Reconstructing signals via stochastic resonance generated by photorefractive two-wave mixing bistability

Guangzhan Cao; Hongjun Liu; Xuefeng Li; Nan Huang; Qibing Sun

doi:10.1364/OE.22.004214

1. Introduction

Stochastic resonance was described as a phenomenon that a weak periodic signal is optimized by the presence of an appropriate level of noise [1]. With the development of stochastic resonance, it has far been found to account for many natural phenomena and shown a large number of applications in different areas of technology including climatic patterns [2, 3], electrical systems [4, 5] and biology [6, 7]. Nowadays, signal processing via stochastic resonance becomes a research focus. Generally, low level optical signals are overwhelmed by the noise, which lead to an extremely low signal-to-noise ratio (SNR). The stochastic resonance systems have been used for amplifying [8, 9] and self-filtering signal [10] to improve the detection sensitivity and output SNR. As we know, the well known conditions for a system to exhibit stochastic resonance are the energy threshold, applied subthreshold modulation and source of noise. One of the classic examples of a system that undergoes stochastic resonance is that of a particle in a double-well potential. With a subthreshold modulation but no noise, the particle will undergo a periodic motion in a small local area of one of the two wells. However, when the external noise is considered, there will be an interaction between the modulation and the nonlinear system. Then the particle will undergo a periodic transition between the two wells, and resulting in an enhancement in the SNR.

For the domain of optics, stochastic resonance has been reported in the context of periodically driven bistable dynamic systems [11]. Several implementations have been proposed to realize a bistable optical system that is switched between its two stable states by a modulation signals aided by noise. Early theoretical and experimental investigations have indicated that any bistable system can exhibit the stochastic resonance effect [12]. Weiss and Fischer proposed that photorefractive two-wave mixing in a ring cavity exhibits the bistability [13]. The configurations permit bistable operation when the photorefractive crystal is oriented such that optical power is transferred from the signal beam to the pump beam during two-wave mixing. Optical bistability, indubitably, has potential applications in optical logic, optical switching, and optical memory elements, and can play a major role in the development of optical communication systems. Extensive work has been done in the area of photorefractive two-wave mixing bistability (PTWMB) [12–14], however, there are still few reports about the application of reconstructing signals via stochastic resonance in the PTWMB system.

In this paper, we theoretically investigate the nonlinear enhancement of noisy signals via all-optical stochastic resonance in a ring cavity firstly. The stochastic resonance effect is generated by the bistable system which is based on a photorefractive two-wave mixer. The optical power is transferred from the pump beam to the signal beam that is different from the works demonstrated in [12, 13]. The influences of the bistable parameters on the stochastic resonance signals are analyzed in detail, and the cross-correlation gain of noisy signals can be improved by using the bistability system which can be optimized by tuning the bistable parameters.

2. Theory model of stochastic resonance in optical bistability

The nonlinear dynamics equation of the stochastic resonance is usually described by the Langevin equation, which is different at various bistability systems [15–17]. Here, we investigate the optical bistability occurs in the ring cavity with a photorefractive two-wave mixer. The ring cavity provides a signal feedback as evident from Fig. 1. The photorefractive crystal is a cerium-doped Sr_0.6Ba_0.4Nb₂O₆ (SBN:60) [18]. To describe the process of nonlinear intensity exchange between the pump and the signal light, we consider near-degenerate two-wave mixing in a unidirectional photorefractive ring cavity. The two waves interact in the photorefractive crystal via the photoinduced reversible refractive-index grating. The coupled wave equations for this case can be expressed as [19]

\begin{array}{l} \frac{d A_{1}}{d z} = - γ \frac{A_{1} A_{2}^{*} A_{2}}{I_{0}} - \frac{α}{2} A_{1}, \\ \frac{d A_{2}}{d z} = γ^{*} \frac{A_{1} A_{1}^{*} A_{2}}{I_{0}} - \frac{α}{2} A_{2}, \end{array}

where α is the absorption coefficient, A_j (j = 1,2) are the complex amplitudes of the pump and signal, I_j = |A_j|², and I₀(z) = I₁(z) + I₂(z) is the total light intensity at z, respectively. The two beams symmetrically incident upon the photorefractive crystal with the incident angles of θ₁ = θ₂ = θ. The complex coupling coefficient γ is given by [20]

