Lummer-Gehrcke interferometer modified for thespectroscopy of thin dielectric films

Cardinal Warde

The obsolete Lummer-Gehrcke interferometer is reexamined, and it is shown how, with a few simple modifi-cations, this instrument can be used to study the optical properties of thin, solid, dielectric films. In partic-ular, the thickness, the wavelength dispersion, and the absorption spectrum of the film can all be obtainedsimultaneously. In the case of anisotropic films, the polarization dependence of these parameters is ob-tained very conveniently. The modified interferometer allows continuous monitoring of the above parame-ters in situ during growth.

I. IntroductionThe first version of this interferometer was devised

by Lummer.1 It consisted of an accurately plane-par-allel plate of glass with one edge beveled so that lightcould be introduced into the plate near the critical angle.Later, it was modified by Lummer and Gehrcke2; thebevel was eliminated, and a prism, cemented to theplate, was used to couple the light into the plate. Theplate was usually about 4.5 mm thick and 10-15 cmlong.

Until about 1927, the instrument was used solely forthe analysis of the fine structure of spectral lines. Theaccessories were usually a monochromator-collimatorarrangement for getting the light in and a telescope forobserving the fringes. After it was discovered that athin aluminum film could be used as a reflector in theuv, the Fabry-Perot interferometer was used instead,and the Lummer-Gehrcke interferometer became ob-solete. Detailed descriptions of the instrument can befound in Candler3 and Born and Wolf. 4

In the work reported here, this interferometer hasbeen modified so it can be used to study the opticalproperties of thin, solid, dielectric films. The modifiedinterferometer offers the advantage that the filmthickness, indices of refraction, and internal reflectionspectra can be monitored simultaneously in situ duringgrowth. Transparent, isotropic films as well as specialcases of weakly-absorbing, and anisotropic films arediscussed. In all of these cases this technique, in asingle experiment, leads to a full characterization of theoptical properties of the material. Anisotropic films ofsolid oxygen at 10 K have been studied with this in-strument, and some of the results are presented.

The author is with Massachusetts Institute of Technology, Department ofElectrical Engineering and Center for Materials Science and Engineering,Cambridge, Massachusetts 02139.

Received 13 March 1976.

II. Modified InterferometerThe first modification is more than an order of mag-

nitude reduction in the thickness of the plate. Platethicknesses of 130-150 Atm were found to be practical.The second modification is the addition of an outputprism so that the absorption spectra of thin films de-posited on the plate could be recorded by the conven-tional technique of internal reflection spectroscopy.5

The third is a change in the fringe-observation opticsso that white light fringes could be analyzed (see Fig.1).

The third modification is accomplished by spatiallyfiltering the interfering beams with two slits orientedparallel to the plane of the plate and focusing the twoemerging beams on the slit of a high resolution spec-trograph. One of the slits is positioned so that only lightnear grazing emergence is allowed to reach the spec-trograph and the other to pass light at about 8-10°above grazing emergence. With slits about mm wide,the resulting diffraction is tolerable, and the angles ofemergence can be measured accurately. Since thestrongest beams come from the area just in front of theinput prism, the angular direction of the emergent lightis well defined, and the angles can be measured to anaccuracy of 0.050. With these modifications twochanneled spectra are produced in the focal plane of thespectrograph.

Making the slits narrow reduces the number ofemergent beams, but because the plates are thin this canbe tolerated. For example, a slit 1/2 mm wide samplesa 14.3-mm length of plate if 03 = 880; it therefore existsabout fifty-five beams if the plate has an index of 1.52and is 150 gm thick. This is greater than the numberof beams that were available with the old instru-ments.

There are, however, two disadvantages to making theplate very thin. The first is a loss of resolving power;

2730 APPLIED OPTICS / Vol. 15, No. 11 / November 1976

X= Xenon arc lamp ES SSLi L 2=Lens | P2 i

P1,P2,P3 ,P4 = Polarizer . ....SS= Spectrograph slitES = 12mm exit slits

S , S2 = Collimating slits 3b . .

SI h - ....... ...... ......... . .

a|.' A\ I l'rA . I~~~~~~~~~~~~P41

Fig. 1. The modified Lummer-Gehrcke interferometer.

