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Far infrared ordinary ray optical constants of quartzK D C u m m i n g s a n d D B T a n n e rDepartment of Physics The hioState U niversity Columbus Ohio 43210(Received 20 March 1979; revised 9 Ju ly 1979)

By using asymmetric Fourier-transform techniques, the room-temperature optical constants of z-cut quartz have been determined between 100 and 500 cm -1

The infrared and far-infrared properties of crystal quartzhave been studied a number of t imes over a period of 50years .1 - 1 7 Th ere a re two reasons for th is a t tent io n . Th ela t t ice dynamics of th is mater ia l a re qui te in te rest ing , par

t icula r ly a t h igh tem pera tures . In addi t ion , quar tz i s one ofthe most useful window materials in the far-infrared, with ahigh t ransparency a t room tempera ture for f requenc iessmal le r than 250 cm - 1 .123 J. Opt. Soc. Am., Vol. 70, No. 1, Jan uar y 1980 0030-3941/80/010123-04 00.50 1980 Optica l Society of America 123

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The optical properties have been determined in variousways by previous investigators. In the far infrared, measurements of the power transmission of crystalline samplesprovide the absorption coefficient (or, equivalently, the extinction coefficient), while analysis of the interference patte rnarising from multiple internal reflections in samples withparallel surfaces gives the index of refraction. This procedure,which has been used by Roberts and Coon,6 Berman andZhukov,10 Plendl et al. 12 and Loewenstein et al.16 atfrequencies smaller than about 350 cm - 1 , has the advantageof being experimentally straightforward and easily analyzed,but does not work in regions of high absorption. At higherfrequencies, in the o ptical phonon region, measureme nts ofthe power reflectance can be followed by either a fit to a seriesof oscillators to find classical dispersion parameters or by aKramers-Kronig analysis to obtain the phase shift and theoptical constan ts. Reflectance measurem ents have beencarried out for quartz by Spitzer and Kleinman, 4 Merten,14Onstott and Lucovsky,15 and Gervais and Piriou17 forfrequencies between 300 and 2000 cm - 1 .

Chamberlain et al.8 and Russell and Bell13 studied theoptical properties of crystal quartz using a third technique,namely, asymmetric Fourier-transform spectroscopy whereinthe sample is introduced into one arm of a Michelson interferom eter. Russell and Bell13obtained the index of refractionand absorption coefficient between 20 and 370 cm -1for boththe ordinary and the extraordinary ray in quartz. In thisLetter we report a measurement of the ordinary-ray opticalconstants of z-cut quartz between1 and 500 cm -1using theasymm etric Fourier-transform technique. A least-squaresfit to a sum of Lorentzian oscillators gives the oscillator parameters for the observed optical phonons in the 350-500cm -1 region, a region that has not previously had much experimental attention.

FIG. 1. Index of refract ion for z cut crystal quartz between 100 and 500c m 1 . The data points are shown at a resolution of 3.2 c m 1 . The scatterin the smo oth port ion of the curve is indicat ive of the signal noise rat io inthe measurement.

FIG. 2. Absorpt ion coeff ic ient for z cut crystal quartz betwee n 100 and500 cm 1 The maxim um value obtained for the absorpt ion coe ff ic ientis 33 000 cm 1 at 440 cm 1 The negative values observed near the stronglines arise because no apod izat ion was used on the interferogram .

Asymm etric interferometry in the far infrared was developed by Chamberlain et al.8,18 and by Bell.19 In this technique,- the sample under investigation is placed in one armof a Michelson interferometer where it affects b oth the amplitude and the phase of the radiation traveling in that arm.Th e interference fringes produced by the interferometer arechanged both in intensity and position by the introduction ofthe sam ple.The interferometer used in these measurementsisthe onebuilt and described by Russell and Bell.21 A beam splitterof 2.5- m thick Mylar and a Golay detector w ith a diamondwindow were used in these measurem ents. With a blackpolyethylene transmission filter, this combination providedfar-infrared energy between 100 and 500 cm -1 .Th e single-crystal z-cut qua rtz plate used in these studieswas obtained from the Karl Lam brecht Corporation. Inz-cutqua rtz th e crystallographic c axis is normal to the surface ofthe c rystal so tha t the optical properties are governed by theordinary-ray complex refractive index. The ordinary-rayindex of refraction of this specimen between75and 550 cm-1is shown in Fig. 1 and the absorption coefficient = 2 k/cover the same frequency region is shown in Fig. 2. The dataare shown at a resolution of 3.2 cm -1 without apodization.The index of refraction near 150 cm -1 is 2.14, in reaso nablygood agreem ent with the data of Russell and Bell.13 The

