Temporal response of the acoustooptic modulator:physical optics model in the low scatteringefficiency limit
Richard V. Johnson
The Green's function model of the acoustooptic modulator proposed by E. I. Gordon et al. is reformulated inFourier transform space to simplify the mathematics and to underscore the physics. Numerical studies ofresponse to sinusoidal video signals and to square pulse trains indicate that the modulator can be approxi-mated by a linear invariant model with a suitably scaled Gaussian impulse response. An angular scatteringwindow analogy is proposed to explain the characteristics of the modulator.
1. Introduction
An acoustooptic modulator (Fig. 1) converts anelectronic video stream into a corresponding temporalmodulation of the optical power. The temporal re-sponse is defined by the overlap of the light and soundfields. The temporal response is primarily governedby the acoustic transit time, as discussed by Hance andParks' and by Korpel et al. 2
A simple and powerful description of this first-ordertemporal response mechanism is the linear invariantthin grating model (Fig. 2). The optical modulation,according to this model, is the convolution of the videosignal stream with the incident light intensity profile.This will be called the thin grating model because itassumes that the sound field has negligible depth (L inFig. 2) along the direction of light propagation. Thismodel has been proposed variously by Rosenthal,3Korpel and associates,2 '4 and Lambert (in the contextof an optical spectrum analyzer).5 One importantconsequence of this model is that good modulation fi-delity (e.g., fast rise time) requires the light beam inci-dent upon the modulator to be focused to minimize theacoustic transit time.
Unfortunately, in its extreme simplicity, this modelignores several important second-order effects:
(1) The scattering efficiency falls off when the inci-dent light beam is focused.
The author is with Xerox Electro-Optical Systems, Pasadena,California 91107.
(2) The scattered light beam profile becomes de-formed with respect to the incident light profile withfocusing.
(3) The actual temporal response is slower than thatpredicted by the thin grating model. An additionalacoustic transit time, arising from the finite sound fielddepth (Fig. 2), must be considered.
All these second-order effects arise because the soundfield is not infinitesimally thin, but rather behaves likea thick diffraction grating.
A more rigorous model of the acoustooptic coupling,capable of describing these second-order effects, is theGreen's function integral, proposed by Gordon andcolleagues.6-10 The Green's function integral resultsfrom the Born approximation to the light wave equationfor propagation in an inhomogeneous medium. TheBorn approximation is the first-order term in a per-turbation series expansion in powers of the sound field.As such, it is strictly accurate only in the limit of lowscattering efficiency, whereas a modulator is normallydriven in the high scattering regime. However, it ren-ders the mathematics far more tractable. Gordon,6Maydan,7 and colleagues8 -10 have demonstrated theBorn approximation to be a most instructive tool inanalyzing modulator performance.
An alternative model compares the acoustoopticcoupling (in the low scattering efficiency limit) to a threeparticle scattering event.1-13 Gordon has applied thisanalogy to reveal the more subtle characteristics of theacoustooptic coupling in modulators and deflectors.6However, Gordon kept his arguments on a heuristiclevel, with no apparent connection to the Green'sfunction integral mathematics. Subsequently, Korpeldemonstrated the intimate duality between these twomodels.14 When the fields in the Green's function in-tegral are Fourier decomposed into plane wave com-
Fig. 1. Block diagram of the acoustooptic modulator.
FINITE SOUND FIELDDEPTH MODEL
THIN GRATING MODEL ( THICK GRATING MODEL)(L-O) (L o)
SOUNDFIELD//C/ ., ... :
.,~~~~~ SOUND.
TRANSDUCERS-"I.
Fig. 2. Definition of the Thin-grating model.
thin grating model. One goal of the present paper is topropose a modified thin grating model which accountsfor finite sound field depth contributions to the acoustictransit time. Response predictions of this model willbe compared against predictions of the more rigorousmodels. Excellent agreement has been found. Thecomputer studies presented in this paper form a usefulsupplement to the rise-time studies of Maydan.7
The second objective of this paper will be to extendGordon's momentum matching arguments to derivegreater insight into the dynamics of the acoustoopticcoupling in a modulator. The concept of an angularscattering window will be proposed. This is -a synthesisof the scattering analogy with communications theoryconcepts.
The paper is divided into four sections. The scat-tering equation for the physical optics model is pre-sented in the second section. Computer studies of thetemporal response predictions of the three models arecompared in the third section. The angular scatteringwindow concept is proposed in the fourth section.
