Temporal response of the acoustooptic modulator:geometrical optics model in the lowscattering efficiency limit

Richard V. Johnson

A phenomenological model of the acoustooptic modulator is proposed. The temporal response of the de-

flected light power is a quadratic invariant function of the video signal amplitude. The two-dimensional

response kernel is defined by an overlap integral of the incident light and sound field profiles, similar to a

convolution operator. When the incident light is focused to reduce risetime, the light throughput efficiency

and the deflected light profile degrade because the modulator is a linear filter with limited angular passband

operating on the propagation angle spectrum of the incident light.

1. IntroductionAn optical modulator converts an electronic stream

of video information into a corresponding modulationof a light beam. One of the most important classes ofoptical modulator is the acoustooptic Bragg cell. Theprimary measures of modulator performance include:

temporal response (e.g., risetime, MTF, dynamiccontrast ratio);

light throughput efficiency, defined as that fractionof the incident light power which is available to theoutput light beam; and

distortion of the intensity profile of the output lightbeam compared with the incident light beam profile.

Each of the above performance measures is governedby the distribution of energy within the light and soundfields, especially by the dimensions of these fields (seeFigs. 1 and 2).

By far the simplest model -of the temporal responsewas proposed in 1961 by Rosenthal' and again in 1966by Korpel et al. of the Zenith Radio Corporation.2 Inthis model the modulator responds to the video signallike a linear invariant system, with an impulse responsedefined by the intensity profile of the incident lightbeam. (For linear invariant system theory, see Ref.3.)

Although this model is incapable of accounting forrisetime contributions from the sound field thickness(the model assumes an infinitesimally thin sound field)and although its scope is limited to the low scatteringconversion regime, it has the considerable virtue ofbeing applicable to arbitrary video signals and arbitrarylight beam profiles with a minimum of computationaleffort.

The author is with Xerox Electro-Optical Systems, Pasadena,California 91107.

Received 16 June 1976.

The present paper, the first of three papers on tem-poral response studies, will extend the Rosenthal thinsound field model to account for sound field depthcontributions, while retaining some of the simplicity ofthe Rosenthal model. The scope of this first paper willbe limited to a phenomenological geometrical opticsdescription of the acoustooptic coupling in the lowscattering efficiency limit. Although lacking the rigorof a physical optics description, this approach most ef-fectively underscores the basic dynamics of the tem-poral response mechanism with a minimum investmentin analytical effort. The second paper will extend theanalysis into a physical optics formulation, again limitedto the low scattering efficiency regime. The third paperwill apply the physical optics model to calculate thehigher order terms in the rescattering series. Thesehigher order terms are necessary to characterize themodulator response in the high scattering efficiencyregime.

II. Review of Acoustooptic Coupling Literature

A large body of literature of the acoustooptic couplingmechanism has evolved since Brillouin first predictedthe effect in 1922.48 Almost universally these studiesconsider only the coupling of a plane wave incident lightfield with an unmodulated plane wave sound field. Theresults of these studies are not directly applicable to theacoustooptic modulator for two reasons. The first isthat the light beam incident upon the modulator is nota plane wave but rather has finite breadth and a typi-cally Gaussian profile. The second reason is that theacoustic carrier must be modulated by the video signalfor the modulator to function. Any analysis which re-stricts the sound field to a pure temporally harmoniccarrier is incapable of modeling the temporal responseof the modulator.

Recently Chu and Tamir9 have extended the planewave coupling studies to include nonplane wave light

February 1977 / Vol. 16, No. 2 / APPLIED OPTICS 507

DEFLECTEDLIGHT BEAM

BEAMSTOP

OELECTRICNSDUCER

ELECTRONIC VIDEOSTREAM

CARRIEROSCILLATOR

DRIVE ELECTRONICS PACKAGE

Fig. 1. Block diagram of the acoustooptic modulator.

