Modeling of the refractive index and extinctioncoefficient of binary composite films

Meenakshi Kar, Bhullan S. Verma, Amitabha Basu, and Raghunath Bhattacharyya

Southwell’s analysis of optical multilayers within the limits of very thin films has been extended toinclude absorption in the multilayer for predicting the effective values of the refractive index ne andextinction coefficient ke of mixed-composition binary homogeneous films over a wide spectral region,including the high-absorption �k � 10�2� region. It has been found that ne in general is a complicatedfunction of the optical parameters �n1, k1, n2, k2� and volume fractions � f1, f2� of the component materialsin a homogeneous layer, and the expression for ne becomes the same as that predicted by the Drude modelin the spectral region where the layers are transparent. Moreover, according to the present analysis, thevolume fractions of the product of the refractive index and the extinction coefficient of the componentmaterials of a binary composite film are additive and the sum equals the product of the effective refractiveindex and extinction coefficient of the composite film. © 2001 Optical Society of America

OCIS codes: 300.1030, 230.4170.

1. Introduction

Mixed-composition thin-film systems �binary, ter-nary, etc.� exhibit different and often superior opticaland physical properties compared with individualsingle-component films.1–7 The improved film prop-erties �of composite films� include refractive-indextailorability �the dependence of refractive index oncomposition�, greater index stability, a modificationof structure �decreased porosity and an increase inpacking density�, a decrease in intrinsic stress,smaller optical scatter, decreased surface roughness,and an increase in laser-induced damage thresholds.

Various models,8 such as linear effective mediumapproximation, Drude’s, Lorentz–Lorentz, and max-imum screening models, have been proposed by dif-ferent researchers to predict the refractive index ofmixed-component films. Following Southwell’s the-ory of digital layers9 �neglecting absorption� and us-ing these models for mixed films, we recently10

established an optical equivalence between digitalfilms and mixed homogeneous layers with equivalentcomposition. However, in all the above-mentioned

models, optical absorption in the component materi-als, which is quite significant in the UV region, hasbeen neglected. In this paper our aim is to general-ize Southwell’s theory by taking into account absorp-tion of the component films for predicting theeffective values of the refractive index and extinctioncoefficient of mixed-composite films.

2. Effective Values of Optical Parameters �ne, ke� ofAbsorbing Binary Composite Films

The characteristic matrix M of a homogeneous thinlayer of complex refractive index N � n � ik �n and kbeing the real refractive index and the extinctioncoefficient of the layer, respectively� and physicalthickness d is given by11

M � � cos �i sin �

NiN sin � cos �

� , (1)

where the phase thickness of layer � equals 2�Nd��for the normal incidence of light of wavelength �.

The matrix above is transformed into9

M � � 12�id

�2�idN 2

�1 � (2)

provided 2�nd�� 1 and 2�kd�� 1.

The authors are with the Thin Film Technology Group, NationalPhysical Laboratory, New Delhi 110012, India. M. Kar’s e-mailaddress is mkar@csnpl.ren.nic.in.

Received 28 September 2000; revised manuscript received 24May 2001.

0003-6935�01�346301-06$15.00�0© 2001 Optical Society of America

1 December 2001 � Vol. 40, No. 34 � APPLIED OPTICS 6301

For the typical case of a very thin layer with anoptical thickness of nd 50 Å the error involved dueto this transformation in each matrix element, m11,m12, m21, and m22 �when as high as the fourth placeof the decimal is considered�, amounts to �0.2% at awavelength of 6000 Å, and the error becomes �0.75%at a wavelength of 3900 Å.

The characteristic matrix M12 of a pair of two verythin layers �say 1 and 2� with complex refractive in-dices N1 � n1 � ik1 and N2 � n2 � ik2 and physicalthicknesses d1 and d2, respectively, is given by thematrix multiplication of the characteristic matrices�with the form as in Eq. �2�� of the individual singlelayers. In this process, neglecting the second-orderterms in thicknesses, one obtains

M12 � � 12�i�d1 � d2�

�2�i�N1

2d1 � N22d2�

�1 � . (3)

The form of Eq. �3� is similar to that of Eq. �2�, andcomparing the two, one gets the effective index Ne �ne � ike and thickness deff of a single-layer equivalentto a pair of very thin layers:

deff � d1 � d2, (4)

Ne2 � N1

2� d1

d1 � d2� � N2

2� d2

d1 � d2� . (5)

The addition of more identical pairs of very thinlayers in sequence to the existing pair of thin filmsresults in an optically equivalent homogeneous filmwhose thickness deff �given in Eq. �4�� increases withan increase in the number of pairs, but the effectiverefractive index Ne remains the same as that given inEq. �5� �Fig. 1�.

Furthermore, the volume fractions f1 and f2 of thecomponent materials in the resultant homogeneouscomposite film equal the thickness ratios of the com-ponent films in the equivalent pair of films. There-fore

f1 �d1

�d1 � d2�, f2 �

d2

�d1 � d2�.

Using these equations and separating the real andthe imaginary parts of Ne in Eq. �5�, we get

ne2 �

12

��n12 � k1

2� f1 � �n22 � k2

2� f2� (1 � �1

� 2�n1 k1 f1 � n2 k2 f2�

�n12 � k1

2� f1 � �n22 � k2

2� f22�1�2) , (6)

ke �1ne

�n1 k1 f1 � n2 k2 f2�. (7)

Equations �6� and �7� reduce to different forms de-pending on the magnitudes of extinction coefficientsand the refractive indices of the component materials.

Case 1: Medium�intermediate absorption region.In this region n and k satisfy

10�3 � k1, k2 � 10�2,

n12 �� k1

2, n1 � 1,

n22 �� k2

2, n2 � 1.

Expressions for ne and ke become

ne2 �

12

�n12f1 � n2

2f2� (1 � �1

� 2�n1 k1 f1 � n2 k2 f2�

n12f1 � n2

2f22�1�2), (8)

ke �1ne

�n1 k1 f1 � n2 k2 f2�, (9)

where ne is now given by Eq. �8�.Case 2: Weak absorption region.For this region we have

k1, k2 � 10�3,

n1 �� k1, n1 � 1,

n2 �� k2, n2 � 1,

so that the expressions for ne and ke become

ne2 � n1

2f1 � n22f2, (10)

ke �1ne

�n1 k1 f1 � n2 k2 f2�, (11)

where ne is given by Eq. �10�.

Fig. 1. �a� Digital-type film with j identical pairs of thin layers with refractive indices N1 and N2 and thicknesses d1 and d2, respectively.�b� Equivalent homogeneous mixed-composition layer.

6302 APPLIED OPTICS � Vol. 40, No. 34 � 1 December 2001

Case 3: Transparent region.In this region k13 0, k23 0. Equations �10� and

�11� reduce to

ne2 � n1

2f1 � n22f2, (12)

ke3 0.

Note that Eqs. �10� and �12� give the same value ofne as predicted by Drude’s model. Furthermore, thevolume fractions of the product of the refractive indexand extinction coefficient of the component materialsof a binary composite film are additive and equal tothe product of the effective refractive index and ex-tinction coefficient of the composite film �Eqs. �7�, �9�,and �11��.

3. Model versus Experimental Observations

It is evident that Drude’s model �Eq. �12�� is a specialcase of the comprehensive analysis of binary compos-ite films, based on Southwell’s theory, and is validover a limited optical region where the materials aretransparent. To predict the effective index ne andthe extinction coefficient ke of a mixed homogeneouslayer over a spectral region where the componentmaterials become absorbing, one can use Eqs. �8�–�11�, depending on the magnitude of the extinctioncoefficients �k1 and k2� and the refractive indices �n1and n2� of the component materials. To the best ofour knowledge, most of the authors who studied thecomposite films of optical materials limited their in-vestigations to predicting the effective index as afunction of the weight ratio�volume ratio�mole frac-tion of the constituent materials or at the most as afunction of wavelength over a limited range wherethe component materials are transparent.

The optical properties of single-component andcomposite films of SiO2 and TiO2 studied over a wholespectral region �0.16–2.0 m� as reported by Demiry-ont12 �Figs. 2�a� and 2�b�, films of SiO2 �100%�, TiO2�100%�, SiO2 �48%� � TiO2 �52%�, volume concentra-tion denoted by percent� will form part of our discus-sion regarding verification of the model predictions.In particular, Eqs. �6�–�12� will be used to predict therefractive index ne and extinction coefficient ke of aSiO2 � TiO2 composite film �concentration of TiO252% by volume�, and the computed values of ne and kewill be compared with the reported experimental da-ta.12 The step-by-step procedure is outlined below.

�1� Figures 2�a� and 2�b� of Ref. 12 were expandedeight times to extract the values of the refractiveindex and extinction coefficient of TiO2 �100%� andSiO2 �100%� layers �with physical thicknesses of0.260 and 0.298 m, respectively� as a function ofwavelength in the spectral region of 0.39–0.60 m.To extrapolate the values of refractive index n1 andextinction coefficient k1 of the TiO2 �100%� layer inthe spectral range from 0.32 to 0.39 m, the extracteddata were fitted to the dispersion relations by usingthe Marquardt regression algorithm as shown inFigs. 2 and 3. However, in the spectral range of0.32–0.60- m the SiO2 �100%� layer, as reported in

Ref. 12, is found to have a constant value of refractiveindex n2 equal to 1.48 and an extinction coefficient ofk2 3 0.

�2� The experimental data of effective refractiveindices ne �exp� and extinction coefficients ke �exp� ofa composite film of SiO2 �48%� � TiO2 �52%�, with aphysical thickness, d � 0.3666 m, were also ex-tracted from the expanded Figs. 2�a� and 2�b� of Ref.12 and fitted to the following dispersion equations:

ne�exp� � a0 �b0

�2 �c0

�12 , (13)

where

a0 � 0.20400650 exp�01,

b0 � 0.23814860 exp�01,

c0 � 0.85318690 exp�06,

ke�exp� �a

1.0 � ��

c�b , (14)

Fig. 2. Experimental data of a refractive index of 100% TiO2 layer�thickness, 0.2666 m� fitted to the Cauchy function with a Mar-quardt regression algorithm:

n1 � A0 �B0

�2 �C0

�12 ;

A0 � 2.35468,

B0 � 1.74498E�2,

C0 � 6.67044 E�6.

1 December 2001 � Vol. 40, No. 34 � APPLIED OPTICS 6303

where

a � 0.29178180 exp�00,

b � 0.12137540 exp�02,

c � 0.35374530 exp�00.

�3� We have computed the effective values of re-fractive index ne and extinction coefficient ke of acomposite layer of SiO2 �48%� � TiO2 �52%� from Eqs.�6� and �7�, using the dispersion relations for n1 andk1 in Figs. 2 and 3 and taking a constant value of n2 �1.48, k2 3 0 in the 0.34–0.70- m spectral region.The values of ne and ke, computed on the basis of thepresent analysis, are plotted in Figs. 4 and 5 forcomparison with the experimental data given by Eqs.�13� and �14� for a composite film of SiO2 �48%� �TiO2 �52%�, as reported in Ref. 12.

4. Accuracy Considerations

The experimental data for n1, k1, n2, and k2 wereextracted from the expanded plots of Figs. 2�a� and2�b� of Ref. 12. Errors were introduced into the datafrom the appreciable line thicknesses in the figuresand from the rapid increase�decrease in the values ofk1, k2�n1, n2 as a function of wavelength. The errorsso introduced influence the values of the effectiverefractive index ne and extinction coefficient ke, cal-culated on the basis of the present analysis. Whenne is expressed as

ne � ne�n1, k1, n2, k2, f1, f2�,

the absolute error �ne is calculated from

�ne � ��ne

�n1�n1�2

� ��ne

�k1�k1�2

� ��ne

�n2�n2�2

� ��ne

�k2�k2�2

� ��ne

�f1�f1�2

� ��ne

�f2�f2�21�2

.

(15)

The incremental coefficients �ne��n1, �ne��k1, and�ne��n2 have been evaluated from Eq. �6� in the re-gion from 0.39 to 0.70 m. In this spectral region�n1 varied from �0.08 to 0.016 and �k1 varied from�0.5 to �0.1 � 10�2 ��n2 � �0.016 remained con-stant�. Furthermore, we assumed that �k2 � 0,�f1 � 0, �f2 � 0. The error thus calculated is shownin Fig. 4 by the error bars. Similarly, the extinction

Fig. 3. Experimental data of the extinction coefficient of 100%TiO2 layer �thickness, 0.266 m�, fitted to a logistic equation, witha Marquardt regression algorithm:

k1 � A11 � � �

C1�B1�1

, A1�0.32667, B1�12.03521,

C1 � 0.37459.

Fig. 4. Comparison of —, the experimentally observed data of theeffective refractive index ne of a TiO2 �52%� � SiO2 �48%� layer withthose predicted by F, the present analysis�model.

Fig. 5. Comparison of —, the experimentally observed data of theeffective extinction coefficient ke of a TiO2 �52%� � SiO2 �48%� layerwith those predicted by F, the present analysis�model.

6304 APPLIED OPTICS � Vol. 40, No. 34 � 1 December 2001

coefficient ke has been expressed as ke � ke �ne, n1, k1,n2, k2, f1, f2�, and the absolute error �ke �given by anexpression similar to Eq. �15�� has been computedfrom Eqs. �7� and �15�. The computed values of theerror bars are shown in Fig. 5.

5. Discussion

The experimental value of the effective refractive in-dex ne �exp� �Fig. 4� of a thick layer �0.3666 m ofTiO2 �52%� � SiO2 �48%�� differs from the value pre-dicted by the model ne �model� by �4%, and thisdifference remains more or less the same in the trans-parent �ke 3 0�-, weak �ke 10�3�-, and medium�10�3 ke 10�2�-absorption regions. In the high-absorption region �ke � 10�2�, ne �exp� shows quali-tative agreement with ne �model�, when uncertaintiesinvolved in the data are taken into account. On theother hand, the experimental values of extinction co-efficient ke �exp� are in qualitative agreement with ke�model� in the whole region, starting from the trans-parent to the high-absorption region down to 0.34 m. However, Eqs. �6� and �7� derived above can beapplied to predicting the refractive index nf of asingle-component film of an optical material with auniform void distribution �porous film�. Introducingthe following transformations in Eqs. �6� and �7�,

ne3 nf

ke3 kf �for the film of single-component optical

material�,

n13 ns

k13 ks �for the solid part of the film�,

n23 nv

k23 kv �for voids�,

one gets

nf2 �

12

��ns2 � ks

2�p � �nv2 � kv

2��1 � p�� � (1 � �1

� 2�ns ks p � nv kv�1 � p��

��ns2 � ks

2� p � �nv2 � kv

2��1 � p��2�1�2),

(16)

nfkf � nsksp � nvkv�1 � p�, (17)

where p is the packing fraction of the film, defined as

p �volume of solid part of the film

total volume of the film �solid � voids�.

Equations �16� and �17� are valid over a wide spec-tral region including the high-absorption region inwhich the extinction coefficient of the film materialexceeds 10�2. We confine our discussion specificallyto the transparent region to test the validity of Eqs.�16� and �17�.

Assuming that ks � 0, kv � 0, Eq. �16� reduces to

nf2 � ns

2p � nv2�1 � p� (18)

for absorption-free porous films. Similar equationshave been derived earlier �see Ref. 13, p. 12� and aregiven below for ready reference and discussion:

nf � �1 � p�nv � pns

�Kinosita and Nishibori13�, (19)

nf2 �

�1 � p��ns2 � 2�nv

2 � p�nv2 � 2�ns

2

�1 � p��ns2 � 2� � p�nv

2 � 2�

�Chopra et al.14�, (20)

nf2 �

�1 � p�nv4 � �1 � p�nv

2ns2

�1 � p�nv2 � �1 � p�ns

2

�Harris et al.15�. (21)

In general, the values of nf for the given values of ns�ns � 1� and p �0 p 1� predicted by Eqs. �18�–�21�differ from one another, the difference becoming max-imum for p � 0.5 and for large values of ns. How-ever, it decreases for small values of ns �1.3 � ns 2.0� as shown in Fig. 6.

6. Conclusion

An analysis of mixed homogeneous �binary compos-ite� films has been attempted, drawing largely fromSouthwell’s theory of optical multilayers. Thisequivalence of the analysis of mixed-composition thinfilms and digital layers appears to hold good, at leastqualitatively, in predicting the effective values of therefractive index and the extinction coefficient of amixed homogeneous film. For a packing fraction of�0.5 or more and large values of the refractive indexand the extinction coefficient of film material �thesolid part of the film�, our analysis predicts valuesthat differ from the values predicted by other pub-lished models.

Thanks are due to the Director, National PhysicalLaboratory, New Delhi �India� for permission to pub-

Fig. 6. Refractive index nf of a single-component porous film �ns �1.40, nv � 1.0� as a function of the packing density p �0 � p � 1�according to various models.

1 December 2001 � Vol. 40, No. 34 � APPLIED OPTICS 6305

lish this paper and also to T. K. Bhattacharya andSuman Bharadwaj for help in preparing this paper.

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