This paper is the second in a trilogy of articles on thetemporal response of the acoustooptic modulator. Thefirst article introduced the geometrical optics model.The third article will discuss coupling in the high scat-tering efficiency limit.
II. Equations of the Physical Optics Model
A. Scattering EquationAn excellent derivation of the scattering equation for
the physical optics model has been presented by Kor-pel.14 Korpel restricted his attention to pure temporalharmonic fields, but his derivation can readily be ex-tended to include nonharmonic fields16"17:
Gout(0out~out = Gin(Oinfin)Ga (Oafa)
ponents, the coupling equation reduces to a set ofphase-matching conditions. These phase-matchingequations bear a striking resemblance to conservationof energy and momentum equations describing athree-particle scattering event.' 3 Both the Green'sfunction integral model and its dual scattering modelwill hereafter be called the physical optics model.
A third temporal response model has recently beenproposed by the author-the phenomenological geo-metrical optics model.15 This model stands in rigormidway between the thin grating and the physical opticsmodels. The geometrical optics model was proposedto clarify the temporal response mechanisms and tointroduce the concept of a quadratic invariant system.However, the analysis of modulator response to an ar-bitrary video signal is just as cumbersome with thismodel as it is with the more rigorous physical opticsmodel. Hence, the geometrical optics model is notrecommended for response calculations.
The more rigorous geometrical and physical opticsmodels are necessarily far more cumbersome to applywhen calculating temporal response than the simpler
whereGoutGinGa = the temporal and spatial frequency
spectra of the scattered light, incidentlight, and acoustic field amplitudes,
OoutOinOa = propagation directions of the planewave components, and
foutfinfa = temporal frequencies.Appropriate coupling constants have been suppressed
for simplicity. The propagation angles OoutOin for thelight fields are measured counterclockwise from the zaxis (defined in Fig. 1), while the acoustic propagationangle Oa is measured counterclockwise from the x axis(also defined in Fig. 1). Attention has been restrictedto the scattering plane defined by the propagation di-rections of the light and sound fields. The fields areassumed to be uniform in the direction orthogonal to thescattering plane.
The scattering equation is subject to the followingphase-matching requirements:
TEMOO laser mode which is brought to a focus inside themodulator. This mode gives the best modulator per-formance.18 The amplitude profile will be Gaussian,with radius w measured to the I/e amplitude point.The Fourier transform of this Gaussian will also be aGaussian:
6i.(Oi.) = exp[-(i. - On)1/0.1],
Fig. 3. The diffraction orders of the scattered light beam. where
00 = X/(7rw), (10)
and 0 is the angular radius of the spectrum profile,measured to the le amplitude point, and Oin is the tiltangle of the incident light beam (see Fig. 3).
The sound field distribution is generally quite intri-cate (Fig. 4), but it is easily specified at the transducerboundary (x = 0). At this boundary, the sound field isassumed to be separable into the following terms:
S(x = 0,z,t) = W(z)V(t) cos(2i7rft),
whereS(x,z,t) = sound field in the modulator medium,
W(z) = spatial profile of the sound field at thetransducer boundary,
V(t) = video signal amplitude, andfc = acoustic carrier frequency.
Define the auxiliary spectral functions
Fig. 4. Stroboscopic schlieren photographs of the sound field.Shown in these photographs are end-on views of two representativeglass modulators, with the transducer on the right. These photo-
graphs were taken by the techniques described in Ref. 22.
k.ot cos(0Out) = kin cos(0in) - ka sin(0 0 ), (4)
where
W = dtV(t) exp(-2rift),
a( aOJa) = J, dzW(z) exp (2YEfcz sinOa) f-_ v I
(12)
(13)
where v(f 0) is the temporal frequency spectrum of thevideo signal, while a(0a,fa) is the scattering windowassociated with the modulator, as discussed in the thirdsection. With these auxiliary functions, the sound fieldspectrum may be written
where the asterisk denotes complex conjugation, t is theDirac delta function, and X is the wavelength of the lightbeam. The propagation angle spectrum profile in(0in)is obtained by Fourier transforming the light beamamplitude profile at focus. For example, consider a
v(fv) = V*(-h) (15)
where the asterisk denotes complex conjugation. Thisspectral symmetry results because the video signal is areal function. Figure 5(b) shows the spectrum of theacoustic carrier (before video modulation). The twoarrows denote Dirac delta functions. The modulationof the acoustic carrier by the video signal can be de-scribed mathematically by a convolution of the fre-quency spectra, giving the acoustic spectrum shown inFig. 5(c). The acoustic spectrum exhibits a doublesymmetry. The spectral profile for positive frequenciesis symmetrical about fa = fc. (This assumes a band-limited video signal: f < fc.) This symmetrical profileis repeated for the negative acoustic frequencies.
