March 15, 2010 / Vol. 35, No. 6 / OPTICS LETTERS 841

Determination of the refractive index of a dielectricfilm continuously by the generalized S-transform

Emre Coskun,* Kıvanç Sel, and Serhat ÖzderDepartment of Physics, Çanakkale Onsekiz Mart University, 17100 Çanakkale, Turkey

*Corresponding author: ecoskun@comu.edu.tr

Received October 22, 2009; revised January 7, 2010; accepted February 2, 2010;posted February 16, 2010 (Doc. ID 118954); published March 12, 2010

The generalized S-transform was improved as a method to determine the refractive index of a dielectric filmcontinuously by using the transmittance spectrum, and the applicability of the method was demonstrated onmica. The result determined from the generalized S-transform method was compared with the results de-termined from the S-transform and the fringe counting methods and published values. The advantage of theproposed method was explained, and the absolute error of the presented method was also calculated.© 2010 Optical Society of America

OCIS codes: 070.4560, 200.4560, 310.6860.

The continuous determination of the optical param-eters of a film over a wavenumber range k0=1/� is abasic requirement to project optical-based devices,and therefore many techniques have been developed.For determining the refractive index n�k0� of thefilms, because of the simple usage, the most commonmethods are based on the transmittance spectrum offilms, such as the fringe counting [1] and envelope [2]methods. In these methods, the continuous determi-nation of n�k0� could be achieved by an interpolationof the analyzed data, since the analysis is based onthe extreme points of the transmittance spectrum,whereas in the Abbe refractometer [3] and waveguide[4] methods, n�k0� is determined at a single wave-number point and in the spectroscopic ellipsometrymethod a fit procedure on the measured data is re-quired (indirect characterization) [5]. In this study,an alternative method introduced from the general-ized S-transform was presented to determine the re-fractive index of a dielectric film by using the trans-mittance spectrum.

When a light beam comes upon the film at a nor-mal direction, the transmittance equation for a ho-mogenous and uniform dielectric film of refractive in-dex n and thickness d, which is surrounded by airhaving a refractive index of n0=1, is given by Eq. (1)[6],

T�k0� = �1 + �n2 − n02

2nn0�2

sin2�2�k0D��−1

, �1�

where D=nd is the optical thickness of the film,which has to be less than the coherence length of theilluminating radiation. The dispersion equation ofn�k0� could be described as a three-term Cauchy for-mula, and it was given by Eq. (2),

n�k0� = A + Bk02 + Ck0

4, �2�

where A, B, and C parameters are characteristic con-stants of the film. Because n is a function of k0, therepetition frequency of the transmittance signal ischanged through the scanned wavenumber range;

therefore the signal is a nonstationary signal, which

0146-9592/10/060841-3/$15.00 ©

is convenient to execute the generalized S-transform.The generalized S-transform is a signal analyzingmethod, and it was improved from the S-transform.The S-transform was first introduced by Stockwell etal. as a method to determine the local spectrum of atime series for geophysical data analysis [7], and itwas adapted to use for determining the birefringenceproperty of liquid crystals in our previous study [8].The aim of this work is to adapt the generalizedS-transform as a suitable method for determining therefractive index of a dielectric film. The generalizedS-transform of the transmittance signal was ob-tained as [9]

SGSp �b,f � = �

−�

+�

T�k0�wGS�b − k0,f,p�exp�− i2�f k0�dk0,

�3�

where wGS�b−k0 , f ,p� is the generalized Gaussianwindow function, and it was given as follows [10]:

wGS�b − k0,f,p� =�f �

p2�exp�−

f 2�b − k0�2

2p2 � . �4�

In this function b is a parameter that controls the po-sition of the generalized Gaussian window on the k0domain, f is the frequency, and p ��0� is a parameterto adjust the width of the generalized Gaussian win-dow. The wGS�b−k0 , f ,p� function must satisfy thenormalization condition and it was verified by Fe-derico and Kaufmann [11]. Using the convolutiontheorem [12], SGS

p �b , f� could be written as follows:

SGSp �b,f� = �

−�

+�

T�x0 + f �WGS�x0,f,p�exp�i2�x0b�dx0

= IFTT�x0 + f �WGS�x0,f,p��, �5�

where T�x0� and WGS�x0 , f ,p� are Fourier transformsof T�k0� and wGS�k0 , f ,p�, respectively. In this form, itis possible to use the fast Fourier transform proce-dure to decrease the computation time. The center

and variance of the w�k0 , f ,p� function in the k0 do-

2010 Optical Society of America

842 OPTICS LETTERS / Vol. 35, No. 6 / March 15, 2010

main were given in Eqs. (6) and (7), respectively,

k0C =

�−�

�

k0�wGS�b − k0,f,p��2dk0

�−�

�

�wGS�b − k0,f,p��2dk0

= b, �6�

��k0�2 =

�−�

�

�k0 − k0C�2�wGS�b − k0,f,p��2dk0

�−�

�

�wGS�b − k0,f,p��2dk0

=p2

2f 2 .

�7�

Thus, the information of the signal is localized in the�k0C−�k0 ,k0C+�k0 interval, and when b�p / �2f �,n�k0� could be approximated as n�k0��n�b�. This ap-proximation allows computing the Fourier transformof the transmittance signal T�x0�, and it was evalu-ated in Eq. (8),

T�x0� = C1��x0 − 2D�b�� + C2��x0� + C3��x0 + 2D�b��,

�8�

where C1, C2, and C3 are constants. The WGS�x0 , f ,p�function is the generalized Gaussian window func-tion in the x0 domain, which was given by Eq. (9),

WGS�x0,f,p� = exp�− 2�2x0

2p2

f 2 � . �9�

The variance of the WGS�x0 , f ,p� function was calcu-lated as ��x0�2= f 2 / �2p2� which demonstrates the x0

domain resolution (spectral resolution). T�x0� andWGS�x0 , f ,p� were inserted in Eq. (5) and usingWGS�x0 , f ,p�=0 for x0�−f, the generalizedS-transform was obtained as

SGSp �b,f � = C1 exp�−

�2D − f �2p2

2f 2 �exp�i2�b�2D − f ��,

�10�

and the normalized modulus of the generalizedS-transform was given as

�SGSp �b,f �� = C1 exp�−

�2D − f �2p2

2f 2 � . �11�

The f value corresponding to the maximum value of�SGS

p �b , f �� for each b was defined as fmax, and it wasgiven by Eq. (12),

fmax�b� = 2D�b�, �12�

and n�k0� was calculated from Eq. (12) by knowingthe thickness of the film. From the localization prop-erty of the generalized S-transform, the proposed ap-proach is valid when bD�b��p /22.

With the experimental work, it could be demon-strated how appropriate is the generalized

S-transform method for determining the refractive

index of a dielectric film. The test sample was se-lected as mica, because it is a natural substance andcleaves perfectly into thin plate. The experimentalsetup consisted of an MS260i spectrograph having1800 lines/mm grating. A 10–250 W quartz tungstenhalogen lamp was used as an input light source. Anunpolarized light coming from the light source waspolarized, and it was incident normally upon themica surrounded by air. The transmitted light pass-ing through the mica was focused on the spectrom-eter by a condenser lens. The light inside the spec-trometer was detected by a CCD camera. Theexperiment was performed by the control of a com-puter. The thickness of the mica was measured as d=41.51 m by a thickness profiler (XP-2 Ambios).The transmittance spectrum of the mica was ac-quired in the 1.4285–1.5384 m−1 wavenumber in-terval with a sampling rate of �k0�210−4 m−1 atroom temperature [Fig. 1(a)], and the generalizedS-transform was executed on the transmittance sig-

Fig. 1. (a) Transmittance spectrum of the mica. (b) Its nor-malized modulus of the generalized S-transform �SGS

p �b , f ���p=8�. (c) Refractive index of the mica determined by thegeneralized S-transform (p=8, solid curve), theS-transform method (dashed curve), fringe countingmethod (circles), and the refractive index result calculatedby using the Cauchy parameters of [13] (dashed-dotted

curve) and [14] (dotted curve).