γ = \frac{ω r_{eff} n_{0}^{3}}{4 c} \frac{E_{q} (E_{0} + i E_{D})}{[E_{0} - Ω τ_{0} (E_{D} + E_{μ})] + i [E_{D} + E_{q} + Ω τ_{0} E_{0}]},

where r_eff is the relevant electrooptic coefficient, n₀ is the ordinary refractive index of the crystal, τ₀ is the characteristic time, Ω = ω₂-ω₁, E₀ is the externally applied electric field parallel to the grating wave vector in the crystal, E_μ, E_D, and E_q are internal electric fields characteristic of drift, diffusion, and maximum space charge, respectively. With A_j = I_j^1/2 exp(iψ_j), ψ_j = k_jz + φ_j, Γ = 2Re(γ) and Γ′ = Im(γ), the Eq. (1) can be rewritten as

\begin{array}{l} \frac{d I_{1}}{d z} = - Γ \frac{I_{1} I_{2}}{I_{0}} - α I_{1}, \frac{d I_{2}}{d z} = Γ \frac{I_{1} I_{2}}{I_{0}} - α I_{2} \\ \frac{d φ_{1}}{d z} = Γ^{'} \frac{I_{2}}{I_{1} + I_{2}}, \frac{d φ_{2}}{d z} = Γ^{'} \frac{I_{1}}{I_{1} + I_{2}} . \end{array}

The solutions of Eq. (3) are

\begin{array}{l} I_{1} (z) = I_{1} (0) \frac{1 + m}{1 + m \exp (Γ z)} \exp (- α z) \\ I_{2} (z) = I_{2} (0) \frac{1 + m}{m + \exp (- Γ z)} \exp (- α z) \\ Δ φ_{1} (z) = - \frac{Γ^{'}}{Γ} \ln (\frac{1 + m}{m + \exp (Γ z)}), \\ Δ φ_{2} (z) = \frac{Γ^{'}}{Γ} \ln (\frac{1 + m}{m + \exp (- Γ z)}) \end{array}

where m is the input intensity ratio m = I₂(0)/I₁(0).

Fig. 1 Principle diagram of the unidirectional photorefractive ring cavity with an injected signal. C, crystal; BS, beam splitters; M_1,2, mirrors.

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The photorefractive two-wave mixing with a signal feedback from the ring cavity is used to produce the optical bistability. The boundary condition for the ring configuration is

A_{2} (0) = T^{1 / 2} A_{i n} + R^{1 / 2} A_{2} (l) \exp (i δ_{0}),

and the output signal amplitude is given by

A_{o u t} = T^{1 / 2} A_{2} (l),

where T and R are the transmissivity and reflectivity of the beam splitter (T = 1-R), respectively. The cavity detuning δ₀ is defined by δ₀ = (ω₂-ω_c)/(c/L), L is the effective cavity length, ω_c = q(2πc/L) is the cavity mode nearest to the frequency of the incident field ω₂ and q is an integer.

The complex amplitude in Eqs. (5) and (6) can be expressed in intensity

I_{i n} = A_{i n} A_{i n}^{*} = T^{- 1} [I_{2} (0) + R I_{2} (l) - 2 \sqrt{R I_{2} (0) I_{2} (l)} \cos (δ_{0} + Δ φ_{2})],

and

I_{o u t} = T I_{2} (l),

where Δφ₂ = φ₂(l)-φ₂(0) is the additional phase due to the nonlinear process and can be obtained from Eq. (4). For convenience, the normalized input intensity and output intensity are simplified as Y = I_in/[TI₁(0)] and X = I_out/[TI₁(0)], respectively. According to Eq. (7), the steady-state equation of optical bistability in the ring cavity can be rewritten as

Y = [m (X) + R X - 2 \sqrt{m (X) R X} \cos (δ_{0} + Δ φ_{2})] .