signal-to-noise ratio is not good enough to see the weakstruc ture at 128 cm -1 Absorption maxima, with corresponding structu re in the refractive index, are seen at263, 393,and 450 cm -1 Th e absorption coefficient isgreater than 100cm -1 for frequencies greater tha n 250 cm -1 ,sothis frequencyis an effective upp er limit for the transm ission ofallbut verythin specimens.The dielectric response of an insulator is often describedby a complex dielectric function which is a sum of Lorentzianoscillators,

124 J. Opt.Soc.Am., Vol.70, No.1, January 1980 JOSA Letters 124

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TABLE I: Osci l lator parameters for low frequency ordinary-ray opt icalphonons in quartz.

quar tz , where the elect r ic f ield i s perpendicu lar to the crys-t a l l o g r a p h i c c ax i s , t h e re a re e ig h t o f t h ese p h o -n o n s .1 3 , 1 4 , 1 7 , 2 2

Th e complex d ielect r ic funct ion i s related to the complexrefract ive index by N2 and can be wri t ten as

FIG.3. Real part of the dielectric function for z-cut crystal quartz between100 and 500 cm -1 The solid line is a fit to a sum of three oscillators usingparameters given in Table I.

wh ere is the con tr ibu t ion of h igh-frequency (elect ron ic)t ransi t ions, the sum run s over the symmetry-al lowed infraredact ive v ibrat ional m odes of the lat t ice, j is the frequency ofth e j t h t r an v er se -o p t i c m o d e ,Sj is i t s s t rength , and j is itsfu ll wid th at half maximu m. For the ord inary ray in crystal

where 1( ) i s the real d ielect ric function an d 1( ) is th efreque ncy-d epen dent c onduct iv i ty . F igures 3 and 4 show 1 )and 1( ) , respect ively . Th e data are shown as po in ts ,an d fits to Eq. (1) as the solid l ines. Th ese fi ts are done as anonlinear least-squares procedure23 directly to1( ) o r t o 1 ( ) . Beca use the wings of the 393- and 450-cm -1 Huesover lap , we have made a s ix-parameter f i t to th is doublet ,whereas the 263-cm -1line was fitted separately.

The parameters obtained by this fi t t ing procedure are givenin Tabl e I. W e list the frequ ency coy, the w idth j , and thes t r en g thSjof each mode in the f i rs t th ree co lumns. Th e nex tthree co lumns of th is tab le g ive the osci l lator s t rengths obtaine d by thre e previous investigators.4,13,17 Our results showno significant difference in the frequencies or widths of thesemod es. On the o ther hand , there are d i f ferences in the oscil lator stre ngth s. Russell and Bell13 have suggested tha t thest rengths ob tained by Spi tzer and Kleinman 4 for the 393 and450-cm -1 mode s should be increased to g ive agreem ent between the static dielectric constant calculated from Eq. (1) andtheir value determined from extrapolation of the far-infraredrefract ive index to zero f requency . Our resu l ts for the 393-cm -1 line are in accord with this suggestion. For the 450-cm-1l ine our resu l ts are smal ler than prev iously found .4,17 H o w ever, this value has a larger unce rtainty because of a low signalfor f requencies greater than 430 cm -1 in the present exper i m en t .

FIG.4. Frequency-dependent conductivity for z-cut crystal quartz between100 and 500 cm -1 The sol id l ine is obtained from a sum of three osci llators using parameters given in Table I.

1H. Rubens, Uber Dispersion Ultraroten Strahlen, Wied Ann. 45,238-261 (1892).2M. Czerny, Zum Raman-Effekt Des Quarzes, Z.Phys.53,317-330(1929).3R. Bowling Barne s, Me asurem ent in the long wavelength infraredfrom 20 to135 , Phys. Rev. 39, 562-575 (1932).4W. G. Spitzer andD.A. Kleinman, Infrared lattice bands of quartz,Phys. Rev. 121, 1324-1335 (1961); D. A. Kleinman and W. G.Spitzer, Theory of the optical properties of quartz in the infrared,Phys. R ev. 125, 16-30 (1962).5R. Geick, Der Brechungsindex Von Kristallinem und Geschmol-zenem Quarz im Spektralbereich um100 , Z. Phys.161,116-122(1961).6S. Roberts and D. D. Coon, Far-infrared p roperties of quartz andsapp hire, J. Opt. Soc. Am. 52, 1023-1029 (1962).