2wINCIDENT LLIGHT wrvt/s~rt 0 BRAGG

1i /--- L TAN 0BRAGG LOBRAGG2W SECO 2W

SOUND FIELD

Fig. 2. Acoustic transit time. The temporal response of the mod-ulator is defined by the dimensions and by the profiles of the incident

light beam and the sound field.

fields incident upon the modulator. Their techniquewas to decompose an arbitrary field distribution, suchas a Gaussian profile, into an equivalent superpositionof plane wave components. The modulator responseto each plane wave component was determined, and theindividual responses summed to determine the totalscattered light field. With this approach they were ableto study the profile changes of the incident and de-flected light fields in the high scattering efficiency re-gime, where a significant fraction of the incident lightenergy is coupled into the deflected light beam. Sucha technique is feasible because the scattering equationis linear in the light field. The Chu and Tamir workassumes an unmodulated acoustic carrier.

A similar decomposition of the sound field into planewave components is inappropriate in the high scatteringefficiency regime because the scattered light field isnonlinearly dependent on the sound field. The math-ematics of coupling a light beam with a modulatedacoustic carrier can be quite formidable, impedinganalysis of the temporal response of the modulator. Tothe author's knowledge, no studies of the temporal re-sponse of the modulator applicable in the high scat-tering efficiency regime have been published.

By restricting the scope of the analysis to the lowscattering efficiency regime, a model of the acoustooptic

mechanism can be constructed which is capable ofhandling realistic field profiles and a modulated soundcarrier. The mathematics is more tractable in this lowscattering regime because the scattered light field islinearly dependent on the sound field. Researchers atBell Telephone Laboratories' 0 -' 3 have demonstratedthe power of this analytical tool to probe the influenceof the field dimensions on the modulator performance.However, Maydan's temporal response studies13 havebeen limited to risetime calculations. The determi-nation of the modulator reponse to an arbitrary videosignal remains unsettled.

The work reported in the present paper is similar tothe Bell Telephone Laboratories work in consideringthe implications of the sound field depth on modulatorperformance. It differs from that work in using aphenomenological geometrical optics approach (i.e., aray analysis) rather than a more rigorous physical opticsmodel. The present model lacks rigor because it failsto account for diffraction-induced changes in the fieldprofiles within the scattering volume, such as thenecking of a Gaussian profile TEMoo laser beam atfocus. The incident light beam diameter can typicallyvary 15% within the scattering volume. The presentmodel also lacks rigor because it lumps all diffractioneffects into a single deflection angle hragg, defined bythe acoustic carrier frequency, rather than a spectrumof deflection angles defined by the video-modulatedcarrier. Chang4 has estimated that the video band-width of a typical modulator can be 25% of the carrierfrequency, implying a corresponding spread of deflec-tion angles.

The disadvantage of the geometrical optics approachis that it lacks the precision of the physical optics model.The present model is capable of generating the broadfeatures of the modulator response but lacks accuracyin the fine details. In compensation, this model has theadvantages of conceptual simplicity and ease of calcu-lation. This model is an ideal vehicle for introducingthe concepts of quadratic invariant system response(Sec. IV) and linear filter distortion of the light beamprofile (Sec. V).

11. Derivation of the Scattering EquationAttention will be restricted to the scattering plane

defined by the propagation directions of the incidentlight and sound fields. The coordinate system is de-fined in Fig. 1. The incident light, deflected light, andsound fields in this model are assumed to be well-colli-mated beams. The incident light and the deflectedlight beams propagate at angle OBragg with respect to-thez axis, where OBragg is the Bragg angle associated withthe acoustic carrier.

sinOnragg = (Xf,)/(2nv), (1)

where X = wavelength of light in vacuo,fc = acoustic carrier frequency,n = index of refraction of the acoustooptic me-

dium, andv = speed of sound in the acoustooptic medi-

um.

508 APPLIED OPTICS / Vol. 16, No. 2 I February 1977

ACOUSTIC ABSORBERSOU NDI FI E D~

1 GHTC EA M

ACOUSTO-OPTICHEAD

c AXIS

PIEZTRAI

RFAMPLIFIER

BALANCEDMIXER

__

z axis

Fig. -3. Ray trace diagram for the deflected light beam.

x axis

- I I*. z axis

Fig. 4. Ray trace diagram relating the deflected light beam to theincident light beam and the sound field.

