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Tandem multipass Fabry-Perot interferometer forBrillouin scattering

J. G. Dil, N. C. J. A. van Hijningen, F. van Dorst, and R. M. Aarts

A new design of a plane mirror Fabry-Perot interferometer with the mirror blanks freely supported by hori-zontal rings is described. Special attention is paid to the surface flatness and dielectric coatings of the mir-rors. It is shown that any two piezoelectrically scanned Fabry-Perot interferometers can be synchronizedin tandem operation using simple optics and standard electronic driving and stabilization equipment. Ap-plications of the tandem system to Brillouin scattering are given. The observation of Brillouin spectra ofsupported thin plastic films and of a clamped metal surface is reported.

1. Introduction

In 1961, Hariharan and Sen' showed that the contrastand finesse of a plane mirror Fabry-Perot interferom-eter are significantly increased by routing the beamsequentially through two different portions of themirrors using a cat's-eye reflector. This highly shar-pened filter was attained at the relatively low cost of adecrease in overall transmission and throughput.

Nine years later, Sandercock2 applied this idea andintroduced a more general form of a multipass inter-ferometer. He used corner-cube prisms to route thebeam three to five times through different areas of themirrors. In this way, a contrast in excess of 109 wasachieved as was necessary for looking at weak signalsfrom opaque surfaces, which cause a tremendousamount of stray scattering.

At the same time, several forms of tandem interfer-ometers appeared. Pine3 constructed a spectrometerconsisting of a plane mirror Fabry-Perot interferometerin series with a confocal one. He used the plane mirrorinterferometer as a fixed filter to reject stray scatteringand scanned the spherical one for resolving the spec-trum of interest. Vacher and co-workers have devel-oped a similar instrument, a new refinement of which

The authors are with N. V. Philips Gloeilampenfabrieken, ResearchLaboratories, 5600 MD Eindhoven, The Netherlands.

Received 26 December 1980.0003-6935/81/081374-08$00.50/0.© 1981 Optical Society of America.

was published recently in this journal.4Cannell5 6 took a different approach to the tandem

system. He showed that when two pressure-scannedFabry-Perot interferometers were placed in series in thesame pressure tank, the contrast and finesse of the re-sulting filter were not only improved, but also a signif-icant increase in the free spectral range was obtained.This latter advantage alleviates the problem of over-lapping orders. The synchronously scanned tandeminterferometer has therefore an interesting and usefulinstrumental profile. If it had not been for the seem-ingly difficult electronic problems, a piezoelectricallyscanned tandem system would have been built at thebeginning of the seventies.

Sandercock7 recently introduced an elegant designfor a piezoelectrically scanned tandem multipassFabry-Perot interferometer. His design makes use ofa mechanical scan where one mirror of each interfer-ometer is mounted on one and the same base plate,which in turn is scanned by one piezostack. The twomirror pairs are set at an angle which determines themirror spacing ratio of the two interferometers.

Independent of Sandercock's work, we have taken adifferent approach to the synchronous scan of a pie-zoelectrically driven tandem system. It is the purposeof the present paper to show that with the aid of somesimple optics, any two Fabry-Perot interferometers canbe scanned synchronously using standard electronicdriving and stabilization circuitry. We also describea new design of a plane mirror Fabry-Perot interfer-ometer. In this design, the mirror blanks are freelysupported by horizontal rings so that the problem ofstress-free mirror mounting is virtually eliminated, andinterchanging of mirror pairs is almost trivial. Finally,

1374 APPLIED OPTICS / Vol. 20, No. 8 / 15 April 1981

0 a 0 fX S , a ir Q )t0 .j .iS:gD tea; D0iaS .0 0 : r: t (b ) 0

70 tmm

Fig. 1. Interferograms of a good pair of 70-mm diam mirrors: fringesare (a) 0.5 m apart and (b) with infinite fringe spacing.

we give several applications of the tandem system toBrillouin spectroscopy.

II. Single-Pass and Multipass Interferometers

We begin this section by considering the single-passtransmission function of a plane mirror Fabry-Perotinterferometer:

A(k,di) =

1 + 4NI sin2 [kd cos(i)]72

where T1 is the overall single-pass transmission, N1 isthe effective single-pass finesse, d is the mirror sepa-ration, and i is the angle of incidence of the illuminatinglight beam of wave number

k 2rn/X. (2)

In Eq. (2), X denotes the free-space wavelength, and nis the refractive index of the medium between the mirrorblanks (n = 1).

The effective finesse N1 contains contributions fromthe reflective finesse and a finesse due to surface aber-rations and roughness of the mirrors.8 It is thereforeimportant to coat and mount the mirrors so that nosurface deformation occurs.

