JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Coating Thickness Measurement by Interferometry*

M. F. BECHTOLDDu Pont Experimental Station, Wilmington, Delaware

(Received May 15, 1947)

Thickness measurements of thin transparent coatings on thick transparent bases by reflectioninterferometry are made possible through use of an immersion medium of the same refractiveindex as the base to intensify interference. A convenient interferometer comprising three essen-tial parts, a "pocket" spectroscope, an incandescent lamp, and a cylindrical immersion tank, isdescribed. Its use for rapid, precise measurements of thickness of single and double layercoatings and films in the range of 0.2 to 50 microns is illustrated with spectra.

I. INTRODUCTION

T HE need for rapid, accurate measurementsof the thickness of thin transparent lam-

inae has long been apparent to those concernedwith the physics and chemistry of surfaces or theproduction and evaluation of protective coat-ings and unsupported films. Thicknesses of 0.2to 10 microns, which are below the sensitiverange of ordinary mechanical calipers, are par-ticularly difficult to measure. The tediousmethods of stripping or sectioning followed bymicroscope examination as well as the limitedmethods of x-ray,1 ultraviolet, and visible lightabsorption have been' used for this purpose.Recently, the thickness of thin silvered films onglass has been measured by multiple-beam wedgeinterference.',3 The ordinary reflection inter-ference method4 for the measurement of un-supported films and the immersion technique ofPfund for showing interference due to coatings6

have been combined in the instrument and pro-cedure described here.

II. THEORY

Analysis of the intensity and phase changes7

in light reflected from the two interfaces of a

* Contribution No. 220 from the Chemical Department,Experimental Station, E. I. du Pont de Nemours andCompany, Inc.

1 H. Friedman and L. S. Birks, Rev, Sci. Inst. 17, 99(1946).

2 Gunn and Scott, Nature 158, 621 (1946).3 Plessner, Nature 158, 915 (1946).4 R. W. Wood, Physical Optics (The Macmillan Com-

pany, New York, 1934), third edition, p. 192.5 H. W. Straub, Alien Property Custodian No. 334, 220

(May 4, 1943).6 A. H. Pfund, J. Opt. Soc. Am. 36, 95 (1946).7 Jenkins and White, Fundamentals of Physical Optics

(McGraw-Hill Book Company, Inc., New York, 1937),first edition, p. 83.

transparent lamina of refractive index 11, im-mersed in a medium of refractive index .Lo, showsthat interference minima will occur when theoptical path difference of the two reflected beamsis equal to a whole number of wave-lengths, or

2,j1t cosO = nX, (1)

where t is the thickness, 0 is the angle (to thenormal) of refracted light in the lamina, n is aninteger, and X is the wave-length in air.

Thus, if white light is incident on the lamina,a number of dark bands appear in the spectrumof the reflected beam, the number and locationof which can be predicted as follows: let be theangle of the incident (or reflected) beam to thenormal; then from Snell's law,

(2)sinO = (,uo/lJl) sin+,

and from a trigonometric relationship,

coso = [1-(-sino)] (

Hence, from Eq. (1),

2,l[1 - (- sino) = nX. (4)

For the usual case in which two or more inter-ference bands are visible in the spectrum at onevalue of 0, i.e., one viewing angle, let

(5)

then, from Eq. (4),

t/kX i =n 873

(6)

VOLUME 37, NUMBER 10 OCTOBER, 1947

(3)

Y 0 2 -1�

k = 21AI 1 - sing)/ I . P i

M. F. BECHTOLD

for any selected band, i. Also let

t/kx, = n (7)

for any other band, s, where X>X,. Now bysubtraction,

(n,-n, ) =t/k[(1/) -(1/Xi)] (8)or

t = ( - ni)k[(X,,Xi)/(Xi - X,) (9)The quantity (-ni) is obviously equal to thetotal number of bands from X, to Xi, inclusive,minus one. Thus, when only two adjacent bandsare used for calculation,

t=k[(X.Xi)/(Xi-X.)]. (10)

In the case of a coating, the refractive indexchange at the coating/base interface is fixed.Consequently, in order to obtain an equivalentchange at the immersion medium/coating inter-face, an immersion fluid of refractive index Auo,equal to that of the base, is chosen. This has theeffect of equalizing the intensity of the totalbeams reflected from the two interfaces so that

…-- … … 3.00

I-0--,;K--,b{--a-____ -- 1 .03Ad 0.93

-- 0.36g = _ 0 . 2 7~~~~~~-02.. Visible Spectrum-

0:4 ' 06 08

,>} (microns)