Bri

The field amplitude Eout(xzt) of the deflected lightdownstream of the modulator can be generated from a

Eout(xzt) fictitious (virtual) profile Dout(xt) located on the z =0 line inside the modulator, again by simple ray pro-

agg jection:

E..t(xzt) = D..t(x - ZOBagg)-

The deflected light is generated by the photoelasticcoupling of the incident light and the sound fields.' 4

The amplitude associated with a given deflected lightbeam ray is proportional to the product of the incidentlight and the sound field amplitudes integrated over thelength of the ray:

Eout(xzt) = (-)

f dzEil(x - Z'Onragg, Z - Z')S(X - Z'OBragg, Z - Z',t), (5)

where p = the elastooptic coupling tensor componentassociated with the scattering geometry; see Fig. 3. Theacoustic strain field S(x,z,t) is assumed to be separableinto a spatial profile function W(z) and a video signalV(t). The profile W is a dimensionless shape functionnormalized to unity, defined by the piezoelectrictransducer which generates the sound field. Thisprofile function is closely analogous to the shape func-tion in antenna theory. To simplify remaining equa-tions, the coupling constants (7rn 3p/X) will be absorbedinto the video signal profile V(t) to give the phase re-tardation induced in the light field by the sound per unitdistance of coupling. This definition of V is closelyrelated to the V parameter of Klein and Cook15 :

(-rf 3 ) S(XZt) = W(Z)V (t - (6)

For the usual modulator field profiles,

W(z) = rect(z/L) = 1 for (z) < L/2,

0 otherwise,

This equation defines the Bragg angle inside the ac-oustooptic medium. The Bragg angle measured outsidethe modulator is given by the same equation, exceptthat the index of refraction of air is substituted for theindex of the modulator medium. The Bragg angle in-side the modulator will be assumed hereafter. Thisangle gives maximum coupling between the incidentand the deflected light beams. In the geometrical opticsmodel the incident light field amplitude Ein(x,z) can begenerated from its amplitude profile Din(x), defined onthe z = 0 line by simple ray projection:

Ein(x,z) = Din(x + zhnragg), (2)

which assumes that the Bragg angle is small enough thatthe sine of the Bragg angle may be replaced by the angleitself. For the usual modulator field profile

D n(x) = exp[-(x/w) 2], (3)

where w = radius of the incident light beam measuredto the l/e2 intensity point. This corresponds to aTEMOO laser mode.

where L = length of the piezoelectric transducer gen-erating the sound field. The desired scattering equa-tion results when the above equations are combined:

D,)ut(x,t) = dz'Dn(x + 2 z'OBragg)W(Z')V (t- x + ZOBragg);! ~~~~~~~~~~~~~~~V

(8)

see Fig. 4 and Sec. IV of Ref. 16.

IV. Temporal Response

A. Quadratic Invariant ResponseThe demodulation of the video signal from the optical

carrier occurs with a square law detector followed by alow pass filter. The measure of modulator response willbe the power P(t) of the deflected light beam. Detailsof the spatial distribution of this power will be consid-ered in detail in a future article. They will be sup-pressed in the present discussion.

P(t) = dx Dout(xt) 12. (9)

February 1977 / Vol. 16, No. 2 / APPLIED OPTICS 509

x axis

(4)

(7)

THE ACOUSTO-OPTIC MODULATOR TEMPORALRESPONSE KERNEL R (tt 2)

L BRAGG - rFOR W 2 , <TQUADRATIC WARIANT RESPONSEMODEL

Fig. 5. Isometric projection of the temporal response kernel. In thelimit of low scattering efficiency, in which the incident light beam islittle affected by the acoustooptic coupling, the power in the deflectedlight beam is a quadratic invariant function of the video signal am-plitude. The modulator's temporal response is completely defined

by a two-dimensional response kernel, shown here in isometricperspective.

of the response kernel profile is given in Fig. 5 for LOBragg

= /27rw, corresponding to comparable risetime contri-butions from the incident light beam thickness w andfrom the sound field depth L. The response kernel iscentered on t, = t2 = 0, with a gradual falloff along thet = t2 axis and a rapid falloff along the t = -t 2 axis.The gradual decay along the t = t2 axis will dominatethe response of the modular, with the rapid decay alongthe t = -t 2 axis introducing small shifts in this re-sponse.

B. Temporal Response in the Thin and the ThickSound Field Limit

In the thin sound field limit, defined by (LOBragg/W)- 0, the response kernel collapses into a single dimen-

sional impulse response function:

R(tl,t2) - Ithin(tl)641 - t2)

as [(L0Bragg)/W] - 0.

