Abstract

Two-dimensional refractive index profiles of ion exchanged channel waveguides in glass have been obtained from the analysis of interferometer data. To obtain depth data, a shallow bevel is produced in the glass by polishing. The refractive index profile information that is contained within the derivative of the phase data is extracted directly using a continuous wavelet transform algorithm. The algorithm used to characterize and smooth the wavelet ridge is discussed in detail.

© 2010 Optical Society of America

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References

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  1. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
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  2. L. R. Wilkins, S. M. Tan, and T. H. Barnes, “Determination of interferometry phase distributions by use of wavelets,” Opt. Lett. 24905–907 (1999).
    [CrossRef]
  3. L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303(2007).
    [CrossRef]
  4. A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle interferometry fringes,” Opt. Eng. 402598–2604 (2001).
    [CrossRef]
  5. A. Federico and G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
    [CrossRef]
  6. A. Dursun, S. Ozder, and F. N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768–1772 (2004).
    [CrossRef]
  7. H. Liu, A. N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
    [CrossRef] [PubMed]
  8. C. Quan, C. J. Tat, and L. Chen, “Fringe-density estimation by continuous wavelet transform,” Appl. Opt. 44, 2359–2365(2005).
    [CrossRef] [PubMed]
  9. M. A. Gdeisat, A. Abid, D. R. Burton, and M. L. Lalor, “Spatial carrier pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
    [CrossRef] [PubMed]
  10. A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007).
    [CrossRef] [PubMed]
  11. C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
    [CrossRef]
  12. C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
    [CrossRef]
  13. C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
    [CrossRef]
  14. M. A. Gdeisat, A. Abid, D. R. Burton, M. L. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
    [CrossRef]
  15. A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133–138 (2009).
    [CrossRef]
  16. R. Oven, “Measurement of two-dimensional refractive index profiles of channel waveguides using an interferometric technique,” Appl. Opt. 48, 5704–5712 (2009).
    [CrossRef] [PubMed]
  17. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  18. Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
    [CrossRef]
  19. A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution,” Appl. Opt. 42, 7066–7071 (2003).
    [CrossRef] [PubMed]
  20. H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principles and applications,” Appl. Opt. 46, 1057–1065 (2007).
    [CrossRef] [PubMed]
  21. C. J. Tay and Y. Fu, “Determination of curvature and twist by digital shearography and wavelet transform,” Opt. Lett. 30, 2873–2875 (2005).
    [CrossRef] [PubMed]
  22. Y. Fu, C. J. Tay, C. Quan, and H. Miao, “Wavelet analysis of speckle patterns with a temporal carrier,” Appl. Opt. 44, 959–965 (2005).
    [CrossRef] [PubMed]
  23. Y. Fu, R. Groves, G. Pedrini, and W. Osten, “Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence,” Appl. Opt. 46, 8645–8655 (2007).
    [CrossRef] [PubMed]
  24. G. H. Kaufman, “Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier,” Opt. Commun. 217, 141–149 (2003).
    [CrossRef]
  25. X. Colonna de Lega, “Processing of non-stationary interference patterns: adapted phase shifting algorithms and wavelet analysis. Application to dynamic deformation measurement by holographic and speckle interferometry,” Ph.D. dissertation 1666 (Swiss Federal Institute of Technology, 1997).
  26. R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their continuous wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590(1997).
    [CrossRef]
  27. L. A. Pars, An Introduction to the Calculus of Variations(Heinemann, 1962).
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  29. R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted ion exchange in glass,” J. Phys. D 37, 2207–2215 (2004).
    [CrossRef]
  30. T. Poszner, G. Schreiter, and R. Muller, “Stripe waveguides with matched refractive index profiles fabricated by ion exchange in glass,” J. Appl. Phys. 70, 1966–1974 (1991).
    [CrossRef]
  31. K. S. Chiang, “Construction of refractive index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3385–391 (1985).
    [CrossRef]
  32. W. S. Dorn and D. D. McCracken, Numerical Methods with Fortran IV Case Studies (Wiley, 1972).
  33. G. D. Smith, Numerical Solution of Partial Differential Equations (Oxford, 1969).

