1. Introduction

In optical planar waveguides, light beams are confined in one dimension to regions only a few micrometers in width. The high optical intensities, which are thus readily obtained, make it relatively easy to observe effects of photorefractivity in wave-guiding layers. Therefore, the high light intensities together with the efficient beam modulation, using surface electrodes, make it particularly interesting to design integrated photorefractivity-based devices [1].

In most recent literature, it is possible to find a lot of examples which lead to possibility of designing integrated optical devices such as linear filters providing for high-resolution spectral filtering [2], beam splitters and couplers [3] or dynamic interconnections for neural networks [4]. Therefore, a numerical model of planar holographic gratings formation in waveguides is a very useful software tool for design purposes.

2. Theoretical model

The mathematical analysis of photorefractive phenomena in waveguides is rather difficult because of the two-dimensional nature of the light and related induced fields. The problem was solved in literature only for some particular cases, for example step-index waveguides in lithium niobate [5]. A detailed review of the main experimental results for waveguide hologram recording was given by Wood et al. [1].

This paper is devoted to the analysis of holographic grating formation on arbitrary graded-index waveguides having a generic electro-optic crystal (full r_ij tensor). Furthermore, the model includes the possibility of having an overlay. At the best of our knowledge, our model takes into account the most generic case dealt with in literature.

The perturbations of the optical dielectric permittivity (tensor ε_ij ) caused by the photorefractive effect are physically related to the distribution of the photo-induced potential, φ. For a generic-mode interaction (i.e. either intermode or intramode), we have derived from the continuity equation (see [5]) the so-called equation of induced oscillations, under the hypotheses of considering only the photogalvanic current (neglecting the diffusion current) and admitting that the its spatially oscillating components depend only on the cut-axis (hereinafter named cut). By assuming the absence of any applied external electric field and both the cut-axis and the direction of propagation, ρ, coincident with the crystallographic axes, the equation of induced oscillations can be written in the very general formulation as follows:

where K_g is the grating vector amplitude, i.e. K_g = |K̄_g |, ${\epsilon }_{\mathit{\text{cut}}}^S$ and ${\epsilon }_{\rho }^S$ are the static (i.e. without any interaction) dielectric tensor components along the cut axis and ρ axis, respectively. Moreover, ${\delta }_{\rho }^{\mathit{\text{ph}}}$ and ${\delta }_{\mathit{\text{cut}}}^{\mathit{\text{ph}}}$ are the photogalvanic current components along the ρ and cut axes, respectively. The physical meaning of the characteristic time t ₀ and function f(t) can be better understood by discussing about φ.

In fact, if we assume the grating vector K̄_g parallel to the propagation direction ρ, φ can be explicated as follows:

where the function φ ₀=φ ₀(cut,t) represents the photo-induced potential for the initial recording section, when conductivity currents are negligible, and its dependence on time and spatial coordinates are simultaneously considered. Furthermore, by neglecting the photoconductivity respect to the dark conductivity σ^d , f(t) is an increasing exponential function (having the Maxwell relaxation time τ_m as a time constant) and t ₀ is equal to τ_m [5]. Remember that τ_m = ${\epsilon }_{\mathit{\text{cut}}}^S$ /${\sigma }_{\mathit{\text{cut}}}^d$ , where ${\sigma }_{\mathit{\text{cut}}}^d$ is the dark conductivity component along cut axis.

The photogalvanic current is strictly related to the photogalvanic tensor β, which depends on the structure considered. For a graded-index waveguide β changes along the cut axis, β = β(cut). However, it is possible to define an effective photogalvanic tensor β̃, which can be evaluated by a quasi-empirical model [5].

For a collinear interaction we can evaluate the photogalvanic current vector as follows

where E^A and E^B are the electric field vectors of two generic guided modes A and B, respectively. In this way, our model is applicable to both an intermode and intramode collinear interaction.

Therefore, by using Eqns. (1), (2) and (3), we can write the equation of induced oscillations as

The numerical solution of Eqn. (4) leads to the evaluation of the perturbations of the dielectric tensor components, Δε_ij , or otherwise, of the permeability tensor components Δb_p by the linear electro-optic effect, as

where ξ_k are the components of the photo-induced electric field, which, in the arbitrary case of X-cut z-propagating, and Y-cut and Z-cut x-propagating, can be written as follows:

Then, performing a simple matrix calculus, we derive Δε_ij in terms of the components Δb_p :

being $b_i^S$ = (${\epsilon }_{\mathit{\text{ii}}}^S$ )^-1 (where ${\epsilon }_{11}^S$ ≡ ${\epsilon }_x^S$ , ${\epsilon }_{22}^S$ ≡ ${\epsilon }_y^S$ , ${\epsilon }_{33}^S$ ≡ ${\epsilon }_z^S$ ) and det(b) = ($b_1^S$ + Δb ₁)($b_2^S$ + Δb ₂)($b_3^S$ + Δb ₃)- Δ$b_4^2$ ($b_1^S$ + Δb ₁)- Δ$b_5^2$ ($b_2^S$ + Δb ₂) + -Δ$b_6^2$ ($b_3^S$ + Δb ₃) + 2Δb ₄Δb ₅Δb ₆

In conclusion, we obtain a numerical expression for all the perturbations of the dielectric tensor, Δε_ij , in the mathematical form of a multidimensional array, by which each information about the holographic grating properties can be derived. It can be noted that, from the numerical solution of rigorous formulation of Eqn.(7), an excess in accuracy could occur.

