Thickness and refractive index dispersion measurement in athin film using the Haidinger interferometer
Elisabeth A. de Oliveira and Jaime Frejlich
A Haidinger interferometer setup was adapted for accurate measurement of thickness and refractive indexdispersion in transparent films using some spectral lines of a commercial argon-ion laser. Experimentalresults are reported and compared with those from other available methods.
1. Introduction
Haidinger, Newton, and Fizeau interferometers arecommon tools for routine testing of optical compo-nents. The first is particularly useful for measuringthe wedge of a parallel transparent plate. In thispaper we report the use of this interferometer for mea-suring the optical thickness and the wavelength dis-persion of the refractive index of a transparent filmcoated on a transparent substrate. The method usesthe interference ring fringes produced by the coatedand uncoated substrates and the interference excessfraction method2 3 for computing their optical thick-nesses. Their difference provides the film thickness.
In this paper we show that the Haidinger setup issuitable for providing us with the necessary accurateexcess fraction data. We also show that the opticalthickness difference between the coated and uncoatedsubstrates may be directly computed from experimen-tal data. Nonstabilized Ar+ laser lines were used inthe experiment.
We also show that the excess fraction method maybe adapted for dealing with dispersive materials, inwhich case the wavelength dependence of the filmrefractive index may also be obtained. To process theexperimental data for dispersive materials, a purelyanalytic method by Tilford4 was adapted and a mixednumerical-analytic method is proposed which is par-ticularly suitable for films and substrates having simi-lar refractive indices where no sensible optical inter-face is developed between them.
The authors are with State University of Campinas, Physics Insti-tute, Optics Laboratory, 13081 Campinas SP, Brazil.
The comparison of this method with other availableones is quite difficult depending on the nature andquality of the film, because different methods usuallymeasure different parameters which are not alwaysstraightforwardly comparable.
II. Method
When a transparent parallel slab is illuminated witha point monochromatic coherent light source, a nonlo-calized circular interference pattern appears (arisingfrom interference of beams reflecting from the firstand the second slab interfaces) whose center lies on aline perpendicular to the slab faces passing throughpoint source A (Fig. 1). The interference pattern oflight projected onto a screen a distance x from the slabis characteristic of the optical thickness of this slab.We may write
2n,1, = (N, + F5 )X, (1)
where S and ns are the geometrical thickness andrefractive index, respectively, of the slab, Ns + Fs isthe number of interference at the center of the circularinterference pattern, and X is the light source wave-length. Fs is the so-called excess fraction. N is theorder of interference of the smallest diameter darkinterference ring in the pattern. If bright fringes wereconsidered instead, a /2 term should be added to theright-hand side of Eq. (1) because of the 7r/2 phase shiftbetween external and internal dielectric reflections.Writing Eq. (1) for the film coated substrate:
(2)2(nSlS + nF1F) = (NSF + FSF)X,
and subtracting Eq. (1) from Eq. (2) we get2 nFIF = (N + f)X, (3)
where N NSF - Ns, f FSF - Fs, and subindices S, F,and SF represent the substrate, film, and coated sub-strate, respectively.
Two questions arise:(A) How to compute the integer term N from the
experimentally measured term f in order to computenFiF from Eq. (3).
(B) How should the excess fraction f in Eq. (3) beexperimentally measured with the required accuracy.
A. Computing Optical Thickness from InterferenceExcess Fractions
This question has been extensively reported23 in thescientific literature because of its relevance to opticalmetrology. In 1977 Tilford4 proposed an interestinganalytic method for solving this problem for a nondis-persive material (length measurement in air). In thispaper we extend Tilford's method for a dispersive ma-terial (mainly glass) resulting in a combined analytic-numerical method which largely reduces computation-al efforts. A parameter characterizing the refractiveindex chromatic dispersion is also simultaneously ob-tained. We assume that the refractive index of thefilm may be described by the Cauchy approach 5 :
nF = A(1 + k 2B), (4)
where k 1/X and A and B are parameters characteriz-ing the film. Substituting Eq. (4) into Eq. (3) andwriting Eq. (3) for different wavelengths Xi (i =
1, . . . r), we get
L( + k2B)ki = Ni + fi with L - 21FA. (5)
Fig. 1. Interference from a parallel transparent slab.
