Method of Stationary Phase for Analysis of FringeFunctions in Hologram Interferometry

Karl A. Stetson

In this paper, a mathematical technique called the method of stationary phase is presented for the purposeof obtaining approximate solutions for characteristic fringe functions in hologram interferometry. Thecases of greatest concern are where holograms are formed of objects that vibrate simultaneously in differ-ent geometrical patterns according to sinusoidal time functions whose frequencies are related by rationalnumbers. The method assumes that the object motion is sufficiently large that contributions to the fringefunction come only from the holographic recordings of the object at positions where its velocity is zero.At such position, the phase of the light scattered from an object point is stationary with respect to time,and the method permits the approximate calculation of the relative intensity of the partial recording atthese positions. The fringe functions become expressions similar to those found in multiple-beam inter-ferometry, and fringe patterns can be understood more easily. The case of two sinusoidal vibrations at afrequency ratio of 2: 1 is considered experimentally and theoretically according to the method presented.

1. Introduction

Recent work by Wilson and Stropel-3 has illustratedthe results that may be obtained when a hologram is re-corded of an object that vibrates simultaneously in twogeometrically different vibration patterns according totwo sinusoidal time functions whose frequencies arerelated by rational numbers. The fringes obtained arequite different from those obtained when the two fre-quencies are related by irrational numbers2' 4 or whenthe two vibrations are of the same frequency and sep-arated in phase.4 In the first case, the fringes may bedescribed by the product of the two Jo fringe functionsdue to each vibration alone, a property resulting fromthe independence of the two object motions.5-7 In thesecond case, the fringes are generated by the zero-orderBessel function of the magnitude of the object motion.The fringes resulting from the Wilson and Strope ex-periments2 and Wilson's theoretical calculations 3 seem todefy any such simple interpretation, and a recently pro-posed alternative analysis,8 based on probability densityfunctions, does not appear to help. Thus, there appearsto be a need for yet another alternative method for cal-culating the characteristic fringe functions of holograminterferometry, one that can offer simple physical insight

The author was with the Division of Optical Metrology, Na-tional Physical Laboratory, England; he is presently with Ce-ramics & Glass Department, Scientific Research Staff, Ford MotorCompany, Dearborn, Michigan 48121.

Received 14 February 1972.

into the fringes due to vibrations whose frequencies arerelated by rational numbers.

The purpose of this paper is to present a mathematicaltechnique that can yield approximate solutions forfringe functions and seems to permit a physical inter-pretation for the fringes obtained by Wilson. Thetechnique is called the method of stationary phase, or,in a variant form, the method of steepest descent.',"With respect to hologram interferometry, the principlemay be stated that if an object is vibrating with suffi-cient amplitude while a hologram of it is being recorded-the motion will destroy the recording except for thosepositions where the object's velocity is zero. At thosepoints on its trajectory, the phase of the light scatteredfrom any object point is stationary with respect to time.The problem then becomes the calculation of thestrength of the hologram recording of the object at eachzero-velocity position relative to all the other zero-velocity positions and as a function of the magnitude ofthe object's vibration, etc. This paper will begin witha derivation of approximate characteristic fringe func-tions by the method of stationary phase. Next, a setof experiments will follow that present a two-dimen-sional map of the characteristic fringe functionsassociated with two geometrically orthogonal vibrationswhose frequencies have the ratio 2: 1. Then thesepatterns will be interpreted by means of the formulationderived by the method of stationary phase.

11. Method of Stationary PhaseIn the absence of beam modulation, the characteristic

fringe function in hologram interferometry may bedefined5 by

August 1972 / Vol. 11, No. 8 / APPLIED OPTICS 1725

ITMT = (1T) exp[Sg(K2,R,t)]dt, (1)

where MT is the characteristic fringe function, T is theexposure time of the hologram, Qt is the function relat-ing object deformations as a function of time of fieldpropagation from the object, K2 is the propagationvector of any plane-wave component in the recon-structed field, R is the space vector, and t is time.If it varies rapidly with time, the contributions to MT

arise mainly from regions where dt/dt = Qts = 0, i.e.,where the phase of the integrand of Eq. (1) is stationarywith respect to time. Call t the time defined by Q2Z =0, and expand 2t in a power series (assuming that 0 <t, < T).