The ratio m(X) is acquired from the second equation of Eq. (4)

m (X) = \frac{1}{2} [a X - 1 + \sqrt{{(a X - 1)}^{2} + 4 a X / b}],

where a and b are defined as a = exp(αl) and b = exp(Γl), respectively.

Figure 2 details the physical mechanism of the stochastic resonance. The nonlinear dynamics of the stochastic resonance is excited by an additive white Gaussian noise [15] and a subthreshold signal. The subthreshold signals will undergo a nonlinear enhancement when an appropriate noise is injected to the stochastic resonance generator. In order to better understand the nonlinear dynamic process of the stochastic resonance, a model for the stochastic resonance in photorefractive unidirectional ring cavity is deduced from the classical stochastic resonance [1, 15–17, 21]. The nonlinear Langevin equation for the normalized transmitted intensity X and input intensity Y is given by

\frac{d X}{d t} = B (Y, X) = - \frac{d V (X)}{d X},

B (Y, X) = Y - [m (X) + R X - 2 \sqrt{m (X) R X} \cos (δ_{0} + Δ φ_{2})],

where the potential function V(X) reads

V (X) = - \int_{0}^{X} B (X) d X .

Having chosen a value for the input pulse sequence intensity Y(t) = Y₀(t) within the region of bistability, we then modulate the value with the additive noise. Both the input and output signals are normalized by the pump intensity, so D is considered to be a normalized intensity. The normalized input intensity Y in Eq. (11) should be substituted by

Y (t) = Y_{0} (t) + D N (t),

where D denotes the noise intensity, N(t) is the additive white Gaussian noise and the noise correlation

〈 N (t) N (t^{'}) 〉 = 2 δ (t - t^{'})

, respectively. According to Eqs. (11) and (12), the time dependence of the transmitted light intensity X can be described by the nonlinear Langevin equation.

Fig. 2 Schematic diagram of the stochastic resonance system.

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3. Results and discussion

3.1. Properties of the optical bistability

In simulations, a laser with output power of a few hundred milliwatts at 514.5 nm is used as the coherent light source. The intensity of the pump beam keeps constant and is assumed much lager than the periodic signal pulses. That is, the normalized value of Y₀(t) is less than 1. The transmissivity of the beam splitter is T = 0.5. M₁ and M₂ are total reflection mirrors as presented in Fig. 1. The electric field was applied antiparallel to the optic axis. The material parameters for the calculation of the complex coupling coefficient γ are according to reference [18]. Moreover, the ring cavity without an injected signal can oscillate over a large range of cavity detuning [13, 22], but here we assume the cavity detuning to be a constant (δ₀ = 0.01) and the boundary conditions are not satisfied to allow the oscillation to occur. To determine the optimum operating parameters for the optical bistability in the unidirectional ring cavity, the output property of the optical bistability with different interaction lengths and electric field intensities are numerically simulated and analyzed as illustrated in Fig. 3. Figure 3(a) shows the relationship between X and Y for various interaction lengths with E₀ = 0 kV/cm. It is found that there is only one stationary X for a fixed Y when the interaction length is 1 mm. The bistability appears with the increase of the interaction length, and three stationary X appear in the bistable region, which are the stable lower branch, the stable upper branch, and the unstable intermediate branch. As seen from Fig. 3(b), the bistability region moves toward to the larger Y direction when the electric field is applied. The bistability region is enlarged by tuning the interaction length and applied electric field. Starting with a weak signal beam, the photorefractive grating transfers a moderate percentage of the pump energy to the signal beam, and the output signal intensity increases slowly with the increase of the input signal intensity at the beginning. As the signal intensity increases further, the photorefractive grating reaches its saturation, the output signal jump to the upper branch. At present, if the signal intensity begins to reduce, the balance in the cavity would not be broken immediately. However, the output signal will jump to the low branch again when the incident light is faint.

