Temporal response of the acoustooptic modulatorin the high scattering efficiency regime
Richard V. Johnson
An efficient and rigorous algorithm is proposed for analyzing the temporal response of the Bragg acoustoop-tic modulator in the high scattering efficiency regime. Computer studies of this model successfully predictpulse profile asymmetries that have been observed experimentally and cannot be predicted by the usualGreen's function (small signal) models of the acoustooptic coupling. The technique of predistorting theelectronic video signal by passing it through a nonlinear electronic network to linearize the modulator re-sponse is effective only for slowly varying video signals. Residual nonlinearities appear for rapidly varyingvideo signals.
1. Introduction
Acoustooptic coupling is the key mechanism in sev-eral commercially important devices including modu-lators, deflectors, and spectral filters. Mathematicalmodels of acoustooptic coupling typically fall into twocategories. The first category consists of coupled-modeequation studies, such as the Raman-Nath equations.14The second category consists of low scattering efficiencymodels such as the Green's function studies of Gordonand colleagues at Bell Laboratories.7-13 Neither cate-gory is suitable for a rigorous study of the temporal re-sponse of a modulator to an arbitrary video signal in thehigh scattering efficiency regime.
Consider the modulator response to a square pulseas reported by Magdin and Molchanov.14 They ob-served an asymmetry in pulse shape, with an overshootthat appears only on the leading edge of the pulse, anda risetime that is shorter than the falltime. This occurswhen scattering into the +1 diffraction order. Whenscattering into the -1 diffraction order, the pulse profileis time reversed (see Fig. 1). Similar observations havebeen made in our laboratory.
The standard temporal response models of themodulator are incapable of predicting this asymmetry.The pulse risetime should always be the mirror imageof the pulse falltime, according to the low scatteringefficiency models. Only a high scattering efficiencymodel, which accounts for the nonlinear dependence of
The author is with Xerox Electro-Optical Systems, Pasadena,California 91107.
the scattered light on the sound field, can successfullypredict the observed pulse asymmetry.
The coupled-mode equation analysis is capable ofmodeling accurately the nonlinear dependence on thesound field. However, the coupled-mode analysis canbe applied rigorously only when the sound field is pe-riodic.
The modulator drive electronics generate a periodicacoustic carrier signal. This periodic carrier is thenamplitude modulated by the video signal stream. Aslong as the video signal is quiescent, the resulting soundfield will indeed be periodic, and the coupled-modeanalysis would be rigorously appropriate. When thevideo is not quiescent, the video modulation destroysthe periodicity of the sound field violating a crucial as-sumption in the derivation of the coupled-mode equa-tions. Therefore, the coupled-mode equations cannotbe applied to the study of the temporal response of amodulator to an arbitrary video signal.
A restricted class of video signals exists, which per-mits rigorous application of the coupled-mode equa-tions. This class is characterized by a periodic videosignal with a repetition rate that is a subharmonic of theacoustic carrier frequency. Figure 1 compares the ex-perimental pulse response observed by Magdin andMolchanov with the theoretical pulse response pre-dicted by this rigorous coupled-mode analysis. Notethe excellent agreement. The rigorous analysis cor-rectly predicts the pulse overshoot at the trailing edgeand the longer risetime when scattering into the -1diffraction order. An efficient algorithm for solving thecoupled-mode equations is discussed in Sec. II. Nu-merical solutions for several important classes of videosignals are presented in Sec. III. This concludes a seriesof three articles on the temporal response of the ac-oustooptic modulator.
Fig. 1. Modulator response to a square pulse video train.