Qt = Qtn TV xutn (t tn,) + P,.tn (t - t)3 + . .. a (2)

where the subscript n indicates that the function or itsderivatives is evaluated at tn.

We now substitute Eq. (2) into Eq. (1) and integratewith respect to time. The first term of Eq. (2) is con-stant with respect to time and can be removed from theintegrand to become a factor times the integral. If theentire function it is multiplied by a scale factor, as thisfactor increases (which would correspond to an in-crease in vibration amplitude) the term with the lowestpower of (t - t,) dominates in the determination of thevalue of the integral. Thus we obtain

1 CTMrn ;: exp(iQtn) - I exp[il7Atn(t - t)2]dt, (3)T j 0

when the second derivative is nonzero. If the first andsecond derivatives are zero simultaneously but thethird derivative survives, we have

1 C T

MTn exp(it.) T 3 exp[ in .."'(t - tn)3Jdt. (4)

Integrals of the form of those in Eqs. (3) and (4) areevaluated in close form for the approximation that Tapproaches infinity."1 Thus, if we assume a sufficientlylong exposure time, we may write

Mn ":: 2r(l/p) expli(0n =[ q7r/2p)] (5)

Tp -d&Qt/dt P

where q = -2-[(-1)P + 1] and the sign of q is that ofdPQ2/dtP when p is even, and where p is the order of thefirst nonvanishing derivative. Each point in the ob-ject's trajectory where its velocity is zero makes aseparate contribution to the characteristic function sothat we obtain

1 1MT t -T,=2r(1/p) expi(Qtg i q/2p)

p 1 dPQ lIdtP ~p -1

If &t is periodic, and many periods are included in theexposure time, the fringe function computed for theentire exposure interval will approach that computedfor a single period of the oscillation. Under these con-

ditions, we may substitute for N the number of sta-tionary-phase points in one period of the motion. It isimportant to realize that when oscillations are consid-ered that combine two or more frequencies that are re-lated by rational numbers the period considered mustbe sufficiently long to include an integral number ofcycles of each oscillation.

The interesting feature of the approximate solutionpresented in Eq. (6) is that it expresses the fringe func-tion as a summation of exponential functions whosearguments Ut. relate to deformations of the object atvarious times during the vibration period. If we pickany small region of the object, the brightness of theimage seen there is determined by the summation ofthese phase factors. The effect becomes quite anal-ogous to multiple-beam interference, in that we seeessentially N images of the object, each with a phasedistribution associated with a deformation of the ob-ject. It is the interference between these multipleimages that creates the fringe patterns we see in thesecases. The term q7r/2p in the exponential function hasan interpretation. The factor q is zero when p is oddand unity when p is even. When p is odd, the objectslows to zero velocity and then continues in the samedirection; therefore the recorded position of the objectcoincides with the central position 0,%. When p is even,the object slows to zero velocity and then returns inthe direction from which it came. Thus the recordedposition is offset by an amount r/2p, and the sign ofthis phase factor will depend upon whether the objectwas accelerating toward or away from the hologram.The higher the order of the first nonvanishing deriva-tive p, the smaller the offset phase. This is reasonablein that the higher the order of the first nonvanishingderivative, the faster the object will approach thestationary-phase point, the longer it will spend in thatregion, and the faster it will move away. The ex-ponent /p in the denominator of Eq. (6) is also inter-pretable. If we multiply 9t by a scale factor, i.e., in-crease the amplitude of motion keeping the time func-tion constant, the amplitude of each image will decreaseas the scale factor to the - 1/p power. Thus the higherthe order for the first nonvanishing derivative, theslower the falloff in amplitude of the correspondingimage. This is also reasonable.

