Abstract

The signal extraction method based on intensity measurements in two close fractional Fourier domains is examined by using the phase space formalism. The fractional order separation has a lower bound and an upper bound that depend on the signal at hand and the noise in the optical system used for measurement. On the basis of a theoretical analysis, it is shown that for a given optical system a judicious choice of fractional order separation requires some a priori knowledge of the signal bandwidth. We also present some experimental results in support of the analysis.

© 2008 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  10. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and the phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
    [Crossref]
  11. K. A. Nugent, "X-ray noninterferometric phase imaging," J. Opt. Soc. Am. A 24, 536-547 (2007).
    [Crossref]
  12. J. Tu and S. Tomura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946-1949 (1997).
    [Crossref]
  13. J. Tu and S. Tomura, "Analytical relation for revovering the mutual intensity by menas of intensity information," J. Opt. Soc. Am. A 15, 202-206 (1998).
    [Crossref]
  14. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms," Opt. Lett. 20, 1181-1183 (1995).
    [Crossref] [PubMed]
  15. M. G. Raymer, M. Beck, and D. F. McAlister, "Complex-wavefield reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
    [Crossref] [PubMed]
  16. A. Semichaevsky and M. Testorf, "Phase-space interpretation of deterministic phase retrieval," J. Opt. Soc. Am. A 21, 2173-2179 (2004).
    [Crossref]
  17. T. Alieva and M. J. Bastiaans, "Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform," J. Opt. Soc. Am. A 17, 2324-2329 (2000).
    [Crossref]
  18. T. Alieva and M. J. Bastiaans, "On fractional Fourier moments," IEEE Signal Process. Lett. 7, 320-323 (2000).
    [Crossref]
  19. T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
    [Crossref]
  20. J. T. Sheridan and R. Patten, "Holographic interferometry and the fractional Fourier transformation," Opt. Lett. 25, 448-450 (2000).
    [Crossref]
  21. J. T. Sheridan, B. M. Hennelly, and D. P. Kelly, "Motion detection, the Wigner distribution function, and the optical fractional Fourier transform," Opt. Lett. 28, 884-886 (2003).
    [Crossref] [PubMed]
  22. D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O' Neill, and J. T. Sheridan, "Paraxial speckle based metrology system with an aperture," J. Opt. Soc. Am. A 23, 2861-2870 (2006).
    [Crossref]
  23. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, "Space-bandwidth product of optical signals and systems," J. Opt. Soc. Am. A 13, 470-473 (1996).
    [Crossref]
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    [Crossref]
  25. M. J. Bastiaans, "Application of Wigner distribution function in optics," in Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbraüker and F.Hlawatsch, eds. (Elsevier Science, 1997), pp. 375-426.
  26. A. W. Lohmann and B. H. Soffer, "Relationship between the Radon-Wigner and fractional Fourier transforms," J. Opt. Soc. Am. A 11, 1798-1801 (1994).
    [Crossref]
  27. D. Dragoman, "Redundancy of phase-space distribution functions in complex field recovery problems," Appl. Opt. 42, 1932-1937 (2003).
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  28. L. Z. Cai and Y. Q. Yang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).
  29. www.edmundoptics.com, Part No. NT45-045.

2007 (1)

2006 (1)

2005 (1)

2004 (1)

2003 (3)

2002 (1)

L. Z. Cai and Y. Q. Yang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

2000 (4)

1999 (1)

G. K. Datta and R. M. Vasu, "Noninterferometric methods of phase estimation for application in optical tomography," J. Mod. Opt. 46, 1377-1388 (1999).

1998 (3)

1997 (2)

J. Tu and S. Tomura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946-1949 (1997).
[Crossref]

M. J. Bastiaans, "Application of Wigner distribution function in optics," in Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbraüker and F.Hlawatsch, eds. (Elsevier Science, 1997), pp. 375-426.

1996 (1)

1995 (2)

1994 (2)

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex-wavefield reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[Crossref] [PubMed]

A. W. Lohmann and B. H. Soffer, "Relationship between the Radon-Wigner and fractional Fourier transforms," J. Opt. Soc. Am. A 11, 1798-1801 (1994).
[Crossref]

1990 (1)

1988 (1)

1985 (1)

1984 (1)

N. Streibl, "Phase imaging by the transport equation of intensity," Opt. Commun. 49, 6-10 (1984).
[Crossref]

1983 (1)

Alieva, T.

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[Crossref]

T. Alieva and M. J. Bastiaans, "On fractional Fourier moments," IEEE Signal Process. Lett. 7, 320-323 (2000).
[Crossref]

T. Alieva and M. J. Bastiaans, "Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform," J. Opt. Soc. Am. A 17, 2324-2329 (2000).
[Crossref]

Barty, A.