Our 70-mm diam mirror pairs were purchased fromICOS in London. Between subsequent stages of pol-ishing, such interferometer mirrors are silver-coated andtested horizontally in a Fizeau interferometer. Duringthese test, each mirror rests on three pieces of aluminumfoil, which in turn lie symmetrically inside a ring servingas the mirror holder. Polishing is stopped when thesilver-coated mirror pair is matched better than X/200(X 500 nm). Their separate flatness is then typicallyof the order of X/50.

The next step is to obtain a dielectric reflectioncoating on the mirrors with minimum absorption andscattering while not affecting the surface figure. Allcoatings have some kind of surface tension and tend todeform even relatively thick pieces of glass. For ex-ample, a stack of ZnS-SiO2 layers deformed our X/200matched mirror pair to a poor X/50, although the

Spectrasil blanks were 18 mm thick. It is a happy co-incidence of nature that the soft dielectric coating ofZnS-MgF2 layers both has very good absorption andscattering properties and does not affect the mirrorflatness.

Our mirrors have a reflectivity of R = 0.94 + 0.05 at500 nm. When mounted horizontally as describedabove, but with one mirror adjustable by means of threepiezostacks, they can be set parallel and on resonance(d = pXo/2, p is an integer). In that position, for aperpendicularly incident (i = 0) parallel beam ofmonochromatic light, the overall transmission is typi-cally T1 = 0.75 ± 0.05. When illuminated over theirtotal aperture with the expanded laser beam, interfer-ograms like the ones shown in Fig. 1 are obtained. Avery good pair should light up completely, so that theareas where the beam is routed for multipassing can bechosen freely. The pair in Fig. 1 has been in use formore than 3.5 yr and still shows no sign of degrada-tion.

There are several experimental pitfalls where themirrors apparently light up over the whole apertureeven though they may suffer from severe aberrations.Therefore, two extra tests should be performed.

First, when measuring the single-pass transmission,one should connect the detector output to an oscillo-scope. Mechanical vibrations, mostly in the 1-Hz to2-kHz region, show up as amplitude modulations in thetransmitted light. These modulations may be due toa voltage ripple on the piezostacks (50 or 60 Hz), me-chanical resonances of the interferometer, or laser fre-quency jitter. They may be as deep as 0.99, thus se-verely reducing the transmission and finesse of the in-terferometer. A mechanically rigid interferometer witha good set of mirrors of R = 0.94 shows <0.10 modula-tion in its transmitted light when the mirrors are set tohalf resonance [d = XO/4N + pXo/2 in Eq. (1)], and dis so small that laser jitter cannot be resolved. A fewscans per second do not bring the freely supportedmirrors into vibration, but the flyback after each scanshould be smooth and take -0.1 sec. The single-passfinesse then takes the value N1 = 47 t 3. We usuallyplace the interferometer on rubber blocks for furthervibration isolation.

The second test which should be done is the investi-gation of the illuminating wave front with, for example,a shear plate. The acceptance angle bi of the interfer-ometer is easily calculated from Eq. (1) and takes theform

(3)

so that all rays incident on the mirrors should fall withinthis angle. When the lenses in the beam expandersuffer from aberrations, or when the laser oscillates ina transverse mode, all incident rays may not fall withinbi, and the mirrors will light up more completely thanthey should.

From the foregoing it is a natural continuation toconstruct a Fabry-Perot interferometer with the mirrorsin a horizontal position. In Ref. 9 it was reported that

15 April 1981 / Vol. 20, No. 8 / APPLIED OPTICS 1375

Bi =I X 11/2

r2 d NJ '

given by Eq. (3). This wedge should be -0.02 rad toprevent scattering from the back mirror surfaces fromentering the last pinhole. In many practical cases, it isan order of magnitude smaller so that ghosts may ap-pear.

Ill. Tandem SystemWhen two plane mirror Fabry-Perot interferometers

are placed in series, the resulting instrumental profileis given by (i = 0)

A(k,d,d') = - (5)

1 + 4N sin2(kd) 1 + 4(N;)2 sin2(kd')

Fig. 2. Design of a Fabry-Perot interferometer with the mirror pairsupported by horizontal rings: s, microspindles for mechanical tilt

adjustment; p, piezostacks; b, nut for varying mirror spacing.

where the primes indicate the corresponding parametersfor the second interferometer. Suppose now that bothinterferometers are at resonance for ko at orders p andp', respectively:

we set the commercially available Burleigh interfer-ometer in an upright position so that the mirrors couldbe supported horizontally on aluminum rings. Al-though almost any mechanically rigid constructionsuffices, the tubelike geometry of Fig. 2 has some dis-tinct advantages. This cylindrical construction is veryrigid and made almost completely of aluminum. Itencloses the air between the mirror blanks so that tur-bulence is not likely to occur. The mirror spacing isvariable from 1 to 50 mm by conveniently rotating thenut b. For smaller mirror spacings, the upper mirrorshould be somewhat larger in diameter. No attempthas been made to maintain parallelism between themirrors when their spacing is changed, since coarsemechanical adjustment of the upper mirror is easilyattained by means of the two microspindles s.