3. Numerical results

To point out the main results of our analysis, let us now consider the real case of a graded-index (Gaussian profile; Δn _emax ≅0.0005 and Δn _omax =0.001 [6]) waveguide on an Y-cut LiNbO₃ crystal, without overlay and at room temperature (T = 300 K), when collinear TE₀-TM₀ mode interaction at the wavelength λ = 632.8 nm is assumed. For the given configuration two non-zero components of photogalvanic current, δ_x = β ₁₅ E_x ^TM₀ E_z ^TE₀* and δ_y = β ₂₄ E_y ^TM₀ E_z ^TE₀* are present. We have to remark that our software simulation tool does not execute any approximation on the field components, although the main field-component approximation (since |E_x ^TM₀| ▯ |E_y ^TM₀| and β ₁₅ = β ₂₄, then δ_x ≅ 0) could be used in this case. Using β̃₂₄ of the order of 10^-13 A/W [5], the simulation gives the results shown in the following figures.

 

Fig. 1. Δε ₁₁ dielectric perturbation in a Y-cut LiNbO₃ graded-index waveguide, x-propagating without overlay at T = 300 K, induced by the collinear TE₀-TM₀ mode interaction at λ = 632.8 nm.

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Fig. 2. Δε ₁₃ dielectric perturbation in the same waveguide.

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Fig. 3. Δε ₂₂ dielectric tensor perturbation in the same waveguide.

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Fig. 4. Δε ₂₃ dielectric tensor perturbation in the same waveguide.

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Both the Δε ₁₂ and Δε ₃₃ components are negligible, less than 10^-12, but not exactly zero because of the above-mentioned excess in accuracy. The decreasing oscillatory behaviour of Δε ₁₃ (Fig. 2) has no a physical meaning but only a mathematical one, also due to its very low numerical values (about 10^-11). The calculated dielectric tensor perturbations can be helpful for designers to know the photorefractive sensitivity of the particular guided-wave structure and mode interaction geometry assumed. For our example, it is clear that the extra-diagonal perturbation Δε ₂₃ and two diagonal perturbations, Δε ₁₁ and Δε ₂₂, are the most important.

As another example for designing tasks, we have compared different types of crystal cuts. Furthermore, we have still considered a graded-index (Gaussian profile; Δn _emax ≅ 0.0005 and Δn _omax ≅ 0.001 [6]) waveguide on LiNbO₃, without overlay at T = 300 K, and a collinear TE₀-TM₀ mode interaction at the wavelength λ = 632.8 nm.

We can note that, for both a X-cut crystal z-propagating and a Z-cut crystal x-propagating, the grating formation is related to the photogalvanic current component along x-axis, δ_x = β ₁₆ E_x ^TM₀ E_y ^TE₀* (using a β̃₁₆, one order lower than β̃₂₄ [7]). Table I shows a comparison among the cuts in the Gaussian profile waveguide. We observe that, in order to point out the results, is useful to show the perturbations Δε_ij in a normalised form with respect to a very low one (10^-8), being negligible those less than 10^-8. The largest perturbation can be obtained in X-cut (Δε ₁₃) and Z-cut (Δε ₃₃), ten times larger than the best case in Y-cut.

Table I. Comparison among LiNbO₃ cuts in Gaussian profile waveguides.

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To conclude our analysis, let us consider another significant example in absence of any overlay, fixing some parameters as the temperature (T = 300 K), the wavelength (λ = 632.8 nm) and the cut (X). The aim has been to underline the Δε_ij dependence on the waveguide fabrication technology. So, we have considered a Ti-diffused waveguide (Gaussian profile; Δn _emax = 0.001 and Δn _omax = 0.001), a proton exchanged (PE) waveguide (step-index profile, Δn_o =0 and Δn_e =0.01, with strongly reduced electro-optic activity, r ≅ r/10 [8]) and an annealed PE (APE) waveguide (exponential profile; Δn _omax = 0.005 and Δn _emax = 0, with moderately reduced electro-optic activity, r ≅ r/2 [8]). A collinear intramode TM₀-TM₀ interaction has been assumed. For such a case, there are two non-zero photogalvanic current components, δ_y = β ₂₁ E_x ^TM₀ E_x ^TM₀* and δ_z = β ₃₁ E_x ^TM₀ E_x ^TM₀* + β ₃₃ E_z ^TM₀ E_z ^TM₀* (using β̃²¹ and β̃³¹ ≅ β̃³³, one and two orders of magnitude lower than β̃²⁴, respectively [7]).

We have still normalised the perturbations with respect to 10^-8. Table II shows the result of comparison among different LiNbO₃ technologies.

Table II. Comparison among different X-cut LiNbO₃ technologies.

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It is clear that, for this intramode interaction geometry, we could choose indifferently either a Ti-diffused or a APE technology, both with the same sensitivity for Δε ₁₃, ten times greater than that of PE:LiNbO₃. Calculations obtained by including an overlay (refractive index 1.5 and 2 μm in depth) have shown similar results as in Tables I and II.

4. Conclusions

In conclusion, the theoretical analysis of the dielectric tensor perturbations during the holographic grating formation in photorefractive graded-index planar waveguides is presented. A powerful software tool has been developed to investigate any kind of guided mode interaction, and results have been presented to select the geometries allowing the maximum sensitivity in different LiNbO₃ crystal cuts and waveguide technologies, with the aim to give a help for design purposes.