(a)

where A = M sinO3a is a constant. Here ml is the orderof the interference; t is the thickness of the plate andfli its index of refraction; ij3 is the index of the sur-rounding medium and O3a the angle of emergence. Thesubscript 1 is a label for the clean substrate.

One way of finding m1 involves knowing the disper-sion of the substrate. The index refraction at twowavelengths is sufficient. If Xc and XD are two wave-lengths at which the index of refraction is known andif maxima of order mic and m1D occur at Xc and XD,then m1c/miD can be found from Eq. (1), and the orderdifference Am 1 = miD - mic can be obtained bycounting fringes. Thus m1 can be calculated explicitly.If the index of refraction is known with sufficient ac-curacy, m1 can be found with an error of i1. It will beshown later how t1 and x11(X) can be determined. In thecase where the dispersion of the substrate is not known,if t 1 can be measured accurately, ml and -a at a partic-ular wavelength can be found from Eq. (1) by measuringthe angular positions of the fringes.

A. Free Spectral RangeIf Eq. (1) is differentiated with respect to ml and Am1

is set equal to unity, it follows that the wavelengthseparation of successive orders, also called the freespectral range (AX)SR, is given by

(A X)SR = 2(,1

2- A2)12

(S)R2t,(77Gl -A2)(2)

(b)

Fig. 2. (a) Ray diagram for a clean plate; (b) ray diagram for a platecovered with a thin dielectric film.

to a first approximation the resolving power is propor-tional to the order of interference and to the effectivenumber of beams. Making the plate thin decreases theorder of interference, but it increases the effectivenumber of beams. However, the second effect is usuallynever big enough to compensate for the first, and theresult is a loss of resolving power. The other disad-vantage is that thin plates are very fragile and must behandled delicately.

Ill. Principle of OperationThe interferometer i's operated so that the input light

is incident on the surfaces of the plate at an angleslightly smaller than the critical angle. When thiscondition is fulfilled the reflectivity is very high, and alarge fraction of the light is reflected back into the plate;however, a series of weak beams emerge from the surfaceof the plate by a process that involves both refractionand diffraction; see Fig. 2(a).

In this new fixed-angle mode, the condition for abright maximum4 can be rewritten as

(mX) = 2t,(nl2 A2)1/2 (1

where G1 is the group index defined by G = w7-X[(dinl)/(dX)]. This implies that (AX)SR is larger atlonger wavelengths and smaller at shorter wavelengths;and in a nondispersive medium it is directly propor-tional to X2. When Eq. (2) is expressed in wavenumbersusing the relation v = 1/X, it is seen that (Av)SR, the freespectral range in wavenumbers, is practically constant;a closer examination reveals that it is essentially thereciprocal of the group path difference between suc-cessive emergent beams. As an example, consider aplate 150 ,4m thick with index of refraction of 1.52 at6000 A; Eq. (2) shows that if dispersion is neglected, theseparation of successive orders is about 10.5 A for 03 =880.

As will be shown later, the interferometer can be usedin this new, constant-angle, white-light mode of oper-ation to study the birefringence of thin films depositedon the plate, so long as the spectral detail associatedwith each order falls within the free spectral range.

B. Angular Dispersion

The angular dispersion is defined by d03/dX; thus, forthis exercise 03 must be treated as a variable. If 773 istaken to be unity and Eq. (1) is differentiated with re-spect to X, it can be shown that

dO3 2i7, X[(dn,)/(dX)] - 2(712 - sin203)

dx X sin2O3(3)

This equation shows that the angular dispersion de-pends only on the plate material and the angle ofemergence. It increases as 03 approaches grazingemergence and does not depend on the thickness of theplate.

November 1976 / Vol. 15, No. 11 / APPLIED OPTICS 2731

.~j.

