1. Introduction

The dynamics of the photorefractive grating and the slow-varying amplitudes of coupled beams involved in producing the grating structure is of importance to many applications such as wave mixing processes, phase conjugation, resonators dynamics, optical image processing, and optical memories [1]. This dynamics has been studied analytically considering a fast phase modulation of one of the beam [2], the undepleted pump-beam approximations [3,4], the linear approximations [5], the steady-state assumption [6], and using numerical techniques [7,8]. In this paper we report a closed-form solution to the grating equations of the two-wave coupling process, resulting in a periodic solution with respect to time that describes the moving grating. The space variation of the solution is almost similar to the steady-state solution of Kukhtarev et al. [6]. The moving grating has been studied extensively in the past normally through interaction of light beam with the sound wave in a medium. For the photorefrative medium, it has been shown that the moving grating will result in a phase shift in the coupling constant [1]. To our knowledge, however, it has not been derived directly from the grating dynamics equations discussed in this paper. Our contribution will link the two results.

There are two dominating dynamics in the photorefrative wave mixing:

(1) “Band-transport” dynamics (x-dynamics)

(2) “Beam-coupling” dynamics (z-dynamics).

Band-transport dynamics is due to migration/recapture of the impurities mainly along the grating vector, x-direction, see Fig. 1. The space-charge dynamics for a small signal (undepleted pump) is given in [1] as

where E_sc is the amplitude of the steady-state space charge field, K is the grating vector along the x direction, ϕ is the phase shift between the steady state space-charge field and the interference pattern, ω_g is the imaginary part of the complex time constant, and the τ_g is the time constant for the space-charge field to reach its steady state. This time constant is responsible for build up of the grating or erasure, the “x-dynamics.”

Due to the beam coupling, there will be an additional grating dynamics along the z direction, perpendicular to the grating vector, see Fig. 1. Since the periodic grating in the x direction must exist before the beam coupling can take place, any dynamics in the z direction is bounded by the dynamics in the x direction.

 

Fig. 1. Two-wave coupling.

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2. Grating Z-Dynamics

Under the assumption that the donor density is larger than the ionized donor density which is larger than the electron density in the conduction band, the two-wave mixing process in a photorefractive medium is described by a set of scalar slow-varying envelope paraxial equations as [5, 9, 10, 11]

where z is the distance perpendicular to the grating vector, t is the time, G(z, t) is the complex grating strength proportional to the index modulation depth, E ₁(z,t), and E ₂(z,t) are the slow-varying amplitudes of the signal and the pump beams, respectively, I ₀=|E ₁(z,t)|²+|E ₂(z,t)|² is the total intensity, τ is the photorefractive response time, Γ is the complex coupling constant which generally depends on the geometry, the wavelength of the beams, and the material constants, and it is independent of z and t [1, 4, 5, 7, 8, 11]. The parameter α is the absorption coefficient in meter^-1 for both beams, and the symbol “*” represents the complex conjugate. The above grating dynamics traces back to the work of Kukhtarev et al. [9]. Equation (2) describes the grating temporal dynamics originated from the charge-transport model of Kukhtarev et al. [9]. Equations (3) and (4) give the spatial dynamics for the signal and pump beams through the coupled-wave theory of Kogelnik [12].

3. Solution by the method of separation of variables

Using the method of separation of variables, we write

where $\varsigma =\frac{t}{\tau }$, G̃(z,ς)=G(z,ςτ), ${\tilde{E}}_1(z,\varsigma )=E_1(z,\varsigma \tau )e^{\frac{\alpha z}{2}}$, ${\tilde{E}}_2(z,\varsigma )=E_2(z,\varsigma \tau )e^{\frac{\alpha z}{2}}$, and Ĩ₀(z,ς)=|Ẽ₁(z,ς)|²+|Ẽ₂(z,ς)|². It can be shown that Ĩ₀(z,ς) is invariant with respect to z.

Considering Eqs. (2)–(7), and separating the space-dependent and the time-dependent parts. we have

where parameters p ₀, p ₁, and p ₂ are constants, and the dot and prime mean the partial derivative with respect to ς and z, respectively.

It can be shown that the term $\frac{{\overline{e}}_1{\overline{e}}_2^{*}}{{\tilde{I}}_0\overline{g}}$ in Eq. (8) is constant; and p ₁ p ₂ and $\frac{p_1}{p_2^{*}}$ are positive real numbers. This implies that the arguments of p ₁ and p ₂ have opposite sign. An expression for g in terms of only space-dependent quantities can be written as

where $I_{12}=\frac{p_2}{p_1^{*}}{\mid e_1\mid }^2+{\mid e_2\mid }^2$. Equation (11) will be used in solving the space-dependent part.