FIG. 1. Location of interference bands in the visiblespectrum of light reflected from a coating of refractiveindex 1.46 () on a base of refractive index 1.50 (0),viewed at an angle of 450 (tk). Intersection of the familyof sloping lines (t = nkX) with horizontal lines shows valueof X at reflection minima.

the interference is complete. The location ofinterference bands viewed at =45' due to acoating of /1i=1.46 on a base of /1o=1.50, im-mersed in a liquid of uo = 1.50, is shown graphi-cally in Fig. 1. The value of k is 0.50; the effectof dispersion is neglected. The family of linesshown were obtained by plotting t vs. X fromEq. (6), letting n vary from 1 to 15. Their inter-sections with the dotted horizontal lines show,for example, that when t= 3 microns, six inter-ference bands occur within the easily observablepart of the visible spectrum. (It is difficult toobserve bands outside the range of 0.425 to0.695 micron for X in the apparatus describedlater.) The number of bands observed does notdecrease with thickness in a regular manner.For example, at = 1.03 micron, only one inter-ference band should be easily observable, whilea thinner coating (0.93 micron) should show twobands. A coating of t=0.36 micron should giveno bands at all, while a thinner coating (0.27micron) should give one band in the middle ofthe spectrum. However, for the 1.03- and the0.36-micron coatings, the use of a slightly dif-ferent value of 9' would be adequate to bring twobands into the observable spectrum to providedata for accurate calculation of t.

Rigorous treatment of the problem of simul-taneously measuring the thicknesses (th, t2) oftwo adjoining laminae would require considera-tion of the fact that the beam transmitted by thefirst lamina encountered contains slight intensitymaxima and has suffered some dispersion. Fur-thermore, the angle of incidence for the secondlamina (02) is equal to angle of refraction in thefirst (r), and the condition for minima for eitherlayer may involve (n+)X instead of n, de-pendent on the phase changes at its two inter-faces. No attempt has been made to incorporatethese complicating factors into general equa-tions; however, calculations of 2 have been madein this paper using a value of 2 for the secondlayer calculated from the values of p1, /12, andkt. Examination of Eq. (10) shows that accuracyin the value of t depends chiefly on the numberand location of the bands, while it is not sensitiveto small inaccuracies in estimates of refractiveindex and of at less than about 50°. Conse-quently, when uO, pu, and /12 are nearly equal,and the same may be used for both layers,

874

COATING THICKNESS MEASUREMENT

good approximations of for both layers can bemade with the one value of k. The two sets ofbands predicted for the case of one thin layer andone thick layer should be easily separable bydifferences in their sharpness and number perAX. Since the intensity of the reflected ray andthe completeness of interference is different foreach layer when ,uo is not equal to either /i or.2, sets of bands due to coatings nearly alike inthickness should be separable by intensity dif-ferences alone. When the refractive index situa-tion is favorable, a third set of bands should bevisible, which arise from interference in reflec-tions from the top and bottom interfaces of thedouble coating.

III. APPARATUS AND PROCEDURE

An apparatus for observing the interferencebands in light reflected at various angles is shownin Fig. 2, and a simplified diagram of the opticalelements is shown in Fig. 3. The three filamentsof a 200-watt 110-volt photo-projection bulb areplaced in line with a slit (6 mm wide) and thecenter of an immersion tank at a distance of 21cm. The light is reflected from the surface of acoating or film held at any angle by the revolvingclamp on top of the immersion tank (7-cm O.D.),and the spectrum of the reflected light is an-alyzed with the pocket spectroscope, which hasbeen rotated to the appropriate angle. To pro-vide for this rotation, the spectroscope ismounted on a wheel of polymethyl metacrylatesheeting that rotates about the immersion tankas an axle. The tank does not turn, since itsbottom is slotted and keyed to a brass baseplate. A fixed wheel of polymethyl methacrylatebearing calibrations in terms of the angle permits easy measurement of the position of thespectroscope on the wheel above it. Thus, theapparatus provides for independent rotation ofthe sample and the spectroscope. The clamppositions the sample so that the incident beam,which enters the cylinder along a radius, is re-,flected at the axis of the cylinder along anotherradius to the spectroscope without refractionsuch as would be encountered with a tank ofrectangular cross section.

The spectroscope used for all results givenhere was manufactured by Zeiss (No. 71112). Itis especially convenient since it contains a readily

FIG. 2. Interferometer.

calibrated wave-length scale; illumination forthis scale is furnished by oblique light from thesource reflected from the exterior of the immer-sion tank down the side arm of the spectroscopefor angles less than 45°. A small mirror (notshown) is used for higher angles. The spectro-scope is positioned with the entrance windowabout 1 cm from the immersion tank and isused substantially fully telescoped. A furtheradvantage of the Zeiss spectroscope for this useis that a reference spectrum, such as a mercuryarc, can be introduced through a side prism andplaced in juxtaposition with the experimentalspectrum for comparison purposes. A*student-

IU.I.50

Slit

illFilament

=1.46

,

Spectroscop]

Eye V

FIG. 3. Optical elements of interferometer.

