For crystal P̂ , the slowly varying envelope of the electric-field component of the optical beam denoted by $\hat{ϕ}$ satisfies the following paraxial wave equation [12]:

\frac{1}{2 \hat{k}} \frac{\partial^{2} \hat{ϕ}}{\partial x^{2}} + i \frac{\partial \hat{ϕ}}{\partial z} - \frac{{\hat{k}}_{0}}{2} ({\hat{n}}_{e}^{3} {\hat{r}}_{33} {\hat{E}}_{sc}) \hat{ϕ} = 0 .

To simplify the analysis, any loss effects are neglected in Eq.(11). Similarly, we have Ê_p = k̃_p $\hat{γ}$ _RN̂_A/(e $\hat{μ}$ ) and

{\hat{n}}_{\infty} = {\hat{s}}_{i} ({\hat{N}}_{D} - {\hat{N}}_{A}) ({\hat{I}}_{\infty} + {\hat{I}}_{d}) / ({\hat{γ}}_{R} {\hat{N}}_{A}),

{\hat{J}}_{\infty} = e \hat{μ} {\hat{n}}_{\infty} [{\hat{E}}_{0} + {\hat{E}}_{p} {\hat{I}}_{\infty} / ({\hat{I}}_{\infty} + {\hat{I}}_{d})] .

Let V and V̂ denote the potential measured between the electrodes of the P and P̂ crystals having width separated by W and Ŵ , respectively. Let S and Ŝ denote the surfaces of the electrodes of the crystals P and P̂ , respectively. First of all, let us consider configuration show in Fig. 1(a), denoted by ⇈, in which the two crystals’ c-axes are oriented in the right-handed screw sense. For the series crystal circuit without an external bias source, we have V + V̂ = 0 and SJ = ŜĴ . If the spatial extent ∆x of the optical wave is much less than the x-width W of the crystal, E ₀ is approximately expressed by E ₀ = V/W [4]. For crystal P̂ , we likewise have Ê ₀ = V̂/Ŵ . Therefore, we find that

{WE}_{0} + \hat{W} {\hat{E}}_{0} = 0 .

Equation (5) implies that the current density J(x, z) is constant everywhere in the crystal, i.e., J _∞ = J(x, z) . For crystal P̂ , we likewise have Ĵ _∞ = Ĵ(x, z). From SJ = ŜĴ , we have SJ _∞ = ŜĴ _∞. Substituting Eqs. (10) and (13) into SJ _∞ = ŜĴ _∞, we find that

Sμ n_{\infty} [E_{0} + E_{p} I_{\infty} / (I_{\infty} + I_{p})] = \hat{S} \hat{μ} {\hat{n}}_{\infty} [{\hat{E}}_{0} + {\hat{E}}_{p} {\hat{I}}_{\infty} / ({\hat{I}}_{\infty} + {\hat{I}}_{p})]

Using of Eqs. (14) and (15), we have found that E ₀ and E ₀ satisfy the following equations:

E_{0} = σ {\hat{E}}_{p} - Γ' E_{p},

{\hat{E}}_{0} = \hat{σ} E_{p} - \hat{Γ}' {\hat{E}}_{p},

where σ = $\hat{δ}$ ÎŴΦ / W , Φ = 1 /[δ(I _∞ + I_p) + $\hat{δ}$ (I _∞ + Î_d)] , Γ′ = δ I _∞ Φ , Γ′ = $\hat{δ}$ Î _∞Φ , $\hat{σ}$ = δ I _∞ WΦ/Ŵ , δ = Sμs_i(N_D - N_A)/(r_RN_AW) , $\hat{δ}$ = Ŝ $\hat{μ}$ ŝ_i(N̂_D - N_A)/(r̂_RN̂_AŴ) . Here σ and $\hat{σ}$ are known as gain coefficients and Γ′ and $\hat{Γ}$ ′ as coupling coefficients. However, the expression of E ₀ (as well as Ê ₀) has different forms under different configuration shown in Fig. 1(b), denoted by ⇅ , in which the c-axes of the two crystals are oriented in opposite screw senses. Because an illuminated PV-PR crystal acts as a current source with current always flowing from its positive electrode [7], for the series crystal circuit without an external bias source, we have V - V̂ = 0 and SJ = -ŜĴ . Arguing previously, we find that