The masks with corner-cube prisms for multipassingare easily mounted on top of and underneath the in-terferometer. The resulting instrumental profile is thefunction of Eq. (1) to the mth power, where m is thenumber of passes.8 Measurements often show that thecontrast of an m-pass interferometer does not increaseas

Cm = (1 + 41) , (4)

kod = pr, kod' = pr, (6)

where p, p' are integers.When the first interferometer is scanned over a

wavelength shift AX,

A = 2Ad/p, (7)

where Ad is the mirror displacement from the originalspacing d of the first interferometer, the second inter-ferometer must transmit the same wavelength A0 + AXso that

A = 2Ad'/p'. (8)

Equations (6)-(8) yield for the relative scan ampli-tudes

(Ad')/(Ad) = d'/d. (9)

When Eqs. (6)-(9) are satisfied, the instrumental profilegiven by Eq. (5) has the shape shown in Fig. 3, which isequivalent to the one given in Cannell's work.6 In thisfigure, we assumed the following practical values for theparameters T, = TI = 0.75, N = N = 50, d'/d = 0.9.The resulting transmission function is not only sharperand has a much higher contrast than a single interfer-ometer but also has a significantly larger free spectralrange, a factor of 10 larger in the case shown. The

but is in practice several orders of magnitude smaller.It may then be argued that the cavity formed by thecorner-cube prisms and the mirrors is responsible forthis effect as it may pull extra light through the system.We have never observed such resonances, and they arenot likely to occur in a significant way since the accep-tance angle and the finesse of this large cavity are small.Moreover, even if these resonances existed, they wouldeasily be observed by pushing slightly against a cor-ner-cube prism.

However, we have observed ghosts which do affect thecontrast significantly. In most cases, these ghosts couldbe identified with resonances between the back surfacesof the mirror blanks. The reasons these ghosts appeardespite accurate spatial filtering is that the mirrorwedge is often chosen too close to the acceptance angle

Log(A(k,d,d))

d'=0.9d

dp)J2

Fig. 3. Instrumental profile of a single-pass tandem interferometerwitir mirror spacing ratio d'Id = 0.9, overall transmission of each in-

terferometer T, = 1 V = 0.75, and effective finesse N1 = N. = 50.


consequence of this latter feature is that broad Stokesand anti-Stokes components will no longer overlap.When spectra are taken near a solid-liquid interface asin Ref. 9, Brillouin peaks both from the solid and fromthe liquid appear. The frequency shifts of the peaksfrom the solid are about three times larger than thosefrom the liquid and therefore show up in higher orderand tend to confuse the spectra. This will not happenwhen such spectra are taken with a tandem system.

The practical realization of the electronic synchro-nization of the two interferometers is quite straight-forward. Equation (6) merely expresses the fact thatthe two interferometers should be locked on the samewavelength, most conveniently the laser line. Duringsubsequent scans, these resonances of both interfer-ometers should occur at exactly the same moment rel-ative to the beginning of each scan. This implies thatboth interferometers are scanned with a ramp voltagewhich originates from the same clock. Electronicallythis means that we start from one low voltage ramp,amplify this in two separate high-voltage amplifiers,which drive the two interferometers, adjust the relativeamplification of these amplifiers according to Eq. (9),and make sure that the laser line resonances of Eq. (6)occur at the same time relative to the beginning of eachscan. Before we show that this electronic synchroni-zation is easily accomplished with standard equipment,we first describe the Brillouin spectrometer with thetandem interferometer.

Consider the geometry of the spectrometer shown inFig. 4. The 514.5-nm line of an argon laser is almostcompletely reflected by the beam splitter B, toward thesample. A small part of the transmitted beam is di-rected by the beam splitter B2 to the spectrum analyzerSA for mode analysis. The beam transmitted by B isthen, via mirror M2, expanded by the lenses 13 and 14 toa parallel beam with a diameter of -15 mm. Thisbeam, which serves for the alignment of both interfer-ometers, can be blocked by the shutter Shl. Thisshutter is just a small piece of metal mounted on thespindle of a fast stepping motor.

