Abstract

The generalized S-transform was improved as a method to determine the refractive index of a dielectric film continuously by using the transmittance spectrum, and the applicability of the method was demonstrated on mica. The result determined from the generalized S-transform method was compared with the results determined from the S-transform and the fringe counting methods and published values. The advantage of the proposed method was explained, and the absolute error of the presented method was also calculated.

© 2010 Optical Society of America

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References

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  1. O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, Opt. Commun. 205, 1 (2002).
    [CrossRef]
  2. R. Swanepoel, J. Phys. E 16, 1214 (1983).
    [CrossRef]
  3. S. A. Khodier, Opt. Laser Technol. 34, 125 (2002).
    [CrossRef]
  4. C. Caliendo, E. Verona, and G. Saggio, Thin Solid Films 292, 255 (1997).
    [CrossRef]
  5. H. Fujiwara, Spectroscopic Ellipsometry Principles and Applications (Wiley, 2007).
  6. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999).
    [PubMed]
  7. R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process. 44, 998 (1996).
    [CrossRef]
  8. S. Özder, E. Coşkun, O. Köysal, and Ö. Kocahan, Opt. Lett. 32, 2001 (2007).
    [CrossRef] [PubMed]
  9. R. G. Stockwell, Ph.D. dissertation (University of Western Ontario, 1999); the S-transform subroutine is available at http://www.cora.nwra.com/~stockwel/.
  10. C. R. Pinnegar and L. Mansinha, Geophysics 68, 381 (2003).
    [CrossRef]
  11. A. Federico and G. H. Kaufmann, Opt. Lett. 33, 866 (2008).
    [CrossRef] [PubMed]
  12. G. B. Arfken, Mathematical Methods for Physicists (Academic, 1995).
  13. S. Y. El-Zaiat, Opt. Laser Technol. 37, 181 (2005).
    [CrossRef]
  14. S. Y. El-Zaiat, Opt. Laser Technol. 29, 995 (1997).

2008 (1)

2007 (1)

2005 (1)

S. Y. El-Zaiat, Opt. Laser Technol. 37, 181 (2005).
[CrossRef]

2003 (1)

C. R. Pinnegar and L. Mansinha, Geophysics 68, 381 (2003).
[CrossRef]

2002 (2)

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, Opt. Commun. 205, 1 (2002).
[CrossRef]

S. A. Khodier, Opt. Laser Technol. 34, 125 (2002).
[CrossRef]

1997 (2)

C. Caliendo, E. Verona, and G. Saggio, Thin Solid Films 292, 255 (1997).
[CrossRef]

S. Y. El-Zaiat, Opt. Laser Technol. 29, 995 (1997).

1996 (1)

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process. 44, 998 (1996).
[CrossRef]

1983 (1)

R. Swanepoel, J. Phys. E 16, 1214 (1983).
[CrossRef]

Arfken, G. B.

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

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999).
[PubMed]

Caliendo, C.

C. Caliendo, E. Verona, and G. Saggio, Thin Solid Films 292, 255 (1997).
[CrossRef]

Coskun, E.

Ecevit, F. N.

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, Opt. Commun. 205, 1 (2002).
[CrossRef]

El-Zaiat, S. Y.

S. Y. El-Zaiat, Opt. Laser Technol. 37, 181 (2005).
[CrossRef]

S. Y. El-Zaiat, Opt. Laser Technol. 29, 995 (1997).

Federico, A.

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry Principles and Applications (Wiley, 2007).

Kaufmann, G. H.

Khodier, S. A.

S. A. Khodier, Opt. Laser Technol. 34, 125 (2002).
[CrossRef]

Kocahan, Ö.

Köysal, O.

S. Özder, E. Coşkun, O. Köysal, and Ö. Kocahan, Opt. Lett. 32, 2001 (2007).
[CrossRef] [PubMed]

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, Opt. Commun. 205, 1 (2002).
[CrossRef]

Lowe, R. P.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process. 44, 998 (1996).
[CrossRef]

Mansinha, L.

C. R. Pinnegar and L. Mansinha, Geophysics 68, 381 (2003).
[CrossRef]

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process. 44, 998 (1996).
[CrossRef]

Önal, D.

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, Opt. Commun. 205, 1 (2002).
[CrossRef]

Özder, S.

S. Özder, E. Coşkun, O. Köysal, and Ö. Kocahan, Opt. Lett. 32, 2001 (2007).
[CrossRef] [PubMed]

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, Opt. Commun. 205, 1 (2002).
[CrossRef]

Pinnegar, C. R.

