In recent years, because of the increasing need toknow the optical characteristics of materials to pro-duce and develop optic and opto-electronic devices,the precise and continuous determination of opticalconstants, suchas refractive indexnðkÞandextinctioncoefficient κðkÞ, of absorbing thin films over a widewavenumber range (k ¼ 1=λ) has gained primary im-portance. For this purpose most common methodsmeasure the transmitted intensities of unpolarizedlight through film and determine the optical con-stants by various analyses, such as the envelopemethod [1] and the fringe counting method [2–4].In these analyses, the optical constants are
obtained fromtheextremesof the transmittance spec-trum. In this respect, nðkÞ and κðkÞ can be determinedby the envelope method, whereas only nðkÞ can be ob-tained by the fringe counting method. On the other
hand, it is not possible to obtain optical constants con-tinuously in the wavenumber range, since thesemethods can analyze only at the wavenumbers corre-sponding to the extremes.Additionally, in thesemeth-ods, the noise in the transmittance signal could resultin an error which would effect the precise determina-tion of the optical constants.
Here we propose an alternative method, improvedfrom the continuous wavelet transform (CWT), forthe continuous determination of nðkÞ and κðkÞ of ab-sorbing thin films from the frequency dependence ofthe transmittance spectrum at the existence of thefringes in the transmittance.
2. Method
Consider a homogeneous and uniform absorbing filmof thickness d and complex refractive index n1 ¼n − iκ, where n is the refractive index and κ is the ex-tinction coefficient of the film, which is depositedonto a finite thickness transparent substrate, whoserefractive index is s. When the light beam is incidentnormally upon this system of ambient medium
D ¼ nd is the optical thickness of the film. Becauseof the wavenumber dependence of the refractive in-dex, the repetition frequency of the transmittancesignal [TðkÞ] varies within the scanned wavenumberinterval similar to nonstationary time-frequencyproblems, and hence the CWT, which was first intro-duced by Grossman and Morlet [5], could be em-ployed [6]. The CWT of the transmittance signal isexpressed as
CWTða; bÞ ¼ 1
a1=2
Z∞
−∞
ψ��k − ba
�TðkÞdk; ð7Þ
where each analyzing wavelet is derived from the di-lations by the scale factor a > 0 and the translationsby factor b of a mother wavelet, ψðkÞ [7]. Applying theconvolution theorem in Eq. (7) yields
CWTða; bÞ ¼ a1=2
Z∞
−∞
ψ̂�ðaxÞT̂ðxÞ expðibxÞdx; ð8Þ
where x is the Fourier space of the k space anda1=2 ψ̂ðaxÞ and T̂ðxÞ are the Fourier transform ofψðk=aÞ=a1=2 and TðkÞ, respectively [8]. Equation (8)enables the use of the fast Fourier transform(FFT) algorithm, and as a result, the calculationscould be much faster [9,10].The appropriate choice of mother wavelet is dic-
tated by the goals of the analysis [7]. In this work,Morlet wavelet was selected for the mother wavelet,because its resolution is better than any other wave-
lets [11]. The Morlet wavelet, consisting of a planewave modulated by a Gaussian, is given as
ψðkÞ ¼ 1
π1=4expðiz0kÞ exp
�−k2
2
�; ð9Þ
and its Fourier transform is given as
ψ̂ðxÞ ¼ ð2πÞ1=2π1=4
exp�−ðx − z0Þ2
2
�; ð10Þ
where z0 is the nondimensional frequency and as-sumed to be equal to 6 to satisfy the admissibilitycondition [12,13]. Based on the localization propertyof the CWT [14], nðkÞ) could be approximated bynðkÞ ≅ nðbÞ, and by this assumption, the Fouriertransform of TðkÞ is evaluated in the following form:
where C1, C2, and C3 are constants. SubstitutingEq. (11) into Eq. (8), and noting that ψ̂ðaxÞ ¼ 0 forx ≤ 0, the normalized modulus of the CWT can beevaluated by
jCWTða; bÞj ¼ C1ð2πÞ1=2π1=4
a1=2 exp�−½4πaDðbÞ − z0�2
2
�;
ð12Þ
which is a matrix in the dimensions of a × b. For afixed b, Eq. (12) peaks at
amax ¼z0 þ ðz20 þ 2Þ1=2
8π1
DðbÞ : ð13Þ
from which the corresponding refractive index is de-termined provided that the film thickness is known.Once nðkÞ is known, κðkÞ can be determined byEqs. (14a) and (14b) [1]:
κðkÞ ¼ −1
4πkd ln�EM − ½E2
M − ðn2 − 1Þ3ðn2 − s4Þ�1=2ðn − 1Þ3ðn − s2Þ
�;
ð14aÞ
EM ¼ 8n2sTmax
þ ðn2 − 1Þðn2 − s2Þ: ð14bÞ
where Tmax is the maximum envelope curve of thetransmittance spectrum.
