# Acousto-optic Figure of Merit Search Acousto-optic ¯¬¾gure of merit search Jonathan...

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Physics Procedia 70 ( 2015 ) 762 – 765

Available online at www.sciencedirect.com

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015 doi: 10.1016/j.phpro.2015.08.129

ScienceDirect

2015 International Congress on Ultrasonics, 2015 ICU Metz

Acousto-optic figure of merit search Jonathan B. Pfeiffer, Kelvin H. Wagner

Dept. of Electrical and Computer Engineering, University of Colorado, Boulder, 80309, USA

Abstract

We present a method for searching for Bragg matched acousto-optic geometries that provide the largest acousto-optic figure of merit M2 for both isotropic and anisotropic diffraction. We apply our method to LiNbO3 and discuss the calculated results. c© 2015 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of 2015 ICU Metz.

Keywords: acousto opitcs; M2; LiNbO3

1. Introduction We present a method for searching for acousto-optic (AO) interaction geometries that will maximize the diffraction

efficiency of an AO device. AO devices are a versatile technology used to electronically, precisely, and rapidly control the intensity, frequency, and propagation direction of a laser beam. Applications include acousto-optic scanners, filters, mode lockers, and modulators [Xu and Stroud (1992)]. In order to maximize the diffraction efficiency of an AO device, we have developed a search method that calculates the acousto-optic figure of merit M2 for every possible Bragg match AO geometry for a fixed acoustic frequency in order to plot and then identify interaction geometries that maximize the diffraction efficiency. In this paper we apply our method to lithium niobate LiNbO3 and discuss our results.

2. M2 calculation

We search for an AO geometry with a large diffraction efficiency by maximizing the AO figure of merit M2 = n3dn

3 i p

2

ρV3a ,

where ni(λ) and nd(λ) are the index of refraction of the incident and diffracted light, ρ is the medium density, Va is the acoustic wave velocity, and p is the effective photoelastic coefficient. For LiNbO3 we need to include the piezoelectric effect when calculating the acoustic velocity Va with the Christoffel equation. The piezoelectric effect increases the velocity of the acoustic waves. M2 is proportional to the inverse of the acoustic velocity cubed V3a , so a faster acoustic wave greatly decreases M2. We also need to include the piezoelectric effect when calculating the effective photoelastic coefficient p. The strain from an acoustic wave can induce a change in the electric field through the piezoelectric effect in the material. The electro-optic effect will then change the index of refraction. LiNbO3 is also so birefringent (δn = 0.083 at 632nm) that we must include the Nelson and Lax local rotary correction when calculating p. For our

E-mail address: pfeiffej@colorado.edu

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015

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Jonathan B. Pfeier and Kelvin H. Wagner / Physics Procedia 70 ( 2015 ) 762 – 765 763

kx

k z

ki

kd

Ka

ne( ) /c

no /c

(a)

kx k z

kdkd

ki ki

Ka

ne( ) /c

no /c

ne( ) /c+Ka

(b)

�1 0

1

kx

�1

0

1

ky

�1

0

1

kz

(c)

Fig. 1: (a) Bragg matching acousto-optic interaction illustrated as a �k-vector triangle in a 2D plane. (b) Solving for the Bragg matching geometry by solving for the intersection of the diffracted �k-surface (blue) with the incident �k-surface (dashed red) displaced by a given acoustic wavevector �KA(green). (c) 3D illustration of the intersection of a sphere (blue) with a displaced ellipsoid (red), showing the set of Bragg matched solutions.

M2 calculations for LiNbO3, we use the same physical parameters as [?], who apply their own acousto-optic figure of merit search to LiNbO3.