CD

0)In

CD

00

to0CD

0

CDU,)

substrateox~~~~~t I, ,l , Al, %1 % ,%I %% . .! .! %. ,,1 11 % I% %% ItY Clean2 X' "A I X It I, It IX * % t ' i i; 't %' i L *t * *% % s ~ % 1p ' wIt % %1 1 X %'' 0. o4 5

4b i &, ~tIstz81izszl¢5..ti4W1%s It9

6 *E|4ib-s|i1t; ; 8 i*^i*%%i~lili~ilii*^la %9~iiiit.kit'llit8

10

14

1.35

1.80 in~C

2.25.c

2.70 3.15 L

16 3.60

Fig. 3. Lummer-Gehrcke inter-ference patterns recorded beforeand during the deposition of a thinfilm of solid oxygen. Note thateach order not only shifts to longerwavelengths but splits into two

components.

Consider the case where the plane of the plate isperpendicular to the slit of the spectrograph. Sinceeach exit slit must have a finite width, the emergentLummer-Gehrcke beams will have a finite angularspread. Then, according to Eq. (3), the maxima willoccur at longer wavelengths for rays emerging throughthe top of the slit than for rays emerging through thebottom of the slit; that is, the maxima will appear asslanted lines in the focal plane of the spectrograph.This can be seen in the clean substrate fringes of Fig. 3.This effect can be eliminated by orienting the plane ofthe plate parallel to the slit of the spectrograph.

C. Thickness and Wavelength Dispersion of the PlateThe condition for a bright maximum with a clean

plate is given by Eq. (1). To find tl and - 1(X) it is nec-essary to record the spectrum at two different angles,03a and 0 3b. If A = 73 sin03 , and B = 73 sinO3b it followsthat

(2t,)2 = (ml)b' -(mX)a (4)

A' - B'

and2 = (mI 2)a 2

771 = +A2. (5)

In practice the data are best handled analytically.The quantities (mX) can be plotted graphically as afunction of X and fitted with least-squares curves. Thentl and 771(X) can be calculated from Eqs. (4) and (5). Itis clear that [(mlX)b 2

- (l X) 2] should be a constantindependent of wavelength. Therefore the most likelyvalue of t1 obtained from Eq. (4) should be used in Eq.(5) and the error computed.

D. Thickness and Index Measurements of Thin Films

Consider a thin, isotropic, nonabsorbing layer ofthickness t2 and index of refraction 772 in contact withthe plate as shown in Fig. 2(b). For light polarized ei-ther perpendicular or parallel to the plane of incidence,the phase changes for rays transmitted across the in-terfaces (12t,k23t, and k21t) are zero; furthermore,since 01 is just below the critical angle 0c13, the phase

changes on reflection (k113r and 'k23r) are also zero. Itcan then be shown that the new condition for a brightmaximum becomes

(mX) = 2t,(n,2 - A2)1/2 + 2t2(q22 - A2)1/2 (6a)

orA(mX)a = [(mX)a - (miX)a] = 2t2(n 2 2 - A2)1/2 . (6b)

For strongly absorbing films, 02 is complex. In thiscase, 02 and the phase changes at the interfaces dependin a complicated manner on 01,711,N 2 and polarization;N 2 is the complex index of refraction of the film definedby N2 = 772(1 + iK 2). Expressions for 02 and the phasechanges can be found in Secs. 13.2 and 13.4.1, respec-tively, of Ref. 4.

For typical, very weakly absorbing dielectric films,772 generally falls in the 1.2-4 range and K2 < 10-2; andfor 03 close to grazing emergence, the above mentionedexpressions show that the phase changes 4012t, 'k21t, and0k23r are negligibly small. Furthermore, under theseconditions the angle of refraction 02 very nearly obeysSnell's law for transparent media, and Eq. (6) remainsa good approximation for the interference condition.

The effect of the new layer is to increase the opticalpath difference. This would result in a shift of the in-terference pattern to longer wavelengths. In general,both the index of refraction and thickness of the newlayer are unknown. Thus the new order of interferencecannot be calculated. However, once the orders of theclean substrate have been determined the shift of thepattern can be followed by recording the new positionsof the fringes as the thickness increases.

As before, it is necessary to record the spectrum attwo different angles. Least-squares fits for (mX) vsX and (X)b vs X can be performed, and the optical pathdifference due to the film can be computed as a functionof wavelength for both angles. It follows that

(2t)2 = [A(MX)bI - [(MX)a' (7)

and

(8)22 = [A(MX),]+ A2n2 (2t 2)2 +A'.