An exemplary simulation was made to clearly de-scribe the CWT method and compare its results withnðkÞ and κðkÞ of the theoretically generated transmit-tance. The transmittance data [Fig. 1(a)], whoseequations are given by Eqs. (1)–(6), were generatedby the initial parameters given in Eq. (15), whichwere defined in the 0:9091–2:0000 μm−1 wavenumber
range. The initial refractive index was described as athree term Cauchy formula, and the extinction coef-ficient was described in the exponential form:
nðkÞ ¼ AþBk2 þCk4;
A ¼ 2:5000; B ¼ 0:0600μm2; C ¼ 4:0 × 10−6μm4;
κðkÞ ¼�103
4π k
�10ð1:5×k2Þ−8;
n0 ¼ 1; s ¼ 1:51; d ¼ 5μm: ð15Þ
The CWT was applied to the transmittance spec-trum in the 0:9102−1:6319 μm−1 wavenumber range,where the fringes of transmittance spectrum exist,and the resulting normalized modulus of the CWTis presented in Fig. 1(b). The dispersion curve ofnðkÞ of the film was calculated from Eq. (13) for eachb, and it is given in Fig. 1(c) as a solid curve. Addition-ally,nðkÞ obtainedby theenvelopeand fringe countingmethods are presented in Fig. 1(c) as circles and as-terisks, respectively. In Fig. 1(c), the deviationsaround both edges of the nðkÞ curve, which was deter-mined from the CWTmethod, originate from the FFT
algorithm, which assumes the data are cyclic; how-ever we are dealing with a finite-length series[7,8,12]. In this work, for all these methods, κðkÞwas determined by using Eqs. (14a) and (14b). InFig. 1(d), κðkÞ obtained from the CWT (dotted curve),envelope (circles) and fringe counting (asterisks)methods are presented. It is clearly observed fromFigs. 1(c) and 1(d) that the continuous determinationsof nðkÞ and κðkÞ by the CWTmethod are well fit to thetheoretically generated data, whereas by the envel-ope and fringe counting methods only the values cor-responding to the extremes of the transmittancespectrumcanbe obtained. TheCauchy equationpara-meters are calculated from nðkÞ, which is obtainedfrom the simulation, and denoted in Table 1, in whichthe results of the CWT method were determined inthe 1:0403–1:5094 μm−1 wavenumber interval. Thecalculated Cauchy parameters by the CWT methodgive very close results with respect to the theoreticalvalues, similar to the other methods.
3. Experimental Work
The test sample, hydrogenated amorphous siliconcarbide (a − Si1−xCx:H) film, was deposited on an or-dinary glass substrate using the plasma enhanced
Fig. 1. (a) Theoretically generated transmittance spectrum of an absorbing thin film with thickness d ¼ 5 μm. (b) Its normalized modulusof the CWT. (c) Refractive index of the film determined by the CWT method (dotted curve), envelope method (circles), fringe countingmethod (asterisks), and the presumed value (solid curve). (d) Extinction coefficient of the film determined by the CWT method (dottedcurve), envelope method (circles), fringe counting method (asterisks), and the presumed value (solid curve).
chemical vapor deposition (PECVD) technique at13:56MHz. The cleaning procedure of the substratewas started by boiling it in detergent for 5 min. Aftereach step, it was rinsed in deionized water and agi-tated by ultrasound at the same time. Next, it wasboiled in trichloroethylene for 5 min and dipped inH2O2 for 5 min. Then, it was dipped in diluted HFand dried by a N2 jet. Finally, the substrate was im-mediately loaded onto the grounded electrode of theparallel plate PECVD reactor, and the depositionchamber was pumped down below 1mTorr. The tem-perature of the grounded electrode was adjusted to25 °C and the substrate was cleaned in situ by H2plasma, using 100mW=cm2 radio frequency (RF)power under the flow of 100 ccmH2 at a pressureof 0:3Torr for 5 min. After the cleaning, the deposi-tion gases 16 ccm silane (SiH4) and 4 ccm ethylene(C2H4), with a total source gas flow rate of 20 ccmand 200 ccm hydrogen (corresponding to H dilutionratio of 91%), are introduced into the chamber underconstant pressure of 0:5Torr and at 30mW=cm2 RFpower. The relative C2H4 concentration (M ¼ ½C2H4=ðSiH4 þ C2H4Þ�) was 0.2.The experimental approach aims to verify the
CWT method and its comparison with the envelopeand fringe counting methods. The experimental set-up consists of an Oriel-MS260i spectrograph, havinga 150 lines=mm grating with a spectral resolution of1:95nm. A 10–250W quartz tungsten halogen lampwas used as an input light source (Fig. 2). The trans-mittance spectrum [Fig. 3(a)] of the film was ac-quired in the 0:9092–1:8765 μm−1 interval at roomtemperature, and it was analyzed by applying theCWT, envelope, and fringe counting methods.