P(t) = 3' dt'Ithjin(t')[V(t - t)]2,

Ithin(t') - ID j, (Wt) 12,

In the low scattering efficiency limit the scattered lightpower P(t) is a quadratic invariant function of the videosignal amplitude V(t), defined as

P(t) = J' f' dtldt2R(t - tt - t)V(tl)V(t2 ). (10)

The two-dimensional kernel R(tl,t2 ) constitutes acomplete specification of the temporal response of theBragg cell to an arbitrary video signal V. This kernelis defined by the spatial profiles of the incident light andsound fields:

R(tlt 2 ) = (_) J dxDi(2vti - x)(OBragg -

W ) Di,,(2t 2 - x)W ( OBragg

For the conventional modulator field profiles, the re-sponse kenel R has a closed form solution:

R(tl,t2 ) -( V2) exp [-v 2exp(t> -t<)2]

X [ erf (fi2vt< + LBragg)

w -\/2w

/erf -/2vt> LOBrEg\ for (t>-t<) LOBragg

\w v'-2w I V

0 otherwise,

(14)

(15)

(16)

where 6 = Dirac delta function; see Fig. 6. In this limitthe modulator responds like a linear invariant system,with an impulse response defined by the intensity pro-file of the incident light beam. This is the responsemodel proposed by Rosenthal' and by Korpel et al. 2

In the opposite limit of the thick sound field, definedby (LOragg/W) -a , the response kernel again collapsesinto a single dimensional impulse response function:

(17)R(tl,t2) - Ithick(t0)5( - t2)

where [(L0Bragg)/W] - al

Ithick(t') - I W(Vt'1/0ragg)12.

The impulse response in this limit issound field intensity profile.

I D n(VtI)l(12) -

where t> = larger of (t1,t 2 ) and t< = smaller of (t1,t2 ).The kernel R exhibits the fundamental symmetryrelation

R(tIt 2 ) = R(tsti), (13)

as can be seen from Eq. (10). For the conventionalmodulator profiles, the kernel also happens to satisfythe symmetry relation R(t1,t2 ) = R(-t 2 ,-t,). A plot

x axis

Sound: V(t.t')

xvt

sound

(18)

defined by the

z axis

Fig. 6. The thin sound field configuration.

510 APPLIED OPTICS / Vol. 16, No. 2 / February 1977

I

THE ACOUSTO-OPTIC MODULATOR TEMPORALRESPONSE KERNEL R(t 1, t2 )

FOR LBRAGG -2

QUADRATIC INVARIANT RESPONSE MODEL

t1 =-t 2 tl A =t 2

Fig. 7. Orthographic projection -of the temporal response kernel.The response kernel in this and in Fig. 5 corresponds to the conven-tional modulator profiles, i.e., a Gaussian incident light beam profile

and a rect-function sound field profile. -

0U,I-2z

z

en

-JwzWWU,

z0a.Wn

W

102RADIUS OF THERESPONSE KERNEL

101 l ALONG THE t t = uJ101 AXIS _

LI

10- °d -

/ RADIUS OF THE <2-if RESPONSE KERNEL I ~- ALONG THE t = -t . AXIS ' 2

10-3 ' z |

102 O P 100 101 12

(LBRAGG)\ W

Fig. 8. Ellipticity of the temporal response kernel-the temporalresponse regimes. A key parameter is the ratio (LOBragg/W), whichis a measure of the relative contributions of the sound field and theincident light field in determining modulator performance. Theimpact of this parameter on the shape of the temporal response kernel,as measured by the radii to the 50% points along the two symmetryaxes, is given in this figure for the conventional modulator profiles.Note that the length greatly exceeds the width in both the thin and

the thick sound field limits.

C. Approximate Temporal Response in theIntermediate Sound Field Thickness Regime

The temporal response model in the thin and thethick sound field regimes is attractive because the re-sponse to an arbitrary video signal can be determinedwith a minimum of. computational effort. A singleconvolution integral is required, which can be mosteasily evaluated in Fourier transform space with the aidof the Cooley-Tukey Fast Fourier Transform (FFT)

algorithm.17 The intermediate regime requires a two-dimensional convolution, which again can be most easilyevaluated in Fourier transform space. However, thelevel of computation which is required is significantlygreater than for a linear invariant system. A linearinvariant model of the modulator which accuratelyapproximates the temporal repsonse in the intermediatesound field thickness regime would be most useful.