Now we consider the space-dependent solution by considering Eqs. (9), (10), and (11) as

By lumping together the same index terms of Eqs. (12), (13) and applying Eqs. (6), (7), we arrive at the following familiar form

where $\tilde{\Gamma }=\frac{\Gamma }{1+i{p}_0}=\tilde{\gamma }+i2\tilde{\beta }$. Equations (14) and (15) are almost the steady-state equations introduced by Kukhtarev et al. [6], and the solutions are

where Ẽ₁(0,ς)=e ₁(0)ē₁(ς) and Ẽ₂(0,ς)=e ₂(0)ē₂(ς) which contain the initial values e ₁(0) and e ₂(0), $r=\frac{{\mid {\tilde{E}}_2(0,\varsigma )\mid }^2}{{\mid {\tilde{E}}_1(0,\varsigma )\mid }^2}$, and the phases φ ₁(z) and φ ₂(z) are given as

It can be shown that r is independent of ς.

Now we turn our attention toward solving the time-dependent part. Using the right-hand side of Eq. (8), the solution for ḡ(ς) can be written as

where ḡ(0) is an initial value. By multiplying Eq. (9) with Eq. (10), we have

${\mid \overline{g}\mid }^2=p_1{p}_2,$

where p ₁ and p ₂ are constant with respect to space and time. Considering the above, the parameter p ₀ must be real. Equation (20) hence rules out the possibility of having an asymptotic steady-state solution in an exponential form for the grating dynamics. It implies that the phase may vary with time not the magnitude.

The solutions for ē₁ or ē₂ (not both) is an arbitrary function of time. Let us apply an arbitrary function Λ(τς) with the unity initial value to ē₂, and then find Ẽ₂ as

where Ẽ₂(0,0)=e ₂(0)ē₂(0) and Ẽ₁=$p_1^{-1}$ ḡ(0)ē₂(0). The function Λ(τς) is equivalent to the modulating function (modulating in time amplitude or phase) that is applied to the beam before its split in forming the pump and probe beams. Thus this modulation is equally applied to both pump and probe beams.

Now considering time and space solutions in one form, we arrive at the following solutions

where G(0,0), E ₁(0,0), and E ₂ (0,0) are the initial values, $\Omega =\frac{p_0}{\tau }$ is the frequency-shift (detuning) parameter.

From the well-documented experimental results in the literature, the grating builds up from zero to a saturation level or decays down to zero from a level. These phenomena are not, however, observed from solutions (23), (24), and (25). The reason is that the set of Eqs. (2), (3), (4) are valid when the grating G assumes its dominance as the fundamental harmonics of the index variation with respect to z. In other words, these equations are valid only at steady state or very slow-variation (adiabatic) state.

4. Oscillation conditions

We apply the above solutions to a unidirectional photorefractive oscillator. Unidirectional photorefractive resonator is a simple optical ring cavity with a photorefractive gain medium [1, 13, 14], see Fig. 2. The two-wave coupling process is the mechanism for achieving optical amplification in a photorefractive medium.

 

Fig. 2. Unidirectional photorefractive resonator. The two-wave coupling configuration inside the photorefractive material is similar to that of Fig. 1.

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The cavity is under oscillation when the following boundary condition is satisfied.

where (1-ρ) and Φ are the magnitude of the field loss (0<ρ≤1) and the phase shift after one round trip, respectively, and d is the thickness of the photorefractive medium. Using Eq. (24) and condition (26), we have

where m=0, ±1,±2, … are integers. In the case of resonance, the initial value for the resonator beam, E ₁, can be arbitrarily small, meaning that r→∞. Therefore, conditions (27) are simplified as

The detuning frequency is obtained from the above condition on the left as (The condition on the right can be satisfied by tuning the round-trip cavity length.)

where γ=Re[Γ] and 2β=Im[Γ]. These parameters are related to γ̃ and β̃ as

$\tilde{\gamma }=\frac{\gamma +2\beta \Omega \tau }{1+{(\Omega \tau )}^2}\mathrm{and}\tilde{\beta }=\frac{\beta -\frac{1}{2}\gamma \Omega \tau }{1+{(\Omega \tau )}^2}.$

From Eq. (29), it is possible to reduce the detuning frequency to zero for a specific loss or gain value. This remains to be experimentally verified. It is also interesting to note that the detuning frequency is independent of the resonant frequency. Note that this conclusion is valid for r→∞. It is also noted that the

Now if r≠∞, we utilize directly conditions (27) which result in the following relations

where I ₁₀=|E₁(0,t)|², I ₂₀=|E₂(0,t)|², ω=ω_p +Ω≅ω_p , ω_p is the pump frequency, D is the round-trip cavity length, and c is the speed of light in the cavity. The relations (30) are simplified in two extreme cases, namely, β=0 (pure diffusion) and γ=0 as shown in the following relations.

