Abstract

The Paul wavelet algorithm was prepared as a method to obtain the birefringence values continuously for a liquid crystal sample by using the transmittance spectrum in visible and near-infrared regions at room temperature. The obtained results determined from the Paul wavelet algorithm are harmonious with the 5CB-coded liquid crystal catalog value and the results determined from the fringe counting method. The controlling on the resolution of the working space is possible with the Paul wavelet, and the presented work attempts to depict the importance of control. Noise effect and absolute errors of the presented method and the fringe counting method are also studied and compared.

© 2011 Optical Society of America

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References

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  1. E. P. Raynes and C. M. Waters, “Supertwisted nematic liquid crystal displays,” Displays 8, 59–63 (1987).
    [CrossRef]
  2. C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
    [CrossRef]
  3. S. Chavali, P. M. Birch, R. Young, and C. Chatwin, “Synthesis and reconstruction of computer generated holograms by a double pass technique on a twisted nematic-based liquid crystal spatial light modulator,” Opt. Lasers Eng. 45, 413–418 (2007).
    [CrossRef]
  4. P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University, 1993).
  5. R. Chang, “Application of polarimetry and interferometry to liquid crystal-film research,” Mater. Res. Bull. 7, 267–278(1972).
    [CrossRef]
  6. S. A. Khodier, “Refractive index of standard oils as a function of wavelength and temperature,” Opt. Laser Technol. 34, 125–128 (2002).
    [CrossRef]
  7. C. Caliendo, E. Verona, and G. Saggio, “An integrated optical method for measuring the thickness and refractive index of birefringent thin films,” Thin Solid Films 292, 255–259 (1997).
    [CrossRef]
  8. H. Fujiwara, Spectroscopic Ellipsometry Principles and Applications (Wiley, 2007).
  9. 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]
  10. S. Özder, E. Coskun, O. Köysal, and O. Kocahan, “Determination of birefringence dispersion in nematic liquid crystals by using an S-transform,” Opt. Lett. 32, 2001–2003 (2007).
    [CrossRef] [PubMed]
  11. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  12. 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]
  13. A. K. Leung, F. Chau, and J. Gao, “A review on applications of wavelet transform techniques in chemical analysis: 1989-1997,” Chemom. Intell. Lab. Syst. 43, 165–184 (1998).
    [CrossRef]
  14. M. J. Fadili and E. T. Bullmore, “A comparative evaluation of wavelet-based methods for hypothesis testing of brain activation maps,” NeuroImage 23, 1112–1128 (2004).
    [CrossRef] [PubMed]
  15. B. Wang and Y. Wang, “Temporal structure of the southern oscillation as revealed by waveform and wavelet analysis,” J. Climate 9, 1586–1598 (1996).
    [CrossRef]
  16. N. Gamage and W. Blumen, “Comparative analysis of low-level cold fronts: wavelet, Fourier, and empirical orthogonal function decomposition,” Mon. Weather Rev. 121, 2867–2878 (1993).
    [CrossRef]
  17. M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395–457 (1992).
    [CrossRef]
  18. 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]
  19. G. B. Arfken, Mathematical Methods for Physicists (Academic, 1995).
  20. 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]
  21. 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]
  22. C. Torrence and G. P. Compo, “A practical auide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).
    [CrossRef]
  23. Merck KgaA, D-64271 Darmstadt, Germany (2002).
  24. 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]

2009 (1)

C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
[CrossRef]

2007 (2)

S. Chavali, P. M. Birch, R. Young, and C. Chatwin, “Synthesis and reconstruction of computer generated holograms by a double pass technique on a twisted nematic-based liquid crystal spatial light modulator,” Opt. Lasers Eng. 45, 413–418 (2007).
[CrossRef]

S. Özder, E. Coskun, O. Köysal, and O. Kocahan, “Determination of birefringence dispersion in nematic liquid crystals by using an S-transform,” Opt. Lett. 32, 2001–2003 (2007).
[CrossRef] [PubMed]

2004 (1)

M. J. Fadili and E. T. Bullmore, “A comparative evaluation of wavelet-based methods for hypothesis testing of brain activation maps,” NeuroImage 23, 1112–1128 (2004).
[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 (3)

S. A. Khodier, “Refractive index of standard oils as a function of wavelength and temperature,” Opt. Laser Technol. 34, 125–128 (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]

