Abstract

We evaluate the extension of the exact nonlinear reconstruction technique developed for digital holography to the phase-recovery problems presented by other optical interferometric methods, which use carrier modulation. It is shown that the introduction of an analytic wavelet analysis in the ridge of the cepstrum transformation corresponding to the analyzed interferogram can be closely related to the well-known wavelet analysis of the interferometric intensity. Subsequently, the phase-recovery process is improved. The advantages and limitations of this framework are analyzed and discussed using numerical simulations in singular scalar light fields and in temporal speckle pattern interferometry.

© 2012 Optical Society of America

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  1. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, J. Opt. Soc. Am. A 28, 983 (2011).
    [CrossRef]
  2. Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, 2001).
  3. A. Federico and G. H. Kaufmann, Advances in Speckle Metrology and Related Techniques, G. H. Kaufmann, ed. (Wiley-VCH, 2011), Chap. 4.
  4. J. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge U. Press, 2004).
  5. C. Gonnet and B. Torresani, Signal Process. 37, 389 (1994).
    [CrossRef]
  6. K. Patorski and K. Pokorski, Appl. Opt. 50, 773 (2011).
    [CrossRef]
  7. A. Federico and G. H. Kaufmann, Appl. Opt. 47, 5201 (2008).
    [CrossRef]
  8. M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Phys. Rev. A 56, 4064 (1997).
    [CrossRef]
  9. F. A. Marengo Rodriguez, A. Federico, and G. H. Kaufmann, Appl. Opt. 47, 1310 (2008).
    [CrossRef]

2011

2008

1997

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

1994

C. Gonnet and B. Torresani, Signal Process. 37, 389 (1994).
[CrossRef]

Ali, S. T.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge U. Press, 2004).

Antoine, J.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge U. Press, 2004).

Depeursinge, C.

Federico, A.

A. Federico and G. H. Kaufmann, Appl. Opt. 47, 5201 (2008).
[CrossRef]

F. A. Marengo Rodriguez, A. Federico, and G. H. Kaufmann, Appl. Opt. 47, 1310 (2008).
[CrossRef]

A. Federico and G. H. Kaufmann, Advances in Speckle Metrology and Related Techniques, G. H. Kaufmann, ed. (Wiley-VCH, 2011), Chap. 4.

Gonnet, C.

C. Gonnet and B. Torresani, Signal Process. 37, 389 (1994).
[CrossRef]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Kaufmann, G. H.

A. Federico and G. H. Kaufmann, Appl. Opt. 47, 5201 (2008).
[CrossRef]

F. A. Marengo Rodriguez, A. Federico, and G. H. Kaufmann, Appl. Opt. 47, 1310 (2008).
[CrossRef]

A. Federico and G. H. Kaufmann, Advances in Speckle Metrology and Related Techniques, G. H. Kaufmann, ed. (Wiley-VCH, 2011), Chap. 4.

Murenzi, R.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge U. Press, 2004).

Patorski, K.

Pavillon, N.

Pokorski, K.

Rodriguez, F. A. Marengo

Seelamantula, C. S.

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Torresani, B.

C. Gonnet and B. Torresani, Signal Process. 37, 389 (1994).
[CrossRef]

Unser, M.

Vandergheynst, P.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge U. Press, 2004).

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Phys. Rev. A

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Signal Process.

C. Gonnet and B. Torresani, Signal Process. 37, 389 (1994).
[CrossRef]

Other

Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, 2001).

A. Federico and G. H. Kaufmann, Advances in Speckle Metrology and Related Techniques, G. H. Kaufmann, ed. (Wiley-VCH, 2011), Chap. 4.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge U. Press, 2004).

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Figures (2)

Fig. 1.
Fig. 1.

Phase recovery for two optical vortices located in combined beams with charges m 1 = + 3 and m 2 = 2 when the simulated phase mask was degraded by random noise uniformly distributed: (a) analyzed interference pattern, (b) phase map recovered by means of the exact complex-wave reconstruction method, (c) phase map recovered using the method reported in [7], and (d) phase map recovered via the wavelet analysis of the interferogram cepstrum.

Fig. 2.
Fig. 2.

Phase recovery in TSPI: (a) analyzed noisy interference pattern, (b) temporal phase recovered by means of the exact complex-wave reconstruction method, and (c) temporal phase recovered via the wavelet analysis of the interferogram cepstrum. (Dash line, black color online) original phase, and (continuous line, red color online) recovered phase.

Equations (6)

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F [ ln ( 1 + E o E r ) ] = F [ ln ( I E r 2 ) ] 1 [ 0 , + ) × [ 0 , + ) ,
E o = [ exp ( F 1 { F [ ln ( I E r 2 ) ] 1 } ) 1 ] E r .
S [ L ] ( a , θ , b ) = 1 a R 2 d 2 x ψ a , θ , b * ( x ) L ( x ) ,
S [ L ] ( a , θ , b ) = a R 2 d 2 k e i k · b F [ ψ a , θ , b * ] ( k ) F [ L ] ( k ) .
S [ L ] ( a , θ 0 , b ) = a R 2 d 2 k e i k · b F [ ψ a , θ 0 , b * ] ( k ) F [ ln ( I / E r 2 ) ] ( k ) .
E o ( b ) = [ exp ( S [ L ] ( a r ( b ) , θ 0 , b ) ) 1 ] E r ( b ) ,

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