III. Experiments with Frequency Ratios of 2:1

With the foregoing theory in mind, it was decided todevise an experiment for mapping the characteristicfunctions for vibrations whose frequencies were in theratio of 2: 1. It was realized that if a plate were sup-ported at its center by a beam that was cut to a smallrectangular cross section, it could be possible to tunethe reasonances of the plate and beam so that it wouldvibrate about a vertical axis at half the frequency atwhich it would vibrate about a horizontal axis. If theplate were set into simultaneous vibration of equal mag-nitude about both axes, then radial distance from thecenter of the plate would correspond to magnitude ofmotion, whereas angular position on the plate would

1726 APPLIED OPTICS / Vol. 11, No. 8 / August 1972

Fig. 1. A reconstruction of a hologram of a rectangular platesupported at its center by a rectangular beam. The plate isvibrating about its vertical axis at the frequency determined by

the sine-wave generator.

correspond to the ratio of the two components. Thenext consideration was how to excite the plate with twosignals that were exactly in a 2:1 ratio. This was doneby using two magnetic transducers; one was a magnetwound with a coil and the other was a soft iron barwound with a coil. In the first transducer, the forcedelivered to a piece of iron fastened to the plate wasprimarily at the same frequency as the current in thecoil. In the second, the force was mainly the firstharmonic of the frequency of the current in the coil.By setting the first transducer at the horizontal axis andthe other at the vertical axis, it was possible to exciteone reasonance at twice the frequency of the other,simultaneously.

The two transducers were connected to the two out-puts of a stereo amplifier, which in turn was connectedto a signal generator. The separate gain controlsallowed control of ratios of the two excitations. Mea-surement of the relative phase of the two vibrations wasaccomplished by detecting the deflections of a laserbeam reflected from a small mirror attached to the backof the plate near a corner. The beam was partiallyintercepted by a knife edge set to 450, after which it wascompletely intercepted by a photocell. The knife edgewas moved across the beam until the photocell indicateda maximum oscillatory signal when the plate wasvibrating. The output of the photocell was displayedon an oscilloscope, and the waveform was composed ofthe sum of the two vibrations of the plate. Inspectionof the waveform indicated whether the two oscillationswere in phase or out of phase (which means that aLissajous figure formed from the two oscillations wouldform a u or an a, respectively), and these two cases couldbe set with considerable precision. Intermediatephases could be obtained, but not set to as high anaccuracy. Different phase relationships between thetwo vibrations were obtained by slightly detuning thetwo reasonances from a perfect 2:1 ratio. The use offorced oscillations guaranteed that the actual vibrationswould still be in the ratio of 2: 1 even if the plate were

excited away from its natural resonance frequencies.Now, however, the relative phases of the two vibrationswould be a function of frequency. The procedurefollowed was to watch the waveform on the oscilloscopeas the frequency of excitation was varied, find theproper phase relationship, and then set the gain controlsof the amplifier until a suitable ratio of amplitudes wasobtained. No attempt was made to control the exactamplitude of the oscillations or even of the ratios of thetwo components, because this would have been quitedifficult by this method. The actual vibrations wereinspected concomitantly by a speckle interferometer,and quasi-focused-image holograms were recorded of theplate on 35-mm film, AGFA-Gevaert Scientia 10E75.

Figures 1 and 2 show photographs of the reconstruc-tions of two holograms made while the plate was vibrat-ing at each of the two resonances alone. A certainamount of bending occurred when the plate was vibrat-ing about the horizontal axis, but the criterion of twoorthogonal vibration patterns is reasonably well met.Figures 3, 4, and 5 show photographs of the reconstruc-tions of three holograms recorded while the plate wasvibrating in both modes simultaneously. The relativephases were 00, 450, and 900, respectively. (Note that0° phase is defined as in phase, and 90'phase is definedas out of phase in terms of the Lissajous figures men-tioned above.) The striking feature of these results isthat in each case the plate is divided into two distinctregions-one where the lower frequency dominates andthe other where the higher frequency' dominates.Where the lower frequency dominates, there are onlytwo stationary-phase points per cycle and the fringesare continuous bands, similar to J fringes. Where thehigher frequency dominates, there are four stationary-phase points per cycle and the fringes are discontinuous.These regions are separated by radii along which thebrightness of the reconstruction is greater than it is toeither side. These singularities denote ratios of thetwo vibrations such that the second derivative of thephase-modulation function is zero at a stationary-phasepoint. In the case of 0 phase, the second and third

Fig. 2. A reconstruction of a hologram of the same rectangularplate made while it was vibrating about its horizontal axis. Herethe vibration frequency is twice that of the sine-wave generator.