3. Bragg-matching In an AO devices the acoustic wave diffracts the incident light by exchanging momentum �kd = �ki± �KA, where �KA is

the acoustic wavevector, �kd and �ki are the diffracted and incident light wavevectors respectively, and the + and − signs correspond to Doppler up-shifting and down-shifting interactions. Momentum mismatch �Δk of the AO interaction is often considered to arise when the wavevectors of the incident light and the acoustic wave do not sum to a wavevector that is a solution of the optical wave equation. In a finite medium, an alternative viewpoint is to ascribe a momentum uncertainty Δk = 2πL , due to the finite length of the interaction, that corresponds to the width of the distribution of Fourier components of the acoustic wave in momentum space. The diffraction efficiency scales as Sinc2

( ΔkL 2π

) where

L is the AO interaction length. The strongest diffraction occurs when the wavevectors are Bragg matched (| �Δk| = 0); thus, we want to design an AO device with an interaction geometry that ideally sets the momentum mismatch �Δk = 0. We only consider Bragg matched AO geometries for our M2 calculations, and we use �k-surfaces as a geometrical tool to solve for the Bragg matched AO geometries. Conservation of energy gives a Doppler shift ωd = ωi ± ΩA, where ωd, ωi±, and ΩA are the angular frequencies of the diffracted, incident, and acoustic waves, but negligibly effects the Bragg matching geometry.

Fig. 1a is a XZ cross-section of an uniaxial �k-surface and shows the acoustic wave �KA deflecting the incident beam �ki from the optical extraordinary momentum surface to the diffracted beam �kd on the optical ordinary momentum surface. The AO interaction illustrated in Fig. 1a is perfectly Bragg matched because the acoustic wavevector �KA and the incident wavevector �ki sum to a diffracted wavevector �kd that lies exactly on the diffracted �k-surface.

To solve for a Bragg matched AO interaction geometry, we first solve for the incident and diffracted �k-surfaces. Next we displace the incident �k-surface by the acoustic wavevector �KA. The intersections of the displaced incident �k-surface with the diffracted �k-surface are possible solutions for the perfectly Bragg-matched diffracted wavevector �kd. To find the incident wavevector �ki, simply subtract the acoustic wavevector �KA from the diffracted wavevector �kd. Fig. 1b illustrates a 2D cross-section of the incident �k-surface being displaced by the acoustic wavevector �KA and the solution for the exact Bragg-matched diffraction occurring where the displaced �k-surface intersects with the incident �k-surface.

In order to solve for the Bragg-matched geometry in three dimensions for isotropic or the ordinary mode of uniaxial crystals, we need to solve for the intersections of a sphere with a sphere. For the extraordinary optical polarization of uniaxial crystals, we need to solve for the intersection of an ellipsoid with an ellipsoid. For the very important

764 Jonathan B. Pfeier and Kelvin H. Wagner / Physics Procedia 70 ( 2015 ) 762 – 765

anisotropic polarization-switching AO interaction, we need to solve for the intersection of a sphere with an ellipsoid. Such quartic intersection problems have been solved numerically in a very general context for computer graphics [Lazard et al. (2006); Wang et al. (2002); Levin (1976)], but we have solved for the special cases of opitcal momentum surfaces of uniaxial crystals such as LiNbO3 where the sphere and ellipsoid have one radius in common. Fig. 1c illustrates our solution for the intersection of a sphere with an ellipsoid displaced from the origin.

To find a Bragg-matched AO interaction, we first choose a propagation direction l̂ of the acoustic wave. We then use the Christoffel equation to solve for the acoustic wave velocity Va, from which we can calculate the acoustic wavevector �KA = 2πVA(l̂) f l̂, where f is the chosen acoustic frequency. With the acoustic wavevector

�KA we can calculate all possible optical Bragg-matched geometries to solve for the incident and diffracted optical wavevectors. Once we know all of the acoustic and optical wavevectors, we can calculate the eigen-polarizations and velocities for each optical and acoustic mode as needed for an M2 calculation.

In order to do an M2 calculation, we first decide the propagation direction of the acoustic wave. For each direction, there will be three different acoustic velocities with their respective polarizations. Each acoustic wave can cause four different types of optical diffraction: (1) ordinary-to-ordinary, (2) extraordinary-to-extraordinary, (3) extraordinary- to-ordinary, and (4) ordinary-to-extraordinary polarization diffraction. The last two diffractions yield identical geome- tries, so we only consider the first three cases. Therefore, there are a total of 9 different ways to diffract the optical light for a given acoustic propagation direction, and thus 9 different M2 surfaces to calculate, each a function of the polar angle, θ, and azimuth angle, φ, of the acoustic propagation direction unit vector l̂(φ, θ).

To find the maximum M2 value, we sweep through all of the possible directi

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