2010 (1)

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

2009 (3)

M. A. Gdeisat, A. Abid, D. R. Burton, M. L. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133–138 (2009).
[CrossRef]

R. Oven, “Measurement of two-dimensional refractive index profiles of channel waveguides using an interferometric technique,” Appl. Opt. 48, 5704–5712 (2009).
[CrossRef] [PubMed]

2007 (7)

H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principles and applications,” Appl. Opt. 46, 1057–1065 (2007).
[CrossRef] [PubMed]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007).
[CrossRef] [PubMed]

Y. Fu, R. Groves, G. Pedrini, and W. Osten, “Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence,” Appl. Opt. 46, 8645–8655 (2007).
[CrossRef] [PubMed]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
[CrossRef]

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303(2007).
[CrossRef]

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

2006 (1)

2005 (3)

2004 (3)

H. Liu, A. N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
[CrossRef] [PubMed]

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted ion exchange in glass,” J. Phys. D 37, 2207–2215 (2004).
[CrossRef]

A. Dursun, S. Ozder, and F. N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768–1772 (2004).
[CrossRef]

2003 (3)

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

G. H. Kaufman, “Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier,” Opt. Commun. 217, 141–149 (2003).
[CrossRef]

A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution,” Appl. Opt. 42, 7066–7071 (2003).
[CrossRef] [PubMed]

2002 (1)

A. Federico and G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
[CrossRef]

2001 (1)

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle interferometry fringes,” Opt. Eng. 402598–2604 (2001).
[CrossRef]

1999 (1)

1997 (2)

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their continuous wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590(1997).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

1991 (1)

T. Poszner, G. Schreiter, and R. Muller, “Stripe waveguides with matched refractive index profiles fabricated by ion exchange in glass,” J. Appl. Phys. 70, 1966–1974 (1991).
[CrossRef]

1985 (1)

K. S. Chiang, “Construction of refractive index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3385–391 (1985).
[CrossRef]

Abid, A.

M. A. Gdeisat, A. Abid, D. R. Burton, M. L. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

M. A. Gdeisat, A. Abid, D. R. Burton, and M. L. Lalor, “Spatial carrier pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
[CrossRef] [PubMed]

Abid, A. Z.

Barnes, T. H.

L. R. Wilkins, S. M. Tan, and T. H. Barnes, “Determination of interferometry phase distributions by use of wavelets,” Opt. Lett. 24905–907 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

Basaran, C.

Burton, D. R.

Carmona, R. A.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their continuous wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590(1997).
[CrossRef]

Cartwright, A. N.

Chen, L.

Chen, M.

Chen, W.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Chiang, K. S.

K. S. Chiang, “Construction of refractive index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3385–391 (1985).
[CrossRef]

Colonna de Lega, X.

X. Colonna de Lega, “Processing of non-stationary interference patterns: adapted phase shifting algorithms and wavelet analysis. Application to dynamic deformation measurement by holographic and speckle interferometry,” Ph.D. dissertation 1666 (Swiss Federal Institute of Technology, 1997).

Darudi, A.

A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133–138 (2009).
[CrossRef]

Davies, P. A.

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted ion exchange in glass,” J. Phys. D 37, 2207–2215 (2004).
[CrossRef]

Dorn, W. S.

W. S. Dorn and D. D. McCracken, Numerical Methods with Fortran IV Case Studies (Wiley, 1972).

Dursun, A.

A. Dursun, S. Ozder, and F. N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768–1772 (2004).
[CrossRef]

Ecevit, F. N.

A. Dursun, S. Ozder, and F. N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768–1772 (2004).
[CrossRef]

Federico, A.

A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution,” Appl. Opt. 42, 7066–7071 (2003).
[CrossRef] [PubMed]

A. Federico and G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
[CrossRef]

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle interferometry fringes,” Opt. Eng. 402598–2604 (2001).
[CrossRef]

Fu, Y.

Gdeisat, M. A.

Groves, R.

Guo, H.

He, X. Y.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

Hosseini, S. M. R. S.

A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133–138 (2009).
[CrossRef]

Hwang, W. L.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their continuous wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590(1997).
[CrossRef]

Kaufman, G. H.

G. H. Kaufman, “Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier,” Opt. Commun. 217, 141–149 (2003).
[CrossRef]

Kaufmann, G. H.

A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution,” Appl. Opt. 42, 7066–7071 (2003).
[CrossRef] [PubMed]

A. Federico and G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
[CrossRef]

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle interferometry fringes,” Opt. Eng. 402598–2604 (2001).
[CrossRef]

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
[CrossRef]

Kim, T.

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Lalor, M. J.

Lalor, M. L.

M. A. Gdeisat, A. Abid, D. R. Burton, M. L. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

M. A. Gdeisat, A. Abid, D. R. Burton, and M. L. Lalor, “Spatial carrier pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
[CrossRef] [PubMed]

Lilley, F.

M. A. Gdeisat, A. Abid, D. R. Burton, M. L. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007).
[CrossRef] [PubMed]

Liu, H.

McCracken, D. D.

W. S. Dorn and D. D. McCracken, Numerical Methods with Fortran IV Case Studies (Wiley, 1972).

Miao, H.

Moore, C.

M. A. Gdeisat, A. Abid, D. R. Burton, M. L. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

Muller, R.

T. Poszner, G. Schreiter, and R. Muller, “Stripe waveguides with matched refractive index profiles fabricated by ion exchange in glass,” J. Appl. Phys. 70, 1966–1974 (1991).
[CrossRef]

Osten, W.

Oven, R.

R. Oven, “Measurement of two-dimensional refractive index profiles of channel waveguides using an interferometric technique,” Appl. Opt. 48, 5704–5712 (2009).
[CrossRef] [PubMed]

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted ion exchange in glass,” J. Phys. D 37, 2207–2215 (2004).
[CrossRef]

Ozder, S.

A. Dursun, S. Ozder, and F. N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768–1772 (2004).
[CrossRef]

Pars, L. A.

L. A. Pars, An Introduction to the Calculus of Variations(Heinemann, 1962).

Pedrini, G.

Poszner, T.

T. Poszner, G. Schreiter, and R. Muller, “Stripe waveguides with matched refractive index profiles fabricated by ion exchange in glass,” J. Appl. Phys. 70, 1966–1974 (1991).
[CrossRef]

Quan, C.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

Y. Fu, C. J. Tay, C. Quan, and H. Miao, “Wavelet analysis of speckle patterns with a temporal carrier,” Appl. Opt. 44, 959–965 (2005).
[CrossRef] [PubMed]

C. Quan, C. J. Tat, and L. Chen, “Fringe-density estimation by continuous wavelet transform,” Appl. Opt. 44, 2359–2365(2005).
[CrossRef] [PubMed]

Qudeisat, M.

M. A. Gdeisat, A. Abid, D. R. Burton, M. L. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

Schreiter, G.

T. Poszner, G. Schreiter, and R. Muller, “Stripe waveguides with matched refractive index profiles fabricated by ion exchange in glass,” J. Appl. Phys. 70, 1966–1974 (1991).
[CrossRef]

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Smith, G. D.

G. D. Smith, Numerical Solution of Partial Differential Equations (Oxford, 1969).

Sun, W.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

Tan, S. M.

L. R. Wilkins, S. M. Tan, and T. H. Barnes, “Determination of interferometry phase distributions by use of wavelets,” Opt. Lett. 24905–907 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

Tat, C. J.

Tay, C. J.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

Y. Fu, C. J. Tay, C. Quan, and H. Miao, “Wavelet analysis of speckle patterns with a temporal carrier,” Appl. Opt. 44, 959–965 (2005).
[CrossRef] [PubMed]

C. J. Tay and Y. Fu, “Determination of curvature and twist by digital shearography and wavelet transform,” Opt. Lett. 30, 2873–2875 (2005).
[CrossRef] [PubMed]

Torresani, B.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their continuous wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590(1997).
[CrossRef]

Watkins, L. R.

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303(2007).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

Wilkins, L. R.

Yang, Q.

Yin, M.