2732 APPLIED OPTICS / Vol. 15, No. 11 / November 1976

U)

.a

E

30

Thus t2 and 772(X) can be determined. To keep the erroras small as possible, 03a and 0 3b should be as differentas possible. So that if 03a is chosen near grazingemergence, 0 3b should be chosen near normal emer-gence. But a disadvantage of the Lummer-Gehrckeinterferometer is that the light intensity falls off veryrapidly as 03 differs from grazing emergence. Fur-thermore, as 03 is decreased the free spectral range isreduced. Thus a compromise has to be made.

In the case where the film is optically anisotropic,there will be two beams in the film. One correspondsto the ordinary ray and the other to the extraordinaryray. Therefore, there will be two sets of fringes. Figure3 shows a region of an interference spectrum before andduring the deposition of solid oxygen on a plate at 10 K.Notice that each order splits into two components whenthe film is deposited indicating that the film is opticallyanisotropic. The two components were observed ex-perimentally to correspond to light polarized paralleland perpendicular to the plane of incidence.

The analysis of the ordinary ray fringes is identicalto that just described for the isotropic case. In thespecial case where the plane of incidence is a principalsection, the ordinary and extraordinary rays both lie inthe plane of incidence, and the analysis of the extraor-dinary fringes is straightforward. The extraordinaryindex of refraction is a function of the angle of the lightin the film, and so it is possible to measure the ex-traordinary index over a range of angles in the film thatcorresponds to those rays near grazing emergence.

In the special case of a uniaxial film with its opticalaxis normal to the substrate, the ordinary and ex-traordinary indices are associated with light polarizedperpendicular and parallel to the plane of incidence,respectively. Let 7720 represent the ordinary index, andlet 772e and 0 2e represent the extraordinary index andthe angle of refraction of the extraordinary ray in thefilm, respectively. For the propagation of light inuniaxial materials it can be shown4 that

1 COS02e+ sin2e29

17

2e 7720 '12E

where 72E is the principal value of the extraordinaryindex of the film. If 72eg and 2eg are the values of 72eand 0 2e corresponding to grazing emergence in vacuum,772eg sinO2eg 1, and Eq. (9) reduces simply to

n2E2 = '202 (1 + 202 - 72eg2)-1. (10)

Thus, if the ordinary index and the extraordinary indexcorresponding to grazing emergence are found, Eq. (10)can be used to calculate the principal value of the ex-traordinary index; hence the birefringence of the filmcould be established.

In applications where the absolute index is not es-sential and only the shape of the dispersion curve isneeded, the spectrum should be recorded near grazingemergence only. In this case A - 1, and since 72 =

where e is the dielectric constant, it follows that

A(mX)g 2t2(E -1)1/2, (11)

where the subscript g indicates grazing emergence. The

results of the classical electron oscillator model6 showthat (E - 1)/(E + 2) = [(4)/(3)]71rna, where a is the mo-lecular polarizability, and n is the number of moleculesper unit volume. Thus,

a(A) 4[(mX)g]2 + 12t'} * (12)

Equation (12) is a direct relationship, between thewavelength dependence of the polarizability and theLummer-Gehrcke measurement.

E. Least Resolvable IndexThe least resolvable film index A772L is that change

in film index which, for fixed film thickness, would shifta given order by an amount equal to the least resolvablewavelength AXL. The least resolvable wavelength isrelated to the resolving power RP of the instrument byRP = X/AXL - mF, where F is the finesse. Neglectingthe dispersion of the plate and of the film and keepingm and t 2 fixed, differentiation of Eq. (6.1) with respectto X leads to

X (22 - A2)1/2A 7t2L 2-- - _

t2 2772F(13)

Similarly, it can be shown that the least resolvable filmthickness T2L is given by

At2L 2F(_ 2 -A)'(14)

2F(X122 - A2)1/2'(4

As an example, consider the case of a film of index 1.3and 4-,m thickness; for a finesse of 30 at 6000 A andgrazing emergence, Eqs. (13) and (14) show that theleast resolvable film index is 0.0016, and the least re-solvable film thickness is 87 A.