The expanded alignment beam is directed via thebeam splitters B3 and B4 , each of which has a reflectivityof 0.01, through the spatial filter which consists of thelenses 15,6 and the pinhole pi. The diameter of thispinhole is given by the product of the acceptance angleof the interferometer as expressed by Eq. (3) and thefocal length of the lens 16. The diameter of the beamof parallel light leaving this lens determines the size Sof the areas selected for multipassing on the interfer-ometer mirrors. This diameter is usually of the orderof 10 mm when corner-cube prisms are used but may bechosen larger when cat's-eye reflectors are employed.

Although two mirrors, M3 and M3, are shown in Fig.4 for convenience, only M3 is necessary for directing thebeam vertically up to the first interferometer FP1. Thisinterferometer is operated in triple pass by means of thecorner-cube prisms Cl and C2. The beam leaving theinterferometer is reflected by the mirror M4 toward thebeam splitter B 5, which reflects a small part, say 0.01 tothe photodiode D.

The parallel beam of light transmitted by the beamsplitter B5 encounters the same interferometer opticsas the beam reflected by B4. When the alignment beamleaves the second triple-pass interferometer FP2 , it isdirected via the mirror M6 toward the reflection gratingGr. This grating is used in Littrov mount and servesto reject the Raman spectra by means of the last spatialfilter. This filter consists of the lenses 19, 110 and thepinhole P3. It also blocks possible ghosts as describedin the last section. The lens 110 finally images the lightonto the sensitive area of the photomultiplier PM, theelectrical signal of which is amplified by standardPrinceton Applied Research (PAR) equipment.

The digital pulses leaving this equipment are fed intothe Burleigh DASI, which is used in combination withthe high voltage amplifier HV2 to scan and servocontrolthe second interferometer FP2. It also accumulates thedata in its multichannel analyzer. This data acquisitionand stabilization system (DAS) has been described ina recent paper' in this journal. It has the basic prop-erty that one can program a window in the ramp at agiven time interval during each scan. A resonance lineof the interferometer is locked in the center of thiswindow and optimized in height by applying test andcorrection voltages to the piezostacks during each scan.The DAS1 has a low voltage ramp output which is am-plified by the separate high-voltage module HV2 whichdrives the interferometer. By varying the amplificationof HV2, the interferometer is programmed to have atilt-free and linear scan over as many as 6 orders. Thelinearity is programmed to be better than 0.004, whichis sufficient for tandem operation.

The low voltage ramp output of the DAS1 is also fedinto another high voltage module HV1 , which drives thefirst interferometer FP1. The laser resonance peak

Fig. 4. Brillouin spectrometer with two Fabry-Perot interferometersFP1,2 piezoelectrically scanned in tandem: i-1ro lenses; B1-5 , beamsplitters; MI-8, mirrors; P 1 5 pinholes; C1_4 corner-cube prisms; D,photodiode; Gr, grating; SA, spectrum analyzer; PM, photomultiplier;PAR, Princeton Applied Research photon counting equipment; DAS,data acquisition and stabilization system; and HVI, 2, high voltage

amplifiers.


from FP1 is detected by the photodiode D whose elec-trical signal is fed into the Burleigh DAS10. TheDAS10 in combination with HV, servocontrols the firstinterferometer. It receives a blanking pulse from DASIafter each scan, and its optimization window is alsotriggered by the corresponding window of DAS1. ThusEq. (6) is satisfied.

The alignment of the tandem system proceeds asfollows. With the shutter Shl in the open position, thefirst interferometer is brought into alignment using theclearly visible alignment beam which passes through thefirst spatial filter. Then this interferometer is scannedand locked into the window of DAS10. The beamleaving the first interferometer and transmitted by thebeam splitter B5 is blocked by the shutter Sh2, thuscompletely isolating the first interferometer from thesecond. The preliminary alignment of the second in-terferometer FP2 with the following optics is establishedby setting the shutter Sh2 in the open position and usingthe beam transmitted by FP1. After this preliminaryalignment, Sh2 is closed to block the light coming fromFP1 but making the way free for the light derived fromthe expanded laser beam by means of the beam splitterB3 and the mirror M8. This light overfloods the pinholeP2 and is still strong enough to be detected by the pho-tomultiplier so that it can serve as an alignment signalfor the DAS1. The amplification of HV2 is then turnedup from zero and the resonance peak from FP2 lockedin the window of DAS1. The proper adjustment of therelative ramp amplitudes to satisfy Eq. (9) is describedlater in this section. Finally, the shutters Sh, and Sh2are triggered by the optimizing window so that onlyduring locking the alignment beam is present and thetwo interferometers are optically isolated. We haveincorporated in the DAS1 an extra feature enabling itto be programmed to activate the shutters during anytime interval of each scan, including flyback.