August 1972 / Vol. 11, No. 8 / APPLIED OPTICS 1727

f2 = 1), and therefore, there are four such points percycle. These two regions are separated by the regionwhere Q ' = t = 0, as mentioned above.

A. First-Order Approximations

Let us assume that 02 >> ,i which implies four sta-tionary-phase points per cycle, and that we are wellaway from the region where U, = 0. Because oneperiod equals -1rw and r(' = ri, we may write

M(2 1,Q2) (r)-l4 expli(Qfjf + 1 2f2 + ( 1)- r/4)]

X E Ilfin + 4Q2 f2 n|1(10)

Fig. 3. A reconstruction of a hologram of the rectangular platemade while it was vibrating simultaneously at both frequencies.The vibrations were O° relative phase, i.e., such that a Lissajousfigure of the two time functions would retrace its path in an

opposite direction.

derivatives both vanish, and it may be noted that inthis case, the brightness of the reconstruction alongthe singularities is greater than in the other cases.

If we examine the regions where the higher frequencydominates, we notice that in the 0 case there is theappearance of horizontal fringes whose visibility goes tozero and reverses phase periodically. In the 90° case,the horizontal fringes are simply modulated in bright-ness and are periodically extinguished with no reversalin phase. In the 450 case, there appear to be twoperiodic modulations of the visibility of the horizontalfringes that are of different spatial frequencies.

In the regions where the lower frequency dominates,we may notice that the vertical fringes are straight inthe 00 case, but curved in the 450 and 90° cases. Theseare the primary observations we may make about theseresults, and in the following section these effects will beanalyzed in detail.

IV. Analysis of Experimental Results

Let us assume that the fringe-locus functions forvibrations about the vertical and horizontal axes are&

2 and Q22 respectively. We may then write

Ut = 1 coscot + 1Q2 cos(2cot - ).

recognizing that the signs of Qt' alternate and using thenotations of fin and f2n to denote the values of thesetime functions at the nth stationary-phase points. If0 2 is sufficiently larger than Q1, we may write thefollowing approximations:

f21 f23 1, f22 f24 -1 and

fil -fi3, fi2 -_t -fi4.

If we substitute these into Eq. (10) and make use of thefact that Q2 is very much greater than Q1, we obtain

M(91,92) = 27Q21 - it [cosf2ifii exp(22 - 7r/4)]

+ coSQlfI2exp[-i(Q2 - r/4)]}. (11)

From inspection of Eq. (11), we see that the ampli-tude of the reconstruction falls off as I21 - 1 (the bright-ness of the reconstruction falls off as Q2-1), and thefringe loci are generated by 2 - 7r/4 . The fringevisibility goes to zero whenever cosQfll = 0 or cosQ1fi2 =0, and thus the vibration pattern 0l is indicated by itsmodulation of the Q2 -7r/4 fringes. When the two vibra-tions are in phase, 0 = 0, and we have fil 0 and f12 1, which means that primarily the visibility of thefringes is affected and the fringes reverse phase with thesign of cos%. When the two vibrations are out ofphase, 4 = 7r/2 , and we have fil fi2 2 -, whichmeans that M(91,02) |21rQ2I (/ cos(Q21 ) COS(~2 -

(7)

When = 0, the two vibrations are in phase and when4) = 7r/2, they are out of phase. By taking derivativesof Eq. (7), we may determine that &A' = 0 when

2Q2 sin(2ct - 0) = - sinwt, (8)

and that

o = -c2[E1 coscot + 4Q2 cos(2wt - )]

= - 2(f21fi + 4 2f 2),

where fi = coswt and f2 = cos(2cwt - ). Inspection ofEq. (8) reveals that when ah >> Q2, the stationary-phasepoints occur at approximately the peak values of fi(where fi = 4 1), and therefore there are two suchpoints per cycle. By contrast, when 2 >> S1, theyoccur at approximately the peak values of 2 (where

Fig. 4. The same as Fig. 3 except that the two vibrations are at450 relative phase.

