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Calculation of effective acou stooptic co eff icientby abbreviated subscript formalismJ ieping Xu and William T. RhodesGeorgia Institute of Technology, School of Electrical Engineering, A tlanta, Georgia 30332-0250.Received 8 December 1987.0003-6935/88/112101-03$02.00/0. 1988 Optical Society of A merica.

In designing a new type of acoustooptic (AO) device, onemust calculate the effective AO coefficient p many times formany different AO interaction configurations (differentacousticwavepropagation directions, polarizations, whetherlongitudinal or shear wave, and different polarization directions of the incident and diffracted light waves). The standard expression for p isin tensor form and isinconvenient forpractical calculations. I t is well extablished that, for quantities containing a pair of symmetrical tensor subscripts, practical calculations are simplified and can be carried out bymatrix-vector (M -V ) multiplications if quantities with abbreviated subscripts are used. In this paper a new abbreviated-subscript expression for p is obtained. With this expression, p can be calculated for any AO interactionconfiguration by two M-V multiplications and one vector-vector multipl ication. In our experience, the compact notation greatly facilitates practical calculations.The standard expression for the effective AOcoefficient isgiven by1

where (d) and (i) are unit vectors along the polarizationdirections of the diffracted and incident light waves, respectively, and kl is the unit tensor of the strain Skl, i-e., Skl =Skl. (The usual summation convention over repeated subscripts is assumed in this and the following expressions.)The calculation of pi jklkl (and hence of p) is in practicecumbersome because (1) for any specific crystal class p i j k l isgenerally given not in tensor form but in the abbreviatedform pIJ and (2) there are two contraction operations withrespect to two tensor subscripts that cannot becarried out bysimple M -V multipl ication. However, it is easily verifiedthat Eq. (1) is equivalent to the following two equations:

The second equation, E q. (3), can be reduced further to theM -V product1 J une 1988 / Vol. 27, No. 11 / APPLIED OPTICS 2101

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where the 3X 6 matrix l(i) is constructed from (i) by thefollowing equation:

It should benoted thatl(i) sconstructed from(i) n the sameway as liK is constructed from li in crystal acoustics.2 Equation (4) can also be written aswherel(d) sconstructed from (d) in the same way asgiven byEq. (5) for l(i) This means that as soon as the types of theacoustic wave (i.e., ) and optical waves [i.e.,(d)and(i)]aregiven, I [throughEq. (4)] and pIJ can both beobtained by aM -V multiplication. Since I is a row vector and pIJ is acolumn vector, the effective AO coefficient p is obtainedthrough Eq. (2) by taking the inner product of these twovectors.Weillustrate the calculational procedure by twoexamples.Consider first a lithium niobate AO device with a longitudinal acoustic wave propagating along the x axis. The propagation direction and polarization direction of the light beamare as shown in Fig. 1. We would like to determine the valueof the angle that makes p maximum and find the maximumvalue of p. Since the acoustic wave is a longitudinal wavealong the x axis, we have [ ] = [1 000 00] T (T denotingtranspose) and

Using the values p12= 0.090,p31 = 0.179,and p41 =-0.l51forlithium niobate, the value of that maximizes p is easilyfound to equal 36.8. Substituting this value of into Eq.(8), we get Pmax = 0.29. [It should be noted that li thiumniobate is also a good piezoelectric and electrooptic medium,and generally there is an indi rect AO effect through bothpiezoelectric and electrooptic effects. The total acoustoop-tic coefficient pIJ isgiven by

where l is the unit vector along the propagation direction ofthe acoustic wave and p'IJ is the indirect AO coefficient.However, for our case,wehave l= (1,0,0). By calculating thematrix (r,Iklk),(lieJ,), it is easy to prove that all elements of p' IJexcept P55, P56, P5, and p'66 equal zero. Since only the firstcolumn of the matrix pI J , as indicated by Eq. (6), isof concernin this case, the indirect AOeffect has no influence.]The second example is a TeO2 AO device with the slowshear wave propagating along [110]. Using the coordinatetransformation rule for [SJ ], it is easy to obtain2

and thus

From Fig.1 it follows that (i) =(d)= [0,-sin,cos]. Equation (4) then yieldsThe propagation direction of the light beam is along [001].The polarization directions of the incident and diffractedlight beams should be e light (i.e., along [110]) and 0 light(i.e., along [110]), respectively. Thus one has

By taking the inner product of I and [p IJ ], one obtains Equations (4) and (2) thus giveThe condition dp/d = 0 yields the following equation involving : If the propagation direction of the light beam is also along[110] (corresponding to a collinear AO interaction), one has

and Eqs. (4) and (2) yield

The result p = 0 indicates that the slow shear wave in a TeO2crystal cannot be used for a collinear AO interaction.

Fig. 1. Workingmodeofalithium niobateAO device withalongitudinal acousticwavepropagating along thex axis.This work was partially supported by the Joint ServicesElectronics Program under contract DAAL03-87-K-0059.J ieping X u is on leave from the Department of A ppliedPhysics of Beijing Polytechnic University.

2102 APPLIED OPTICS / Vo l. 27, No. 11 / 1 J une 1988

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1 J une 1988 / Vol . 27, No. 11 / APPLI ED OPTI CS 2103

References1. I. C. Chang, "Acoustooptic Devices and Applications," I EEETrans. Sonics Ultrason. SU-23, No. 1, 2 (1976).2. B. A. Auld,Acoustic Fields and Waves in Solids,Vol. 1 (Wiley,New Y ork, 1973).