Fig. 6. Temporal frequency spectrum of the scattered light field.
Figure 6 shows the modulation of the light beam in-troduced by the acoustooptic coupling. The spectrumof the incident light beam is presented in Fig. 6(b),where again the arrows denote Dirac delta functions.Figure 6(a) is a repeat of Fig. 5(c). Figure 6(c) illus-trates the spectrum of the scattered light. The doublesymmetry exhibited in Fig. 5(c) has been destroyed inthe acoustooptic coupling. This will be discussed ingreater detail in Sec. IV. One symmetry remains in thisspectrum:
(16)
Because of this symmetry, attention can be confinedto the positive optical frequencies.
The same physical arguments which are illustratedin Figs. 5 and 6 for the temporal frequency spectrum canbe repeated for the propagation angle spectrum. Fromsuch arguments, one finds that the two components ofthe scattered light spectrum shown in Fig. 6 not only areDoppler shifted in temporal frequency with respect to
the incident light (and with respect to each other), butRUM they are also Doppler shifted in propagation direction
(Fig. 3). The scattered light component which isDoppler upshifted in temporal frequency with respectto the incident light beam is called the +1 diffraction
I 7 _order. Similarly, the component which is Doppler+ff downshifted is called the -1 diffraction order.
The modulator is usually designed for Bragg regime(i.e., thick grating) operation, which selectively couplesthe incident light energy into only one of the two scat-tered light orders (Fig. 7). This dominant diffractionorder is separated from the unscattered light and fromthe remaining vestigial diffraction order by spatial fil-tering (Fig. 1). This dominant diffraction order is the
-a desired modulated light beam.The following assumptions will simplify the scatter-
nd field. ing equation which results from the coupling of the in-cident light and sound fields. First, the acousticfrequencies are typically 6 orders of magnitude slowerthan the optical carrier frequency fin, so Dopplerbroadening of the optical wavelength X induced by the
M sound field is negligible. Second, the modulator usuallyhas an identical index of refraction for both the incidentand the scattered light fields, i.e., nin = nout n. Thisis called isotropic Bragg scattering. It will be assumed
+(' 0 fin hereafter. The alternative case of anisotropic Braggscattering is discussed by Dixon in Ref. 19. The as-sumption of isotropic Bragg scattering allows a Braggangle OB to be defined for acoustic frequency fa:
sinOB = Xfa/(2nv).
From the temporal response analysis, one finds
fa = dfc + Jv.
Hence two shift angles need to be defined:
OB = S0C + 0°,
where
0, = Xf0 /(2nv),
0v = Xfv/(2nv).
(17)
(18)
(19)
(20)
(21)
Finally, all angles associated with a modulator aretypically a fraction of a degree (on the order of 10 mrad).Therefore, all functions of angles will be expanded ina Taylor's series,and all terms higher than second order
Fig. 7. Bragg regime modulator. When the sound field is thick, theincident light energy is selectively coupled into only one of the several
will be dropped. [Equations (19)-(21) have anticipatedthis approximation.]
The resulting coupling equation is
G..t(O0utJou = G+1(000tfut) + G_1(0OutJ0out (22)
where
G±1(0O.tJ.u. = /2v(fv)a(00,J)e6i.(0i.) (23)
subject to the phase-matching conditionsc
fout = fl~ + f (24)
f = 4-c + f,, (25)
Oin = out T 20 - 20,,, (26)
Oa-= out T Oc - v. (27)
C. Hierarchy of Coupling Models
All three models of the acoustooptic coupling (i.e.,the physical optics, geometrical optics, and thin gratingmodels) lead to precisely the same coupling equations,namely, Eqs. (22)-(27). All three models differ solelyin the detailed structure of the scattering window a.
The scattering window for the physical optics modelis derived from a strict application of the above equa-tions:
a(Ooutf,,) = 4' dzW(z) exp(i * arg), (28)
where
-2 rzarg = (+f, + f,)(Oout F C- 0,,) (29)
V
The argument in the exponent is quadratic in the videofrequency rather than linear. Because of this quadraticdependence, the physical optics model is awkward toanalyze. The quadratic dependence arises because thephysical optics model explicitely describes Dopplerbroadening of the acoustic wavelength induced by thevideo modulation. This is exhibited in Eq. (29) by theterm (fc + f) rather than (fc).
