## Modeling of holographic gratings in graded-index photorefractive planar waveguides

Optics Express, Vol. 10, Issue 20, pp. 1133-1138 (2002)

http://dx.doi.org/10.1364/OE.10.001133

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### 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

2. T. W. Mossberg, “Planar holographic optical processing devices,” Optics Letters **26**, 414–416 (2001). [CrossRef]

3. K. Itoh, K. Ikewaza, W. Watanabe, Y. Furuya, Y. Masuda, and T. Toma, “Fabricating micro-Bragg reflectors in 3-D photorefractive waveguides,” Optics Express **2**, 503–508 (1998). [CrossRef] [PubMed]

4. O. Matoba, K. Ikewaza, K. Itoh, and Y. Ichioka, “Modification of photorefractive waveguides in lithium niobate by guided beam for optical interconnections,” Opt. Review **2**, 438–443 (1995). [CrossRef]

## 2. Theoretical model

*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.

*K*

_{g}is the grating vector amplitude, i.e.

*K*

_{g}= |

*K̄*

_{g}|,

*cut*axis and

*ρ*axis, respectively. Moreover,

*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*.

*K̄*

_{g}parallel to the propagation direction

*ρ*,

*φ*can be explicated as follows:

*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

5. G. Glazov, I. Itkin, V. Shandarov, E. Shandarov, and S. Shandarov, “Planar hologram gratings in photorefractive waveguides in LiNbO_{3},” J. Opt. Soc. Am. B **7**, 2279–2288 (1990). [CrossRef]

*τ*

_{m}=

*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.

*ε*

_{ij}, or otherwise, of the permeability tensor components Δ

*b*

_{p}by the linear electro-optic effect, as

*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:

*ε*

_{ij}in terms of the components Δ

*b*

_{p}:

^{-1}(where

*b*) = (

*b*

_{1})(

*b*

_{2})(

*b*

_{3})- Δ

*b*

_{1})- Δ

*b*

_{2}) + -Δ

*b*

_{3}) + 2Δ

*b*

_{4}Δ

*b*

_{5}Δ

*b*

_{6}

*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

*n*

_{emax}≅0.0005 and Δ

*n*

_{omax}=0.001 [6

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.

_{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

5. G. Glazov, I. Itkin, V. Shandarov, E. Shandarov, and S. Shandarov, “Planar hologram gratings in photorefractive waveguides in LiNbO_{3},” J. Opt. Soc. Am. B **7**, 2279–2288 (1990). [CrossRef]

*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.

*n*

_{emax}≅ 0.0005 and Δ

*n*

_{omax}≅ 0.001 [6

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.

_{3}, without overlay at T = 300 K, and a collinear TE

_{0}-TM

_{0}mode interaction at the wavelength λ = 632.8 nm.

*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

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]

*n*

_{omax}= 0.005 and Δ

*n*

_{emax}= 0, with moderately reduced electro-optic activity, r ≅ r/2 [8

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]

_{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]).

^{-8}. Table II shows the result of comparison among different LiNbO

_{3}technologies.

*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

_{3}crystal cuts and waveguide technologies, with the aim to give a help for design purposes.

## References and links

1. | V. E. Wood, P. J. Cressman, R. L. Holman, and C. M. Verber, “Photorefractive effects in waveguides” in Photorefractive Materials and their Applications II,” |

2. | T. W. Mossberg, “Planar holographic optical processing devices,” Optics Letters |

3. | K. Itoh, K. Ikewaza, W. Watanabe, Y. Furuya, Y. Masuda, and T. Toma, “Fabricating micro-Bragg reflectors in 3-D photorefractive waveguides,” Optics Express |

4. | O. Matoba, K. Ikewaza, K. Itoh, and Y. Ichioka, “Modification of photorefractive waveguides in lithium niobate by guided beam for optical interconnections,” Opt. Review |

5. | G. Glazov, I. Itkin, V. Shandarov, E. Shandarov, and S. Shandarov, “Planar hologram gratings in photorefractive waveguides in LiNbO |

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, |

8. | I. Savatinova, S. Tonchev, R. Todorov, M. N. Armenise, V. M. N. Passaro, and C. C. Ziling, “Electrooptic Effect in Proton Exchanged LiNbO |

**OCIS Codes**

(090.2900) Holography : Optical storage materials

(130.2790) Integrated optics : Guided waves

(130.3730) Integrated optics : Lithium niobate

(230.7390) Optical devices : Waveguides, planar

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 16, 2002

Revised Manuscript: October 2, 2002

Published: October 7, 2002

**Citation**

Vittorio Passaro and Daniele Marseglia, "Modeling of holographic gratings in gradedindex photorefractive planar waveguides," Opt. Express **10**, 1133-1138 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-20-1133

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### References

- V. E. Wood, P. J. Cressman, R. L. Holman and C. M. Verber, �??Photorefractive effects in waveguides�?? in Photorefractive Materials and their Applications II,�?? 62, 45-100, Springer-Verlag, Berlin (1988).
- T. W. Mossberg, �??Planar holographic optical processing devices,�?? Opt. Lett. 26, 414-416 (2001). [CrossRef]
- K. Itoh, K. Ikewaza, W. Watanabe, Y. Furuya, Y. Masuda, T. Toma, �??Fabricating micro-Bragg reflectors in 3-D photorefractive waveguides,�?? Opt. Express 2, 503-508 (1998), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-12-503">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-12-503</a> [CrossRef] [PubMed]
- O. Matoba, K. Ikewaza, K. Itoh and Y. Ichioka, �??Modification of photorefractive waveguides in lithium niobate by guided beam for optical interconnections,�?? Opt. Review 2, 438-443 (1995). [CrossRef]
- G. Glazov, I. Itkin, V. Shandarov, E. Shandarov and S. Shandarov, �??Planar hologram gratings in photorefractive waveguides in LiNbO3,�?? J. Opt. Soc. Am. B 7, 2279-2288 (1990). [CrossRef]
- J. G. P. dos Reis, H. J. A. da Silva, �??Modelling and simulation of passive optical devices,�?? <a href="http://www.it.uc.pt/oc/ocpub/jr99cp01.pdf">www.it.uc.pt/oc/ocpub/jr99cp01.pdf</a>.
- A. M. Prokhorov , Y. S. Kuz'minov, Physics and chemistry of cristalline lithium niobate, Adam Hilger Series on Optics and Optoelectronics, 275-327 (1990).
- I. Savatinova, S. Tonchev, R. Todorov, M. N. Armenise, V. M. N. Passaro and C. C. Ziling, �??Electrooptic Effect in Proton Exchanged LiNbO3 and LiTaO3 Waveguides,�?? J. Lightwave Technol. 14, 403-409 (1996). [CrossRef]

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