Abstract

We present the continuous wavelet transform (CWT) method for determining the dispersion curves of the refractive index and extinction coefficient of absorbing thin films by using the transmittance spectrum in the visible and near infrared regions at room temperature. The CWT method is performed on the transmittance spectrum of an aSi1xCx:H film, and the refractive index and extinction coefficient of the film are continuously determined and compared with the results of the envelope and fringe counting methods. Also the noise filter property of the method is depicted on a theoretically generated noisy signal. Finally, the error analyses of the CWT, envelope, and fringe counting methods are performed.

© 2008 Optical Society of America

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  1. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222(1983).
    [CrossRef]
  2. O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thickness measurement of dielectric films by wavelength scanning method,” Opt. Commun. 205, 1-6 (2002).
    [CrossRef]
  3. K. H. Yang, “Measurements of empty cell gap for liquid-crystal displays using interferometric methods,” J. Appl. Phys. 64, 4780-4781 (1988).
    [CrossRef]
  4. R. Chang, “Application of polarimetry and interferometry to liquid crystal-film research,” Mater. Res. Bull. 7, 267-278(1972).
    [CrossRef]
  5. A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
    [CrossRef]
  6. M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
    [CrossRef]
  7. S. D. Meyers, B. G. Kelly, and J. J. O'Brien, “An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves,” Mon. Weather Rev. 121, 2858-2866 (1993).
    [CrossRef]
  8. G. B. Arfken, Mathematical Methods for Physicists (Academic, 1995).
  9. L. Angrisani, P. Daponte, and M. D'Apuzzo, “A method for the automatic detection and measurement of transients. Part I: the measurement method,” Measurement 25, 19-30 (1999).
    [CrossRef]
  10. P. S. Addison, “Wavelet transforms and the ECG: a review,” Physiol. Meas. 26, R155-R199 (2005).
    [CrossRef] [PubMed]
  11. D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
    [CrossRef]
  12. C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61-78 (1998).
    [CrossRef]
  13. M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395-457 (1992).
    [CrossRef]
  14. O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol. 14, 790-795 (2003).
    [CrossRef]

2005 (1)

P. S. Addison, “Wavelet transforms and the ECG: a review,” Physiol. Meas. 26, R155-R199 (2005).
[CrossRef] [PubMed]

2003 (1)

O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol. 14, 790-795 (2003).
[CrossRef]

2002 (2)

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thickness measurement of dielectric films by wavelength scanning method,” Opt. Commun. 205, 1-6 (2002).
[CrossRef]

1999 (1)

L. Angrisani, P. Daponte, and M. D'Apuzzo, “A method for the automatic detection and measurement of transients. Part I: the measurement method,” Measurement 25, 19-30 (1999).
[CrossRef]

1998 (1)

C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61-78 (1998).
[CrossRef]

1993 (1)

S. D. Meyers, B. G. Kelly, and J. J. O'Brien, “An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves,” Mon. Weather Rev. 121, 2858-2866 (1993).
[CrossRef]

1992 (2)

D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
[CrossRef]

M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395-457 (1992).
[CrossRef]

1988 (1)

K. H. Yang, “Measurements of empty cell gap for liquid-crystal displays using interferometric methods,” J. Appl. Phys. 64, 4780-4781 (1988).
[CrossRef]

1984 (1)

A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

1983 (1)

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222(1983).
[CrossRef]

1972 (1)

R. Chang, “Application of polarimetry and interferometry to liquid crystal-film research,” Mater. Res. Bull. 7, 267-278(1972).
[CrossRef]

Addison, P. S.

P. S. Addison, “Wavelet transforms and the ECG: a review,” Physiol. Meas. 26, R155-R199 (2005).
[CrossRef] [PubMed]

Afifi, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Angrisani, L.

L. Angrisani, P. Daponte, and M. D'Apuzzo, “A method for the automatic detection and measurement of transients. Part I: the measurement method,” Measurement 25, 19-30 (1999).
[CrossRef]

Arfken, G. B.

G. B. Arfken, Mathematical Methods for Physicists (Academic, 1995).

Cerisier, J. C.

D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
[CrossRef]

Chang, R.

R. Chang, “Application of polarimetry and interferometry to liquid crystal-film research,” Mater. Res. Bull. 7, 267-278(1972).
[CrossRef]

Compo, G. P.

C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61-78 (1998).
[CrossRef]

Daponte, P.

L. Angrisani, P. Daponte, and M. D'Apuzzo, “A method for the automatic detection and measurement of transients. Part I: the measurement method,” Measurement 25, 19-30 (1999).
[CrossRef]

D'Apuzzo, M.

L. Angrisani, P. Daponte, and M. D'Apuzzo, “A method for the automatic detection and measurement of transients. Part I: the measurement method,” Measurement 25, 19-30 (1999).
[CrossRef]

Ecevit, F. N.