In both the thin and the thick sound field regimes, theresponse kernel took the form R(t,,t 2 ) - I(t1)(t, - t2 ).The response kernel deviates most strongly from thislimiting form when LOhragg =

1/2wrw. Even here, how-ever, the width of the response kernel along the tl = t2

axis (as measured to the 50% points) is only one-thirdof the width along the t, = -t 2 axis (see Figs. 5, 7, and8). This suggests that the temporal response may beapproximated by

P(t) = 3| dt'R(t - tat -t)[V(t)]2 (19)

with R defined by Eq. (11). For the conventionalmodulator profiles, analytical expressions for the re-sponse to various video signals are difficult or impossibleto obtain because of the error functions appearing in Eq.(12). Since the gross characteristics of the impulseresponse, such as its width, are more important for de-termining the modulator response than the fine details,such as the sharpness of the shoulders of the impulseresponse function, an additional approximation may beinvoked. This approximation is to replace the R(t',t')impulse response with an equivalent Gaussian, whichis scaled to give the correct 3-dB down frequency in theMTF. The scaling factor is given in Fig. 9. The mod-ulator response to a square pulse video train in thisapproximation is shown in Figs. 10 and 11.

In a quadratic invariant system, a pure harmonicvideo signal V(t) = cos(27rfvt) induces an output powerP(t), which contains a dc term and a pure harmonic

0I-0U-

zW

U)

I-enWW,

2.0

1.8

1.6

1.4

1.2

1.0

00 3.01.0 2.0

( LO BRAGG )

- -W I

Fig. 9. The response time adjustment factor. The temporal re-sponse of the modulator for the conventional profiles is closely ap-proximated by the modulator response in the thin sound field limit,but only if the temporal intervals are scaled by a factor which accountsfor risetime contributions arising from the sound field depth. This

figure corresponds to Fig. 5 in Ref. 13.

February 1977 / Vol. 16, No. 2 / APPLIED OPTICS 511

MAXIMUM DEFLECTEDPOWER MAX:

4 1/f

LIGHT MODULATOR IMPULSERESPONSE FUNCTION

IIL9 TIMEr- U/ -1 I-

f = PULSE REPETITION RATEb = PULSE DUTY CYCLE

MINIMUM DEFLECTED LIGHT POWER PMIN:

L o BRAGG

- _ ~DC RESPONSE TERM

X ~ONE DIMENS-\KRESPONSE KERNEL

R( t 1 , t 1 ) MODEL

GAUSSIAN RESPONSENSAPPROXIMATION

VIDEO SIGNAL FREQUENCYTWO DIMENSIONALRESPONSE KERNELR (t1,t2 ) MODEL

MAXDYNAMIC CONTRAST RATIO = -MIN

Fig. 10. The dynamic contrast ratio for a square pulse video train-thin sound field limit. The quadratic invariant temporal responsein the limit of a thin sound field (LOBragg/W - 0) reduces to a linearinvariant response in which the deflected light power is defined bythe overlap of the incident light intensity profile with the acoustic

video stream power flowing past the light.

Fig. 12. The acoustooptic modulator MTF. Two linear invariantmodels of the modulator's temporal response are proposed in the text.In this figure the MTFs associated with these approximate models

are compared with the MTF for the quadratic invariant model.

LO BRAGG - r T _W = 2 4W~~

ONE DIMENSIONALz RESPONSE KERNE LO R (t 1j 1 ) MODEL

THE RISETIMESARE / L 1O COMPARALE / 11 T TWO DIMENSIONAL- COMPARABLE 9%//11 11 RESPONSE KERNEL. WITHIN 7°/, 3%/ R(tl,t 2) MODELa -J/ 1

1,1

0

a:U,

z000

zin

0.20 0.25 0.30 0.35 0.40 0.45

VIDEO FREQUENCY - RISETIME PRODUCT

Fig. 11. Dynamic contrast ratio vs video frequency for a square pulsetrain.