E_{0} = - σ {\hat{E}}_{p} - Γ' E_{p},

{\hat{E}}_{0} = - \hat{σ} E_{p} - \hat{Γ}' {\hat{E}}_{p},

In the region where I(x, z) varies with x, from Eqs. (2) and (4), we have

J = eμn (E_{sc} + \frac{k_{B} T}{e} \frac{\partial In n}{\partial x} + E_{p} \frac{I}{I + I_{b}})

As described above, J _∞ = J(x, z). Therefore, form Eqs. (10) and (20), we have

n_{\infty} (E_{0} + E_{p} \frac{I_{\infty}}{I_{\infty} + I_{p}}) = n (E_{sc} + \frac{k_{B} T}{e} \frac{\partial In n}{\partial x} + E_{p} \frac{I}{I + I_{p}})

In turn, by entirely ignoring the diffusion effect, the expression of E_sc can be obtained from Eq. (21) as follows:

E_{sc} = E_{0} \frac{I_{\infty} + I_{p}}{I + I_{p}} + E_{p} \frac{I_{\infty} - I}{I + I_{p}} .

Similarly, we can obtain the result for P̂ as follows:

{\hat{E}}_{sc} = {\hat{E}}_{0} \frac{{\hat{I}}_{\infty} + {\hat{I}}_{d}}{\hat{I} + {\hat{I}}_{d}} + {\hat{E}}_{p} \frac{{\hat{I}}_{\infty} - \hat{I}}{\hat{I} + {\hat{I}}_{p}} .

Although the expression of E_sc (as well as E_sc) has the same form as that for a biased PV-PR crystal[12, 18], the value of the field depends on the parameters of the two crystals, including I _Ȟ and Î _∞. On the other hand, E_sc and E_sc are not independent. They couple each other by the parameters σ, $\hat{σ}$ , Γ′ and $\hat{Γ}$ ′.

Then the evolution equations of the two solitons formed in the crystal P and P̂ can be established by substituting Eqs. (22), (23) into Eqs. (1), (11). Adopting the following dimensionless coordinates and variables; i.e., let ξ = z/(kx ₀ ²), $\hat{ξ}$ = ẑ/(k̂x̂ ₀ ²) , s = x/x ₀ , s = x/x ₀ and ϕ = (2η ₀ I_p/n_e)^1/2 U, $\hat{ϕ}$ = (2η ₀ Î_d/n̂_e) Û. Here x ₀ ana x ₀ are an arbitrary spatial widths. The dynamical evolutions of the normalized envelope U and Û can be determined as follows:

i U_{ξ} + \frac{1}{2} U_{ss} - [β (1 + ρ) - α (ρ - {∣ U ∣}^{2}) - g_{0} - ig] \frac{U}{1 + {∣ U ∣}^{2}} + iϑU = 0

i {\hat{U}}_{ξ} + \frac{1}{2} {\hat{U}}_{ss} - \hat{β} (\hat{ρ} + 1) \frac{\hat{U}}{1 + {∣ \hat{U} ∣}^{2}} - \hat{α} \frac{(\hat{ρ} - {∣ \hat{U} ∣}^{2}) \hat{U}}{1 + {∣ \hat{U} ∣}^{2}} = 0

where ρ = I _∞/_{I_p} , g ₀ = kx ₀ ²Γ₀ , g = kx ₀ ²Γ/2 , β = χr ₃₃ E ₀ , α = χr ₃₃ E_p , χ = (k ₀ x ₀)² (n_e ⁴/2) , ϑ = kx ₀ ² ϑ ₀/2 and $\hat{ρ}$ = Î _∞/Î_d , $\hat{β}$ = $\hat{χ}$ Ê ₀ , $\hat{α}$ = $\hat{χ}$ Ê_p , $\hat{χ}$ = (k̂ ₀ x̂ ₀)² (n̂_e ⁴/2). From PR theory[30] , we can obtain the expressions for g and g ₀ as g = [χr_effE_s [E ₀ ² + E_dE_ds)]/(E_ds ² + E ₀ ²) and g ₀ =(χr_eff E ₀ E _s ²)/(E_ds ² + E ₀ ²) , where E_s = eN_Aλ ₀/(4πℰ ₀ 𡌠_r sinθ) is the saturation field, E_d = 4πk_BT sinθ/(λ ₀ e) is the diffusion field, and E_ds = E_d + E_s.