O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol. 14, 790-795 (2003).
[CrossRef]

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thickness measurement of dielectric films by wavelength scanning method,” Opt. Commun. 205, 1-6 (2002).
[CrossRef]

Farge, M.

M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395-457 (1992).
[CrossRef]

Fassi-Fihri, A.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Forget, B.

D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
[CrossRef]

Grossman, A.

A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

Kelly, B. G.

S. D. Meyers, B. G. Kelly, and J. J. O'Brien, “An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves,” Mon. Weather Rev. 121, 2858-2866 (1993).
[CrossRef]

Köysal, O.

O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol. 14, 790-795 (2003).
[CrossRef]

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thickness measurement of dielectric films by wavelength scanning method,” Opt. Commun. 205, 1-6 (2002).
[CrossRef]

Lagoutte, D.

D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
[CrossRef]

Marjane, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Meyers, S. D.

S. D. Meyers, B. G. Kelly, and J. J. O'Brien, “An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves,” Mon. Weather Rev. 121, 2858-2866 (1993).
[CrossRef]

Morlet, J.

A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

Nassim, K.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

O'Brien, J. J.

S. D. Meyers, B. G. Kelly, and J. J. O'Brien, “An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves,” Mon. Weather Rev. 121, 2858-2866 (1993).
[CrossRef]

Önal, D.

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thickness measurement of dielectric films by wavelength scanning method,” Opt. Commun. 205, 1-6 (2002).
[CrossRef]

Özder, S.

O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol. 14, 790-795 (2003).
[CrossRef]

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thickness measurement of dielectric films by wavelength scanning method,” Opt. Commun. 205, 1-6 (2002).
[CrossRef]

Plagnaud, J. L.

D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
[CrossRef]

Rachafi, S.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

San, S. E.

O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol. 14, 790-795 (2003).
[CrossRef]

Sidki, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Swanepoel, R.

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222(1983).
[CrossRef]

Torrence, C.

C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61-78 (1998).
[CrossRef]

Villain, J. P.

D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
[CrossRef]

Yang, K. H.

K. H. Yang, “Measurements of empty cell gap for liquid-crystal displays using interferometric methods,” J. Appl. Phys. 64, 4780-4781 (1988).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395-457 (1992).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61-78 (1998).
[CrossRef]

J. Appl. Phys. (1)

K. H. Yang, “Measurements of empty cell gap for liquid-crystal displays using interferometric methods,” J. Appl. Phys. 64, 4780-4781 (1988).
[CrossRef]

J. Atmos. Terr. Phys. (1)

D. Lagoutte, J. C. Cerisier, J. L. Plagnaud, J. P. Villain, and B. Forget, “High latitude ionosphere turbulence studied by means of the wavelet transform,” J. Atmos. Terr. Phys. 54, 1283-1293 (1992).
[CrossRef]

J. Phys. E (1)

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222(1983).
[CrossRef]

Mater. Res. Bull. (1)

R. Chang, “Application of polarimetry and interferometry to liquid crystal-film research,” Mater. Res. Bull. 7, 267-278(1972).
[CrossRef]

Meas. Sci. Technol. (1)

O. Köysal, S. E. San, S. Özder, and F. N. Ecevit, “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol. 14, 790-795 (2003).
[CrossRef]

Measurement (1)

L. Angrisani, P. Daponte, and M. D'Apuzzo, “A method for the automatic detection and measurement of transients. Part I: the measurement method,” Measurement 25, 19-30 (1999).
[CrossRef]

Mon. Weather Rev. (1)

S. D. Meyers, B. G. Kelly, and J. J. O'Brien, “An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves,” Mon. Weather Rev. 121, 2858-2866 (1993).
[CrossRef]

Opt. Commun. (2)

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, “Thickness measurement of dielectric films by wavelength scanning method,” Opt. Commun. 205, 1-6 (2002).
[CrossRef]

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Physiol. Meas. (1)

P. S. Addison, “Wavelet transforms and the ECG: a review,” Physiol. Meas. 26, R155-R199 (2005).
[CrossRef] [PubMed]

SIAM J. Math. Anal. (1)

A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

Other (1)

G. B. Arfken, Mathematical Methods for Physicists (Academic, 1995).

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Figures (5)

Fig. 1
Fig. 1

(a) Theoretically generated transmittance spectrum of an absorbing thin film with thickness d = 5 μm . (b) Its normalized modulus of the CWT. (c) Refractive index of the film determined by the CWT method (dotted curve), envelope method (circles), fringe counting method (asterisks), and the presumed value (solid curve). (d) Extinction coefficient of the film determined by the CWT method (dotted curve), envelope method (circles), fringe counting method (asterisks), and the presumed value (solid curve).

Fig. 2
Fig. 2

Experimental setup for the measurement of the transmittance of an absorbing thin film.

Fig. 3
Fig. 3

(a) Measured transmittance spectrum of the a Si 1 x C x :H absorbing thin film. (b) Its normalized modulus. (c) Refractive index of the film determined by the CWT (dotted curve), envelope method (circles), and fringe counting method (asterisks). (d) Extinction coefficient of the film determined by the CWT method (dotted curve), envelope method (circles), and fringe counting method (asterisks).