II. Solving the Scattering Equation
This section presents an efficient algorithm for cal-culating the light field structure that results when afocused incident light beam is scattered off a periodicpropagating sound field. The section is divided intofour subsections. In the first, the light propagationequation is reduced to a denumerably infinite set ofcoupled first-order differential equations, when theincident light beam is a plane wave and the sound fieldis periodic. This is a brief outline of the analysis ofKlein et al. 1 to which the reader is referred for a morecomplete discussion. The set of coupled-mode equa-tions is completely characterized by a Hermitian cou-pling matrix. The solution of the coupled-modeequations requires the diagonalization of this matrix.In the second subsection, this matrix is derived for thespecial case of a pure harmonic (unmodulated) soundfield, corresponding to the Raman-Nath coupled-modeequations. This is compared with the coupling matrixfor a modulated carrier in the third subsection. Finally,in the fourth subsection, the analysis is extended frompurely plane wave incident light beams to focused lightbeams.
geneous perturbation driven by the photoelastic cou-pling. The sound field is assumed to be periodic instructure. The angular frequency a* and the waveconstant K* are associated with the fundamental periodof the sound field. Since At'(x,t) is periodic, it can beexpanded into a Fourier series:
ju'(x,t) = 2:j cosU(w*t - K*x) + j]. (3)
Consider a plane wave light beam incident upon thesound field with propagation direction 0 and angularfrequency w. The scattered light will be a superpositionof plane waves of the form
E(r,t) = 1 {n(z) exp i[(w + nw*)t-K * r] , (4)
where
K. r = ,uoK(z cosO + x sino) + nK*x. (5)
Klein et al. 1 have demonstrated how to reduce this setof equations to an infinite set of coupled-mode differ-ential equations with constant coefficients
2iL (dtj/dz) + 2Vk [j - k exp(iak)
+ (j + k exp(-ibk)] = 4jNU - 2a)tj, (6)
where the coupling parameters (vj,N,,a) are definedby
v = K11jL/cosO,
N = K*2L/(4r 1oK cosO),
(7)
(8)
a = -( 11oK/K*) sinO, (9)
where K is the wave constant of the light field, and L isthe depth of the sound field. [Equations (1)-(9) havedirect counterparts in Ref. 1, save that Klein et al. useQ = 4N for the coupling parameter that scales thesound field depth.] These coupled-mode equations canbe expressed in the compact matrix notation
The genesis for the following temporal responsestudies is the wave equation that describes light prop-agation through an optically inhomogeneous medium
v2E = [xt)/c] 2(( 2E/1t 2 ), (1)
where E is the electric field of the light beam, (x,t) isthe index of refraction of the acoustooptic medium, andc is the speed of light in vacuo. The index of refractionfor scattering off a propagating sound wave has theform
11(x,t) = 110 + g(w*t - K*x),
Note that the coupling matrix Aj,k is Hermitian, as-suming the scattering medium to be free of absorp-tion.
B. Raman-Nath Coupled-Mode Equations
Consider a sound field structure that has a pureharmonic profile
,u'(x,t) = cos(co*t - K*x). (12)
The Fourier series expansion of A' has the especiallysimple form
(2)
where AO is the index of refraction that exists in theabsence of the sound field, and A' is a small inhomo-
Fig. 2. Coupling matrix for the Raman-Nath cou-pled-mode equations. The dashed entries indicatezero coupling. The ellipsis indicates that the matrixis infinitely large. Practical calculations requirethat the matrix be truncated to finite size, such as
the 7 X 7 array shown on the right.
Then the coupling matrix has the form shown in Fig. 2.The matrix is infinitely large, as indicated by the ellipsisin Fig. 2. Only three diagonals are nonzero. All otherdiagonals contribute zero coupling, which is indicatedin the figure by a dash. The main diagonal consists ofterms Dj:
Dj = 4irjN(j - 2a). (15)
These main diagonal terms exist in the absence of thesound field. They govern the phase shift that occursnaturally in propagation of the. plane wave components.The coupling induced by the sound field is confined tothe two adjacent diagonals containing elements v.