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[Crossref]

T. Alieva and M. J. Bastiaans, "On fractional Fourier moments," IEEE Signal Process. Lett. 7, 320-323 (2000).
[Crossref]

T. Alieva and M. J. Bastiaans, "Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform," J. Opt. Soc. Am. A 17, 2324-2329 (2000).
[Crossref]

M. J. Bastiaans, "Application of Wigner distribution function in optics," in Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbraüker and F.Hlawatsch, eds. (Elsevier Science, 1997), pp. 375-426.

Beck, M.

Cai, L. Z.

L. Z. Cai and Y. Q. Yang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

Clarke, L.

Datta, G. K.

G. K. Datta and R. M. Vasu, "Noninterferometric methods of phase estimation for application in optical tomography," J. Mod. Opt. 46, 1377-1388 (1999).

Dorsch, R. G.

Dragoman, D.

Dutta, G. K.

Ferreira, C.

Gopinathan, U.

Gureyev, T. E.

Hennelly, B. M.

Ichikawa, K.

Jayashree, M.

Kelly, D. P.

Lohmann, A. W.

Mayer, A.

McAlister, D. F.

Mendlovic, D.

Nugent, K. A.

O' Neill, F. T.

Paganin, D.

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[Crossref]

D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[Crossref]

Patten, R.

Raymer, M. G.

Roberts, A.

Roddier, F.

Semichaevsky, A.

Sheridan, J. T.

Soffer, B. H.

Stankovic, L.

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[Crossref]

Streibl, N.

N. Streibl, "Phase imaging by the transport equation of intensity," Opt. Commun. 49, 6-10 (1984).
[Crossref]

Takeda, M.

Teague, M. R.

Testorf, M.

Tomura, S.

J. Tu and S. Tomura, "Analytical relation for revovering the mutual intensity by menas of intensity information," J. Opt. Soc. Am. A 15, 202-206 (1998).
[Crossref]

J. Tu and S. Tomura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946-1949 (1997).
[Crossref]

Tu, J.

J. Tu and S. Tomura, "Analytical relation for revovering the mutual intensity by menas of intensity information," J. Opt. Soc. Am. A 15, 202-206 (1998).
[Crossref]

J. Tu and S. Tomura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946-1949 (1997).
[Crossref]

Vasu, R. M.

M. Jayashree, G. K. Dutta, and R. M. Vasu, "Optical tomographic microscope for quantitative imaging of phase objects," Appl. Opt. 39, 277-283 (2000).
[Crossref]

G. K. Datta and R. M. Vasu, "Noninterferometric methods of phase estimation for application in optical tomography," J. Mod. Opt. 46, 1377-1388 (1999).

Ward, J. E.

Yang, Y. Q.

L. Z. Cai and Y. Q. Yang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

Zalevsky, Z.

Appl. Opt. (4)

IEEE Signal Process. Lett. (1)

T. Alieva and M. J. Bastiaans, "On fractional Fourier moments," IEEE Signal Process. Lett. 7, 320-323 (2000).
[Crossref]

IEEE Trans. Signal Process. (1)

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[Crossref]

J. Mod. Opt. (1)

G. K. Datta and R. M. Vasu, "Noninterferometric methods of phase estimation for application in optical tomography," J. Mod. Opt. 46, 1377-1388 (1999).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

M. R. Teague, "Image formation in terms of transport equation," J. Opt. Soc. Am. A 2, 2019-2026 (1985).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and the phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
[Crossref]

K. A. Nugent, "X-ray noninterferometric phase imaging," J. Opt. Soc. Am. A 24, 536-547 (2007).
[Crossref]

J. Tu and S. Tomura, "Analytical relation for revovering the mutual intensity by menas of intensity information," J. Opt. Soc. Am. A 15, 202-206 (1998).
[Crossref]

A. Semichaevsky and M. Testorf, "Phase-space interpretation of deterministic phase retrieval," J. Opt. Soc. Am. A 21, 2173-2179 (2004).
[Crossref]

T. Alieva and M. J. Bastiaans, "Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform," J. Opt. Soc. Am. A 17, 2324-2329 (2000).
[Crossref]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O' Neill, and J. T. Sheridan, "Paraxial speckle based metrology system with an aperture," J. Opt. Soc. Am. A 23, 2861-2870 (2006).
[Crossref]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, "Space-bandwidth product of optical signals and systems," J. Opt. Soc. Am. A 13, 470-473 (1996).
[Crossref]