4. Results and Discussion
The CWT method was applied in the 0:9100–1:5411 μm−1 wavenumber interval of the measuredtransmittance spectrum of the a − Si1−xCx:H film,and the obtained normalized CWTmodulus is shown
in Fig. 3(b). The film thickness (d) was 2:462 μm,which was measured by a thickness profiler (XP-2Ambios). The dispersion of nðkÞwas determined fromthe normalized CWT modulus and is depicted inFig. 3(c) as a solid curve (the obtained Cauchy para-meters by using the least squares fitting to the Cau-chy formula are A ¼ 2:2977, B ¼ 0:5589 μm2,C ¼ −2:372 × 10−5 μm4). nðkÞ were also determinedfrom the envelope (the Cauchy parameters areA ¼ 2:8076, B ¼ 0:1525 μm2, C ¼ 4:184 × 10−6 μm4
and fringe counting (the Cauchy parameters areA ¼ 2:0150, B ¼ 0:7606 μm2, C ¼ −8:18 × 10−5 μm4)methods and are shown in Fig. 3(c) as circles and as-terisks, respectively. In Fig. 3(c), nðkÞ values obtainedby all these methods nearly cover each other aroundthe 1:136–1:350 μm−1 wavenumber interval, wherethe deviations at the edges of the nðkÞ curve are ex-cluded for the CWT method. From nðkÞ, κðkÞ valueswere determined for these three methods and areshown in Fig. 3(d) in which the CWT method resultis a solid curve, the envelope method result is circles,and the fringe counting method result is asterisks. Itis observed that κðkÞ takes the values between 10−3
and 10−2, which shows that, in this wavenumber in-terval, the absorption of the film could be consideredas a weak and medium absorption regime, respec-tively, and as a result, the envelope method analyseswere performed by applying the equations derivedfor this regime (Appendix A).
To the best of our knowledge, here we applied theCWT to the analysis of the transmittance spectrumin absorbing thin film for the first time, as far as thescanned literature is concerned. The determined nðkÞand κðkÞ values by use of the CWT method were con-tinuous, which is an important advantage of theCWT method for optical characterization of the ab-sorbing thin films with respect to the other methods.
Evaluation errors are inevitable for any methodsfor the determination of optical constants, and hence,the errors of the CWTmethod should also be studied.Using Eq. (13), the relative error ðΔn=nÞ of nðkÞ forthe CWT method is evaluated as Eq. (16):
ΔnCWT
nCWT¼
��Δamax
amax
�2þ�Δd
d
�2�1=2
: ð16Þ
The scale is discretized as follows [12]:
aj¼πx2
2jdj; dj¼ log2ðx2=x1ÞN
; j¼0;1;…;N; ð17Þ
where the lower and upper boundaries of the Fouriertransform of the transmittance signal are chosen asx1 and x2 (i.e., where T̂ðxÞ → 0); and N is the totalnumber of the scale. The relative error of the amaxis given as
Δamax
amax¼
�x2x1
�1=N
− 1: ð18Þ
Table 1. Cauchy Coefficient Values of the Simulation Determined fromthe CWT, Envelope, and Fringe Counting Methods and the Theoretical
Values
Cauchy Equation :nðkÞ ¼ Aþ Bk2
þCk4A (Coefficientof Refraction)
B ðμm2Þ(Coefficient ofDispersion) C ðμm4Þ
Theoretical 2.5000 0.0600 4:00 × 10−6
CWT 2.4996 0.0604 −4:98 × 10−7
Envelope 2.5052 0.0601 −1:08 × 10−6
Fringe Counting 2.4742 0.0789 −1:30 × 10−6
Fig. 2. Experimental setup for the measurement of the transmit-tance of an absorbing thin film.