F. Internal Reflection Spectra

To record internal reflection spectra with this in-strument, 01 should be increased to a value just abovethe critical angle 013c. When this condition is fulfilledthe Lummer-Gehrcke beams disappear. However, theinput beam will, in general, have some finite angularspread; and if the beam is centered about the criticalangle, the interference spectrum and the absorptionspectrum can be recorded simultaneously and withoutfurther need to adjust 01.

If the system is operated in this fashion, because ofthe conservation of energy, the internal reflectionspectra will be modulated with an intensity ripplecomplementary to the Lummer-Gehrcke interferencepattern (see Fig. 4). Thus, absorption structure nar-rower than the wavelength separation between succes-sive orders of interference cannot be successfully re-corded by this technique.

The spectrum can be recorded as a function of po-larization of the input light. For thin, weakly absorbingfilms, good approximate expressions5 exist for the am-plitudes of the components of the evanescent field in thefilm. If the thickness and the indices of refraction ofthe film are known, these expressions can be used inconjunction with the observed absorption data to arriveat the absorption coefficient of the film as a function ofwavelength and polarization. Note that the modified

November 1976 / Vol. 15, No. 11 / APPLIED OPTICS 2733

Cl)C: Zero 11 trans.o 0

6400 6300 6200 6100 6000 5900 5800 5700 5600

Wavelength (A)Fig. 4. The internal reflection spectra of a thin film of solid oxygenfor light polarized parallel and perpendicular to the plane of in-

cidence.

Lummer-Gehrcke interferometer gives all three piecesof information with the same apparatus at the sametime.

IV. FabricationModified Lummer-Gehrcke interferometers were

constructed according to the guidelines just proposed.The plates were made from Corning number one coverglass slides. These slides have a nominal thicknessbetween 0.13 mm and 0.16 mm. Corning cover glassslides are made of Corning 0211 glass, also called mi-crosheet. This is a glass of the soda lime family. Mi-crosheet is described by Corning Glass Works as a thin,optical-quality glass sheet with an excellent surfacefinish. The surfaces of microsheet are fairly smoothwith an rms roughness of approximately 10 A. This isbetter than most commercially polished optical flatsthat generally exhibit rms surface roughness of the orderof 25-30 A.7 Generally the surfaces of thin sheets ofglass are wavy and not close enough to plane parallel forprecise optical experiments.7 But there are alwaysexceptions to the general rule, so a search was made forsmooth, flat, plane parallel plates.

The plates were tested for parallelism, flatness, sur-face roughness, and waviness by standard optical in-terference techniques that involve the examination ofthe shape and irregularities of Haidinger fringes and ofNewton rings. About one in five hundred was accept-able. From about a dozen of these, two exceptionallyflat plates plane parallel to about 2 sec of arc were found.Prisms made of the same material were cemented8 tothem as shown in Fig. 1.

V. PerformanceThe first row of fringes in Fig. 3 is a typical Lum-

mer-Gehrcke spectrum of a clean plate. It was recordedat 30 above grazing emergence for a plate 143 Am thickat 10 K. The spectra of a hollow cathode iron-neonlamp was used for calibration. The plane of the platewas perpendicular to the slit of the spectrograph. Theexit slits were opened to show the angular dispersion ofthe instrument. The spectrograph has a lateral dis-persion of about 3.9 A/mm and a resolving power of70,000.

The indices of refraction of the plate at room tem-perature at three standard wavelengths as supplied bythe manufacturer are 1.5231 at Xc = 6562.8 A; 1.5264at XD = 5892.9 A; and 1.5331 at XF = 4861.3 A. Theorder was first determined from the interference spec-trum at room temperature; it was then traced to 10 Kby photographing the fringes at regular intervals as thetemperature fell. The order near 6000 A is about 550,and the finesse in this region is about 16. This leads toa resolving power of about 8800.