When the video signal varies slowly compared withthe acoustic carrier, i.e., when f << f, this Dopplerbroadening of the acoustic wavelength can be neglected.The argument in Eq. (29) becomes linear in the videofrequency f. One consequence is that the scatteringequation [Eq. (23)], can be Fourier transformed backinto normal space to yield the coupling equation whichdefines the geometrical optics model [Eq. (8) in Ref.15].
The thin grating model results from the more rigorousmodels in the limit of an infinitesimally thin soundfield:
W(z) - (z) implies a(Outfv) 1 (30)
(thin grating model).
The physical implications of these mathematicalexpressions will be discussed in the fourth section of thispaper.
Ill. Computer Studies of the Temporal Response
A. Quadratic Invariant System Formalism
Consider an optical communications system with asquare law detector. The detector intercepts the far-field light distribution &out(Ooutt) and integrates overthe intensity profile. The detector window is assumedto capture all the dominant diffraction order but noneof the remaining orders. The output of the detector willbe a voltage proportional to the optical power P(t),rather than the scattered light amplitude gout:
P(t) = f d 0outl d6Out(,out,t) 12. (31)
Because of this square law nonlinearity, the modula-tion/detection configuration cannot be rigorouslymodeled as a linear invariant system. Rather, in thelow scattering efficiency limit, the scattered light powerP(t) is a quadratic invariant function of the video signalamplitude V(t), defined by
where the asterisk denotes complex conjugation.The temporal response kernel R (t 1 ,t 2 ) constitutes a
complete specification of the modulator response. Itis analogous to an impulse response function for a linearinvariant system. Two independent coordinates arerequired to define R(tbt 2) to specify the modulationproducts which occur in a quadratic invariant sys-tem.
The temporal response kernel can be derived from thescattering equation given previously. However, thecalculations are more easily executed in Fourier trans-form space:
P(t) = Jf JO dfidf2p(f,f 2)v(f1)v*(f2) exp[-27ri(fi - t],
(33)where R (t1,t2) and P(f1,f2) are a Fourier transform pair.The kernel p(flf2) is analogous to an MTF of a linearinvariant system. This kernel is defined by an overlapintegral, similar to a convolution operator, relating theincident light and sound field spectra din and a:
Fig. 8. Light throughput efficiency. Fast temporal response can
be achieved by focusing the incident light beam, but the coupling
efficiency degrades. The A parameter, defined in the text, is ameasure of the degree of focusing.
is obeyed by the geometrical optics model when the fieldprofiles are symmetrical, but is not obeyed by thephysical optics model, even for symmetrical field pro-files.
B. Modulator Parameters
The computer studies which follow assume specificfield profiles, namely, TEMoO incident light [Eq. (9)]and a rect function sound field profile:
These are the profiles normally encountered in an ac-oustooptic modulator.
The modulator configuration is completely charac-terized by two scaling parameters:
A = L0,/w,
B = v/(ffw).
(45)
(46)POWER SPILLOVER BEAM STOP
DEFLECTED\ ,l,,,
2ec
Fig. 9. Overlap of the deflected and undeflected light beams. In-
complete separation of the diffraction orders degrades the extinctionof the modulated light beam. The B parameter, defined in the text,
is a measure of the separation of the diffraction orders.
1000:1
0;7 100:1
z0
10:1
1 I I I I I I0.0 1.0 2.0 3.0 4.0 5.0 6.0
B PARAMETER
Fig. 10. The static contrast (extinction) ratio.
which is required to insure that the optical power P(t)is a real function. The additional symmetry condi-tion
R(tl,t 2) = R(-t 1,-t 2 ), (42)
P(ff2) = P(-fl,-f2), (43)
The A parameter is identical to the diffraction anglespread ratio defined by Maydan 7 save for a factor of 1/27r.
This ratio determines the severity of the second-ordereffects listed in Sec. I. For example, Fig. 8 shows theroll-off in scattering efficiency associated with focusingof the incident light beam.
The B parameter governs the overlap of the diffrac-tion orders, i.e., the ability to separate cleanly thedominant diffraction order from the unscattered light(Fig. 9). The presence of unscattered light in themodulated light beam path will degrade the ability toextinguish (turn off) the modulated light beam (Fig.10).
A third parameter is related to these two-the waveintercept number N defined by Maydan7:
N = A/B. (47)
The parameter N demarks thin grating operation fromthick grating operation. It is a scaling parameter for thetransducer length L. The Q parameter of Klein andCook is equal to 47rN.20
The limit B = 0 corresponds to the geometrical opticsmodel of the acoustooptic coupling (i.e., fc - o). Thelimit A = 0 corresponds to the thin grating model of thecoupling (i.e., L - 0).