In this letter soliton formation of the signal beam in the crystal P is studied under the condition |U|² = I/I_p ≪ 1, thus allowing the approximation that (l + |U|²) ≈ 1 - |U|² As a result, Eq. (24) becomes

i U_{ξ} + \frac{1}{2} U_{ss} - (P_{d} + iG) U + (Q_{d} + ig) U {∣ U ∣}^{2} = 0

where G = g - ϑ, P_d = (1 + ρ)θ + ρα - g ₀ and Q_d = (1 + ρ)(ρ + β) - g ₀. Although the two dynamical evolution equations have a similar form to that for a single PR crystal[4, 18], they couple each other by the coupling coefficients σ, $\hat{σ}$ , Γ′ and $\hat{Γ}$ .

3. Two types of soliton pairs and coupling effects

Let us consider the effects of the interaction between the two solitons in a separate Holographic-Hamiltonian soliton pair on the intensity profiles of the two solitons. In order to provide some relevant examples, we use a SBN (strontium barium niobate) crystal as P and LiNbO₃ crystal as P̂ . The two crystals have the following parameters: n_e = 2.33, n ₀ = 2.36, ℰ_r = 880 , r ₃₃ = 220pm/V , E_p = 0V/m , N_A = 1.2×10¹⁷ cm ^-3 , θ = 2° , φ = 87° , ϑ ₀ = 0.27cm ^-1 , n̂_e = 2.2 , r̂ ₃₃ = 30pm/V , Ê_p = -10⁵ V/m , δ = $\hat{δ}$ , W = Ŵ , I_p = mÎ_d and r_eff = (n_e/n ₀) r ₃₃cosθcos(θ/2)cosφ. The arbitrary scales are x ₀ = x ₀ = 40μm and the free-space wavelengths are λ ₀ = $\hat{λ}$ ₀ = 0.5μm . For these set of values, we have χ = 3.72×10⁶ m/V, $\hat{χ}$ = 8.88×10"^-5 m/V , α = 0 and α = -8.8781.

When the LiNbO₃ crystal is illuminated, it acts as a source to bias the SBN crystal and the light-induced current will flow from it to the other. As a result, the two solitons, supported separately by the two crystals, will affect each other. Because σ , $\hat{σ}$ , Γ′ and $\hat{Γ}$ ′ depend on the parameters of the two crystals, the soliton profile in one crystal not only depends on the parameters of that crystal, but also depends on the parameters of the other crystal. In other words, the character of any soliton in a soliton pair depends on the parameters of the two crystals. When the input optical beam intensity of one crystal changes, not only will the soliton profile in that crystal change, but also the soliton profile in the other crystal will change. However, the effect of the input intensity of pump beam on the light-induced current is too weak to affect the other soliton. That is, changing the input intensity of the Hamiltonian soliton can affect the holographic one whereas changing the input intensity of pump beam can only affect the holographic soliton and cannot affect the Hamiltonian one. The collisionless and unilateral effect may be useful in some applications.

3.1 Dark-dark soliton pair

We begin our analysis by considering dark-dark soliton pair, i.e., both crystals support dark soliton. First, this dark soliton solution in the crystal P of Eq.(26) has been found as the following [18]:

U = D \tanh (Hs) \exp {id In [\cosh (Hs)]} \exp (- i Ω ξ)

where D = (G/ g)^1/2 , H = [2G/ (3d)]^1/2 , d = [3Q_d + (9Q _d ² + 8g ²)^1/2]/(2g) and Ω = 2G/ (3d) + P_d. Then, we derive the dark soliton solution in the crystal P̂ by expressing the beam envelop Û in the usual fashion: Û = $\hat{ρ}$ ^1/2 ŷ(ŝ)exp(iv̂ $\hat{ξ}$ ), where v̂ represents a nonlinear shift of the propagation constant and ŷ(ŝ) is a normalized real function bounded between 0 ≤ ŷ(ŝ) ≤ 1 and denotes the normalized field profile. Using the boundary condition of the dark soliton ŷ (0) = 0 , ŷ(∞) = 0 and ŷ(ŝ → ±∞) = 1, the profile of the dark soliton in crystal P̂ can be obtained from Eq.(24) as:

{[- 2 \hat{G}]}^{1 / 2} \hat{s} = \pm \int_{\hat{y} (\hat{s})}^{0} \frac{d \hat{y}'}{{[({\hat{y}}^{' 2} - 1) - \frac{1 + \hat{ρ}}{\hat{ρ}} In (\frac{1 + \hat{ρ} {\hat{y}}^{' 2}}{1 + \hat{ρ}})]}^{1 / 2}}

where Ĝ = $\hat{α}$ ± $\hat{β}$ .