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 (2)

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

A. K. Leung, F. Chau, and J. Gao, “A review on applications of wavelet transform techniques in chemical analysis: 1989-1997,” Chemom. Intell. Lab. Syst. 43, 165–184 (1998).
[CrossRef]

1997 (1)

C. Caliendo, E. Verona, and G. Saggio, “An integrated optical method for measuring the thickness and refractive index of birefringent thin films,” Thin Solid Films 292, 255–259 (1997).
[CrossRef]

1996 (1)

B. Wang and Y. Wang, “Temporal structure of the southern oscillation as revealed by waveform and wavelet analysis,” J. Climate 9, 1586–1598 (1996).
[CrossRef]

1993 (2)

N. Gamage and W. Blumen, “Comparative analysis of low-level cold fronts: wavelet, Fourier, and empirical orthogonal function decomposition,” Mon. Weather Rev. 121, 2867–2878 (1993).
[CrossRef]

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 (1)

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

1987 (1)

E. P. Raynes and C. M. Waters, “Supertwisted nematic liquid crystal displays,” Displays 8, 59–63 (1987).
[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]

1972 (1)

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

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).

Birch, P. M.

S. Chavali, P. M. Birch, R. Young, and C. Chatwin, “Synthesis and reconstruction of computer generated holograms by a double pass technique on a twisted nematic-based liquid crystal spatial light modulator,” Opt. Lasers Eng. 45, 413–418 (2007).
[CrossRef]

Blumen, W.

N. Gamage and W. Blumen, “Comparative analysis of low-level cold fronts: wavelet, Fourier, and empirical orthogonal function decomposition,” Mon. Weather Rev. 121, 2867–2878 (1993).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bullmore, E. T.

M. J. Fadili and E. T. Bullmore, “A comparative evaluation of wavelet-based methods for hypothesis testing of brain activation maps,” NeuroImage 23, 1112–1128 (2004).
[CrossRef] [PubMed]

Caliendo, C.

C. Caliendo, E. Verona, and G. Saggio, “An integrated optical method for measuring the thickness and refractive index of birefringent thin films,” Thin Solid Films 292, 255–259 (1997).
[CrossRef]

Chang, R.

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

Chatwin, C.

S. Chavali, P. M. Birch, R. Young, and C. Chatwin, “Synthesis and reconstruction of computer generated holograms by a double pass technique on a twisted nematic-based liquid crystal spatial light modulator,” Opt. Lasers Eng. 45, 413–418 (2007).
[CrossRef]

Chau, F.

A. K. Leung, F. Chau, and J. Gao, “A review on applications of wavelet transform techniques in chemical analysis: 1989-1997,” Chemom. Intell. Lab. Syst. 43, 165–184 (1998).
[CrossRef]

Chavali, S.

S. Chavali, P. M. Birch, R. Young, and C. Chatwin, “Synthesis and reconstruction of computer generated holograms by a double pass technique on a twisted nematic-based liquid crystal spatial light modulator,” Opt. Lasers Eng. 45, 413–418 (2007).
[CrossRef]

Chen, Y.

C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
[CrossRef]

Compo, G. P.

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

Coskun, E.

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]

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]

De Gennes, P. G.

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University, 1993).

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]

Fadili, M. J.

M. J. Fadili and E. T. Bullmore, “A comparative evaluation of wavelet-based methods for hypothesis testing of brain activation maps,” NeuroImage 23, 1112–1128 (2004).
[CrossRef] [PubMed]

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]

Fujiwara, H.

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

Gamage, N.

N. Gamage and W. Blumen, “Comparative analysis of low-level cold fronts: wavelet, Fourier, and empirical orthogonal function decomposition,” Mon. Weather Rev. 121, 2867–2878 (1993).
[CrossRef]

Gao, J.

A. K. Leung, F. Chau, and J. Gao, “A review on applications of wavelet transform techniques in chemical analysis: 1989-1997,” Chemom. Intell. Lab. Syst. 43, 165–184 (1998).
[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]

Jiang, I.

C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
[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]

Khodier, S. A.

S. A. Khodier, “Refractive index of standard oils as a function of wavelength and temperature,” Opt. Laser Technol. 34, 125–128 (2002).
[CrossRef]

Kocahan, O.