Fig. 3 Photorefractive optical bistability. Normalized output intensity X versus normalized input intensity Y for (a): E₀ = 0 kV/cm, l_eff = 1 mm (red), 3 mm (green) and 5 mm (blue) ; (b) l_eff = 2 mm, E₀ = 0 kV/cm (red), 2 kV/cm (green) and 4 kV/cm (blue), respectively.

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In terms of the potential function V(X), we get two minima of V(X) from Eq. (13) when the system exhibits optical bistability as illustrated in Fig. 4. The dynamic stochastic resonance system has two stable stationary states corresponding to the two minima of the potential, which generally is a double-well potential.

Fig. 4 Potential V is plotted versus the input and output light intensity Y and X.

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3.2. Numerical analysis of the stochastic resonance

The term stochastic resonance has been adopted to describe an interesting statistical property of periodically modulated and noise-driven bistable dynamical systems. Under proper conditions, an increase in the input noise level results in a raise in the output signal-to-noise ratio. A continuous wave (CW) pump beam with a constant intensity and a train of periodic signal pulses with an additive white Gaussian noise are incident into the bistable system. The signal is a Gaussian pulse with a normalized peak intensity, which can be expressed as

Y_{0} (t) = y_{0} \exp [- 2 {(t / t_{g})}^{2}],

where y₀ = 0.05 is located in the bistable region illustrated by the green curve in Fig. 3(b), t_g = t_p /1.777 and t_p is the pulse width, respectively. The observation time is about 20 s, and the sampling period is 0.001 s in the calculations.

Figure 5 depicts the ability to reconstruct signals of the optical stochastic resonance system with different levels of the applied noise. The signals are effectively extracted from the noise background via the stochastic resonance system and the trailing edge appears simultaneously. Nevertheless, as the noise intensity is raised further, the output signals gradually become indistinguishable and distorted compared with to the pure input signals. This is because the output signals become increasingly dominated by the noise.

Fig. 5 Transmission property of the optical stochastic resonance system. (a)-(d) Input signals Y with the white Gaussian noise intensity D = 0, D = 0.4, D = 1.2 and D = 2.0, respectively; (a′)-(d′) are the output signals X corresponding to the input signals. Insets: detail showing the signals at t = 5 s. In simulations, E₀ = 2 kV/cm, l_eff = 2 mm.

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To describe the energy transfer of the nonlinear dynamical process, we use the cross-correlation as a quantitative measure of the similarity between input signals Y(t) and output signals X(t). The normalized cross-correlation coefficients are defined as [23]

C_{Y_{0} X} = 〈 Y_{0} | X 〉 = \frac{〈 (Y_{0} - 〈 Y_{0} 〉) (X - 〈 X 〉) 〉}{{[〈 {(Y_{0} - 〈 Y_{0} 〉)}^{2} 〉 〈 {(X - 〈 X 〉)}^{2} 〉]}^{1 / 2}},

and

C_{Y_{0} Y} = 〈 Y_{0} | Y 〉 = \frac{〈 (Y_{0} - 〈 Y_{0} 〉) (Y - 〈 Y 〉) 〉}{{[〈 {(Y_{0} - 〈 Y_{0} 〉)}^{2} 〉 〈 {(Y - 〈 Y 〉)}^{2} 〉]}^{1 / 2}} .

The cross-correlation gain of the bistable system is then given by

C_{g} = C_{Y_{0} X} / C_{Y_{0} Y} = 〈 Y_{0} | X 〉 / 〈 Y_{0} | Y 〉 .

C_Y ₀_X is close to 0 when the output signal is severely distorted, while it is close to 1 under the condition that the output signal X are very similar with the pure input signal Y₀. C_Y ₀_Y has the same variation trend as C_Y ₀_X.