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted ion exchange in glass,” J. Phys. D 37, 2207–2215 (2004).
[CrossRef]

Appl. Opt. (9)

A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution,” Appl. Opt. 42, 7066–7071 (2003).
[CrossRef] [PubMed]

H. Liu, A. N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
[CrossRef] [PubMed]

Y. Fu, C. J. Tay, C. Quan, and H. Miao, “Wavelet analysis of speckle patterns with a temporal carrier,” Appl. Opt. 44, 959–965 (2005).
[CrossRef] [PubMed]

C. Quan, C. J. Tat, and L. Chen, “Fringe-density estimation by continuous wavelet transform,” Appl. Opt. 44, 2359–2365(2005).
[CrossRef] [PubMed]

M. A. Gdeisat, A. Abid, D. R. Burton, and M. L. Lalor, “Spatial carrier pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
[CrossRef] [PubMed]

H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principles and applications,” Appl. Opt. 46, 1057–1065 (2007).
[CrossRef] [PubMed]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007).
[CrossRef] [PubMed]

Y. Fu, R. Groves, G. Pedrini, and W. Osten, “Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence,” Appl. Opt. 46, 8645–8655 (2007).
[CrossRef] [PubMed]

R. Oven, “Measurement of two-dimensional refractive index profiles of channel waveguides using an interferometric technique,” Appl. Opt. 48, 5704–5712 (2009).
[CrossRef] [PubMed]

Electron. Lett. (1)

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

IEEE Trans. Signal Process. (1)

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their continuous wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590(1997).
[CrossRef]

J. Appl. Phys. (1)

T. Poszner, G. Schreiter, and R. Muller, “Stripe waveguides with matched refractive index profiles fabricated by ion exchange in glass,” J. Appl. Phys. 70, 1966–1974 (1991).
[CrossRef]

J. Lightwave Technol. (1)

K. S. Chiang, “Construction of refractive index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3385–391 (1985).
[CrossRef]

J. Phys. D (1)

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted ion exchange in glass,” J. Phys. D 37, 2207–2215 (2004).
[CrossRef]

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A. Dursun, S. Ozder, and F. N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768–1772 (2004).
[CrossRef]

Opt. Commun. (2)

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

G. H. Kaufman, “Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier,” Opt. Commun. 217, 141–149 (2003).
[CrossRef]

Opt. Eng. (3)

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle interferometry fringes,” Opt. Eng. 402598–2604 (2001).
[CrossRef]

A. Federico and G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
[CrossRef]

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C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Coordinate system used in work.

Fig. 2
Fig. 2

Schematic of interferometer and sample.

Fig. 3
Fig. 3

Typical interferometer image. Dashed line indicates location of bevel edge.

Fig. 4
Fig. 4

Ridge and ridge path (solid curve) and Gaussian fits (dashed curves) at representative values of b.

Fig. 5
Fig. 5

Mean of absolute error versus number of intervals for various noise levels. ( S.D. = standard deviation of Gaussian noise). □, k = 0.61 ; ○, k = 0.75 ; solid curves, S.D. = 0.3 ; dashed curves, S.D. = 0.1 ; dashed–dotted curves, noise S.D. = 0 .

Fig. 6
Fig. 6

(A) Model RI profile, (B) recovered RI profile, no ridge smoothing, (C) recovered index profile using Eq. (14), (D) recovered RI profile using Eq. (17). For (C) and (D), noise was added to the intensity data. (RI contours 0.01, 0.02, 0.03, 0.04, and 0.05). Dashed line in (A) indicates the region over which MAD was evaluated.

Fig. 7
Fig. 7

Measured RI profile of Cu + doped 7740 glass (RI contours 0.0025, 0.005, 0.0075, 0.01, 0.0125, and 0.015).

Fig. 8
Fig. 8

Measured RI profile of Ag + doped B270 glass (RI contours 0.01, 0.02, 0.03, 0.04, and 0.05).

Fig. 9
Fig. 9

Comparison between interferometer RI profile (solid curve) and RI profile from prism coupling measurements (dashed curve). ▪, effective RI of modes.

Fig. 10
Fig. 10

Convergence of alternating direction implicit method.