875

M. F. BECHTOLD

type diffraction grating spectroscope is alsosatisfactory for thickness measurements.

Non-colored photographs of the spectra weremade by placing a shielded photographic plateat distance of 6.5 cm back of the spectroscopeeyepiece with the telescope setting at about 13,the slit opening in the range of 0.4 to 2.5, andexposure time of 15-40 seconds. A "Daylite"filter was used (only for pictures, except Fig.

0 n 0 M 0 "v I ) 0 c

o + O o N o r eI I

C> so ra l

1 1- 1- 1-

0 I 1 0

0 I

0

00~

D

FIG. 4. Photographs of reflection spectraof various laminae.

A. Polyallyl methacrylate (u=about 1.51), on a thickpolymethyl methacrylate base (= 1.50) at 01=47.5°.(kl=0.487; t=4.41 microns.)

B. Two layers of a synthetic resin (,ul=about 1.46,2 -about 1.48), on a thick polymethyl methacrylate base,

(juo=1.50) at ¢01=47.5O. (kl=0.525; t=1.56 micron; k2=0.452, t2= 1.25 micron.)

C. One layer of a synthetic resin (=about 1.48), onCellophane ( 2 =about 1.52), in a liquid (uo= 1.52) at4'x =52.50. (kj =0.582; 4=2.53 microns; k2=0.518, 2=23.6microns.)

D. Polystyrene (u=1.59), on a thick polymethylmethacrylate base (o=1.50) at =53.50 . (k=0.478; t1=0.29 micron.)

4C) between the source and the immersion tankin order to lower the intensity of light in theyellow-red end of the spectrum; conditions weresuch that the intensity minima of the filter didnot interfere with the interference spectra. Con-siderable contrast visible to the unaided eye waslost in the photographic process; in addition,the wave-length scale could not be brought intofocus and was blocked off during exposure. Con-sequently, it was much easier to observe thespectrum with the unaided eye, and to makecalculations therefrom, than it was to read thedata from photographs. The wave-length dataaccompanying Fig. 4 were obtained by readingthe wave-length scale with the spectroscopetelescoped directly before and after taking of thepictures.

When an estimate of refractive index of alamina from its composition is not sufficientlyaccurate, and the lamina is available in un-supported form, the apparatus of Fig. 2 can beused for the measurement by choice of an im-mersion liquid (such as a mixture of Nujol andAroclor) so that minimum light is reflected fromthe surface down the properly aligned spectro-scope. (A refined procedure for this purpose hasbeen described by Billmeyer.5 ) Independentmeasurement of the immersion liquid establishesthe surface refractive index of the lamina. Anapparatus for measuring the Brewster angle'also provides data for calculation of the surfacerefractive index.

IV. RESULTS

Calibration

A calibration experiment was performed bymounting a section of a glass bubble (1= 1.52)on a frame and observing the location of inter-ference bands when examined at two angleswith air (o=l), then water (o=l. 33) in theimmersion tank. Thus, the glass film is found toaverage 1.35 micron thick, independent of theimmersion medium or the angle of viewing. Thisexperiment and others show that at this thick-ness the average deviation of a single deter-mination from the mean is about 2 percent.

8 F. W. Billmeyer, Jr., Phys. Rev. 71, 489 (1947).9 A. H. Pfund, J. Opt. Soc. Am. 31, 679 (1941).

876

COATING THICKNESS IMEASURVMENT 7

Non-Uniform Coatings

Ordinary techniques of applying coatings, suchas dipping, spraying, or puddling yield obviouslynon-uniform coatings, since the interferencecolors observed upon immersion are not thesame in all areas. When the coatings are toothick to show interference colors, there haspreviously been no convenient method of de-termining the presence of non-uniformity ofthickness. A spectrum indicating the presence ofa mild non-uniformity in a coating of polyallylmethacrylate on polymethyl methacrylate withinthe area viewed with the spectroscope is shown inFig. 4A. In other areas the changes in thicknessvaried from none at all to complete absence ofthe coating. The thickness in the center of thespot examined is calculated to be 8X0.487X(0.691 XO.429)/0.262 = 4.41 micr6 ns.