According to Eqs. (16)–(18), the cross-correlation coefficients and cross-correlation gain as a function of the noise intensity D are numerically calculated and analyzed. The normalized intensity of the input signal is set at Y₀ = 0.05, and the noise intensity D is gradually increased from zero in our simulations. Figure 6(a) clearly shows that C_Y ₀_Y and C_Y ₀_X decrease as the noise intensity increases, and C_Y ₀_Y decays much faster than C_Y ₀_X when D is small. As seen from Fig. 6(b), the cross-correlation gain increases to a maximum value of 4 firstly, and then begins to drop dramatically. The reason is that the stochastic resonance system becomes increasingly dominated by the noise, which leads to the distortion of extracted signals. Too little noise disables the reversal of the potential well, but too much noise leads to excessive distortion. Therefore, there exists an optimal noise level for acquiring the maximum cross-correlation gain that can be realized by adjusting the input noise intensity. This is the signature of the stochastic resonance effect, which is caused by the interaction between the random noise and weak signal in the potential well.

Fig. 6 Cross-correlation coefficients (a) and the cross-correlation gain (b) as a function of the noise intensity D. The black curves are under the particular conditions of fixed noise distribution. In simulations, E₀ = 2 kV/cm, l_eff = 2 mm.

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In addition, the impacts of the interaction length and applied electric field on the stochastic resonance are also numerically calculated and shown in Fig. 7. It is found that the cross-correlation varies with the interaction length and there exist an optimal interaction length of 2 mm for obtaining the maximum cross-correlation gain. Moreover, the cross-correlation can be maximized by tuning of the applied electric field with an optimal interaction length of 2 mm. According to Eq. (2), the complex coupling coefficients γ at different applied electric fields E₀ are acquired as shown in Fig. 8 (a). Figure 8(b) presents the curve of -Γ′/ Γ versus E₀ calculated from Eq. (4)(d). The dependence of stochastic resonance on the parameter of E₀ can be clearly seen from Figs. 7(b) and 8(b). It is found that the maximum of the cross-correlation gain and the value of -Γ′/ Γ can be achieved at nearly the same E₀. As a result, the stochastic resonance effect performed by tuning the parameter of E₀ is mainly depending on -Γ′/Γ.

Fig. 7 The cross-correlation coefficients versus (a) the interaction length l_eff with E₀ = 2 kV/cm and (b) the applied electric field E₀ with l_eff = 2 mm,(Y₀ = 0.05, D = 1.2), respectively. Insets: cross-correlation gain. and fixed noise intensity.

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Fig. 8 The complex coupling coefficient (a) and the value of﹣Γ′/ Γ (b) versus the applied electric field E₀, respectively.

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Here, the nonlinear interaction strength of the bistable system is optimized to achieve the stochastic resonance, i.e., the stochastic resonance can be controlled by bistable parameters with fixed noise intensity. The generalized parameter-tuning stochastic resonance is introduced based on the tuning of the bistable parameters, which has the incomparable flexibility and is highly effective in the applications of signal processing [24, 25]. The pulse width and the pulse-repetition rate of signal also have impacts on stochastic resonance effect. In our simulations, the pulse-repetition rate was limited to very low level due to the lower response time of the crystal. Hence, faster crystals and higher-intensity beams can be used to significantly increase the maximum rate. That is, the bistable parameters are related to the potential turnover, and thus maximum energy transfer and optimal signal reconstruction can be realized by carefully design theses parameters.

4. Conclusion

In this paper, we report, the first time to our knowledge, the theoretically study of reconstructing signals by all-optical stochastic resonance in a PTWMB system. The optical bistability are obtained when optical power is transferred from the pump beam to the signal beam in a unidirectional photorefractive ring cavity. We also briefly investigate the dependence of the stochastic resonance on the effective interaction length l_eff and applied external electric field E₀ in the PTWMB system. The parameters l_eff and E₀ of the PTWMB are related to the cross-correlation gain of the stochastic resonance. The parameter-tuning stochastic resonance has great prospect in the applications of signal processing, and it provides a general method for reconstructing signals in nonlinear communications systems.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61078029, 61178023 and 61275134.

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Reconstructing signals via stochastic resonance generated by photorefractive two-wave mixing bistability

Abstract

1. Introduction

2. Theory model of stochastic resonance in optical bistability

3. Results and discussion

3.1. Properties of the optical bistability

3.2. Numerical analysis of the stochastic resonance

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (18)

Optics Express