Equations (27)

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I ( x , y ) = A ( x , y ) + B ( x , y ) cos ( ω x + ϕ ( x , y ) ) ,
ϕ ( x , y ) = Δ ϕ ( x , y ) + ϕ 0 ( x , y ) = [ 4 π λ x tan θ W Δ n ( z , y ) d z + 4 π λ ( n s n o ) x tan θ ] + ϕ 0 ( x , y ) ,
ϕ ( x , y ) = 4 π λ 0 W Δ n ( z , y ) d z + ϕ 0 ( x , y ) .
Δ n ( z = x tan θ , y ) = λ 4 π tan θ d d x [ Δ ϕ ( x , y ) Δ ϕ ( x , Y ) ] ,
W ( s , b ) = 1 s + I ( x ) h * ( x b s ) d x ,
h ( x ) = 1 2 π σ 2 exp ( x 2 2 σ 2 ) exp ( j ω c x ) ,
W ( s , b ) = s B 2 1 + j ϕ σ 2 s 2 exp [ ( ( ω + ϕ ( b ) ) s ω c ) 2 σ 2 2 ( 1 + ϕ 2 σ 4 s 4 ) ] exp [ j ϕ σ 4 s 2 ( ( ω + ϕ ( b ) ) s ω c ) 2 2 ( 1 + ϕ 2 σ 4 s 4 ) ] exp [ j ( ϕ ( b ) + ω b ) ] .
W ( s , b ) = s B 2 exp [ ( ( ω + ϕ ( b ) ) s ω c ) 2 σ 2 2 ] exp [ j ( ϕ ( b ) + ω b ) ] .
ϕ ( b ) = tan 1 ( Im ( W ( s ^ , b ) ) Re ( W ( s ^ , b ) ) ) ω b ,
s ^ ( b ) = ω c ω + ϕ ( b ) ,
F = 0 b MAX | W ( s ^ , b ) | 2 d b + C 0 b MAX ϕ 2 d b ,
log e ( | W ( s , b ) | s ) = log e ( B 2 ) ( s ( ω + ϕ ( b ) ) ω c 2 ) 2 σ 2 = α ( s β c ) 2 ,
J ( s ^ ) = 0 y MAX 0 b MAX ( s ^ ( b , y ) β ( b , y ) c ) 2 + γ ( s ^ ( b , y ) b ) 2 + μ ( s ^ ( b , y ) y ) 2 d b d y ,
γ 2 s ^ b 2 + μ 2 s ^ y 2 s ^ β 2 ( b , y ) = c β ( b , y ) .
s ^ b | b = 0 = 0 , s ^ b | b = b MAX = 0.
s ^ y | y = 0 = 0 , s ^ y | y = y MAX = 0.
γ ( b ) 2 s ^ b 2 + γ ( b ) s ^ b + μ 2 s ^ y 2 s ^ β 2 ( b , y ) = c β ( b , y ) .
γ ( b ) = γ o γ o f exp ( ( b b s L ) 2 ) ,
Δ n ( z , y ) = Δ n MAX f ( z ) 1 2 [ erf ( y + W m / 2 d L ) erf ( y W m / 2 d L ) ] ,
f ( z ) = 1 1 + exp ( z d α ) ,
y i = log e ( | W ( s i , b ) | s i ) .
β 0 c s ^ 0 ,
F = i = 1 M ( y i α + ( s i β 0 c ) 2 + 2 Δ β ( s i β 0 c ) s i ) 2 .
Δ β = M [ i = 1 M y i ( s i β 0 c ) s i + ( s i β 0 c ) 3 s i ] [ i = 1 M y i + ( s i β 0 c ) 2 ] [ i = 1 M ( s i β 0 c ) s i ] 2 [ i = 1 M ( s i β 0 c ) s i ] 2 2 M i = 1 M ( s i β 0 c ) 2 s i 2 ,
α = [ i = 1 M y i + ( s i β 0 c ) 2 ] + 2 Δ β [ i = 1 M ( s i β 0 c ) s i ] M .
s ^ i , j n + 1 { β i j 2 + 2 γ i Δ b 2 + 2 μ Δ y 2 } + s ^ i + 1 , j n + 1 { γ i 2 Δ b γ i Δ b 2 } + s ^ i 1 , j n + 1 { + γ i 2 Δ b γ i Δ b 2 } = c β i j + μ Δ y 2 ( s ^ i , j + 1 n + s ^ i , j 1 n ) ,
s ^ i , j n + 2 { β i j 2 + 2 γ i Δ b 2 + 2 μ Δ y 2 } s ^ i , j + 1 n + 2 μ Δ y 2 s ^ i , j 1 n + 2 μ Δ y 2 = c β i j s ^ i + 1 , j n + 1 { γ i 2 Δ b γ i Δ b 2 } s ^ i 1 , j n + 1 { + γ i 2 Δ b γ i Δ b 2 } .

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