For this work, N ¼ 1000, x2 ¼ 13:38 μm, and x1 ¼4:83 μm values were determined from T̂ðxÞ and therelative error of the scale was calculated asΔamax=amax ¼ 0:0010. The main factor affecting therelative error of amax was the N value, which couldbe 500–1000; but even for any consideration of N,the order of the relative error of amax was about10−3. The thickness deviation was determined asΔd ¼ 45nm. So, the relative error of the thicknesswas obtained as Δd=d ¼ 0:0183. The relative errorof n was calculated as ΔnCWT=nCWT ¼ 0:0193 byusing Eq. (16), which means that the relative errorof n for the CWTmethod depends mostly on the errorin the thickness measurement. The error calcula-tions of the envelope and fringe counting methodswere also studied, and the mathematical equationsare presented in Appendixes A xB. For the envelopemethod, under the assumption that ΔT=T ¼ΔTmax=Tmax ¼ ΔTmin=ΔTmin ¼ ΔTs=Ts are equal,Δnenv=nenv depends only on ΔT=T, because onlythe transmittance, among the measured parameters,affects nðkÞ calculations (Appendix A). The measure-ment error of the transmittance was calculated aboutΔT=T ¼ 0:0090, resulting in a relative error by theenvelope method as Δnenv=nenv ¼ 0:0167. On the
other hand, for the fringe counting method, errorsin d and k parameters affectΔnfrn=nfrn (Appendix B);the calculated relative error in the wavenumber wasΔk=ðk2 − k1Þ ¼ 0:0067, so that the relative error inthe refractive index wasΔnfrn=nfrn ¼ 0:0240. Similarto the CWTmethod, the error in the film thickness isthe dominant one in the fringe counting evaluation.The result of the relative error of the refractive indexfor the CWT method was of the same order with therelative error results of both the envelope and fringecounting methods. Additionally, the variation of therelative error of the refractive index with the relativeerror of the measured parameters was investigated.Δn=n as a function of Δd=d are depicted in Fig. 4 forthe CWT and the fringe counting methods, since themajor source of error in n is the error in d for thesetwo methods. For the envelope method, Δn=n is alsoplotted in Fig. 4, as a function of ΔT=T atk ¼ 1:1697 μm−1 value. Moreover, it should bepointed out that the error rate of the CWT methodis similar to the other two methods.
Another advantage of the CWT method is its noiseimmunity, which could be demonstrated in the simu-lation work by adding a random noise to the trans-mittance data given in Fig. 1(a). The resultant
Fig. 3. (a) Measured transmittance spectrum of the a − Si1−xCx:H absorbing thin film. (b) Its normalized modulus. (c) Refractive index ofthe film determined by the CWT (dotted curve), envelopemethod (circles), and fringe countingmethod (asterisks). (d) Extinction coefficientof the film determined by the CWT method (dotted curve), envelope method (circles), and fringe counting method (asterisks).
transmittance data with a random noise, the maxi-mum magnitude of which was modified to be10% of the transmittance signal, is presented in
Fig. 5(a). The noisy signal was analyzed by theCWT, envelope, and fringe counting methods andthe determined nðkÞ is depicted in Fig. 5(b) togetherwith the presumed values. As presented in Fig. 5(b),nðkÞ of the CWT method almost perfectly matchedwith the presumed values, which shows that theCWT method successfully filters the noise from thedata as a bandpass filter, since the CWTmethod ana-lyzes by using the frequency information of the trans-mittance signal. On the other hand, nðkÞ of theenvelope and fringe counting methods deviate toomuch from the presumed values, respectively, dueto the fact that they analyze by using the magnitudesof the extreme points of the transmittance signal. Ad-ditionally, the random noise of 20% and 30% was alsoadded to the presumed data to investigate the sus-tainability of the CWT method; the results are pre-sented in Fig. 5(c). It is observed that, for thevalues of noise level over 10%, the obtained resultsdeviate increasingly from the presumed values. How-ever, even the CWT analysis of a 30% noisy signalgives better refractive index values than that ob-tained by the other two methods applied to a 10%noisy signal. As a result, it is demonstrated thatthe CWT method is also an appropriate method for
Fig. 4. Variations of the relative error of n with the relative errorof d for the CWTand fringe counting methods (x represents d); andwith the relative error of T for the envelope method (x representsT) at k ¼ 1:1697 μm−1 value.
Fig. 5. (a) Simulated noisy transmittance signal of an absorbing film. (b) Refractive index of the noisy signal determined by the CWTmethod (dotted curve), envelope method (circles), fringe counting method (asterisks), and presumed data (solid curve). (c) Refractive indexof the CWT method for the presumed transmittance data with random noises, maximum magnitude of which were modified to be 10%(dotted curve), 20% (dash–dot curve), 30% (dashed curve) of the transmittance signal, and the presumed data (solid curve).
noisy signals, whereas the envelope and fringecounting methods are not sufficiently applicable.
