OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 20 — Oct. 7, 2002
  • pp: 1133–1138
« Show journal navigation

Modeling of holographic gratings in graded-index photorefractive planar waveguides

Vittorio M. N. Passaro and Daniele Marseglia  »View Author Affiliations


Optics Express, Vol. 10, Issue 20, pp. 1133-1138 (2002)
http://dx.doi.org/10.1364/OE.10.001133


View Full Text Article

Acrobat PDF (153 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

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,” 62, 45–100, Springer-Verlag, Berlin (1988).

].

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

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

], beam splitters and couplers [3

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]

] or dynamic interconnections for neural networks [4

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]

]. Therefore, a numerical model of planar holographic gratings formation in waveguides is a very useful software tool for design purposes.

2. Theoretical model

This paper is devoted to the analysis of holographic grating formation on arbitrary graded-index waveguides having a generic electro-optic crystal (full rij 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.

d2φdcut2Kg2ερSεcutSφ=f(t)t0εcutS(iKgδρph+δcutphcut)
(1)

where Kg is the grating vector amplitude, i.e. Kg = |g |, εcutS and ερS are the static (i.e. without any interaction) dielectric tensor components along the cut axis and ρ axis, respectively. Moreover, δρph and δcutph 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 g parallel to the propagation direction ρ, φ can be explicated as follows:

φ=φ(cut,ρ,t)=φ0(cut,t0)f(t)exp(jKgρ)
(2)

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 σd , f(t) is an increasing exponential function (having the Maxwell relaxation time τm as a time constant) and t 0 is equal to τm [5

5. 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]

]. Remember that τm = εcutS /σcutd , where σcutd is the dark conductivity component along cut axis.

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

δph=βEAEB*exp(jKgρ)
(3)

where EA and EB 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

d2φ0dcut2Kg2ερSεcutSφ0=t0εcutS(jKg[βEAEB]ρ+cut[βEAEB]cut)
(4)

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

Δbp=k=13rpkξkp=1,,6
(5)

where ξk 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:

ξ=(ξ1,ξ2,ξ3)={(φx,0,jKgφ)[X-cut](jKgφ,φy,0)[Y-cut](jKgφ,0,φz)[Z-cut]
(6)

Then, performing a simple matrix calculus, we derive Δεij in terms of the components Δbp :

{Δε11=(b2S+Δb2)(b3S+Δb3)Δb42det(b)ε11SΔε12=Δε21=(b3S+Δb3)Δb6Δb4Δb5det(b)Δε13=Δε31=Δb4Δb6(b2S+Δb2)Δb5det(b)Δε22=(b1S+Δb1)(b3S+Δb3)Δb52det(b)ε22SΔε23=Δε32=(b1S+Δb1)Δb4Δb5Δb6det(b)Δε33=(b1S+Δb1)(b2S+Δb2)Δb62det(b)ε33S
(7)

being biS = (εiiS )-1 (where ε11SεxS , ε22SεyS , ε33SεzS ) and det(b) = (b1S + Δb 1)(b2S + Δb 2)(b3S + Δb 3)- Δb42 (b1S + Δb 1)- Δb52 (b2S + Δb 2) + -Δb62 (b3S + Δb 3) + 2Δb 4Δb 5Δb 6

In conclusion, we obtain a numerical expression for all the perturbations of the dielectric tensor, Δεij , 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

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.

]) 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, δx = β 15 Ex TM0 Ez TE0* and δy = β 24 Ey TM0 Ez 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 |Ex TM0| ▯ |Ey TM0| and β 15 = β 24, then δx ≅ 0) could be used in this case. Using β̃24 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 LiNbO3,” J. Opt. Soc. Am. B 7, 2279–2288 (1990). [CrossRef]

], 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.
Fig. 2. Δε 13 dielectric perturbation in the same waveguide.
Fig. 3. Δε 22 dielectric tensor perturbation in the same waveguide.
Fig. 4. Δε 23 dielectric tensor perturbation in the same waveguide.

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

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.

]) waveguide on LiNbO3, without overlay at T = 300 K, and a collinear TE0-TM0 mode interaction at the wavelength λ = 632.8 nm.

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

table-icon
View This Table
| View All Tables

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 Δεij 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, Δno =0 and Δne =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 LiNbO3 and LiTaO3 Waveguides,” J. Lightwave Technol. 14, 403–409 (1996). [CrossRef]

]) 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

8. 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]

]). A collinear intramode TM0-TM0 interaction has been assumed. For such a case, there are two non-zero photogalvanic current components, δy = β 21 Ex TM0 Ex TM0* and δz = β 31 Ex TM0 Ex TM0* + β 33 Ez TM0 Ez TM0* (using β̃21 and β̃31β̃33, one and two orders of magnitude lower than β̃24, respectively [7

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).

]).

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.

table-icon
View This Table
| View All Tables

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.

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,” 62, 45–100, Springer-Verlag, Berlin (1988).

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]

5.

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]

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 LiNbO3 and LiTaO3 Waveguides,” J. Lightwave Technol. 14, 403–409 (1996). [CrossRef]

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


Sort:  Journal  |  Reset  

References

  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,�?? 62, 45-100, Springer-Verlag, Berlin (1988).
  2. T. W. Mossberg, �??Planar holographic optical processing devices,�?? Opt. Lett. 26, 414-416 (2001). [CrossRef]
  3. 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]
  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]
  5. 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]
  6. 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>.
  7. A. M. Prokhorov , 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 LiNbO3 and LiTaO3 Waveguides,�?? J. Lightwave Technol. 14, 403-409 (1996). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited