Relation between light scatteringand the microstructure of optical thin films

Angela Duparr6 and Samer Kassam

Special substrate-film designs are used to measure roughness-induced scattering and scattering from thevolume of optical thin films separately. So theoretical models of surface roughness and volumescattering become applicable to the experimental data, and quantitative information on thin-filmmicrostructure can be derived. Measuring total integrated and angle-resolved scattering on oxide,fluoride, and chalcogenide films of different film thicknesses yields the evolution law of microstructuralgrowth, which for the majority of investigated films roughly follows a square-root dependence on filmthickness. Packing densities of fluoride films calculated from volume-scattering data are found to agreewith results from quartz-crystal monitoring.

Key words: Light scattering, optical thin films, thin-film microstructure.

1. Introduction

Light scattering in optical thin films has become afield of great interest, because when high-qualityoptical coating components are to be manufactured itis necessary to understand the mechanisms respon-sible for scattering losses, which, in general, limit theoptical performance. Moreover, light scattering hasbeen widely recognized as being of use in the analysisof film microstructures since the study of film mor-phology by means of scattering benefits from thenoncontact and nondestructive nature of those mea-surements. A number of publications in the lastdecade have witnessed the growing interest in thisaspect.'- 6

In the past, however, the majority of scatteringtheories and measurements on optical thin films wererestricted to interface roughness scattering, eventhough in the seventies there were already a fewinteresting approaches to volume-scattering effects aswell, such as those of Guenther et al. 7 and Carniglia.8

Nowadays increasing attention is being paid to vol-ume-scattering effects in thin films 5 '91 since measure-ments of volume scattering along with suitable theo-

A. Duparr6 is with the Optical Coatings Department, FraunhoferInstitution for Applied Optics and Fine Mechanics, Schillerstrasse1, 0-6900 Jena, Germany. S. Kassam is with the Physikalisch-Astronomische Fakultdt, Institute of Solid State Theory andTheoretical Optics, Friedrich-Schiller-Universitdt, Max-Wein-Platz 1, 0-6900 Jena, Germany.

Received 13 November 1992.0003-6935/93/285475-06$06.00/0.o 1993 Optical Society of America.

retical models can be successfully used for examiningintrinsic film morphology. As an essential precondi-tion, however, one must be able to measure volumescattering separately.

On the other hand, in investigations of roughnessscattering, in which the aim is the easiest possibleestimation of roughness and morphological param-eters, it often seems desirable to avoid the superposi-tion of scattered fields arising from the individualinterfaces of a transparent film sample as well as toexclude contributions from the film volume. Theevaluation of scattering that originates only from thetop surface, by metallizing the optical film surface,can provide helpful information in connection withthe application of roughness-scattering theories ofsingle interfaces.'12" 3

Here we concentrate on the derivation of micro-structural information from measurements of puresurface scattering as well as pure volume scattering ofsingle layers by applying appropriate theoretical mod-els. In Section 2, relations are given that permit thecalculation of film packing densities and the quantita-tive description of the law of microstructural evolu-tion with increasing film thickness. In Section 3,methods of using special substrate-film designs thatenable roughness-induced scattering and volume scat-tering to be measured separately are presented.Mainly total integrated scattering (TIS) and a num-ber of angle-resolved scattering (ARS) measurementsare performed at X = 633 nm on oxide, fluoride, andchalcogenide films. Results of measurement andderived structural information with respect to the

1 October 1993 / Vol. 32, No. 28 / APPLIED OPTICS 5475

evolution law of morphological growth and to packingdensity are evaluated in Section 4.

2. Theoretical Models

A. Surface Scattering

The expression for the angular dependence of scatter-ing from a single surface, which is represented in ourinvestigations by the metallized film, is given by12"14"15

dP,P dQ = Fsg8(k), (1)

where dP, is the intensity scattered from the surfaceinto the solid angle dfl normalized to the incidentintensity P0. F is the optical factor, which dependson only the properties of the ideal surface as well ason the conditions of illumination and observation,but is independent of the surface roughness statistics.g8(k) represents the power spectral density of surfaceroughness for isotropic roughness and normal inci-dence.

Considering the roughness caused by the filmmicrostructure, the correlation length Ts will be smallcompared with the wavelength X for investigations inthe visible region. In this case an easy approxima-tion with respect to TIS can be applied12:

TIS = Fs ,4 (2)

where F is an optical factor, and os is the rmsroughness. Any effects of substrate surface smooth-ing16 will be neglected so that TIS, simply followsfrom subtracting the TISo level that is measured forcomparison on metallized bare substrates from thetotal TIS value.'3 On the basis of Eq. (2), an expres-sion for the evolutionary exponent of film growth K,

can be derived that describes the development of themean lateral dimension of the characteristic struc-tural features (most frequently columns or columnar-like structures if the films are deposited by conven-tional evaporation methods), represented by thecorrelation length Ts, with increasing film thickness din terms of a power law:17

B. Volume Scattering

Recently we reported on a theory of light scatteringfrom the volume of columnar-structured single-layeroptical films.10 The expression obtained for angle-resolved volume scattering can be written in a formsimilar to Eq. (1):

dP==~f =~V~) (5)

where now F, is the optical factor of volume scatter-ing and g, is the power spectral density of the volumefluctuations, which are assumed to be homogeneousand isotropic in the lateral direction. Assuming aGaussian autocovariance function with respect to theplane perpendicular to the sample normal and thatthe correlation length T, (describing the mean lateralextension of the columns within the film volume) issmall compared with the wavelength, the total inte-grated volume scattering TISV can be approximatedby

TIS FVT2oVUIV2

(6)

where Fk' = F 'mv; v 1 for m m*, where m* ismaterial dependent, m is an integer with m = 2nd/X,and u, is the rms fluctuation strength. The opticalfactor F' of the total volume scattering is numeri-cally calculable. u, can be directly related to thepacking density p of the film by the use of anexpression suggested by Elson18:

oI2= (n= 2 - n2)2p (1 - p). (7)

nvoid and n are the refractive indices of the voids andthe film material, respectively. Analogously to thederivation of the evolutionary exponent within theframework of surface roughness scattering in Subsec-tion 2.A, a formula for this structural parameter canalso be obtained from volume scattering. From acombination of Eq. (6) with TU c dKV we find that'0

ln([m 'TIS(m)]/[mTIS'(m))]}I - 2 ln(m/m')

Combining Eq. (2) and expression (3), and on thecondition that cr(d) x T(d), we get the evolutionaryexponent'3

ln[TIS(d)/TIS5(d ')]Ks- 41n(d/d') (4)

where TIS(d) and TISs(d') are the surface-scatteringvalues measured on films with thicknesses d and d',respectively. So the growth behavior of, for in-stance, the columnar diameter of a certain film typecan be estimated from surface roughness measure-ments made on at least two differently thick films.

TISv(m) and TIS,(m') are the total volume-scatteringvalues measured on films with optical thicknessm(X/2) and m'(X/2), respectively.

Finally it should be emphasized that within thelimit T << considered here, only the product U,2T'

2

occurs as an independent variable in the formula forangular scattering and, hence, the information in-volved in Eq. (5) does not exceed that of TIS,. Thesame fact is valid with respect to the roughnessscattering in Subsection 2.A: with Ts << , the product(FS2TS2 becomes inseparable. We come back to thisproblem in the discussion in Section 4.

5476 APPLIED OPTICS / Vol. 32, No. 28 / 1 October 1993

Ts c dKs. (3) (8)

3. Experimental

A. Separate Estimation of Volume Scattering and SurfaceRoughness Scattering

Using specific substrate-film designs enables volumescattering and roughness scattering to be measuredseparately, as is shown schematically in Fig. 1. Formeasuring pure volume scattering, the sample designconsists of a high reflecting substrate covered withthe thin film under study, the optical thickness ofwhich must be a multiple of half of the wavelength inthe case of normal incident light. As a result of thissample design, the electrical-field strength is simulta-neously canceled at both interfaces in a good approxi-mation, and, hence, interface roughness scattering issuppressed. Whereas multilayer dielectric mirrorswere utilized as high reflectors in Ref. 19, by now Allayers deposited onto the substrate before the opticalfilm was deposited have been recognized to be bestsuitable for convenient experimental employment.

As for the second problem, the measurement ofpure surface roughness scattering, it can be solved byovercoating the optical film with a thin opaque metal-lic layer that reproduces the film roughness profile.This method has been widely used with respect to theassessment of surface roughness of single layers aswell as multilayer stacks. 3 20 22 In our experiments,Al films of 70- to 100-nm thicknesses were depositedonto the front film surface. In Fig. 2, a cross-sectional TEM micrograph of a PbF2 film (on a BK-7substrate) overcoated with a 70-nm Al film revealsthe good reproduction of the PbF2 surface by the Alcover layer. Only a slight smoothing effect is percep-tible, which occurred in most of our investigations.To test the influence of the Al cover layer on theroughness of a bare BK-7 glass substrate (rememberthat an aluminized glass sample is needed for estimat-ing the TISo value), the roughness of BK-7 surfaceswith and without an Al layer was measured by opticalprofilometry with a Zygo Maxim 3D instrument.The rms roughnesses of 1.4 and 1.3 nm were obtainedfor the bare and the aluminized glass, respectively.This, again, indicates a weak surface smoothingcaused by the Al.

,- surface (PbF2 + Al)

.- Al-layer

._ PbF2 filmcross-section

200nm

Fig. 2. Cross-sectional TEM picture (C-Pt replica) of an Al-overcoated PbF 2 film.

B. Measurement Technique and Descriptionof the Samples

TIS measurements were performed at the He-Nelaser wavelength X = 632.8 nm with the experimentalarrangement shown in Fig. 3 and are described indetail elsewhere.23 24 Backscattered light collectedby a Coblentz sphere within an angular range of 2° to840 was measured while the sample relative to thelaser beam (1.6-mm spot size) was laterally shifted byan automatic translation device. All samples werescanned over a distance of at least 10 mm, and theminimum TIS values of the scans were taken tocalculate the structural quantities presented in Sec-tion 4.

ARS measurements were carried out at X = 632.8nm on a few distinct points of the sample with thescatterometer schematically shown in Fig. 4 anddescribed in detail in Ref. 25. The scattering wasmeasured perpendicular to the plane of incidence.The examples of ARS curves in this paper refer toapproximately normal incident light (00 1) and spolarization.

A variety of film materials and deposition condi-tions were employed for preparing the samples understudy. BK-7 glasses with normal polish were used assubstrates, some of which were covered with a 200-nmAl layer before deposition of the dielectric film with aview to measuring volume scattering.

surface scattering

He-Ne-laser (633nm)He-Cd-laser (325nm),

-Bf (nd=m /2) Al-layer f

Fig. 1. Sample configurations used for volume-scattering andsurface-scattering measurements at normal incidence. f, dielec-tric film; s, substrate.

reference sicdetector

sample shifting system sample photomultiplier

&-..I \

,hopper diaphgm olentzsphere

-t ~ ~~~~~~~mirrorInat

lock-inPC amplifier -

Fig. 3. TIS measurement setup.

1 October 1993 / Vol. 32, No. 28 / APPLIED OPTICS 5477

volume scattering

I

rs .. reference signala ... analyzerngf... neutral glass filterpm... photomultiplier

sample

9,,

I 5-

:ngf ] d d I

I I

pm

Fig. 4. ARS measurement setup.