The coupled mode equations which correspond to thismatrix are the Raman-Nath equations.3
The solution of the coupled mode equations proceedsmost efficiently by diagonalizing the coupling matrix.1 5
Since the coupling matrix has infinite dimensions, itmust be truncated to keep the problem tractable.Physically, this is equivalent to ignoring the scatteringinto diffraction orders higher than some specified limit.This truncation need not introduce significant error ifenough coupled modes are included. As a guideline, theRaman-Nath equations can be solved to excellent ac-curacy (e.g., 1 part in 106) for typical Bragg modulatorparameters by truncating to a 7 X 7 matrix (i.e., con-sidering only the -3 to the +3 diffraction orders), asindicated in Fig. 2. This matrix can be diagonalizedwith minimal computer resources. This technique isfar more efficient than brute force numerical integrationschemes, such as the Runga Kutta, which have typicallybeen employed in the literature.2
C. Modulated Carrier Equations
Consider a sound field consisting of a harmonicacoustic carrier which is amplitude modulated by avideo signal. Let the video signal V(t) have the form
V(t) = Vo[1 + M cos(w*t)], (16)
where M is the modulation depth. Assume that thevideo signal frequency is the third subharmonic of theacoustic carrier. For example, the carrier could be 90MHz, with a 30-MHz video. The propagating soundwave will induce an index perturbation g'(x,t) of theform
S
.
.
- D,7 v m
m - D-6 Y
I v m D. -
. , .- -v * - m v
. . . I.
. . . . I
. . . .
. . . . .
. . . . .
. . . . .
D. - m ,
m . .1 m.
v m D 2-
*m v m )3- m v
D,2- m v mD
- DM v t
S
Fig. 3. Coupling matrix for a modulated acoustic carrier. The videomodulation splits the basic Raman-Nath matrix into finer structureand spreads the coupling among more diagonals. A direct relationexists between the spectrum of the modulated sound carrier and the
distribution of coupling terms among the diagonals.
The coupling matrix will now have the form shown inFig. 3. The main diagonal terms Dj remain unaltered.The adjacent coupling diagonals of Fig. 2 have split intoa more complex structure in response to the videomodulation. Note that the index j now corresponds tothe video subharmonics w*, rather than the diffractionorders associated with wc as was the case for theRaman-Nath coupling equations. In the present ex-ample, three diagonals must be retained for every di-agonal of the Raman-Nath matrix. Thus, truncationto a 21 X 21 array is required to achieve numerical ac-curacy comparable with the 7 X 7 Raman-Nath matrix
of Fig. 2. The coupling matrix can be diagonalized bythe same techniques used to diagonalize the Raman-Nath coupling matrix shown in Fig. 2.15
A modification of the definition of the coupling pa-rameters (N,a) is appropriate. Klein et al.' definedthese with respect to the fundamental sound field fre-quency w*, because they were analyzing the acous-tooptic scattering off of a periodic sound carrier, whichwas not necessarily a pure harmonic (e.g., a square ortriangular profile carrier). Modulation of this carrierwas not considered. The emphasis in the present articleis on a video-modulated carrier with carrier frequencyWc. The carrier frequency will generally not be thefundamental frequency * of the modulated sound fieldbut rather an overtone of this fundamental frequency.Rather than defining the coupling parameters (N,a)with respect to the fundamental w*, these parameterswill be defined with respect to the acoustic carrier'sfrequency wc. This requires a slight revision of thecoupled-mode equations
Dj = 47rjN(] - 2a) 4irjN(j/J - 2a), (20)
where J is the overtone of the video fundamental thatcorresponds to the carrier frequency. In the exampleof Eq. (17), J = 3.
D. Focused Incident Light Beam
The discussion up to now has assumed that the lightbeam, which is incident upon the sound field, is a planewave. In a typical modulator configuration, the lightbeam will be focused to reduce the risetime. This fo-cused light beam contains a spectrum of plane wavecomponents, each propagating at a different angle. Theanalysis of the scattering of a focused light beam is asimple extension of the analysis of plane wave lightscattering. The key to the analysis is the linearity ofthe coupling Eq. (10) in the light field E. Because ofthis linearity, the principle of superposition applies.The incident light beam is decomposed into its planewave components by Fourier analysis. The scatteringresponse to each plane wave component is calculatedby diagonalizing the appropriate coupling matrix asdiscussed above. These scattered light fields, whichcorrespond to the component plane waves, are thensummed to determine the resulting scattered light fieldcorresponding to the focused light beam. Chu andTamir have published this analysis for scattering of aTEMOO Gaussian profile light beam off of an unmodu-lated acoustic carrier. 6-18
In the present analysis, TEMoo light is assumed to befocused to a waist inside the modulator. The size of thewaist w is scaled by configuration parameter A (whichwas introduced in Refs. 12 and 13 and differs fromMaydan's ratio a by a factor of r/4 in Ref. 8):
A = K*L/(2,uoKw), (21)
where w is the incident light beam radius to the 1/e2
intensity point at focus.