L(B) = E Li(B)/r.i=l
(11)
Substituting the latter L(B) instead of L into Eq. (5)and knowing Ni from Eq. (9), a set of fi is obtained:
1i = L(B)(1 + kB)ki - Ni. (12)
A figure of merit W(B) is defined by comparing these fivalues with the experimental fi:
Equation (5) is ready to be handled using the analyticmethod of Tilford just noting that length L and wave-number k in Tilford's paper should now be substitutedby L -- 21FA and ki(l + k0B), respectively, with all theother terms and definitions adjusted accordingly.The synthetic wavelength, for example, is correspond-ingly formulated as
Xs= 1 Iiki(l + Bk?), (6)
where Ii are integers. A result for L is then obtainedfollowing Tilford's procedure, which value now de-pends on the chosen B parameter in Eq. (4):
L = L(B). (7)
The above result is substituted into Eq. (5) so thatcomputed values for Ni + fi (which we label Gi + ei)now result:
W(B) - ( -f)2i=l
(13)
The whole procedure above is repeated for a new valueB + AB in a step-by-step scanning of B. The mini-mum (or minima) values for W(B) are recorded and thecorresponding values L = L(B) and B are thus found.An independent experimental refractive index mea-surement should be performed for a given wavelengthfrom which data the parameter A in Eq. (4) may becomputed so that the film thickness may be found:
IF = L/(2A). (14)
As described in Tilford's work, his analytic methodprovides a good approach for L only if an initial esti-mate L' is available such that
IL - L'I < (0.5 - )s, (15)
where 6 is defined from experimental uncertainties inexcess fractions Afi and wavelengths AXi as follows:
L(B)(1 + k,2B)ki = Gi + ei (8)
for each Xi (i = 1, . . . ,r) used in the experiment. Gi + eiare now compared with the experimental values fi andintegers Ni are chosen so that
(9)
Substituting the Ni + fi values obtained from Eq. (9)instead of G + e in Eq. (8), we get a value of
Li(B) = (Ni + fi)/(l + kB)ki (10)
for each wavelength Xi. An average value is then com-puted:
(16)=-6 IIIAfi + L E IIlkAXi < 0.5.i=1 i=1
The final precision in L = L(B) is crudely estimated tobe
AL = 6Xs. (17)
The actual experiment was performed using theeight stronger lines of an argon-ion laser: X = 0.4579,0.4658, 0.4727, 0.4765, 0.4880 0 4965, 0.5017, and0.5145 gm. Other values are iJ= 1, AXi 5 X 10-5
-m, and Afi 0.03. From these data we may computethe maximum value L which may be processed using
this method by making = 0.5 in Eq. (16). Suchinformation is displayed in Table I for different experi-mental conditions.
Note from Table I and our experimental conditionsthat we are not able to compute the thickness of a 1-mm thick glass plate unless a highly stabilized laser isused. Glass slab thickness differences may always becomputed up to the limits in Table I.
B. Experimental Measurement of the Excess FractionsThe excess fractions FSF and Fs are the fundamental
data for this method and its performance is closelyrelated to the accuracy with which these data are mea-sured. The Haidiner interferometer allows measuringFs and FSF easily and with good accuracy. With Rpbeing the radius of thepth dark ring on the screen (Fig.1), we may write
p + F - 1 = (Sl/b2 ;)R' + ( 2/b4X)R 4 + ... ,
which may be approached byp + F - 1 (Sl/b2X)RI for ( 2/b4X)R < 0.01,
where
(18)
(19)
S1 [ (b/2) I1- (1 + a/]b)2 '
S2 - (b/8) [1 1 + 21(4n2 - 3)/(bn3 )1(1+ a/b)4 J
a 2t 21/n,
b X+x, (20)where n is the refractive index of the slab and X, x, andt are described in Fig. 1. The excess fraction F may becomputed from Eq. (19) by linear regression from R' vsp data.
Ill. Experimental Setup
The modified Haidinger interferometer used in thiswork is shown in Fig. 2. The argon-ion laser beampasses through a spatial filter to provide a clean diver-gent coherent beam. The numerical aperture of theobjective in the filter should be large enough to that alarge number of interference rings on the screen arevisible. The objective should be as close to the sampleas possible to illuminate a small area, and a reasonablysmall beam splitter cube should be used in the setup.A kinematic homemade support is used to hold thesample, ensuring its replacement in the same position.
(b)
Fig. 2. Experimental setup: (a) scheme of the modified Haidingerinterferometer and (b) photograph of the actual setup. A 40X, N.A.= 0.65 microscope objective was used with a 5-/im diam pinholeplaced 20 mm from the sample surface. Sample-screen distancewas 450 mm and the diameter of the tenth interference ring in the
screen was 90 mm.
The mirror and mechanical stages before the objectiveare adjusted so that the laser beam can be interfero-metrically centered relative to the objective. The lat-ter is then screwed off its support and the plate samplein the kinematic support is adjusted to be orthogonal'to the laser beam. The beam splitter cube and screenare now adequately positioned. The objective andcorresponding pinhole are then placed in their posi-tions in the spatial filter. Slight adjustment of thesample support may be necessary to keep it orthogonalto the optical axis in the system. If the laser is proper-ly adjusted it may be successively tuned to differentwavelengths without modifying the whole optical
Table I. Maximum Value for L (m) that may be Processed
\X~ 0.005 0.01 0.02 0.03 0.05(Mm) \
5 X 10-5 268 245 198 151 582 X 10-8b 0.67 X 106 0.61 X 106 0.5 X 106 0.38 X 106 0.15 X 1065 X 10-9c 2.7 X 106 2.5 X 106 2 X 106 1.5 X 106 0.6 X 106
a Nonstabilized argon-ion laser.b Temperature controlled etalon argon-ion laser lines.6c Stabilized CO 2 laser lines. 4
Fig. 3. Photograph of the circular interference pattern as projectedon the screen.
alignment in the interferometer. The beam splittercube should be AR coated so that no interference ringsare visible on the screen. The interference patternsare directly recorded on a piece of Photolite paper (Fig.3) supported at the screen, one for each wavelength forboth the uncoated and coated substrates (sixteen suchrecordings for the eight wavelengths we used). Thesedata are analyzed and R' is plotted vs p for eachrecording, with the excess fraction computed by linearregression for each one. R vs p are actually plottedfor both the horizontal and vertical directions in eachrecording, and the excess fraction is then independent-ly computed and compared. Both should agree towithin experimental precision, otherwise after reviewthese data are eventually rejected. During the experi-ments a maximum temperature variation of 0.50C isacceptable and a maximum of 20C variation should betolerated between the uncoated and the coated sub-strate data collection of our samples.