If we had locked on the resulting resonance peak ofthe two interferometers as Sandercock7 does, the opti-mizing cycle of the test and correction voltages wouldhave become rather long. In that case, an extra dif-ferential drift step would have to be included in the fi-nesse optimization.

Our setup is flexible; the free spectral ranges of thetwo interferometers may be set quite arbitrarily sincesynchronization is guaranteed electronically. Thealignment of both interferometers is established inde-pendently as is locking to the laser line. We employ twotriple-pass interferometers in tandem as this is a goodcompromise in terms of contrast, finesse, and overalltransmission.

One may argue that the three spatial filters employedare cumbersome and introduce optical losses. This isnot true. They are helpful in aligning the spectrometerwith the clearly visible alignment beam. Once theselenses and pinholes are set into position, they should notbe touched again. In our setup, all lenses are standardachromatic doublets of 100-mm focal length. The di-ameter of the pinholes is then 100 pm, which allows forinterferometer spacings up to 10 mm [Eq. (3)]. Theoverall transmission of the spectrometer from 15 to 1o

is 0.10, which includes the 0.60 grating efficiency. Thetotal setup starting from mirror M2 and including thephotomultiplier can be mounted on a base plate of 0.25m 2 .

The alignment of the sample, the focusing, and thecollecting lenses 11,2 with the spectrometer may nowproceed while the spectrometer is scanning and lockedon the laser line. The axial alignment of the lens 12 isestablished by placing a diffuse scattering surface at thecommon focus of 1 and 12 and causing the scatteredbeam collected by 12 to pass without vignetting throughthe first spatial filter. After this axial alignment, thelens 12 should only be allowed to move along the opticalaxis. For optimum collection efficiency, the focallengths of 11 and 12 should be chosen so that the focalarea Sf seen from 12 with a solid angle dQ satisfies therelation of constant throughput:

r (bi)2 = SfdQ.4

(10)

In Eq. (10), as before, bi is the acceptance angle of theinterferometer, and S is the surface area of the parallelbeam of light incident on the interferometer mirrors.

As a test sample, a strong Brillouin scatterer is used,for example, ethanol. With the aid of a microscopepointed at the pinhole Pl, the focal region in the sampleis focused onto this pinhole. This light mainly containsthe two Brillouin components. The proper adjustmentof the relative ramp amplitudes is readily achieved bymaximizing the Brillouin peaks appearing on the mul-tichannel analyzer during each scan. Both conditionsof Eqs. (6) and (9) are now satisfied, and the spec-trometer is ready for operation. When the temperaturein the laboratory does not change more than a few de-grees, the DAS1 and DAS10 keep the interferometersin tune over any desired measuring time.

Although it is not necessary to perform a calibrationmeasurement after each change of mirror spacings orbefore a new series of measurements, it is useful and farfrom time consuming to do so. Such a measurementalso yields the absolute mirror spacings of both inter-ferometers. A change in either one can be measuredmechanically very accurately. The spacing ratio is thenknown in advance, and the corresponding relative rampamplitudes can be adjusted by looking at the laser res-onances on the multichannel analyzer.IV. Applications

As explained in the last section, a tandem Fabry-Perot interferometer becomes a powerful optical filterwhenever the problem of overlapping orders arises.Although sophisticated numerical techniques have beendeveloped to deconvolute the instrumental profile fromthe spectrogram, this deconvolution does not necessarilygive a unique answer. In many cases, the very sharpfilter described in the last section makes numericaldeconvolution obsolete.

Overlapping orders are, for example, present in theBrillouin spectra taken from viscous polymeric liquidsor soft plastics. In addition to the normal Brillouinpeaks, many of these substances given spectra with anextra central component which may be wider than theBrillouin shifts. For example, Wang and co-workers1


ar

aa,2

Ca)

.6a,a.U,

PPG ki

(b)

A ~

4 8 12 16 20shift[GHzl

Fig. 5. Brillouin spectrum from PPG taken with (a) a single inter-ferometer and (b) the tandem system with mirror spacing ratio d'/d

- 0.8.

have done beautiful experimental and theoretical workon the structural relaxation in polypropylene glycols(PPG). Although they carefully selected the freespectral ranges for their experiments, the problem ofoverlapping Stokes and anti-Stokes componentssomewhat limited their data. We have illustrated thisoverlapping in Fig. 5 where the upper part shows aspectrum of PPG taken with the second triple-pass in-terferometer alone (mirrors removed from P,),whereas the lower part was taken with the tandemsystem. The scattering geometry is shown in the inset:ki and k are the wave vectors of the incident andscattered light. The polarization is indicated by a dotor an arrow, depending on whether the light is polarizedperpendicular to or in the scattering plane. Note thesmall ghost at 19.22 GHz, which is due to the strongstray scattering of the liquid. This ghost is not suffi-ciently suppressed by the first interferometer. Thesecond interferometer has a somewhat larger contrastso that the ghost at 16 GHz (d' 0.8d) is completelysuppressed. It should be emphasized that the spectraof Fig. 5 only serve to illustrate the higher resolution incombination with the larger dynamic range of the tan-dem system.