Köysal, O.

S. Özder, E. Coskun, O. Köysal, and O. Kocahan, “Determination of birefringence dispersion in nematic liquid crystals by using an S-transform,” Opt. Lett. 32, 2001–2003 (2007).
[CrossRef] [PubMed]

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]

Leung, A. K.

A. K. Leung, F. Chau, and J. Gao, “A review on applications of wavelet transform techniques in chemical analysis: 1989-1997,” Chemom. Intell. Lab. Syst. 43, 165–184 (1998).
[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.

S. Özder, E. Coskun, O. Köysal, and O. Kocahan, “Determination of birefringence dispersion in nematic liquid crystals by using an S-transform,” Opt. Lett. 32, 2001–2003 (2007).
[CrossRef] [PubMed]

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]

Prost, J.

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University, 1993).

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]

Raynes, E. P.

E. P. Raynes and C. M. Waters, “Supertwisted nematic liquid crystal displays,” Displays 8, 59–63 (1987).
[CrossRef]

Saggio, G.

C. Caliendo, E. Verona, and G. Saggio, “An integrated optical method for measuring the thickness and refractive index of birefringent thin films,” Thin Solid Films 292, 255–259 (1997).
[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]

Shih, C.

C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
[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]

Torrence, C.

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

Tsai, C.

C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
[CrossRef]

Verona, E.

C. Caliendo, E. Verona, and G. Saggio, “An integrated optical method for measuring the thickness and refractive index of birefringent thin films,” Thin Solid Films 292, 255–259 (1997).
[CrossRef]

Wang, B.

B. Wang and Y. Wang, “Temporal structure of the southern oscillation as revealed by waveform and wavelet analysis,” J. Climate 9, 1586–1598 (1996).
[CrossRef]

Wang, Y.

B. Wang and Y. Wang, “Temporal structure of the southern oscillation as revealed by waveform and wavelet analysis,” J. Climate 9, 1586–1598 (1996).
[CrossRef]

Waters, C. M.

E. P. Raynes and C. M. Waters, “Supertwisted nematic liquid crystal displays,” Displays 8, 59–63 (1987).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Wu, S.

C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
[CrossRef]

Young, R.

S. Chavali, P. M. Birch, R. Young, and C. Chatwin, “Synthesis and reconstruction of computer generated holograms by a double pass technique on a twisted nematic-based liquid crystal spatial light modulator,” Opt. Lasers Eng. 45, 413–418 (2007).
[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 auide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).
[CrossRef]

Chemom. Intell. Lab. Syst. (1)

A. K. Leung, F. Chau, and J. Gao, “A review on applications of wavelet transform techniques in chemical analysis: 1989-1997,” Chemom. Intell. Lab. Syst. 43, 165–184 (1998).
[CrossRef]

Displays (1)

E. P. Raynes and C. M. Waters, “Supertwisted nematic liquid crystal displays,” Displays 8, 59–63 (1987).
[CrossRef]

J. Climate (1)

B. Wang and Y. Wang, “Temporal structure of the southern oscillation as revealed by waveform and wavelet analysis,” J. Climate 9, 1586–1598 (1996).
[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. (2)

N. Gamage and W. Blumen, “Comparative analysis of low-level cold fronts: wavelet, Fourier, and empirical orthogonal function decomposition,” Mon. Weather Rev. 121, 2867–2878 (1993).
[CrossRef]

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]

NeuroImage (1)

M. J. Fadili and E. T. Bullmore, “A comparative evaluation of wavelet-based methods for hypothesis testing of brain activation maps,” NeuroImage 23, 1112–1128 (2004).
[CrossRef] [PubMed]

Opt. Commun. (3)

C. Shih, Y. Chen, S. Wu, C. Tsai, and I. Jiang, “Voltage-controlled optical switch in planar nematic liquid crystal film,” Opt. Commun. 282, 3977–3981 (2009).
[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]

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]

Opt. Laser Technol. (1)

S. A. Khodier, “Refractive index of standard oils as a function of wavelength and temperature,” Opt. Laser Technol. 34, 125–128 (2002).
[CrossRef]

Opt. Lasers Eng. (1)

S. Chavali, P. M. Birch, R. Young, and C. Chatwin, “Synthesis and reconstruction of computer generated holograms by a double pass technique on a twisted nematic-based liquid crystal spatial light modulator,” Opt. Lasers Eng. 45, 413–418 (2007).
[CrossRef]

Opt. Lett. (1)

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]

Thin Solid Films (1)

C. Caliendo, E. Verona, and G. Saggio, “An integrated optical method for measuring the thickness and refractive index of birefringent thin films,” Thin Solid Films 292, 255–259 (1997).
[CrossRef]

Other (5)

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

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University, 1993).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

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

Merck KgaA, D-64271 Darmstadt, Germany (2002).