The temporal response kernel P(flf2) is shown inisometric and orthographic projection in Figs. 11-13 forvarious combinations of the A and B parameters.
C. Modified Thin Grating Model
The thin grating model underestimates the modu-lator response time because it ignores the contributionof the finite sound field depth to the acoustic transittime. The thin grating model can be modified to in-clude this transit time contribution by defining an ef-fective incident light beam radius w. This effectivelight beam radius is chosen so that the rise time pre-diction of the thin grating model matches the rise timeprediction of the more rigorous models (see below).The effective light beam is fatter and better collimatedthan the actual incident light beam.
D. Modulator Response to a Sinusoidal Video Signal
Consider the following video signal:
V(t) = 1 + M cos(2irft),
Fig. 11. The modulator's frequency response kernel (A = 1/47r se-quence). The modulator's temporal response to an arbitrary videosignal (in the limit of low scattering efficiency) is completely specifiedby a 2-D frequency response kernel, shown here in isometric projec-tion.
A = 7r/2
B = 1/2
(48)
where M is the modulation depth. The correspondingpower spectrum is
The first term is the dc bias, the second term is therectification of the video signal, the third term is thefundamental video harmonic, and the fourth term is thesecond harmonic. No other harmonic componentsexist. The deflected light power will have a similartemporal spectrum, but the distribution of opticalpower among these harmonics will differ. The modu-lation transfer functions for the several harmoniccomponents are shown in Figs. 14 and 15.
The class of sinusoidal video signals has special sig-nificance in linear invariant system analysis. Knowl-edge of the system response to all possible sinusoidalsignals is sufficient information to predict the systemresponse to any arbitrary signal. This is not true of aquadratic invariant system. However, the response toa sinusoidal video is still a useful measure of modulatorperformance.
Fig. 12. The modulator's frequency response kernel (A = /27r se-quence). This kernel is similar to an MTF. However, two inde-pendent frequency variables are required to characterize the harmoniccoupling which is introduced by the square law nonlinearity in the
optical detector.
(A 72
f = f)
ORIENTATION NO. 1
z0
I
0
.0 0.5RISETIME - REPETITION RATE PRODUCT
1.0
Fig. 14. Modulator response to a sinusoidal video signal with dcoffset (A = 1/4ir sequence).
B = 21
0
C
:5E0zORIENTATION NO. 2
Fig. 13. The modulator's frequency response kernel (orthographicprojection). Orientation 1 refers to scattering into the +1 diffraction
order; orientation 2 refers to scattering into the -1 diffractionorder.
0.0 0.5RISETIME - REPETITION RATE PRODUCT
1.0
Fig. 15. Modulator response to a sinusoidal video signal with dcoffset (A = /27r sequence).
E. Modulator Response to a Square Pulse Video Train
Another important class of video signals is the set ofbinary digital signals, as exemplified by the square pulsetrain with 50% duty factor. Key measures of modulatorperformance are rise time (the time interval requiredby the scattered light to grow from 10% to 90% of itsfinal quiescent power) and the modulation depth. Theresults are presented in Figs. 16-18.
IV. Angular Scattering Window Concept
The dynamics of the acoustooptic coupling are gov-erned by Eqs. (3) and (4). These are called the mo-mentum matching equations because of their strikingsimilarity to conservation of momentum in a scatteringevent. This suggests a billiard ball analogy: the inci-dent light is a billiard ball traveling at angle Oin prior tothe scattering event. When the light enters the mod-ulator, it collides with the sound field's billiard ball andcaroms off at angle 0 out. The billiard ball analogy is apowerful and appealing heuristic model of the dynamicsof the acoustooptic coupling. Unfortunately, a fieldmost closely approximates a billiard ball when it is wellcollimated, whereas the most interesting properties ofa modulator arise precisely because the fields are notperfectly collimated. What is needed is a more so-phisticated analogy which retains the heuristic powerof the billiard ball model without suffering from itsconceptual limitations.
This analogy can be drawn from a synthesis of com-munications theory and the quantum mechanics ofscattering processes. In this model, the modulator isa scattering center with a strong dependence upon thepropagation direction of the incident light beam. Amore suggestive name might be a scattering windowwhich operates in propagation angle space. The inci-dent light beam is analogous to a quantum mechanicalparticle spanning a spread of momentum directions, asdistinct from a classical billiard ball which has only onemomentum direction. In communications terminology,the modulator is a bandpass filter operating upon thepropagation angle spectrum of the incident light beam,rather than its temporal frequency spectrum.