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7A. Hadni, J . Claudel , E. Decamps, X. Gerbaux, and P . S tr imer,S pec t re s D 'Abso rp t ion de Monoc r i s taux Dans L ' in f ra rougeLo in ta in (50 -1600 ) , A la Tempe ra tu re de L 'He l ium Liqu ide :Iodu re de Ces ium, Qua r tz , Ge rman ium e t Ni t ra te de Neodyme ,C. R. Acad. Sci. 255, 1595-1597 (1962).8J . E. Cham berla in , J . E. Gibbs, and H. A. Gebb ie , Ref ractom etryin the F a r In f ra -Red Us ing a Two-Beam In te r fe rom e te r , Na tu re198, 874-875 (1963).9J . E. Cham berla in and H. A. Gebbie , De term inat ion of the Refractive Index of a Solid Using a Far Infra-red Ma ser, N atur e 206,602-603 (1965).10L. V. Berm an and Z. G. Zhukov, Opt ical Con stant s of Crysta l l ineQua rtz in the Far Infrared Region, Opt. Spektro sk. 21 , 735-740(1966) [Opt. Sp ectrosc . 21 , 401-404 (1966)] .11O. M. Clark , Te mp erat ure Depen dence of the Infrared Spectra ofSelected Dielectr ics II , Air Force Cam bridge Research La borato r ie s Repo r t AF CRL 66 -20 (unpub l i shed ) .12 J . N. P lendl, L. C. Mansur, A. Hadni, F . Brehat , P . Henry , G. Morlot ,F .Naudin , and P . S tr imer, Low Tem peratu re Far Infrared Spectraof SiO 2 P o lym orphs , J . P hys . Chem. S o l id s 28 , 1589 -1597(1967).13E. E. Russell and E. E. Bell , Mea surem ent of the optical constan tsof crysta l quartz in the far infrared with the asymmetric Fourier-transform me thod , J . Opt. Soc. Am. 57, 341-348 (1967).14L. Merten, Zur Ultrarot-Dispersion Zweiachsiger und EinachsigerKris ta l le , Z. Natu rforsch 23a, 1183-1193 (1968).15J . Onstot t and G. Lucovsky, Direc tional Dispersion of Ext raor di

na ry Op t ica l P honons in -Qua r tz in the F requency Domain f rom380 to 640 cm - 1 , J . Phys. Chem. Solids 31, 2171-2184 (1970).16E. V. Loew enste in , D. R. Smith , and R. L. Morg an, Opt ical Constan ts of Far Infrared Material s. 2: Crystalline Solids, Appl. Op t.12 ,398 -406 (1973) .17F. Gervais and B. P ir iou, Tem pera ture Depen dence of Transv erseand Long i tud ina l Op t ic Modes in the an d phase o f Qua r tz ,Phys. Rev. B 11 , 3944-3950 (1975).18J . E . Chamber la in and H. A. Gebb ie , Dispe rs ion M easu rem en tson Polyte traf luor ethylen e in the Far Infrare d, Appl. Opt. 5 ,393-396 (1966).19E. E. Bell , Mea surem ent of the Far Infrared Optical Propert ies ofSolids with a Michelson Interferometer Used in the AsymmetricMode: Pa rt I, Mathem atical Formulation, Infrared Phys. 6 , 57-74(1966).20 For an e lementary descrip tion, see R. J . Bell,Introductory FourierTransform Spectroscopy (Academic, New York, 1972), Cha p. 8 .2 1 E . E. Russell and E. E. Bell , Me asur em ent of the Far InfraredOptical Propert i es of Solids with a Michelson Interferometer Usedin the Asymm etr ic Mode : P a r t I I, the Vacuum In te r fe rome te r ,Infrared Phy s. 6 , 75-84 (1966).22A. N. Mirgorodskii and A. N. Lazarev, Calcu la t ion of In tensit iesin an -qua rtz C rysta l as a Method of Checking on the Shapes ofthe Vibra tions, Opt. Spektro sk. 34, 895-901 (1973) [Opt. Spectrosc. 34, 514-517 (1973)].23 Phil ip R. Bevington,Data Reduction and Analysis for the PhysicalSciences (McGraw-Hill , New York, 1969), Chap. 11.

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