Our thinking about tandem systems was initiated byour measurements near a liquid-solid interface as de-scribed in Ref. 9. In these experiments, laser light wasfocused near a liquid-glass interface so that Brillouinpeaks of the liquid as well as of the glass were observed.Only the Brillouin spectrum of the liquid was of interest.The larger shifts corresponding to the transverse andlongitudinal Brillouin doublets of the solid appeared inhigher order and tended to confuse the spectrum ofinterest. By careful setting of the mirror spacing of thesingle interferometer we used, most of these doubletscould be arranged to fall under the laser resonancepeaks and in the center of the free spectral range.However, for some scattering geometries it was notpossible to separate the peaks of the substrate from thespectrum of interest. This is a repeating problem inlight-scattering experiments performed in the vicinityof interfaces.

A relatively new problem of industrial importance isthe adhesion and stiffness of thin plastic films on hardglass substrates. The adhesion of such a film onto itssubstrate can be investigated by means of the Brillouinscattering technique described in Ref. 9. The hyper-sound stiffness is directly calculated from the frequencyshifts of the Brillouin peaks. Therefore, Brillouinscattering is a unique tool to investigate the elasticproperties of such supported thin films, and we shall seethat here lies a powerful application of the tandemsystem.

In this paper we want to show an interesting featureof supported thin plastic films. These films act asacoustic waveguides for the thermal acoustic phononsin complete analogy with the planar optical waveguidesin integrated optics. However, unlike the electro-magnetic waves, the acoustic modes can have transverseas well as longitudinal polarization. The transversemodes polarized in the scattering plane are coupled tothe longitudinal ones by means of the acoustic boundaryconditions at the interfaces. These so-called general-ized Lamb modes12 are observed in the backscatteringgeometry illustrated in the upper spectrum of Fig. 6. Inthis geometry, the transverse acoustic phonons polar-

shift IGHz]

Fig. 6. Brillouin spectra of polycarbonate films of varying thick-nesses on Pyrex substrates taken with the tandem system; the scat-

tering geometries are shown in the insets.


ized perpendicular to the scattering plane should notbe observed according to the standard Brillouin scat-tering selection rules.

We have used polycarbonate (PC) films supportedby Pyrex substrates. From a light-scattering point ofview, these films have the advantage that their opticalcontrast with the substrate is negligible (nPyrex = 1.47and npc = 1.59), whereas the acoustic contrast, whichbecomes apparent in a Brillouin scattering experiment,is significant.

Consider now Fig. 6. When the PC film is 5.5 Amthick, the zero-order transverse Lamb mode appearsalready in the spectrum as the small peak at 2.40 GHz.The strong peak at 13.15 GHz corresponds to the lon-gitudinal Brillouin peak of bulk PC. With decreasingfilm thickness, the discrete acoustic modes becomeapparent and more separated. Especially the narrowtransverse modes with the lower shifts are most distinct.It is interesting to note that a Brillouin spectrum isreadily observed, even from a film 0.1 ,m thick. Thedata accumulation time of this spectrum is 30 min withan incident laser power of 0.5 W.

The last two spectra of Fig. 6 were taken in a forwardscattering geometry (see inset) so that the transversemodes polarized perpendicular to the scattering planeshould be observed. The first of this series shows thebulk spectra of both the film and the Pyrex substrate.In this symmetric scattering geometry, the frequencyshifts v of the transverse and longitudinal peaks, (tp ,tg)and (Lp,Lg), of the plastic and glass, respectively, di-rectly yield the transverse and longitudinal hypersoundvelocities v = 2nr-vA/v ki. The two transverse peaks arenow depolarized. With decreasing film thickness, adiscrete set of these depolarized peaks due to the so-called Love modes12 in the polycarbonate films shouldbe observable. As the last spectrum of Fig. 6 shows,only the zero-order depolarized mode is present. Whenthis spectrum was taken, a polaroid was placed in thedirection of observation to block the polarized scatteringfrom Lamb modes.