Considering the configuration ⇈ shown in Fig. 1(a), we take $\hat{ρ}$ = I _∞/I_d = 1, ρ = I _∞/I_p = 0.03 and m = 10 in the first place, and then we have σ = $\hat{Γ}$ ′ = $\hat{ρ}$ /(mρ + m + $\hat{ρ}$ + 1) = 1/12.3 , Ĝ = -8.2886 , D = 0.5643 , H = 1.4777 , d = 0.0902 . With these values, the normalized intensity profiles of the two dark solitons in the crystals P and P are obtained by solving Eqs. (27) and (28), as shown in Fig. 2(a) curve (1) and Fig. 2(b) curve (1), respectively. When the input intensity of the crystal P̂ increases but the other parameters remain unchanged, such as p̂ increase from 1 to 100, we get σ = $\hat{Γ}$ ′ = 50/57, Ĝ = -1.0903, D = 0.6247, H = 5.9221, d = 0.0077 . We can see that not only the Hamiltonian screening-photovoltaic dark soliton in crystal P̂ changes as shown in Fig. 2(b) curve (2), but also the holographic screening dark soliton in crystal P̂ changes as shown in Fig. 2(a) curve (2). It should be noted that when the input intensity of pump beam of the crystal P increases but the other parameters remain unchanged, i.e., m increase from 10 to 100 but $\hat{ρ}$ keeps 1 , only the holographic screening dark soliton in crystal P changes as shown in Fig. 2(a) curve (3), but the Hamiltonian screening-photovoltaic dark soliton in crystal P̂ does not change as shown in Fig. 2(b) curve (3). The two curves are calculated at σ = $\hat{Γ}$ ′ = 1/105, Ĝ = -8.7936, D = 0.5639, H = 0.5535, d = 0.6418. The above results imply that, for a dark-dark soliton pair, the Hamiltonian screening-photovoltaic dark soliton can affect the profile of the holographic screening dark soliton by the light-induced current, whereas the holographic dark soliton cannot affect the profile of the Hamiltonian dark soliton.

Fig. 2. Dark-dark separate spatial soliton pairs in a series PR crystal circuit with the orientations of the two crystals’ c-axes being same. (a) Holographic screening dark soliton profiles in the crystal P; (b) Hamiltonian screening-photovoltaic dark soliton profiles in the crystal P̂.

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3.2 Bright-dark soliton pair

For a bright-dark soliton pair, let us assume that the bright soliton forms in the crystal P and the dark one forms in the crystal P̂. For the bright soliton, the optical beam intensity is expected to vanish at infinity (s → ±∞), i.e., I _∞ = 0 and then ρ = I _∞ /I_p = 0 . From Eq.(26), we can obtain the bright soliton solution as the following[18]:

U = F \sec h (Bs) \exp {ib In [\sec h (Bs)]} \exp (- ivξ)

where F = [3G/(2g)]^1/2 , B = (G/b)^1/2 , b = [-3Q_b, + (9Q_b ² + 8g ²)^1/2]/(2g) , v = (b ² - 1)G/ (2b) + P_b , P_b = β - g ₀ and Q_b = α + α - g ₀. The dark soliton profiles in the bright-dark soliton pair can be obtained by use of a similar way to above and determined by the following equation:

{[- 2 \hat{G}]}^{1 / 2} \hat{s} = \pm \int_{\hat{y} (\hat{s})}^{0} \frac{dy'}{{[(y^{' 2} - 1) - \frac{1 + \hat{ρ}}{\hat{ρ}} In (\frac{1 + \hat{ρ} y^{' 2}}{1 + \hat{ρ}})]}^{1 / 2}}