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Figures (4)

Fig. 1
Fig. 1

Optical configuration for LC birefringence measurement.

Fig. 2
Fig. 2

(a) Simulated transmittance of an LC. (b) Its normalized modulus of the Paul wavelet. (c) Birefringence values determined by Paul wavelet analysis (dotted curve), determined by fringe counting method (asterisks), and presumed values (solid curve).

Fig. 3
Fig. 3

(a) Measured transmittance spectrum of the 5CB-coded nematic LC. (b) Normalized modulus of the CWT of 5CB-coded nematic LC sample. (c) Birefringence values determined by Paul wavelet analysis with m = 64 (dotted curve) and m = 1 (dashed curve), determined by fringe counting method (asterisks).

Fig. 4
Fig. 4

(a) Simulated 10% noisy transmittance signal of an LC. (b) Birefringence values determined by Paul wavelet analysis (dotted curve), fringe counting analysis (analysis, asterisks; fitted values, dashed curve), and presumed value (solid curve).

Equations (18)

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T ( k 0 ) = cos 2 χ sin 2 φ sin 2 ( φ χ ) sin 2 h 2 ,
T ( k 0 ) = cos 2 h 2 .
CWT ( a , b ) = 1 ( a ) 1 / 2 ψ * ( k 0 b a ) T ( k 0 ) d k 0 ,
CWT ( a , b ) = ( a ) 1 / 2 ψ ^ * ( a x 0 ) T ^ ( x 0 ) exp ( i 2 π b x 0 ) d x 0 ,
ψ a , b ( k 0 ) = 1 ( a ) 1 / 2 2 m m ! [ 1 i ( k 0 b a ) ] ( m + 1 ) 2 π [ ( 2 m ) ! / 2 ] 1 / 2 ,
ψ ^ ( a x 0 ) = 2 m [ m ( 2 m 1 ) ! ] 1 / 2 ( a x 0 ) m exp ( a x 0 ) ξ ( a x 0 ) ,
T ^ ( x 0 ) = π 2 δ ( x 0 2 π d Δ n ) + π δ ( x 0 ) + π 2 δ ( x 0 + 2 π d Δ n ) .
CWT ( a , b ) = ( a ) 1 / 2 ψ ^ * ( a x 0 ) T ^ ( x 0 ) exp ( i b x 0 ) d x 0 .
CWT ( a , b ) = ( a ) 1 / 2 π 2 δ ( x 0 2 π d Δ n ) 2 m [ m ( 2 m 1 ) ! ] 1 / 2 ( a x 0 ) m exp ( a x 0 ) exp ( i 2 π b x 0 ) d x 0 .
| CWT ( a , b ) | = ( a ) 1 / 2 4 2 m [ m ( 2 m 1 ) ! ] 1 / 2 a m ( 2 π d Δ n ) m exp ( a 2 π d Δ n ) exp [ i 2 π b ( 2 π d Δ n ) ] ,
a max = 2 m + 1 4 π d Δ n ,
A = 0.2000 , B = 2.0000 × 10 1 μm 2 , C = 4.0000 × 10 4 μm 4 , Δ n = A + B k 0 2 + C k 0 4 , d = 30 μm .
T ( k 0 ) = f ( k 0 ) cos 2 h 2 .
T ^ ( x 0 ) = f ( b ) T ^ ( x 0 ) ,
| CWT ( a , b ) | = f ( b ) | CWT ( a , b ) | ,
Δ ( Δ n PW ) Δ n PW = [ ( Δ a max a max ) 2 + ( Δ d d ) 2 ] 1 / 2 .
a j = π x 02 2 j d j , d j = log 2 ( x 02 / x 01 ) N , j = 0 , 1 , , N ,
Δ a max a max = ( x 02 x 01 ) 1 / N 1 .

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