C. R. Pinnegar and L. Mansinha, Geophysics 68, 381 (2003).
[CrossRef]

Saggio, G.

C. Caliendo, E. Verona, and G. Saggio, Thin Solid Films 292, 255 (1997).
[CrossRef]

Stockwell, R. G.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process. 44, 998 (1996).
[CrossRef]

R. G. Stockwell, Ph.D. dissertation (University of Western Ontario, 1999); the S-transform subroutine is available at http://www.cora.nwra.com/~stockwel/.

Swanepoel, R.

R. Swanepoel, J. Phys. E 16, 1214 (1983).
[CrossRef]

Verona, E.

C. Caliendo, E. Verona, and G. Saggio, Thin Solid Films 292, 255 (1997).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999).
[PubMed]

Geophysics (1)

C. R. Pinnegar and L. Mansinha, Geophysics 68, 381 (2003).
[CrossRef]

IEEE Trans. Signal Process. (1)

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process. 44, 998 (1996).
[CrossRef]

J. Phys. E (1)

R. Swanepoel, J. Phys. E 16, 1214 (1983).
[CrossRef]

Opt. Commun. (1)

O. Köysal, D. Önal, S. Özder, and F. N. Ecevit, Opt. Commun. 205, 1 (2002).
[CrossRef]

Opt. Laser Technol. (3)

S. Y. El-Zaiat, Opt. Laser Technol. 37, 181 (2005).
[CrossRef]

S. Y. El-Zaiat, Opt. Laser Technol. 29, 995 (1997).

S. A. Khodier, Opt. Laser Technol. 34, 125 (2002).
[CrossRef]

Opt. Lett. (2)

Thin Solid Films (1)

C. Caliendo, E. Verona, and G. Saggio, Thin Solid Films 292, 255 (1997).
[CrossRef]

Other (4)

H. Fujiwara, Spectroscopic Ellipsometry Principles and Applications (Wiley, 2007).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999).
[PubMed]

R. G. Stockwell, Ph.D. dissertation (University of Western Ontario, 1999); the S-transform subroutine is available at http://www.cora.nwra.com/~stockwel/.

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

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Figures (1)

Fig. 1
Fig. 1

(a) Transmittance spectrum of the mica. (b) Its normalized modulus of the generalized S-transform | S GS p ( b , f ) | ( p = 8 ) . (c) Refractive index of the mica determined by the generalized S-transform ( p = 8 , solid curve), the S-transform method (dashed curve), fringe counting method (circles), and the refractive index result calculated by using the Cauchy parameters of [13] (dashed-dotted curve) and [14] (dotted curve).

Equations (13)

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T ( k 0 ) = [ 1 + ( n 2 n 0 2 2 n n 0 ) 2 sin 2 ( 2 π k 0 D ) ] 1 ,
n ( k 0 ) = A + B k 0 2 + C k 0 4 ,
S GS p ( b , f ) = + T ( k 0 ) w GS ( b k 0 , f , p ) exp ( i 2 π f k 0 ) d k 0 ,
w GS ( b k 0 , f , p ) = | f | p 2 π exp ( f 2 ( b k 0 ) 2 2 p 2 ) .
S GS p ( b , f ) = + T ̂ ( x 0 + f ) W GS ( x 0 , f , p ) exp ( i 2 π x 0 b ) d x 0 = IFT { T ̂ ( x 0 + f ) W GS ( x 0 , f , p ) } ,
k 0 C = k 0 | w GS ( b k 0 , f , p ) | 2 d k 0 | w GS ( b k 0 , f , p ) | 2 d k 0 = b ,
( Δ k 0 ) 2 = ( k 0 k 0 C ) 2 | w GS ( b k 0 , f , p ) | 2 d k 0 | w GS ( b k 0 , f , p ) | 2 d k 0 = p 2 2 f 2 .
T ̂ ( x 0 ) = C 1 δ ( x 0 2 D ( b ) ) + C 2 δ ( x 0 ) + C 3 δ ( x 0 + 2 D ( b ) ) ,
W GS ( x 0 , f , p ) = exp ( 2 π 2 x 0 2 p 2 f 2 ) .
S GS p ( b , f ) = C 1   exp ( ( 2 D f ) 2 p 2 2 f 2 ) exp ( i 2 π b ( 2 D f ) ) ,
| S GS p ( b , f ) | = C 1   exp ( ( 2 D f ) 2 p 2 2 f 2 ) .
f max ( b ) = 2 D ( b ) ,
Δ n n = [ ( Δ f max f max ) 2 + ( Δ d d ) 2 ] 1 / 2 .

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