Considering the configuration ⇅ shown in Fig. 1(b), we take $\hat{ρ}$ = Î _∞/Î_d = 1 and m = 10 in the first place, and then we have σ = $\hat{Γ}$ ′ = $\hat{ρ}$ /( $\hat{ρ}$ + 1 + m) = 1/12 , Ĝ = -8.1383, F = 0.6914 , B = 1.7652, b = 0.095. With these values, the normalized intensity profiles of the holographic screening bright soliton in the crystal P and the Hamiltonian screening-photovoltaic dark soliton in the crystal P̂ are obtained by solving Eqs.(29) and (30), as shown in Fig. 3 (a) curve (1) and Fig. 3(b) curve (1), respectively. When the input intensity of the crystal P̂ increases but the other parameters remain unchanged, such as $\hat{ρ}$ increase from 1 to 100, not only does the dark soliton in the crystal P̂ change as shown in Fig. 3(a) curve (2), but also the bright soliton in the crystal P changes as shown in Fig. 3(b) curve (2). The two curves are calculated at σ = $\hat{Γ}$ ′ = 100/111, Ĝ = -0.8798, F = 0.7686 , B = 6.4372, b = 0.0099. It should be noted that when the input intensity of pump beam of the crystal P increases but the other parameters remain unchanged, such as m increase from 10 to 100 but $\hat{ρ}$ keeps 1, only the bright soliton in the crystal P changes as shown in Fig. 3(a) curve (3), and the dark soliton in the crystal P̂ does not change as shown in Fig. 3(b) curve (3). The two curves are calculated at σ = $\hat{Γ}$ ′ = 1/102 , Ĝ = -8.7911 , F = 0.6906 , B = 0.6774, b = 0.6427 . The above results imply that, for a bright-dark soliton pair, the Hamiltonian screening-photovoltaic dark soliton can affect the profile of the holographic screening bright soliton by the light-induced current, whereas the bright soliton cannot affect the profile of the dark soliton.

Fig. 3. Bright-dark separate soliton pairs in a series PR crystal circuit with the orientations of the two crystal’s c-axes being reverse. (a) Holographic screening bright soliton profiles in the crystal P; (b) Hamiltonian screening-photovoltaic dark soliton profiles in the crystal P̂.

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It is worthy of note that the current in the series PR crystal circuit will be zero if the incident light of the crystal P is a dark solitonlike one-dimensional laser beam and the incident light of the crystal P̂ is a bright solitonlike one-dimensional laser beam. Thus the dark-bright soliton pair that means a holographic screening dark soliton forms in the crystal P and a Hamiltonian screening-photovoltaic bright soliton forms in the crystal P̂ cannot exit. The current will also be zero if the two incident lights are bright solitonlike one-dimensional laser beams, so the bright-bright soliton pairs cannot exit either. One can check these from the above formulae.

4. Conclusions

In this paper we derived the coupled equations that completely describe nonlinear propagation of optical beams in an unbiased series PR crystal circuit with two laser beams, signal beam and pump beam, in one crystal and one in the other in detail. We showed theoretically that one dissipative holographic soliton and one Hamiltonian soliton can form as a separate soliton pair in a serial crystal circuit in which one unbiased non-photovoltaic PR crystal and one unbiased PV-PR crystal are connected electronically by electrode leads in a chain. There are two types of soliton pair i.e., dark-dark and bright-dark. Because the two PR crystals in this unbiased series PR crystal circuit are connected electronically, the PV-PR crystal acts as a source to bias the other crystal when it is illuminated. And as a result, the character of any soliton in the soliton pair depends on the parameters of the two crystals. When the input intensity of one crystal changes, not only will the profile of the soliton formed in that crystal change but also the profile of the soliton formed in the other crystal will change. However, only the Hamiltonian soliton can affect the holographic one by the light-induced current because the effect of the input intensity of pump beam on the current is too weak to affect the Hamiltonian soliton. These properties of the soliton pair may be useful in some domains, such as a unidirectional optical coupler, and optical switch and so on.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under grant 10174025 and 10574501

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Separate spatial Holographic-Hamiltonian soliton pairs and solitons interaction in an unbiased series photorefractive crystal circuit

Abstract

1. Introduction

2. Theoretical model

3. Two types of soliton pairs and coupling effects

3.1 Dark-dark soliton pair

3.2 Bright-dark soliton pair

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (30)

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