The scattering window is defined quantitatively bythe function a(0afa) defined in Eq. (13).
A. Scattering Windows for a Quiescent Video Signal
To simplify the discussion, consider first the class ofquiescent video signals, i.e., signals which vary slowlycompared with the modulator rise time. A scatteringwindow is associated with each diffraction order.Cohen and Gordon have proposed an experimentaltechnique for mapping these window profiles.8 Thetechnique consists of illuminating the modulator witha well collimated light beam (approximating a planewave field as closely as possible), rotating the modulator
a, 2.0 0a 1.9 =4B=5
1. . . . . . .
B=41.7-
01.6 B=3
1.5
1.4
a, 1.3
01.2 B=2
0 ~ ~ ~ ~ ~ 5
:E 1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0A PARAMETER
Fig. 16. Response time adjustment factor. The thin grating model(A = 0) of the acoustooptic coupling can yield reasonably accuratetemporal response predictions if the true incident light beam radiusw is replaced by an effectively larger radius. The larger radius ac-counts for the acoustic transit time contribution of the finite soundfield depth L. The required resealing is shown in this figure. Res-caling is assumed in the A = 0 curves in Figs. 14, 15, 17, and 18.
100
L::E
I
a
z0
4
C0
80
60
40
20
N A = 0, B = 0
--- A = / 4, B = CA = / 4, B =(OVERLAPPINC
----- A =7r/4, B =
I I I I0.0 0.2 0.4
RISETIME -0.6 0.8 1.0 1.2
REPETITION RATE PRODUCT
0 AND-
G)
Fig. 17. Modulator response to a square pulse video train (A = /47r
Fig. 19. The Cohen-Gordon technique for mapping the angularscattering windows.
ai
I,
AAni
-2 0
c -8c 0
-1 DilAngular
ifraction Order Scattering Window .if1
UI
-28C -8
c 0Incident Light Beam Propagat
Fig. 21. The scatting windows for a F
as shown in Fig. 19, and observingations in the optical power in the dinvestigation.
Typical windows are shown inBragg regime (thick phase gratina Raman-Nath regime (thin phasA Bragg modulator can couple on'order at a time and only when thof the modulator is carefully ttcouping. Peak response occurs rlight enters the modulator normalrather when the light is cocked Bragg angle Oc. A Raman-Nath into both diffraction orders with cPeak response for the Raman-]occurs when the incident light entbut this modulator is far moremisalignment.2 0 Note the differabcissas in Figs. 20 and 21.
Cohen and Gordon have show:low scattering efficiency, the windto the far-field diffraction patte(with angles rescaled by the inde:window width is inversely proporthe sound field, i.e., to the lengtf
Fig. 20. The scattering windows for a Bragg modulator.
+1 Diffraction Order A Bragg modulator has a thick sound field and a narrowgular Scattering Window window profile. A Raman-Nath modulator has a thin
sound field and a wide window profile. A Bragg mod-ulator can be distinguished from a Raman-Nath mod-ulator by the relative scaling of the diffraction order
c 2 lc separation to the width of the scattering window. De-ec 20c in fining the window width to the first minima, the re-
sulting ratio is Maydan's wave intercept number N in-troduced previously.
Please note that the angular scattering window isdefined with respect to a well collimated (plane wave)incident light beam. Typical modulator practice is to
|in _ focus the incident light to minimize the acoustic transitec 20 c in time. This produces a spread of propagation angles.
t Direction The scattered light angular spectrum is defined by thetaman-Nath modulator. incident light spectrum multiplied by the angular re-
sponse window of the modulator. This is diagrammedg the resulting fluctu- in Fig. 22 for a Bragg modulator..iffraction order under The scattering window defines a fundamental limit
to the degree of focusing which can be attempted whileFigs. 20 and 21 for a still enjoying high coupling efficiency. Focusingg) modulator and for broadens the angular spread of the incident light beam.e grating) modulator. If faster modulator response is attempted by focusingly into one diffraction harder, an appreciable fraction of the incident lighte angular orientation energy is pulled outside of the angular passband of thetred to optimize the modulator (Fig. 23). The result is a loss in couplingnot when the incident efficiency and a distortion of the scattered light profile.to the sound field, but The modulator acts like a spatial filter, scattering ait a slight angle-the beam which is more collimated than the incident lightmodulator will scatter beam, with a larger effective radius inside the modula-comparable efficiency. tor..4ath modulator alsoers at the Bragg angle, B. Scattering Window for Rapid Video Modulationforgiving of angular The scattering window which corresponds to a dcence in scaling of the video signal will be called the quiescent window. When
the video varies rapidly compared with the modulatorn that, in the limit of rise time, the single window concept discussed abovelow profile is identical must be replaced by a spectrum of windows, one for eachrn of the sound field temporal frequency component of the video signalx of refraction). The spectrum. Consider a sinusoidal video signal. The
tional to the depth of scattering [Eq. (23)] describes coupling through twoL of the transducer. distinct windows for such a video signal. One window
Fig. 22. Scattering in a Bragg modulator. Thefar-field diffraction profile of the scattered light isdefined by the product of the incident light dif-
fraction profile with the modulator's scatteringwindow.