The data obtained are processed according to theanalytic-numerical method described above. Thecomputational work was performed on a VAX 11. Theprocess uses three successively decreasing syntheticwavelengths [Eq. (6)] Xs 8.5, 3.5, and 1.5 gm toobtain increasing accuracy in the final value for L =2A1F, accounting for the fact that the uncertainty di-rectly depends on the size of Xs [Eq. (17)]. Table IIreports the final results computed from experimentaldata for the excess fractions obtained from differentsamples, including a glass sample resulting from wet
chemical etching of a glass substrate. The index ofrefraction for each material was independently mea-sured by the Abeles7 8 or Brewster angle method for the6328-A He-Ne laser line to get parameter A for theCauchy formula [Eq. (4)] to compute IF for the film[Eq. (14)].
IV. Comparative Results
In this work we measured films thin enough for thesecond term in Eq. (16) to be negligible so as not to belimited by the uncertainty in laser wavelength. In thiscase the viability and precision of the method are onlydetermined by the experimental uncertainty Afi in themeasured excess fractions. Such uncertainties are ap-proximately ±0.03 in our case so that the final error inL may be estimated from Eq. (17):
AL 0.03 Xs 0.03,um. (21)
Six samples were measured using this method and theresults were compared with those obtained from otheravailable methods (interference microscopy,2 trans-mission spectrophotometry,9 and Talystep1 0) as shownin Table III. Note that excessively large differencesexist for the KMR-747 samples. The agreement isbetter for the positive AZ-1350J films and for the glasssample. Note also that the methods that directly mea-sure geometrical thickness (Talystep and interferencemicroscopy) always produce higher values than thosemeasuring optical thickness. Indices of refractionwere measured with better than 1% precision which isdirectly translated to the final results. This is muchsmaller than the differences in thickness for most ofthe samples in Table III. The contribution of opticalinterfaces in this work was neglected because it wasfound to be smaller than that from experimental un-certainties in f. The influence of any hypotheticaltransition layer was also disregarded because our datashowed that the agreement among different methodsimproves when the films are more homogeneous as wasthe case for the glass film. We suppose that in somecases a disorderly settlement of large molecules (suchas for the KMR-747) during film formation throughspinning and solvent evaporation may result in voidsin the bulk film or nonhomogeneous refractive indexdistribution, producing incomparative results. The Bparameter in the Cauchy formula computed with theHaidinger interferometer closely agrees either with theliterature or with independent measurements for allthe samples.
Table 11. Experimental Results Obtained for Some Thin Film Samples
WSample Description L (Mm) B (um2) (X 10-3) A 1F (Am)
KMR-747 and AZ-1350J are trademarks for a negative photoresist from Kodak and a positive photoresist from Shipley, respectively.Samples 1-5 were prepared by spin coating the photoresist over a cleaned glass substrate from a wasted high resolution Kodak photographicplate type 1A. Sample 6 is a plain type 1A glass substrate.
a Measured using the Abeles or Brewster method and plotting the refractive index vs X-2.b Real-time reflectivity measurement during wet etching."
V. Conclusions
We have showed that a simple setup based on theHaidinger interferometer is able to provide preciseexcess interference fraction data for interferometriclength-measurement. This method also allows simul-taneous measurement of the refractive index disper-sion. Good agreement was found with other availablemethods, both for length and chromatic dispersion.Lack of agreement for length measurements in somecases may arise from differences in what those meth-ods actually do measure. This method does indeedmeasure the accurate optical thickness variation of atransparent substrate, which fact may be used for mea-surement of film thickness (particularly suited whenfilm and substrate have similar refractive indices),chemical etching of the substrate, thermal expansion,or even accurate difference measurements among sim-ilar samples.
References
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We acknowledge the research center of TELEBRASin Campinas (CPqD) for their kind measurementswith the Talystep, the Laboratory of Electronic andDevices (LED) of the Faculty of Engineering of Cam-pinas for allowing us to use the interferential micro-scope, and to Marco Aurelio de Paoli of Instituto deQuimica of this University for their spectrophotomet-ric measurements.
This work was supported by the Financiadora deEstudos e Projetos (FINEP) and the Conselho Na-cional de Desenvolvimento Cientifico e Tecnol6gico(CNPq). We also acknowledge Enrique Landgravefrom Centro de Investigaciones en Optica, LEON,Mexico who largely contributed to the success of thisresearch.
