^{1,a)}and Ole Sigmund

^{1}

### Abstract

A method for modeling the interaction of the mechanical field from a surface acoustic wave and the optical field in the waveguides of a Mach–Zehnder interferometer is presented. The surface acoustic wave is generated by an interdigital transducer using a linear elastic plane model of a piezoelectric, inhomogeneous material, and reflections from the boundaries are avoided by applying perfectly matched layers. The optical modes in the waveguides are modeled by time-harmonic wave equations for the magnetic field. The two models are coupled using stress-optical relations and the change in effective refractive index introduced in the Mach–Zehnder interferometer arms by the stresses from the surface acoustic wave is calculated. It is then shown that the effective refractive index of the fundamental optical mode increases at a surface acoustic wave crest and decreases at a trough. The height and the width of the waveguides are varied for a silicon on insulator sample, and it is shown that the difference in effective refractive index between the waveguides can be increased 12 times for the right choice of waveguide size such that the optical modulation is improved. The difference is four times bigger if the waveguides are kept single moded. It is furthermore shown that the difference increases more than ten times when the waveguides are buried below the surface, where the mechanical stresses have their maximum, and in the case where two interdigital transducers are used the difference is increased 1.5 times.

This work is supported by the European FP6 research project ePIXnet—*European Network of Excellence on Photonic Integrated Components and Circuits*. The authors thank the partners from the joint research group of ePIXnet *Photonic Switches and Modulators based on Surface Acoustic Waves* Paulo V. Santos and Markus Beck from Paul-Drude-Institut für Festkörperelektronik, Berlin, Germany and Mike van der Poel from Department of Photonics Engineering, Technical University of Denmark, for valuable input related to the model presented.

The authors are grateful to Jakob S. Jensen from the Department of Mechanical Engineering and Martin P. Bendsøe from the Department of Mathematics, Technical University of Denmark, for helpful discussions related to the presented work.

The support from Euro-horcs/ESF European Young Investigator Award (EURYI) through the grant *Synthesis and Topology Optimization of Optomechanical Systems* as well as from the Danish Center for Scientific Computing (DCSC) is gratefully acknowledged.

I. INTRODUCTION

II. THE ACOUSTO-OPTICAL MODEL

A. Problem description

B. The piezoelectric model

1. PMLs

C. The optical model

III. RESULTS

A. Simulation of SAWs

B. Simulation of optical waves

C. Increasing the difference in effective refractive index between the waveguides

1. Changing the waveguide size

2. Other device structures

IV. CONCLUSION

### Key Topics

- Surface acoustic waves
- 82.0
- Refractive index
- 33.0
- Electrodes
- 31.0
- Optical waveguides
- 31.0
- Piezoelectric materials
- 29.0

## Figures

Light modulation in a MZI with a SAW generated by an IDT. (a) Three dimensional geometry of a MZI with a propagating SAW ([van der Poel (Ref. 9)]. (b) A 2D cross section through the waveguide arms of the MZI, which is used in the simulations. The SAW is absorbed in PMLs at the boundaries. The dimensions indicated in are in microns given by , , , , , , , , , , , , and .

Light modulation in a MZI with a SAW generated by an IDT. (a) Three dimensional geometry of a MZI with a propagating SAW ([van der Poel (Ref. 9)]. (b) A 2D cross section through the waveguide arms of the MZI, which is used in the simulations. The SAW is absorbed in PMLs at the boundaries. The dimensions indicated in are in microns given by , , , , , , , , , , , , and .

Generation of a SAW in a GaAs sample by the piezoelectric model. The position of the IDT is indicated. (a) The color indicates the displacement and the shape of the surface is deformed with the unified displacements and . (b) The graph shows the absolute amplitude along the material surface as function of

Generation of a SAW in a GaAs sample by the piezoelectric model. The position of the IDT is indicated. (a) The color indicates the displacement and the shape of the surface is deformed with the unified displacements and . (b) The graph shows the absolute amplitude along the material surface as function of

Results for the SAW in the GaAs substrate. (a) Displacements and as function of depth at position . (b) The absolute amplitude normalized with the square root of the electrical power as function of the frequency for 10 and 20 electrode pairs.

Results for the SAW in the GaAs substrate. (a) Displacements and as function of depth at position . (b) The absolute amplitude normalized with the square root of the electrical power as function of the frequency for 10 and 20 electrode pairs.

Generation of SAWs in a SOI sample by the piezoelectric model. The results are given to the right of the IDT. (a) The color indicates the displacement and the shape of the surface is deformed with the unified displacements and . (b) The graph shows the absolute amplitude along the material surface as function of .

Generation of SAWs in a SOI sample by the piezoelectric model. The results are given to the right of the IDT. (a) The color indicates the displacement and the shape of the surface is deformed with the unified displacements and . (b) The graph shows the absolute amplitude along the material surface as function of .

The -component of the time averaged power flow of the fundamental mode in the waveguides when no stresses are applied.

The -component of the time averaged power flow of the fundamental mode in the waveguides when no stresses are applied.

Change in the effective refractive index of the fundamental mode in the two waveguides normalized by the square root of the power as function of the SAW phase . corresponds to a wave crest in the left waveguide and a trough in the right.

Change in the effective refractive index of the fundamental mode in the two waveguides normalized by the square root of the power as function of the SAW phase . corresponds to a wave crest in the left waveguide and a trough in the right.

Results for a study of the height of the waveguides with (- - -) indicating the results for the original waveguide geometry. (a) Difference in effective refractive index of the fundamental mode between the two waveguides as function of for 6 and 12 electrode pairs. (b) The effective refractive index for the three lowest order modes in the waveguides normalized to the value of the fundamental mode in the left waveguide for the original geometry as functions of .

Results for a study of the height of the waveguides with (- - -) indicating the results for the original waveguide geometry. (a) Difference in effective refractive index of the fundamental mode between the two waveguides as function of for 6 and 12 electrode pairs. (b) The effective refractive index for the three lowest order modes in the waveguides normalized to the value of the fundamental mode in the left waveguide for the original geometry as functions of .

Results for a study of the waveguide width for the optimal height . (a) Difference in effective refractive index of the fundamental mode between the two waveguides as function of . (b) The effective refractive index for the four lowest order modes in the waveguides normalized to the value of the fundamental mode in the left waveguide for the original geometry as functions of .

Results for a study of the waveguide width for the optimal height . (a) Difference in effective refractive index of the fundamental mode between the two waveguides as function of . (b) The effective refractive index for the four lowest order modes in the waveguides normalized to the value of the fundamental mode in the left waveguide for the original geometry as functions of .

## Tables

The elastic stiffness constants and the density for the materials used in the piezoelectric model.

The elastic stiffness constants and the density for the materials used in the piezoelectric model.

The piezoelectric stress constants and the permittivity constants for the materials used in the piezoelectric model.

The piezoelectric stress constants and the permittivity constants for the materials used in the piezoelectric model.

Stress-optical constants (Ref. 19).

Stress-optical constants (Ref. 19).

Results for the difference in effective refractive index for different combinations of IDT numbers and single and double electrode fingers, .

Results for the difference in effective refractive index for different combinations of IDT numbers and single and double electrode fingers, .

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