## Abstract

A numerical model is presented for the evaluation of the dielectric permittivity tensor changes as induced by guided modes during the formation of holographic gratings in arbitrary photorefractive graded-index planar waveguides. Comparisons among lithium niobate waveguides with different cuts and technology are shown.

© 2002 Optical Society of America

## 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, Δ*ε*_{ij}
, or otherwise, of the permeability tensor components Δ*b*_{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 Δ*ε*_{ij}
in terms of the components Δ*b*_{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*) = (${b}_{1}^{S}$ + Δ*b*
_{1})(${b}_{2}^{S}$ + Δ*b*
_{2})(${b}_{3}^{S}$ + Δ*b*
_{3})- Δ${b}_{4}^{2}$ (${b}_{1}^{S}$ + Δ*b*
_{1})- Δ${b}_{5}^{2}$ (${b}_{2}^{S}$ + Δ*b*
_{2}) + -Δ${b}_{6}^{2}$ (${b}_{3}^{S}$ + Δ*b*
_{3}) + 2Δ*b*
_{4}Δ*b*
_{5}Δ*b*
_{6}

In conclusion, we obtain a numerical expression for all the perturbations of the dielectric tensor, Δ*ε*_{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; Δ*n*
_{emax} ≅0.0005 and Δ*n*
_{omax} =0.001 [6]) waveguide on an Y-cut LiNbO_{3} crystal, without overlay and at room temperature (T = 300 K), when collinear TE_{0}-TM_{0} 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 *β̃*_{24} of the order of 10^{-13} A/W [5], the simulation gives the results shown in the following figures.

Both the Δ*ε*
_{12} and Δ*ε*
_{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 Δ*ε*
_{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 Δ*ε*
_{23} and two diagonal perturbations, Δ*ε*
_{11} and Δ*ε*
_{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; Δ*n*
_{emax} ≅ 0.0005 and Δ*n*
_{omax} ≅ 0.001 [6]) waveguide on LiNbO_{3}, without overlay at T = 300 K, and a collinear TE_{0}-TM_{0} 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 *β̃*_{16}, one order lower than *β̃*_{24} [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 Δ*ε*_{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 (Δ*ε*
_{13}) and Z-cut (Δ*ε*
_{33}), ten times larger than the best case in Y-cut.

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 Δ*ε*_{ij}
dependence on the waveguide fabrication technology. So, we have considered a Ti-diffused waveguide (Gaussian profile; Δ*n*
_{emax} = 0.001 and Δ*n*
_{omax} = 0.001), a proton exchanged (PE) waveguide (step-index profile, Δ*n*_{o}
=0 and Δ*n*_{e}
=0.01, with strongly reduced electro-optic activity, r ≅ r/10 [8]) and an annealed PE (APE) waveguide (exponential profile; Δ*n*
_{omax} = 0.005 and Δ*n*
_{emax} = 0, with moderately reduced electro-optic activity, r ≅ r/2 [8]). A collinear intramode TM_{0}-TM_{0} 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 *β̃*^{21} and *β̃*^{31} ≅ *β̃*^{33}, one and two orders of magnitude lower than *β̃*^{24}, respectively [7]).

We have still normalised the perturbations with respect to 10^{-8}. Table II shows the result of comparison among different LiNbO_{3} technologies.

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 Δ*ε*
_{13}, ten times greater than that of PE:LiNbO_{3}. 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 LiNbO_{3} crystal cuts and waveguide technologies, with the aim to give a help for design purposes.

## References and links

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**6. **J. G. P. dos Reis and H. J. A. da Silva, “Modelling and simulation of passive optical devices,” www.it.uc.pt/oc/ocpub/jr99cp01.pdf.

**7. **A. M. Prokhorov and Y. S. Kuz’minov, *Physics and chemistry of cristalline lithium niobate*, Adam Hilger Series on Optics and Optoelectronics, 275–327 (1990).

**8. **I. Savatinova, S. Tonchev, R. Todorov, M. N. Armenise, V. M. N. Passaro, and C. C. Ziling, “Electrooptic Effect in Proton Exchanged LiNbO_{3} and LiTaO_{3} Waveguides,” J. Lightwave Technol. **14**, 403–409 (1996). [CrossRef]