TIME

Fig. 13. Step function video-modulator risetime. The step func-tion response and the risetimes for one of the two approximate models

is compared with the response for the quadratic invariant model.

term cos(4 ir fat). The MTF is defined as the ratio ofthe pure harmonic power response to the dc response.Note that, in contrast with a linear invariant system, thedc response for a quadratic invariant system is a func-tion of the video frequency f The results of a com-puter study of the MTF and the risetime for the two-dimensional R(t,,t 2 ) model, the one-dimensional R(t, t)model, and the Gaussian impulse model are reportedin Figs. 12 and 13. Note that the Gaussian model givesa closer fit than the one-dimensional R(tl,tl) model.This should not be surprising since a numerical fit isrequired in the Gaussian model, but no fit is needed inthe R (t l,ti) model. A table of the modulator responseto common video signals for a Gaussian impulse is in-cluded (Table I).

512 APPLIED OPTICS / Vol. 16, No. 2 / February 1977

____7

Table 1. Temporal Response of the Acoustooptic Modulator for Common Video Signals: Thin Sound Field Limit

r+o° P(t) = deflected light powerP(t) = dt' I(t - t') [ V(t')] 2 V(t) = video signal amplitude

I(t) = modulator impulse response function

u = speed of sound in the acoustooptic mediumI(t) = exp[-2(vt/w) 2 ] w = radius of the incident light beam to the

1/e2 intensity point

V(t) P(t) Comments

step (t) =lfor t> 0 10 otherwise - [1 + erf(V2vt/w)] Risetime = 1.282 (w/v) (see Fig. 13)

cos(27rf~t) -8l + cos(47rft) MTF = exp[ (7rfvw/v)2] (see Fig. 12)

exp [- (7rfvw/v)2]}

rect (t/T) = 1 for ItI < t - erfkI2v( + -TI/w2 2(L 2 2/

0 otherwise - 1 - T) )-erf /2v t-T

+oo +oo 1E rect[(fVt +j)/b , -(erf/ 2v(f t + j + 1/2b)If1,w)] Dynamic contrast ratio

square pulse train with -erf[y'2v(ft + j- 1/2b)/(fvw)]} erf[v(1 + b)(v'2fw) -erf[v(1 - b)/(\/2fvw)Ipulse repetition rate f erfrvb/(./2fw)and duty cycle b ( 1V

(see Fig. 11)Note: To include sound field depth contributions, replace w in the above with (see Fig. 9)

W W~ - 0.0 7 4 0( L0a2) + 37

where L = transducer length and Bragg = Bragg angle associated with the acoustic carrier.

0

I

=

0=I-

Le BRAGGW

Fig. 14. Light throughput efficiency. Fast modulator response canbe realized by reducing the incident light beam radius w, but only atthe expense of a degraded coupling efficiency into the deflected lightbeam and distorted profile of the deflected light compared with the

incident light beam profile.

I-U,

ZwI-

NORMALIZEDLL913RAGG tr g;

If

/// BEA \\\ \\

PROFILE >

DISTANCE

Fig. 15. Deflected light beam profile distortion. The loss in lightthroughput efficiency and the distortion of the deflected light beamprofile are evidence of a filtering action of the modulator on thepropagation angle spectrum of the incident light beam. As the inci-dent light is more strongly focused, its propagation angle spectrumwidens and starts to exceed the passband associated with the modu-lator. The modulator's angular response is defined by the angularspectrum of the sound field. Note that the relative sizes shown in thisfigure will be reversed a large distance downstream of the modulator

because of diffraction effects.

February 1977 / Vol. 16, No. 2 / APPLIED OPTICS 513

V. Light Throughput Efficiency and Profile Distortion

The modulator response is fastest when the incidentlight beam diameter is the smallest. Focusing of theincident light beam to achieve fast response, however,degrades the light throughput efficiency and distortsthe profile of the incident light beam. An extremelyuseful model of this degradation is to picture the mod-ulator as a linear bandpass filter operating on thepropagation angle spectrum of the incident light beam.Maximum modulator response occurs when the lightenters at the Bragg angle associated with the soundcarrier. However, because of the finite extent of theincident light profile, a spectrum of propagation anglesis associated with the incident light. Focusing of theincident light beam broadens this spectral range. Whenthis spectral range exceeds the angular passband asso-ciated with the modulator, then the modulator can nolonger couple those plane wave components whose di-rection deviates too strongly from the optimum Braggangle. In the low scattering efficiency limit, this pass-band is defined by the sound-field angular spectrum.This can be seen from Eq. (8) with a quiescent videosignal:

Dout(x) X dz'Di.(x + 2z'0Brngg)W(Z'). (20)

This has the form of a convolution operator. In Fouriertransform space, the deflected light spectrum is givenby the product of the incident light spectrum and thesound field spectrum. Analytical solutions exist for thelight throughput efficiency and the deflected lightprofile for the conventional modulator profiles:

D,ut(x) [erf (X + LBrag) _ erf (X - LOBragg)] (21)

E (LOBragg) =\/ (LWr) erf (V/2 LOBragg)

(2L2 ) [ exp (-2L20Bragg

2 )], (22)

E = light throughput efficiency.

These solutions are plotted in Figs. 14 and 15.The basic concepts presented in this section were

given previously by Gordon'0 and Cohen and Gordon."1

However, they did not use the extremely suggestivelanguage of linear filter response, nor did they haveaccess to Eq. (8). The angular filter concept extendseven into the high scattering efficiency regime (witnessthe work of Chu and Tamir9 ), but the angular responsebegins to shrink as the sound field strength increases.Details will be given in a future paper.

VI. SummaryThe phenomenological geometrical optics model of

the acoustooptic modulator presented in this paper iscapable of accounting for sound field depth contribu-tions as well as incident light beam contributions to themodulator performance. This model can be applied toarbitrary field profiles and video signals with a mini-mum of computational effort. The temporal responsein the low scattering efficiency regime is a quadratic

invariant function of the video signal amplitude, withthe modulator response completely defined by a two-dimensional response kernel. This response kernel isdefined in turn by an overlap integral of the incidentlight and sound field profiles, similar to a convolutionintegral. For the conventional modulator field profiles(Gaussian incident light beam and rect function soundfield), the modulator responds to a good approximationlike a linear invariant system with a Gaussian impulseresponse. This linear system operates on video drivepower to generate deflected light power. The risetimefor the conventional modulator profiles is approxi-mately

Tr - 1.282 l [ - 0.0740 (LOBr0ag) + 0.1237 (LOBrag)2]

to within 10% over 0 < [(LOBragg)/W] < 3. The degra-dation in light throughput efficiency and distortion ofthe deflected light beam profile which occurs with fo-cusing of the incident light beam results from a filteringof the incident light propagation angle spectrum by amodulator with a limited angular passband.

The author gratefully acknowledges the contributionsand support of Len DeBenedictis and John Lucero.The author also wishes to thank a referee for ac-quainting him with the Rosenthal work and for severalconstructive criticisms.

References1. A. H. Rosenthal, IRE Trans. Ultrason. Eng. 8, 1 (1961).2. A. Korpel, R. Adler, P. Desmares, and W. Watson, Proc. IEEE

54, 1429 (1966).3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,

New York, 1968), Chap. 2.4. I. C. Chang, IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).

5. N. Uchida and N. Niizeki, Proc. IEEE 61,1073 (1973).6. C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE

53, 1604 (1965).7. R. A. Adler, IEEE Spectrum 4,42 (1967).

8. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,

1965), Chap. 12.9. R. S. Chu and T. Tamir, J. Opt. Soc. Am. 66, 220 (1976).

10. E. I. Gordon, Proc. IEEE 54,1391 (1966).11. M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 44, 693

(1965).12. R. W. Dixon and E. I. Gordon, Bell Syst. Tech. J. 46, 367

(1967).13. D. Maydan, IEEE J. Quantum Electron. QE-6, 15 (1970).

14. J. F. Nye, Physical Properties of Crystals (Clarendon Press,Oxford, 1957).

15. W. R. Klein and B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14,

123 (1967).16. A. Korpel, in Applied Solid State Science, Vol. 3 (Academic, New

York, 1972), pp. 71-180.17. W. T. Cochran et al., IEEE Trans. Audio Electroacoust. AU-15,

45 (1967).

514 APPLIED OPTICS / Vol. 16, No. 2 / February 1977