BaF2, MgF2, and PbF2 films were evaporated fromresistance-heated boats at substrate temperatures ofT = 25 and 150 C. Electron-beam evaporation wasused to prepare ZnS and ZnSe (T, = 25 C) as well asZrO2 films (300 C). Nb 2 O5 and Ta2O5 films weremagnetron sputtered at room temperature. Witheach material, we investigated films of different thick-nesses, two at least. In the case of volume scatter-ing, the optical thickness had to be a multiple of halfthe wavelength, as described in Subsection 3.A.Most samples served for determining the evolution-ary exponent of structural growth, either from TISSor from TIS, measurements as well as, in selected

8 /degrees(a)

t 10

10"4

0 10 20 30 40 50 60 70 80 90

9/degrees(b)

Fig. 5. Measured ARS (s polarization, = 633 nm): (a) surfacescattering of Al-overcoated films (upper curve, 3 PbF2 film; lowercurve, BK-7 substrate); (b) volume scattering of a 5 PbF 2 film.

cases, from both types of scattering. We confinedour investigations of angle-resolved volume and sur-face scattering to the BaF2 and PbF2 samples, like-wise the TIS, measurements for the derivation ofpacking densities.

4. Results and Discussion

Figures 5 and 6 represent angle-resolved surface-scattering and volume-scattering measurements onPbF2 and BaF2 layers, the optical thicknesses ofwhich were nd = 3, and 5X, respectively, for volumescattering of PbF2 . The shapes of the scatteringcurves prove that the assumption of T8,0 << involvedin the approximate TIS formulas of Eqs. (2) and (6) isreasonable. There is only a weak dependence onscattering angle 0, which means that the curves arenearly parallel to the horizontal axis, with the follow-ing two exceptions: the interference effects inherentin the volume-scattering curves, and the enhancednear-angle scattering, which is particularly obvious inthe volume- as well as surface-scattering behavior ofthe PbF2 films (Fig. 5). We believed this enhancedscattering at low angles to be caused by some degreeof long correlated microroughness of the aluminizedsubstrate [near-angle scatter also occurs for the alumi-nized bare BK-7 in Fig. 5(a)] as well as by particulatesin the film volume and on its surface.

It should be emphasized that neither for BaF2 norfor PbF2 does the angular dependence of volumescattering show perfect scattering interference effectsthat would result in sharp and deep minima at 0 _

a- aod0~la?~

0 10 20 30 40 50 60

9 / degrees(a)

70 80 90

a ci 31 1CL-d 10-3 Ia

-4_10 1 105

o 10 20 30 40 50 60 70 80 909/ degrees

(b)

Fig. 6. Measured ARS on BaF2 films (s polarization, 633nm): (a) surface scattering of an Al-overcoated 3X film, (b) volumescattering of a 3 film.

5478 APPLIED OPTICS / Vol. 32, No. 28 / 1 October 1993

'd ch pot d bs d

1 I m7 ' v II ,~ , , I

rs He --- .

d ... diaphragmch... chopperpot... polarizerbs... beam splitter

Table 2. Packing Densities Obtained from TIS, Measurements(X = 633 nm) on PbF2 and BaF2 Films

Packing Densityp (%)

Fixed Substrate Rotating SubstrateFilm T = 250 C T = 1500 C

BaF2 67 76PbF 2 98 99.9

noted that, in electron-microscopy studies of evapo-I' ''> 2 4 6 8 10 12 rated TiO2 films27 and ZrO2 films,28 an approximate

o 2 4 6 8 10 12 14 mm dependence of the columnar dimension was ob-sample shifting / mm tained as well.