As a last application we discuss some new measure-ments on liquid metals. In our former paper,9 we pro-posed a scattering experiment where a Brillouin spec-trum is taken from liquid mercury resting against a glasssubstrate. The mercury surface is then nearly clampedso that no scattering from the stress-free surface ripplesas in our early work on liquid metals1 3 is allowed tooccur. Figure 7 shows the relevant Brillouin spectrumwith the scattering geometry shown in the inset. Theincident and scattered light are polarized in the scat-tering plane. The data are accumulated for 2 h with0.3-W incident power. This spectrum has the followingnew features. The strong peak next to the laser line isdue the scattering from the combined surface ripplesof the glass substrate and the mercury. The frequenciesof these so-called Stoneley waves12 '14 have about halfof the values of the corresponding frequencies of theRayleigh waves on the stress-free glass surface and are-1-2 orders of magnitude higher than those of theripplons on a free liquid mercury surface. The shift ofthis peak varies with cos(Gi) + cos(05) [i and O. are the

k r.

BK 7 -Hg

shift 1GHz]

Fig. 7. Brillouin spectrum taken at a clamped liquid mercury sur-face; scattering geometry is given in the inset. The sharp peak closeto the laser line is due to the Stoneley surface phonons at the glass-mercury interface. The small jump at the tail of this peak is due tothe bulk phonons in the mercury. The peak of the broken line isobserved when instead of the mercury the back surface of the glass

is coated with aluminum.

angles that ki and k, make with the interface (Fig. 7)]and is slightly dependent on the elastic constants of thedifferent glasses we used as substrates (fused silica,BK7, SF59, and gadolinium gallium garnet). As Fig.7 shows, it rests on a central component whose origin ispresumably due to the isobaric density fluctuations inthe mercury. On the tail of the surface peaks, oneclearly discerns a discontinuous jump in the data. Thisjump corresponds to the cutoff frequency of the Bril-louin spectrum of a clamped liquid mercury surface aspredicted in Ref. 9. The tail of this latter spectrumdecreases in intensity because it rests on a weak anddiminishing central component due to the glass sub-strate.

We have also taken spectra from the glass demi-spheres with their back surfaces coated with a 40-nmthick aluminum layer. The resulting shifts originatingfrom the Stoneley surface waves are a factor of 2.5 largerthan in the case of liquid mercury (broken line in Fig.7).

This concludes the section on applications of thetandem Fabry-Perot interferometer to Brillouin scat-tering. For further applications to surface phononscattering on opaque solids and spin wave scattering,the reader is referred to Sandercock's work.7

V. Conclusions

We have shown that the construction of a multipassFabry-Perot interferometer is quite simple when themirror blanks are supported by horizontal rings. Whenthe mirrors are coated with a stack of ZnS-MgF 2 layers,the mirror flatness is not affected, and only low scat-tering and absorption losses are present.

It is also shown that the operation of any two Fabry-Perot interferometers piezoelectrically scanned in


tandem is not difficult with standard electronic scan-ning and stabilization equipment when some simpleoptical alignment procedures are followed. The mainideas behind tandem synchronization are that bothinterferometers are scanned with a ramp originatingfrom one clock and that the interferometers are opti-cally isolated during the small time interval of lockingto the laser line.

Applications of the tandem system to Brillouinscattering are given. The extra feature of a tandeminterferometer is its relatively large free spectral rangeresulting in less overlap of orders. Full merit of thissystem is obtained when broad spectra are to be ob-served or when a range of peaks over a wide frequencyregime is expected. The system has potential uses forresearch areas in polymer science.

Finally, we have reported the observation of a Bril-louin spectrum from a clamped liquid-metal surface,a spectrum which we were not able to resolve with asingle interferometer. This experiment also yielded asimple scattering geometry for observing the surfacephonons of the Stoneley type.

We are very grateful to Burleigh Instruments, Inc.,Fishers, N.Y., for their cooperation in this research.One of the authors (O.G.D.) would like to thank HarryYates from Imperial College Optical Systems in Lon-don, England, for his stimulating cooperation in re-search on optimum dielectric coatings and stress-freemounting of his beautifully X/200 matched mirror pairs.Finally, thanks are due to G.M.M. van de Hei of ourlaboratories, who made the thin plastic films on sub-strates.

References1. P. Hariharan and D. Sen, J. Opt. Soc. Am. 51, 398 (1961).2. J. R. Sandercock, in Light Scattering in Solids, Proceedings,

Second International Conference, Paris, M. Balkanski, Ed.(Flammarion, Paris 1971), p. 9.