1728 APPLIED OPTICS / Vol. 11, No. 8 / August 1972

(9)

and = 7r/2, and discuss each in the region where Q2dominates and where Q£ dominates. To begin, let usrewrite Eq. (8) as

sin2wt sinqk + [cosw cost

+ (/422)] inwt - sino = 0. (14)

Now let us consider the case where 2 dominates and4 = r/2 , i.e., the motions are out of phase. This im-plies that sin = 1 and cos4) = 0. Substituting theseinto Eq. (14) gives

Fig. 5. The same as Fig. 3 except that the two vibrations are at90° relative phase, i.e., such that a Lissajous figure of the two

time functions would form a figure eight.

7r/4). This is the product of two cosine functions timesan envelope function, and the fringes will be modulatedwith cos(9 1/2

1) and will exhibit no phase reversal.When the phases lie between these two cases, the fringevisibility will follow the two functions cosgif1 l andcosQJf12. These three situations are illustrated quiteclearly in Figs. 3, 5, and 4, respectively. It is of interestto note that if the function Q2 is known, as it is in thiscase, determination of the two periodicities of the fringevisibility in Fig. 4 will allow approximate determina-tion of the ratio of f/f12. From this it is possible tocalculate the relative phase angle of the two vibra-tions.

Let us now consider the case when £ I>> £2, i.e., thecase where there are only two stationary-phase pointsper cycle. Again the period is rc and r(s) = 7r- sothat we may write

M(21,£) = ) (2 x)i 2 exp{i[lfi + 2f2 + (-1)-,r4]} (12)M(01'02) = (2r)-I 1: £2Qfj + 42f2.1I(12

If Q, is sufficiently greater than 2, we may write thefollowing approximations:

fil -fi2 and f2l f22.

Substituting these into Eq. (12) and making use of thefact that £1 is very much greater than 2, we obtain

M(Q,£2) = (27r£2)i exp(i£2f 2l) cos( 1 - r/4). (13)

Because the phase factor exp(i2f 2l) is not apparent tothe observer, the fringes resemble those due to thevibration £2 alone. The curvature of the fringes,shown in Figs. 4 and 5, is described by second-orderapproximations, which will be described in the nextsection.

B. Second-Order Approximations

In this section we will examine the behavior of theseapproximate solutions as we relax the conditions thatone motion is very much greater than the other. Be-cause it is difficult to do this for an arbitrary phaseangle, we shall limit the discussion to two cases, = 0

sincwt = -(gi/8122) i [(91/822)2 + 2]7-

If 2 is sufficiently larger than £1, we may write

sinwt 4 -2- Q1/8822,

coswt 4±2-I i 21/822,

cos(2cot - r/2 ) = 2 sinwt cosot (1 - 212/3222).

From these relationships we may write

1l. + 22f2. + (-1)'2r/4

n = 1, 1/2l + £22 + 32/32222 - /4,

n = 2, -Q 1 /2 ' - 22 - 31/32£22 + w/4,

= 3, -/2 + 22 + 3/3292 - 7r/4,n = 4, 1/2 - 22 - 3/3222 + r/4.

The denominator of Eq. (10) may be written as

(15)

(16)

(17)

(18)

(19)

Q0f£j + 4Q2f2ni [1 -4 1/8(2)I£2]/2 Q2, (20)

where the sign is minus for n = 1 and n = 2 and is plusfor n = 3 and n = 4. Substitution into Eq. (10) andfactoring gives

M(Q,£2) (2/7rQ2)1 cos[Q2(1 + 312/32Q22)

-7r/4] {cos£/2 - [/ 8 (2 )2Q2] sin£j/2i}. (21)

Inspection of Eq. (21) will indicate that the fringesgenerated by the fringe-locus function 2 will exhibit anincreased spatial frequency with increasing Q1, due tothe factor (1 + 212/32Q22). This will have the effect ofcurving the fringes toward the horizontal axis in Fig. 5.Because they are curved in this direction already (seeFig. 2), the effect is not as striking as it would be if theoriginal fringes were straight. The other effect thatmay be noticed is that the modulation of the 2 fringesby the factor cosQ1/21 is not complete owing to thesecond imaginary term within the parentheses. Thismeans that the vertical black fringes that interruptthe horizontal fringes in Fig. 5 are not, themselves, com-pletely black.