The above relations are plotted in Figures 3–6 for α=d=τ=1,β=-10,γ=10, ρ=0.95, I ₂₀=1, $\frac{{\omega }_{p}D}{c}-2\pi m=1$. For the pure diffusion case (Fig. 3), the resonator intensity (I ₁₀) has the maximum value at the zero detuning frequency as expected from the past documented experimental results [13, 14, 15]. It has also been observed that the detuning frequency wonders around zero with positive and negative values in the range of few Hz. In the case of γ=0 (Fig. 4), it is however interesting to note that the resonator intensity is at its minimum when the detuning frequency is zero. It reaches its maximum at higher positive frequencies while the resonator is not exited for negative detuning frequencies.

 

Fig. 3. Resonator intensity (I ₁₀) vs. detuning frequency (Ω) when β=0, the pure diffusion case.

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Fig. 4. Resonator intensity (I ₁₀) vs. detuning frequency (Ω) when γ=0.

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Fig. 5. Detuning frequency (Ω) vs. resonator loss (1-ρ) when β=0, the pure diffusion case.

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Fig. 6. Detuning frequency (Ω) vs. resonator loss (1-ρ) when γ=0.

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The detuning frequency has the resonator loss dependency, as seen in Figs. 5, 6, where it could result to zero frequency when there is a heavy loss in the resonator cavity in the case of pure diffusion (see Fig. 5). In the other case when γ=0 (see Fig. 6), the detuning frequency can reach zero when the cavity loss is small. Figures 7, 8 indicate an interesting result where the detuning frequency dependency is inversed when the parameters are switched (from β=0, γ≠0 to β≠0, γ=0). We investigate this further for possibility of finding a situation where the detuning frequency could be less sensitive to cavity length variations. Using Eqs. (30), we have

 

Fig. 7. Detuning frequency (Ω) vs. cavity length D when β=0, the pure diffusion case.

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Fig. 8. Detuning frequency (Ω) vs. cavity length D when γ=0.

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Taking partial derivatives of Eq. (33) with respect to D and Ω with variations δD and δΩ, we have

$\frac{{\omega }_p+\Omega }{c}\delta D+\left[\frac{D}{c}-\frac{\tau (\alpha d-2\ln \rho )({\gamma }^2+4{\beta }^2)}{2{(\gamma +2\beta \Omega \tau )}^2}\right]\delta \Omega =0.$

By choosing

$D=\frac{c\tau (\alpha d-2\ln \rho )({\gamma }^2+4{\beta }^2)}{2{(\gamma +2\beta \Omega \tau )}^2},$

the detuning frequency becomes less sensitive to the variations of D, hence more stable resonator. This would have an interesting practical application where the frequency stability of resonator with respect to length variation is important. As evident from Eq. (33), it is also noted that the cavity length variation and the pump frequency variation are not easily distinguishable. Therefore the cavity with high sensitivity to the length variation would require an ultra frequency-stable pump beam to be useful for any sensing applications.

5. Conclusion

We have presented a closed-form solution for the two-wave coupling process, resulting in a periodic solution with respect to time that describes a moving grating. We have also shown that the temporal dynamics does not have any asymptotically stable equilibrium points of an exponential form. We have developed oscillation conditions resulting in a new frequency detuning expression where it could be used in designing a photorefractive resonator with minimal sensitivity to the cavity length variation. The significance of these new results is outlined below:

(1) The set of closed-form solutions (23–25) has an interesting part (the arbitrary function on Λ(t)) not being considered in the previous works. The arbitrary function Λ(t) is part of any solutions, steady–state or not. Let G, E ₁, E ₂ be a set of solutions to the system of Eqs. (2), (3), (4), then it is easy to verify that G, Λ(t)E ₁, Λ(t)E ₂ is also a set of valid solutions with any Λ(t). It is interesting to note that the grating G does not see Λ(t) at all. This shows further that the Eqs. (2), (3), (4) may be valid to describe the grating dynamics only in the steady state.

(2) Direct application of these solutions resulted in new oscillation conditions (29–32) in a unidirectional ring.

(3) In the non–diffusion case of γ=0, the resonator intensity is at its minimum when the detuning frequency is zero. It reaches its maximum at higher positive frequencies while the resonator is not excited for negative detuning frequencies.

(4) We found a situation where the detuning frequency could be less sensitive to cavity length variations. This would have an interesting practical application where the frequency stability of resonator with respect to length variation is important.