SOFT FOCUS (SLOW TEMPORAL RESPONSE):
HARD FOCUS(FAST TEMPORAL RESPONSE):
SCATTERED LIGHT SPECTRUM
Fig. 23. Focusing the incident light beam. When the incident lightbeam is well collimated (top sequence in this illustration), its far-fielddiffraction pattern (shaded Gaussian) is narrow compared with theangular passband defined by the modulator's angular scatteringwindow (top center sketch in this figure). As a result, the scatteredlight profile is negligibly distorted with respect to the incident lightprofile (top right illustration). When the incident light is focusedhard to minimize the modulator response time (bottom sequence),the spread of the corresponding diffraction pattern is increased(shaded Gaussian, bottom sequence). The angular passband of themodulator filters out a significant fraction of the incident light power(bottom center illustration) and distorts the scattered light profile(solid line, bottom right illustration) compared with the incident light
profile (dashed line).
corresponds to the positive video frequency +fv, whilethe other window corresponds to the negative videofrequency -f Figure 24 shows these windows for thegeometrical and the physical optics models. Eachwindow is displaced with respect to the quiescent videowindow. This will be called a splitting of the quiescentwindow induced by the video modulation.
The corresponding scattered light propagation anglespectrum is shown in Fig. 25.
'in
Physical Optics ModelSinusoidal Video
fv = 1/3fc
A n~~~~~~~~~~./NIA/\I U,
/IN~~~\I~
Fig. 24. Splitting the quiescent window by video modulation.
1. Thin Grating ModelThe scattered light profile is split into two component
profiles. The shape of each component profile isidentical to the shape of the incident light profile. Thephase difference between these two components variescontinuously in time, so that the resulting intensityprofile grows and collapses in time (Fig. 26) in responseto the video drive signal. The separation between thecomponent profiles is proportional to the video signal
Fig. 25. Splitting the scattered light beam profile by video modu-lation. A sinusoidal video signal is assumed in Figs. 21-26.
0
z
Fig. 26. The resulting evolution of the scattered light's far-field.diffraction profile.
I_ Physical Optics
X Model n ~~~~~Deflector> '~~~~~'/ i AN, ~ Operation
JGeometrical /I\
Acoustic Carrier Frequency
Fig. 27. Carrier frequency response of an acoustooptic deflector.
frequency. As the video frequency increases, the twocomponent profiles separate with less and less overlap.One result is that the far-field diffraction pattern of thescattered light beam expands as the video frequencyincreases. This phenomenon is called FM blur. Asecond consequence is that the two component profilesno longer cancel completely when they are 180° out of
phase. The fidelity of the temporal response of themodulator is directly related to the degree of overlap ofthe two component profiles. As Korpel has observed,this response roll-off mechanism is the dual in Fouriertransform space of the acoustic transit time associatedwith the incident light beam diameter.' 4
2. Geometrical Optics ModelThe geometrical optics model exhibits the same basic
splitting of the quiescent light profile. However, eachcomponent profile is not identical in shape to the inci-dent light profile, but rather corresponds to a light beamwhich is broader inside the modulator and better colli-mated. Furthermore, as the video frequency is in-creased, this distortion of the scattered light profilebecomes more and more severe. This additional re-sponse degradation mechanism is the Fourier transformdual of the acoustic transit time associated with the fi-nite sound field depth.