Now let us consider the alternate case where = 0.This sets sin = 0 and cos = 1, so that substitutioninto Eq. (14) leads to stationary-phase points definedby the conditions

sinci = 0, i.e., coswt = 1, and coswt = -I/4£2- (22)

From this we obtain

cos2cot = 2 cos 2wt - 1 = 1, or Q£2/822 - 1. (23)

From these relationships we may write

August 1972 / Vol. 11, No. 8 / APPLIED OPTICS 1729

Ulfi + 2f2. + (-1),r/4 =

n = 1, £2 + 2 -/4,

n= 2, -2 - £212/8£2 + r/4, (24)

n = 3, -1 + 2 - 7r/4,

n = 4, -Q2 - 12/822 + 7r/4.

The denominator of Eq. (10) becomes

I£,fi. + 42f2n1 [1 F (1/8£2)I/2£2 (25)

for n = 1 and n = 3 (the minus sign is for n = 1) and

I£2jfin + 42f2nl -i (1 + Q12/162 2)/2£2 (26)

for n = 3 and n = 4. Substitution into Eq. (10) gives,after factoring,

M(E2,,£2) ([1 + (£212/16£22)I exp{ -i[2 2 + (012/802)

- (7r/4)]} + [cos2i - i(£1/8£2) sin£l]exp{i[Q2 -

(7r/4)] })/(2IQ2)', (27)

Examination of Eq. (34) reveals that the spacing of thefringes generated by Q, will decrease as the factor 022/212

increases, but the visibility of the fringes will persist.This variation in fringe spacing causes the curvature ofthe fringes toward the vertical axis in Fig. 5 in the regionof lower frequency dominance. It can be seen on thefigure that the black fringes in this region remain quiteblack up to the boundary between the two regions ofdominance.

To consider the final case, in which 4 = 0 in theregion of £2, dominance, we examine Eqs. (22) with thepresumption that 1 is considerably greater than 2.

The only stationary-phase points that remain are thosedefined by

coswt = 1, (35)

which leads to cos2cot = + 1, and + 1, for the two sta-tionary-phase points. From these two relationshipswe may write

which may be rewritten as

M(£ 1 ,£ 2) [exp(-iQ12/8Q 2 )/(2irQ 2 ) {1( + f212/16£22)

X exp[-i(2 - 7r/4)] + cos£2[1

+ (Q£2/64£2 2)tan22l exp[i(2 - 7r/4)]}. (28)

Inspection of Eq. (28) indicates that the loci of thefringes generated by £2 are essentially unaffected by thepresence of the subharmonic vibration £1. Their visi-bility, however, is affected primarily by the factorcos£i, which is responsible for the phase reversal of thefringes, and to a lesser degree by the other factors of theexponentials within the brackets.

Now let us consider the alternate cases, where 2

dominates. To consider the case where 4) = 7r/2 , wemust examine Eq. (15) under the presumption that £2is considerably larger than £22. This presumption re-quires that only the plus sign be used, and we have,after factoring,

Žifin + 22f2n + (-l)'7r/4 = 2 + (-1)(Q - 7r/4), (36)

and the denominator of Eq. (12) becomes

I£2,fi. + 42f2nI 1 (1 F 22/£1)/£1I. (37)

Substituting these into Eq. (12) and factoring gives

M(£ 1 ,£2) (2/irQ1) exp(i£2)[cos(l - 7r/4)

- i(2£ 2/£1) sin(l - 7r/4)]. (38)

Inspection of this equation leads to the conclusion thatthe fringes generated by 1 remain undeviated by thepresence of the first harmonic vibration. Their visi-bility, however, decreases with increasing values of thefactor 22/1 within the brackets. This behavior isquite noticeable in Fig. 3, in the region of dominance ofthe lower frequency.