Fig. 4
Fig. 4

Variations of the relative error of n with the relative error of d for the CWT and fringe counting methods (x represents d); and with the relative error of T for the envelope method (x represents T) at k = 1.1697 μm 1 value.

Fig. 5
Fig. 5

(a) Simulated noisy transmittance signal of an absorbing film. (b) Refractive index of the noisy signal determined by the CWT method (dotted curve), envelope method (circles), fringe counting method (asterisks), and presumed data (solid curve). (c) Refractive index of 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).

Tables (1)

Tables Icon

Table 1 Cauchy Coefficient Values of the Simulation Determined from the CWT, Envelope, and Fringe Counting Methods and the Theoretical Values

Equations (28)

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T ( k ) = A x B C x + F x 2 ,
A = 16 ( n 2 + κ 2 ) s ,
B = [ ( n + 1 ) 2 + κ 2 ] ]. ( n + 1 ) ( n + s 2 ) + κ 2 ] ,
C = [ ( n 2 1 + κ 2 ) ( n 2 s 2 + κ 2 ) 2 κ 2 ( s 2 + 1 ) ] 2 cos θ κ [ 2 ( n 2 s 2 + κ 2 ) + ( s 2 + 1 ) ( n 2 1 + κ 2 ) ] 2 sin θ ,
F = [ ( n 1 ) 2 + κ 2 ] [ ( n 1 ) ( n s 2 ) + κ 2 ] ,
x = exp ( α d ) , α = 4 π k κ , θ = 4 π k D .
CWT ( a , b ) = 1 a 1 / 2 ψ * ( k b a ) T ( k ) d k ,
CWT ( a , b ) = a 1 / 2 ψ ^ * ( a x ) T ^ ( x ) exp ( i b x ) d x ,
ψ ( k ) = 1 π 1 / 4 exp ( i z 0 k ) exp ( k 2 2 ) ,
ψ ^ ( x ) = ( 2 π ) 1 / 2 π 1 / 4 exp [ ( x z 0 ) 2 2 ] ,
T ^ ( x ) = C 1 δ ( x 4 π D ( b ) ) + C 2 δ ( x ) + C 3 δ ( x + 4 π D ( b ) ) ,
| CWT ( a , b ) | = C 1 ( 2 π ) 1 / 2 π 1 / 4 a 1 / 2 exp [ [ 4 π a D ( b ) z 0 ] 2 2 ] ,
a max = z 0 + ( z 0 2 + 2 ) 1 / 2 8 π 1 D ( b ) .
κ ( k ) = 1 4 π k d ln { E M [ E M 2 ( n 2 1 ) 3 ( n 2 s 4 ) ] 1 / 2 ( n 1 ) 3 ( n s 2 ) } ,
E M = 8 n 2 s T max + ( n 2 1 ) ( n 2 s 2 ) .
n ( k ) = A + B k 2 + C k 4 , A = 2.5000 , B = 0.0600 μ m 2 , C = 4.0 × 10 6 μ m 4 , κ ( k ) = ( 10 3 4 π k ) 10 ( 1.5 × k 2 ) 8 , n 0 = 1 , s = 1.51 , d = 5 μ m .
Δ n CWT n CWT = [ ( Δ a max a max ) 2 + ( Δ d d ) 2 ] 1 / 2 .
a j = π x 2 2 j d j , d j = log 2 ( x 2 / x 1 ) N , j = 0 , 1 , , N ,
Δ a max a max = ( x 2 x 1 ) 1 / N 1.
n env = [ N + ( N 2 s 2 ) 1 / 2 ] 1 / 2 ,
N = 2 s T max T min T max T min + s 2 + 1 2 ,
Δ n env n env = [ ( n / T max n env ) 2 Δ T max 2 + ( n / T min n env ) 2 Δ T mn 2 + ( n / s n env ) 2 Δ s 2 ] 1 / 2 .
s = 1 T s + ( T s 2 1 ) 1 / 2 , Δ s = [ ( s T s ) 2 Δ T s 2 ] 1 / 2 ,
Δ n env n env = 1 2 ( { B [ 1 + A ( A 2 s 2 ) 1 / 2 ] 4 s ( A 2 s 2 ) 1 / 2 } 2 + ( 8 s ) 2 [ 1 + A ( A 2 s 2 ) 1 / 2 ] 2 ( 1 T max 4 + 1 T min 4 ) ) 1 / 2 Δ T ,
A = 2 s ( T max T min ) T max T min + s 2 2 + 1 2 ,
B = 8 ( T max T min ) T max T min + 4 s .
n frn = N cyc ( k 2 k 1 ) d ,
Δ n frn n frn = [ 2 ( Δ k ( k 2 - k 1 ) ) 2 + ( Δ d d ) 2 ] 1 / 2 .

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