3. A. S. Pine, Phys. Rev. B: 5, 2997 (1972).4. H. Sussner and R. Vacher, Appl. Opt. 18, 3815 (1979).5. D. S. Cannell and G. B. Benedek, Phys. Rev. Lett. 25, 1157

(1970).6. D. S. Cannell, Adv. Chem. Phys. 24, 28 (1973).7. J. R. Sandercock, in Proceedings, Seventh International Con-

ference on Raman Spectroscopy, W. F. Murphy, Ed. (NorthHolland, Amsterdam, 1980), p. 364.

8. C. Roychoudhuri and M. Hercher, Appl. Opt. 16, 2514 (1977).9. J. G. Dil and N. C. J. van Hijningen, Phys. Rev. B: 22, 5924

(1980).10. W. May et al., Appl. Opt. 17, 1603 (1978).11. Y. H. Lin and C. H. Wang, J. Chem. Phys. 70, 681 (1979).12. B. A. Auld, Acoustic Fields and Waves in Solids, Vol. 2, (Wiley,

New York, 1973), pp. 95-103.13. J. G. Dil and E. M. Brody, Phys. Rev. B: 14, 5218 (1976).14. E. L. Albuquerque, J. Phys. C: 13, 2623 (1980).

Books continued from page 1373Electromagnetic Theory of Gratings. Edited by R. PETIT.

Springer-Verlag, Heidelberg, 1980. 284 pp. $38.35.

Although diffraction gratings were invented in the eighteenthcentury by David Rittenhouse and improved over the years by variousscientists, very little was done toward calculating their efficiencies.During the last decade and a half, steps have been taken to fill thishiatus by a group at the University of Aix-Marseille III, headed byR. Petit. This is not to say that Petit and his group were the first, orare the only ones, to calculate efficiencies, but they have mounted aconcerted effort to solve the problem that has met with much suc-cess-and they have written the first book on the subject. This bookcan be considered as an overview of the work done to date, in calcu-lating efficiencies of plane gratings. Rigorous calculations for concavegratings are still beyond the state of the art. Most of the contents areby Petit and his colleagues, but the final chapter was written by threeresidents of Australia, a country where there is also a group workingon the problem of grating efficiencies.

The book contains seven chapters: a tutorial introduction byPetit; some mathematical aspects of the grating theory by M. Cad-ilhac; integral methods by D. Maystre, differential methods by P.Vincent; the homogeneous problem by M. Neviere; experimentalverifications and applications of the theory by Maystre, Neviere, andPetit; and theory of crossed gratings by R. C. McPhedran, G. H.Derrick, and L. C. Botten.

The first two chapters, tutorial introduction and some mathe-matical aspects ... , provide an explanation of how the problem isreduced to mathematics and some of the techniques required to obtainthe solutions. It should be mentioned here that in this book the or-ientation of the electric vector is with respect to the grooves; Ell isparallel to the grooves and E 1 is perpendicular to them. The usualorientation convention, e.g., as used in calculating reflectance usingFresnel equations, is to refer the direction of the electric vector to theplane of incidence so that El, perpendicular to the plane of incidence,would be parallel to the grooves when the grating is used in the con-ventional manner.

Integral methods (IM) shows how to reduce the problem of gratingdiffraction to the solution of a linear integral equation or a system ofcoupled linear integral equations. This technique appears to be themost general for a rigorous solution of the problem, although not thesimplest for computing or for providing an intuitive understanding.The differential method (DM) can also be used in rigorous calcula-tions of grating efficiencies by the solution of partial differentialequations with suitable boundary conditions. However, the generalconditions are more restrictive than for the IM. The IM is capableof dealing with materials having very high conductivity, but appar-ently large conductivities cause numerical problems in the algorithmsused in computing with the DM. In fact, the DM will not work forperfectly conducting surfaces, and it is necessary to resort to a con-formal mapping method that is competitive with the IM for gratingswith these surfaces. One might ask, why bother with perfectly con-ducting surfaces since there are, in fact, none. The authors point outthat the properties of gratings with metal surfaces can be calculatedquite accurately assuming perfectly conducting surfaces for wave-lengths >4,m.

The homogeneous problem deals with grating anomalies; phe-nomena connected with the excitation of surface waves along a peri-odic structure. This section also includes grating couplers used inintegrated optics and a theory of the grating coupler is presented.One of the unexpected results arising from this study is the fact thata grating with a highly reflecting coating can be made to absorb theincident wave completely if the ratio of grating spacing to groovedepth is adjusted properly.

To obtain experimental verifications .. . of the calculations, onemust measure the efficiency of the grating in all possible orders,preferably with polarized radiation, the groove profile must be de-

continued on page 1402


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