V. Conclusion

sincot = (£2/8£2)[ -1 + (1 + 32Q22/Q12)]. (29) It has been shown that the method of stationaryphase has yielded an approximate solution that is

If £1 is sufficiently larger than £2, we may approximate physically interpretable for the characteristic fringethis by function in hologram interferometry of an object

sinwt 2£2/1 (30) vibrating in two modes at frequencies whose ratios are2:1. It is reasonable to expect that the technique

from which we may write would be applicable to most cases of interest in this

coswt + 4[1 - 2(£2/£1)2] and cos(2cwt - 7r/2) area. Aside from the benefit of yielding a physicallyinterpretable solution, this method has the added ad-

+ 4£22/£1. (31) vantage that it becomes more accurate the larger the

From these relationships we may write object motion, whereas the method used by Wilson be-comes more difficult to use under the same conditions.

1l. + £22f2. + (- 1)7/4 Thus it would be reasonable to expect that the applica-= (- )n'l{£2 + 2(£22/Q)[1 - 4(£22/212)] - 7r/4}, (32) tion of numerical computation to this technique would

greatly increase the speed and extend the range of theand the denominator of Eq. (12) becomes display devised by Wilson.3

I£2ifin + 4Q2f2n - = (1 - 7£22/212)/011. (33) The final conclusion that may be drawn is that thecase of vibrations at a frequency ratio of 2:1 shown by

Substituting these into Eq. (12) gives, after factoring, Wilson,2 '3 that of a circular plate vibrating at its funda-mental and first diametral mode, would be easier to

M(£21 ,£22) = [(1 + 7£222/£212)/(7£21/2)I] cos(£2 { 1 . interpret if the examples shown had a higher excita-

+ 2(Q22/£21)[1 - 4(£22/Q12

)1} - 7/4). (34) tion amplitude. This experiment was pefonmed,

1730 APPLIED OPTICS / Vol. 11, No. 8 / August1972

Fig. 6. A reconstruction of a hologram of a circular plate madewhile it was vibrating in its fundamental and first diametral

modes simultaneously with frequencies whose ratios were 2: 1.

therefore, and the result is shown in Fig. 6. The tworegions of dominance are clearly discernable with thislevel of excitation, and it is reasonably obvious from themodulation of the fringes in the region dominated by thehigher frequency that the phase between the two vibra-tions is in the neighborhood of 45°. It is interestingthat the pattern is not perfectly symmetrical to eitherside of the diametral mode line. Apparently a smallamount of the orthogonal diametral mode has entered

into this vibration so that the diametral vibration is notuniphase. This is further substantiated by the factthat the upper left matches the lower right, and viceversa. One final observation is that in the examplespresented by Wilson it is not clear what the two com-bining vibration patterns are, whereas with this highervibration amplitude they are more apparent.

The author is greatly indebted to J. R. Bell and to G.F. Miller of the Mathematics Divsion of NPL for sug-gesting the use of the method of stationary phase tosolve these problems. In addition, gratitude is ex-pressed to E. Archbold of NPL for his help with theseexperiments and to A. D. Wilson of IBM, Endicott,New York, for fruitful discussion of this topic and help-ful criticisms of the manuscript.

References

1. A. D. Wilson, J. Opt. Soc. Am. 60, 1068 (1970).2. A. D. Wilson and D. H. Strope, J. Opt. Soc. Am. 60, 1164

(1970).3. A. D. Wilson, J. Opt. Soc.Am. 61, 924 (1971).4. N.-E. Molin and K. A. Stetson, J. Phys. E. (J. Sci. Instrum.)

2,609 (1969).5. K. A. Stetson, Optik 29,386 (1969).6. K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).7. N.-E. Molin and K. A. Stetson, Optik 33, 399 (1971).8. K. A. Stetson, J. Opt. Soc. Am. 61, 1359 (1971).9. J. Jeffreys and B. Jeffreys, Methods of Mathematical Physics

(University Press, Cambridge, 1950), Chap. 17, art. 17.05, p.505.

10. J. Irving and N. Mullineux, Mathematics in Physics and En-gineering (Academic Press, New York, 1959), Chap. 9, art. 12,p. 566.

11. I. S. Gradshteyn and I. M. Ryshik, Tables of Integrals, Series,and Products, (Academic Press, New York, 1965), art. 3.712,p. 399.

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August 1972 / Vol. 11, No. 8 / APPLIED OPTICS 1731