Two-Layer Coatings

An apparently uniform two-layer coating wasprepared on a polymethyl methacrylate panel byconstant-speed withdrawal, then drying, afterdipping in solutions of synthetic resin of slightlydifferent composition. The spectrum of thiscoating (Fig. 4B) shows two sets of two inter-ference bands of distinctly different intensity.It was established by separate experiments onsimilar coatings in single-layer form that thelighter set of bands is caused by the layer nextto polymethyl methacrylate, and the darker setby the top layer. In addition, the sum of thick-nesses measured in single-layer form was equalto the total thickness in two-layer form. Calcula-tion from Fig. 4B gives t=0.525(0.609X0.497)/0.102=1.56 micron for the top layer andt2=0.452(0.546XO.456)/0.090 =1.25 micron forthe lower. In coatings of this kind, cases of par-tial and complete coincidence of the two sets ofbands have been observed. The failure to ob-serve a third set of bands corresponding to theover-all coating thickness may be due to tooclose approach of the refractive index of the bot-tom layer to that of the base, resulting in poorequalization of intensity of the two beamsinvolved.

Another example of measurement of thicknessof two laminae simultaneously is seen in Fig.4C, which is the spectrum of a commercial coated

In air.

0 =45°; k =0.372X(microns) diff. t(microns)

0.4400.060 1.36

0.5000.080 1.35

0.580

0 =600; k =0.400

0.4700.075 1.37

0.5450.105 1.35

0.650

In water.

.= 45 °; k=0.419X(microns) diff. t(microns)

0.4400.075 1.27

0.5150.100 1.33

0.615

0 =60°; k =0.504

0.4350.080 1.41

0.5150.125 1.33

0.640

Cellophane film. In this case the lines due to thecoating are obviously the three broad ones, whilethe Cellophane film itself (or the Cellophane plustwo thicknesses of coating) is responsible for thefine lines. The sample of coated film, which waswet with oil and held clamped between twomicroscope slides, showed very non-uniformcoating thickness; considerable scanning wasdone in order to find an area giving bands suffi-ciently straight and continuous to measure.Calculation of coating thickness yields t/ = 2.53microns. Although conditions were nearly idealto prevent interference due to the Cellophanebase, since the immersion fluid has nearly thesame refractive index as ordinarily ascribed toCellophane sheeting, there is some doubt as towhether the fine lines observed are due to baseor base plus coatings because the Cellophane it-self may reflect because of its double refraction.It is thought that in this case, the fine bandsare from the base itself, although this point wasnot proven experimentally. Twenty-two lineswere counted in the range of =0.65 to 0.50micron, equivalent to a thickness of 23.6

87 7

878 . F. BECHTOLD

microns. By making use of high refractive indeximmersion fluids and values of p at 60-75°, allof which tend to make for fewer bands per AX,laminae at least 50 microns thick can bemeasured.

Thin Coatings

For every set of AO, gi values, there is a mini-mum thickness of the lamina which yields apair of interference bands at a given angle ofviewing. Since within the range of 0.7 to 0.4micron there is no set of numbers bearing theratio of 2/1, first- and second-order interferencebands will never be visible as a pair at one valueof within this range. Consequently, bands forwhich n = 2 and 3 constitute the first observablepair. To illustrate use of this analysis, the thin-nest coating of polystyrene on polymethylmethacrylate that will show two interferencebands at the lowest convenient angle ( = 300) is0.43 micron, which gives bands at 3 = 0.4 micronand 2=0.6 micron. At k=53.5', this minimumthickness is 0.57 micron. The order of the singleband from a coating of this kind, whose spectrumis shown in Fig. 4D, is in doubt, since if it is ofsecond order, the third-order band may or maynot be visible at 0.406 micron. However, rota-tion to k=48.50 caused the single band to shiftto 0.665 micron; no other band appeared. If it

is of second order, the third-order band shouldhave been easily observed at 0.444 micron at thisangle. Consequently, the conclusion is drawnthat the band is of first order, and the thicknessof the coating is 0.29 micron. In another area onthe same coating, the single band was located at0.665 micron at 0=30', 0.565 at 45°, and 0.500at 53.50, which data yield thickness of 0.24 mi-cron at each position. By observation at severalangles and analysis of the data in this fashion, itis possible to ascertain the order and then thethickness of coatings showing single bands downto the minimum thickness showing one band; at

= 30° this thickness is 0.143 micron whenAi= 1.59, O = 1.50.

Acknowledgment

The author is indebted to Professor A. H.Pfund of the Johns Hopkins University for theessential suggestion of the use of an immersionfluid, and to Dr. F. W. Billmeyer, Jr., of thedu Pont Plastics Department, who showed thatA, of Eq. (1) must be the refractive index of thelamina with respect to air, regardless of the im-mersion medium, and made the calibration ex-periment to prove this point. Thanks are ex-tended to Mr. Alexander Marshall, who assistedin the preparation of the photographs.

878