March 15, 2010 / Vol. 35, No. 6 / OPTICS LETTERS 843

nal. The obtained �SGSp �b , f �� matrix was presented in

Fig. 1(b) in mesh form. n�k0� was calculated continu-ously from Eq. (12) by using �SGS

p �b , f ��, and it was de-picted in Fig. 1(c) as a solid curve (the three-termCauchy formula parameters were calculated as A=1.4896, B=0.0428 m2, and C=5.096810−12 m4

by a least-squares fitting procedure). The transmit-tance spectrum was also analyzed by the fringecounting method, and the determined n�k0� was pre-sented in Fig. 1(c) as circles. The determined n�k0�using the generalized S-transform method was foundto be in accordance with the fringe counting methodresult, and the published values by El-Zaiat [13,14](dotted and dashed-dotted curves in Fig. 1(c) were re-trieved by using Cauchy parameters from [13,14], re-spectively). Additionally, the generalized S-transformmethod is capable to obtain n�k0� continuously differ-ent from the fringe counting and any other methods.For showing the importance of the control of spectralresolution, the S-transform �p=1� was also applied tothe transmittance signal, and the correspondingn�k0� was found, which was depicted as a dashedcurve in Fig. 1(c). The high frequency of the transmit-tance signal of mica caused the insufficiency of theS-transform method depending on fixed spectral res-olution, which was overcome in the generalizedS-transform method by introducing the p parameter�p=8�. The calculations were performed in a MATLABenvironment by using the S-transform subroutinegiven in [9] at a computation time of around 0.5 s ona standard personal computer. It is worth noting thatthe computation time is independent on parameter p,but changes quadratically with the number of trans-mittance data as determined from the simulationstudies.

The error calculation of the presented method wasalso studied. By Eq. (12), the relative error ��n /n�from the generalized S-transform method was calcu-lated as in Eq. (13),

�n

n= ���fmax

fmax�2

+ ��d

d �2�1/2

. �13�

From the discretization procedure of f in the analyz-ing subroutine [9], the relative error in fmax is deter-mined as �fmax/ fmax�1/ jmax, where jmax is the indexof fmax, and the mean relative error is calculated as�fmax/ fmax=0.0357. Also, if the absolute error in thefilm thickness is assumed to be �d=1 m, the meanrelative error of the refractive index by the general-ized S-transform analysis is �n /n =0.0431,

GST GST

which is consistent with the mean relative error bythe fringe counting method ��nfrn/nfrn=0.0400�.

In this work, an algorithm based on the general-ized S-transform was applied to the analysis of thetransmittance spectrum of a homogenous and uni-form dielectric film having smooth plane parallel sur-faces. By neglecting the absorption over the studiedwavenumber range, a continuous dispersion curve ofthe refractive index was experimentally verified onmica, in comparison with the fringe counting,S-transform, and the previously published results.Since the fringe counting method uses successivepeak values of the transmittance signal, naturally itis not possible to obtain a continuous dispersioncurve; this insufficiency is overcome by theS-transform method. However, if the frequency of thetransmittance signal is high as in the mica sampleused in this work, the S-transform method gives un-dulating results. This problem is solved in the gener-alized S-transform method by adjusting the spectralresolution with the parameter p. The extracted re-fractive index of mica by the generalized S-transformis in agreement with the findings of El-Zaiat [13,14].

This work was supported by the Turkish Scientificand Technological Research Council (TUBITAK-TBAG no. 105T136).References

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