Total integrated surface-scattering measurements on Packing densities of BaF2 and PbF2 films with 3Xfilms on BK-7 substrates. Film thicknesses d = 690 nm optical thickness were determined from TIS, measure-curve) and d' = 410 nm (lower curve). Both samples were ments with Eqs. (6) and (7). The voids were as-ated with Al.metwihEs(6an(7.Tevisera-

sumed to be filled with water. The average lateralstructural dimensions T was obtained from electron

was theoretically calculated for PbF2 in Ref. 10. micrographs. Different evaporation conditions withresult can be explained by a nonideal columnar respect to substrate heating and rotation resulted inture being present, so that the maxima will be different values of the packing density, as can be seenr smeared. in Table 2. It seems to be plausible that both)m TIS and TIS, measurements, evolutionary substrate rotation and enhanced temperature giveients have been calculated for the different types rise to increasing packing densities. In the case oftical film by means of Eqs. (4) and (8), respectively. fixed, unheated substrates, the results were com-

example, the TIS measurement on two MgF2 pared with packing densities estimated by conven-is demonstrated in Fig. 7. The minima of the tional quartz-crystal monitoring, and good agreementscans have been put into Eq. (8) after TISo = was found (97% and 69% for PbF2 and BaF2, respec-

10-4, the value of the aluminized BK-7 sub- tively). However, we consider the necessity of per-3, had been subtracted. forming electron-microscopy studies to get the Tv

ble 1 summarizes the K, and the K, parameters parameters unsatisfactory; hence we are about toned. Regarding the sputtered oxide films, no develop a method of TIS, investigation that willtural development with increasing film thick- permit the product a,2 'i 2 to be separated. This cancould be observed. This is in accordance with be accomplished when an additional wavelength (theearly amorphous structure displayed by electron He-Cd laser with X = 325 nm), for which T,,, << X

)scopy inspections we also made. does not hold any longer, is used. Then, by means ofth respect to the evaporated F2, chalcogenide, an equation calculated numerically in Ref. 12 andZrO2 films, the results indicate a preferred with X = 325 nm and X = 632.8 nm, two independenthological evolution of the average dimension of equations for the determination of ur, and T wouldltructural features with roughly the square root become available. Eventually, when Ts and the evolu-m thickness, except for the BaF2 films. It is tionary exponent are known, the r value within theesting to discover similar evolution laws for layer volume can be easily obtained.

quite different film materials and different substratetemperatures (in this experiment we used ambienttemperture for BaF2 , PbF2 , ZnS, and ZnSe; 150 'C forMgF2 ; and 300 'C for ZrO2 ). Hence there will be filmstructures belonging to zone 1 of the structure zonemodel,26 whereas others more likely fall into zone 2,as, for instance, the PbF2 films, even though they aredeposited onto unheated substrates because of thelow melting point of PbF2. It should be further

Table 1. Evolutionary Exponent K Estimated from TIS, and TIS,Measurements (X = 633 nm)

Evaporated Sputtered

5. Concluding RemarksSurface roughness scattering and volume-scatteringmeasurements on single-layer films of different mate-rials have proved to be suitable tools for quantita-tively deriving microstructural parameters such aspacking density and the evolutionary exponent of filmgrowth. So far, our theoretical description of vol-ume scattering and hence our evaluation of measureddata from the film bulk are limited to columnarstructures. It is intended to extend the investiga-tion to other structures, as well as to focus on thescattering behavior of structural anisotropies.

The authors thank J. Neubert for the ARS measure-ments; A. Kohler, L.O.T., Darmstadt, for the measure-ment with the Zygo Maxim 3D instrument; and H.Muller for the electron micrograph. The topic ofthis paper was the subject of a paper presented at the

1 October 1993 / Vol. 32, No. 28 / APPLIED OPTICS 5479

1.2 -M0xU)0.8

0.4 -

Fig. 7MgF2 I(upperoverco;

50 0aEThisstrucrathe

Frcexpoiof optAs arfilmstwo E

1.5 >

stratE

Talobtai:strucnessthe nmicrc

Wi.andmorpthe sof filinter

K PbF 2 BaF2 MgF2 ZnS ZnSe ZrO2 Ta 2O5 Nb2O5

Ka 0.53 0.27 0.43 0.49 0.58 0 0KU 0.46 0.29 0.42

Fifth Topical Meeting on Optical Interference Coat-ings, Tucson, Arizona, 1-5 June 1992.

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