0
TIME
Fig. 4. Modulator response to a triangular video train.
Ill. Numerical Solutions
The first signal to be studied is a square pulse trainof 50% duty duty cycle shown in Fig. 1. The top pulseprofiles (a) correspond to the coupled-mode theory andscattering into the -1 diffraction order. Note the ov-ershoot on the trailing edge. The risetime is slower thanthe falltime by almost 40%. The modulator configu-ration is defined by the parameters A = 1 and N = 1.The video pulse repetition rate is 1/15th of the acousticcarrier frequency. The bottom two pulse profiles areoscilloscope traces taken by Magdin and Molchanov.14Pulse (b) corresponds to scattering into the -1 dif-fraction order, while pulse (c) corresponds to scatteringinto the +1 diffraction order.
The next video signal to be studied is a triangularpulse train suitably predistorted by a nonlinear elec-tronic network. The combination of electronic pre-distortion and transfer function of the modulator shouldachieve linear modulator response, at least for slowvideo signals. The numerical solution is presented inFig. 4. The result is a reasonably linear response withthe tips of the triangle blurred by the finite responsetime of the modulator. However, a close inspection ofthe computer data shows that the right-hand slopedrops 12% faster than the left-hand slope, even thoughthe original video signal was perfectly symmetrical.The asymmetry, which is observed in the modulatorresponse in Figs. 1 and 4, can only occur because of re-sidual nonlinearities in the acoustooptic coupling. Thisasymmetry will appear only with rapid video modula-tion and cannot be eliminated by simple electronicpredistortion. The modulator configuration is again N= 1 and A = 1. The video repetition rate is 1/10th theacoustic carrier frequency.
The final video signal to be considered is a pure har-monic video, again predistorted by a nonlinear elec-tronic network to linearize the modulator response. Astandard test of the linearity of a system is to measurethe harmonic content of its output when a pure har-monic signal is applied to the input. If the system weretruly linear and shift invariant, the output would consistsolely of the desired harmonic. If any nonlinearitiesexist, their presence would be betrayed by the existence
Fig. 5. Modulator response to a pure harmonic video signal.
of spurious harmonics in the signal output. Indeed, astandard measure of the linearity of a system is thedynamic range that separates the desired harmonicoutput from the next largest spurious harmonic. Figure5 shows the numerical results for A = 1 and N = 1 anda video repetition rate that is 1/10th of the acousticcarrier frequency. The numerical results show thatresidual nonlinearities exist in the system consisting ofan electronic predistortion network and an acoustoopticmodulator. The dynamic range between the desiredharmonic output and the next largest harmonic outputis less than 10:1 as shown in Fig. 5. These residualnonlinearities arise because the modulator output de-pends not only on the instantaneous value of the videosignal but also on past and future values within a tem-poral window, which is defined by the Bragg angle walkoff mechanism discussed in Ref. 12. All video signalshave been scaled to drive the modulated light beamfrom zero intensity to saturation.
IV. Conclusion
An efficient and rigorous analytical technique hasbeen proposed, which enables the temporal response ofthe acoustooptic modulator to be studied in the high
scattering efficiency limit. This analysis has success-fully predicted profile asymmetries that have been ob-served experimentally but cannot be modeled with a lowscattering efficiency analysis. The technique of pre-distorting a video signal by passing it through a non-linear electronic network to linearize the modulatorresponse has been shown to be effective only when thevideo signal varies slowly compared with the acousticcarrier. When the video varies rapidly, residual non-linearities in the acoustooptic coupling, which cannotbe compensated by a simple nonlinear network, willdistort the modulator response.
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