3. Physical Optics ModelThe physical optics model exhibits the same basic
profile splitting and distortion as the geometrical opticsmodel. In addition, the physical optics model predictsan asymmetry in light energy scattered into the videosidebands which is not predicted by the geometricaloptics model. This asymmetry becomes more severeas the video frequency is increased. This asymmetryis called vestigial sideband modulation of the opticalcarrier. 21 However, the existence of this asymmetry isobscured in optical communications applications whendemodulation is effected by a square law detector.Note that the computer response curves presentedpreviously show almost identical behavior for both thegeometrical and the physical optics models. Theasymmetry is more dramatically exhibited by acous-tooptic deflectors, as shown in Fig. 27. This figureshows the variation in scattering efficiency which resultswhen the acoustic carrier frequency is slowly varied.The figure assumes a well collimated incident lightbeam which enters the acoustooptic deflector at a fixedangle.
V. Conclusion
The physical content of the acoustooptic coupling isbest understood in Fourier transform space with thevarious fields represented by plane wave spectra. Themodulator acts upon the incident light beam like abandpass filter, limiting the spread of the propagationdirections which scatter efficiently. The modulatoreffectively presents a scattering window to the lightbeam.
The temporal response calculations which are derivedfrom the rigorous coupling models are cumbersome andcomputer-intensive to perform. An approximate linearinvariant model, with a suitably scaled impulse re-sponse, has been proposed which gives reasonably ac-curate response predictions for several important classesof video signal. This approximate model of the mod-ulator response is much easier to apply than the morerigorous quadratic invariant models.
References1. H. V. Hance and J. K. Parks, J. Acoust. Soc. Am. 38, 14 (1965).2. A. Korpel, R. Adler, P. Desmares, and W. Watson, Appl. Opt. 5,
1667 (1966).3. A. H. Rosenthal, IRE Trans. Ultrason. Eng. 8, 1 (1961).4. J. Randolph and J. Morrison, "Spatial and Temporal Response
of Acousto-Optics Devices," presented at Electro-Optics 1971,New York Colliseum (September 1971).
5. L. B. Lambert, IRE Nat. Conv. Rec. 10, Part 6 (1962).6. E. I. Gordon, Appl. Opt. 5, 1629 (1966).7. D. Maydan, IEEE J. Quantum Electron. QE-6, 15 (1970).8. M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 44, 693
(1965).9. R. W. Dixon and E. I. Gordon, Bell Syst. Tech. J. 46, 367
(1967).10. D. M. Henderson, IEEE J. Quantum Electron. QE-8, 184
(1972).11. R. A. Adler, IEEE Spectrum 4, 42 (1967).
12. C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE53, 1604 (1965).
13. R. W. Damon, W. T. Maloney, and D. H. McMahon, in PhysicalAcoustics, Vol. 7 (Academic, New York, 1970), pp. 277-280
14. A. Korpel, in Applied Solid State Science, Vol. 3 (Academic, NewYork, 1972), pp. 100-103.
15. R. V. Johnson, Appl. Opt. 16, 507 (1977).16. C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).17. H. Weyl, Ann. Phys. 60, 481 (1919).18. J. A. Lucero, J. A. Duardo, and R. V. Johnson, SPIE Proc. 90,32
(1976).19. R. W. Dixon, IEEE J. Quantum Electron. QE-3, 85 (1967).20. W. R. Klein and B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14,
123 (1976).21. P. F. Panter, Modulation, Noise, and Spectral Analysis
(McGraw-Hill, New York, 1965), Chap. 5.22. J. Lapierre, D. Phalippou, and S. Lowenthal, Appl. Opt. 14,1949
(1975).
Road Lighting Lantern and Installation DataPhotcmetrics, Classification and Performance
A report in English on road lighting lantern and installation data --photometrics, classification, and performance -- has been published bythe Commission Internationale de 1'Eclairage (Publication No. 34).The Publication is the result of active cooperation between differentcountries and has been produced by the member- of CIE TechnicalCommittee 4.6 (Road Lighting) which has representation from 28countries. This report is one of a series of CIE publications whichtreat of road lighting, the fundamental document being PublicationNo. 12.2 (1977) Recommendations for the lighting of roads for motorizedtraffic. The report is concerned with the data described in CIEPublication No. 30 (1976) Calculation and measurement of luminanceand illuminance in road lighting. The report has three parts: lanternphotometric data, lantern classification and performance data, installationperformance data. There are 15 figures.
Copies of this document, CIE Publication No. 34, may be obtained postpaidat $10 each from Jack L. Tech, Secretary, U. S. National Committee, CIE,National Bureau of Standards, Washington, D. C. 20234. Payment shouldaccompany the order and should be made payable to U.S. National Committee,CIE. Canadians may obtain copies by sending a check payable to TheReceiver General of Canada, Credit National Research Council with theirorder to Publications Distribution Office, National Research Council ofCanada, Ottawa, Ontario, KA OR6.