The other rows in Fig. 3 show how each order splitsinto two components when a thin film of solid oxygenis deposited on the plate. The spectra were recordedevery 2 h, and the final thickness is about 3.6 Am. Thecircles in the figure show how a typical order shifts tolonger wavelength as the thickness increases. Noticealso that the separation between components of thesame order increases with increasing film thickness.This is consistent with Eq. (6.1) from which it followsthat this separation is proportional to the thickness ofthe film.

In the wavelength region around 6000 A, the finesseof the long wavelength components of the 3.6-gm filmis about 13, and the order is about 558. The estimatedindex of refraction of oxygen is about 1.3. Thus, usingEqs. (13) and (14) the least resolvable index A772L0.004, and the least resolvable thickness At2L 0.028gim.

The low values of the finesses in these experimentsare partly due to the optical imperfections of the plateand to the nonuniformity of the thickness of the oxygenfilm. The walls of the standard Pyrex Dewar and thebubbles in the liquid nitrogen and liquid helium in thepath of the Lummer-Gehrcke beams also helped todegrade the finesse. However, because of the improvedreflectivity, the finesse could have been improved byoperating closer to grazing emergence (about 10 away).In the absence of the cell windows, Dewar walls and thebubbling liquids, finesses of 25-30 were typical for thebest clean plates operated at room temperature and 10above grazing emergence. It is therefore conceivablethat with an excellent plate and some care, changes inindex as small as 10-3 could be successfully recorded bythis technique.

Figure 4 shows the internal reflection spectra of solidoxygen for light polarized parallel and perpendicularto the plane of incidence. The spectrum was recordedfor 01 centered about the critical angle. Notice that asexpected the spectra are modulated.

2734 APPLIED OPTICS / Vol. 15, No. 11 / November 1976

VI. ConclusionsIt has been shown how the Lummer-Gehrcke inter-

ferometer can be modified into a powerful tool for thesimultaneous determination of thickness, wavelengthdispersion, and the absorption spectra of thin dielectricfilms. For anisotropic films, both the ordinary and theextraordinary index can be calculated from the data.The thickness and index data for very weakly absorbingfilms can be handled by making certain approximationsthat reduce the interference condition to that of thetransparent film. Although the plate is considerablythinner than it was in the original instrument, the re-solving power remains sufficiently high that changes inindex in the third decimal place can be recorded, if thesystem is not limited by the resolution of the spectro-graph.

It has also been shown that the interference spectrumat grazing emergence bears a simple relationship to themolecular polarizability of the film material. Forweakly absorbing films, the polarization and wavelengthdependence of the internal reflection spectra, combinedwith the thickness and index measurements, lead verynaturally to a full characterization of the optical prop-erties of the film.

The modified interferometer has been used to studythe optical properties of thin films of solid oxygen, anddetailed results are being prepared for publication.

The author acknowledges with thanks the very en-lightening discussions with Robert G. Wheeler and EricSamuel.

Part of this research was carried out at Mason Lab-oratory, Yale University, New Haven, Connecticut06520.

References1. 0. Lummer, Verh. Dtsch. Phys. Ges. 3, 85 (1901).2. 0. Lummer and E. Gehrcke, Ann. Phys. 10, (3) (1903), 457.3. C. Candler, Modern Interferometers (Hilger and Watts, Ltd,

London, 1951), pp. 321-344.4. M. Born, and E. Wolf, Principles of Optics (Pergamon, London,

1975).5. N. J. Harrick, Internal Reflection Spectroscopy (Interscience, New

York, 1967).6. J. M. Zimam, Principles of the Theory of Solids (Cambridge U.

P., London, 1964), p. 229.7. H. E. Bennett and J. M. Bennett, Phys. Thin Films 4, 1 (1967).8. Cement used was UV-71 obtained from Summers Laboratories,

Fort Washington, Pa.

THE PHYSICS OF QUANTUM ELECTRONICSCodirectors S. F. Jacobs & M. 0. ScullyOptical Sciences Center

an announcement of a short course

Physics of Quantum Electronics

Aug. 8-19, 1977, Telluride, Colo.

Physics of Quantum Electronics, Rt 2, Box 732D, Tucson, Az. 85715

November 1976 / Vol. 15, No. 11 / APPLIED OPTICS 2735