Appendix A: Error Calculation of the Envelope Method
The refractive index equation for the region of weakand medium absorption for the envelope method isgiven as [1]
nenv ¼ ½N þ ðN2 − s2Þ1=2�1=2; ðA1Þ
N ¼ 2 sTmax − Tmin
Tmax Tminþ s2 þ 1
2; ðA2Þ
where Tmin is the maximum envelope curve of thetransmittance spectrum. The relative error of n is gi-ven by
Δnenv
nenv¼
��∂n=∂Tmax
nenv
�2ΔT2
max þ�∂n=∂Tmin
nenv
�2ΔT2
mn
þ�∂n=∂snenv
�2Δs2
�1=2
: ðA3Þ
The equation of the substrate refractive index andthe change of the s with dTs are given by
s ¼ 1Ts
þ ðT−2s − 1Þ1=2; Δs ¼
��∂s∂Ts
�2ΔT2
s
�1=2
;
ðA4Þwhere Ts is the maximum value of transmittance.Δn=n is calculated as in Eq. (A5) by assuming thatΔT=T ¼ ΔTmax=Tmax ¼ ΔTmin=Tmin ¼ ΔTs=Ts:
Δnenv
nenv¼12
0BB@�B
�1þ A
ðA2−s2Þ1=2
�− 4s
ðA2−s2Þ1=2
�2
þð8sÞ2�1þ A
ðA2−s2Þ1=2
�2�
1T4
maxþ 1
T4min
�1CCA
1=2
ΔT;
ðA5Þ
A ¼ 2 sðTmax − TminÞTmax Tmin
þ s2
2þ 12; ðA6Þ
B ¼ 8ðTmax − TminÞTmax Tmin
þ 4 s: ðA7Þ
Appendix B: Error Calculation of the Fringe CountingMethod
The refractive index equation for the fringe countingmethod is given by [4]
nfrn ¼ Ncyc
ðk2 − k1Þd; ðB1Þ
where k2 and k1 are consecutive maximum or mini-mum wavenumber values of transmittance and Ncycis the total fringe number between the k2 − k1 inter-val. The Δn is calculated as
Δnfrn
nfrn¼
�2
� Δkðk2 � k1Þ
�2þ�Δd
d
�2�1=2
: ðB2Þ
This research was supported by the Turkish Scien-tific and Technological Research Council (TUBITAK-TBAG 105T136).
References1. R. Swanepoel, “Determination of the thickness and optical
constants of amorphous silicon,” J. Phys. E 16, 1214–1222(1983).
2. O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thicknessmeasurement of dielectric films by wavelength scanningmethod,” Opt. Commun. 205, 1–6 (2002).
3. K. H. Yang, “Measurements of empty cell gap for liquid-crystaldisplays using interferometric methods,” J. Appl. Phys. 64,4780–4781 (1988).
4. R. Chang, “Application of polarimetry and interferometry toliquid crystal-film research,” Mater. Res. Bull. 7, 267–278(1972).
5. A. Grossman and J. Morlet, “Decomposition of Hardy func-tions into square integrable wavelets of constant shape,”SIAM J. Math. Anal. 15, 723–736 (1984).
6. M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, andS. Rachafi, “Paul wavelet-based algorithm for optical phasedistribution evaluation,” Opt. Commun. 211, 47–51 (2002).
7. S. D. Meyers, B. G. Kelly, and J. J. O'Brien, “An introduction towavelet analysis in oceanography and meteorology: with ap-plication to the dispersion of Yanai waves,”Mon. Weather Rev.121, 2858–2866 (1993).
8. G. B. Arfken,Mathematical Methods for Physicists (Academic,1995).
9. L. Angrisani, P. Daponte, and M. D’Apuzzo, “A method for theautomatic detection and measurement of transients. Part I:the measurement method,” Measurement 25, 19–30 (1999).
10. P. S. Addison, “Wavelet transforms and the ECG: a review,”Physiol. Meas. 26, R155–R199 (2005).
11. D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, andB. Forget, “High latitude ionosphere turbulence studied bymeans of the wavelet transform,” J. Atmos. Terr. Phys. 54,1283–1293 (1992).
12. C. Torrence and G. P. Compo, “A practical guide to waveletanalysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).
13. M. Farge, “Wavelet transforms and their applications to tur-bulence,” Annu. Rev. Fluid Mech. 24, 395–457 (1992).
14. O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novelapproach for the determination of birefringence dispersionin nematic liquid crystals by using the continuous wavelettransform,” Meas. Sci. Technol. 14, 790–795 (2003).