B. M. Hennelly and J. T. Sheridan, "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A 22, 917-927 (2005).
[Crossref]

A. W. Lohmann and B. H. Soffer, "Relationship between the Radon-Wigner and fractional Fourier transforms," J. Opt. Soc. Am. A 11, 1798-1801 (1994).
[Crossref]

Opt. Commun. (1)

N. Streibl, "Phase imaging by the transport equation of intensity," Opt. Commun. 49, 6-10 (1984).
[Crossref]

Opt. Laser Technol. (1)

L. Z. Cai and Y. Q. Yang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

Opt. Lett. (4)

Phys. Rev. E (1)

J. Tu and S. Tomura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946-1949 (1997).
[Crossref]

Phys. Rev. Lett. (2)

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex-wavefield reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[Crossref] [PubMed]

D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[Crossref]

Other (2)

www.edmundoptics.com, Part No. NT45-045.

M. J. Bastiaans, "Application of Wigner distribution function in optics," in Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbraüker and F.Hlawatsch, eds. (Elsevier Science, 1997), pp. 375-426.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation (top view) of the AF of the one-dimensional signal illustrating sampling considerations. For illustration, the figure is greatly exagerated and simplified. The dots indicate samples of the FT of the two intensity measurements at FRT orders 0 and Δ α . The two arrows indicate the values A ( ν ̃ , 0 ) and A ( ν ̃ , ν ̃ Δ α ) of the two samples at ( ν ̃ , 0 ) and ( ν ̃ , ν ̃ Δ α ) .

Fig. 2
Fig. 2

Experimental results for 8 mm focal length lens (diameter 3 mm ): (a) spatial frequency ν x , (b) spatial frequency ν x with the y direction averaged out; (c) spatial frequency ν y ; (d) spatial frequency ν y with the x direction averaged out. The dashed–dotted lines in (b) and (d) indicate the predicted spatial frequency corresponding to an 8 mm focal length lens.

Fig. 3
Fig. 3

Phase calculated from the estimated spatial frequency for the 8 mm focal length lens.

Fig. 4
Fig. 4

Experimental results for the 20 cm focal length lens: (a) spatial frequency ν x ; (b) spatial frequency ν x with the y direction averaged out; (c) spatial frequency ν y ; (d) spatial frequency ν y with the x direction averaged out. The dashed–dotted lines in (b) and (d) indicate the predicted spatial frequency corresponding to a 20 cm focal length lens.

Fig. 5
Fig. 5

Collimated beam input: (a) spatial frequency ν x ; (b) spatial frequency ν x with the y direction averaged out; (c) spatial frequency ν y ; (d) spatial frequency ν y with the x direction averaged out.

Equations (43)

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f α ( x ) = f ( x 0 ) K α ( x , x 0 ) d x 0 ,
K α ( x , x 0 ) = exp ( i α 2 ) i sin α exp [ i π ( x 2 tan α 2 x x 0 sin α + x 0 2 tan α ) ] .
W ( x , ν ) = f ( x + x 2 ) f * ( x x 2 ) exp ( i 2 π ν x ) d x ,
A ( x ¯ , ν ¯ ) = f ( x ̂ + x ¯ 2 ) f * ( x ̂ x ¯ 2 ) exp ( i 2 π ν ¯ x ̂ ) d x ̂ .
A ( x ¯ , ν ¯ ) = W ( x , ν ) exp [ i 2 π ( ν ¯ x ν x ¯ ) ] d x d ν .
A ( 0 , ν ¯ ) = m 0 ( x ) exp ( i 2 π ν ¯ x ) d x ,
m 0 ( x ) = W ( x , ν ) d ν = I 0 ( x ) .
A ( x ¯ , ν ¯ ) x ¯ x ¯ = 0 = 2 π i m 1 ( x ) exp ( i 2 π ν ¯ x ) d x ,
m 1 ( x ) = ν W ( x , ν ) d ν = ν I 0 ( x ) .
I α ( x ) = W ( x cos α ν sin α , x sin α + ν cos α ) d ν .
I ̃ α ( ν ̃ ) = I α ( x ) exp ( i 2 π ν ̃ x ) d x .
I ̃ α ( ν ̃ ) = A ( x ¯ , ν ¯ ) x ¯ = ν ̃ sin α , ν ¯ = ν ̃ cos α .
I ̃ 0 ( ν ̃ ) = I ̃ α 0 ( ν ̃ ) α 0 = 0 = A ( x ¯ , ν ¯ ) x ¯ = 0 , ν ¯ = ν ̃ ,
I ̃ Δ α ( ν ̃ ) = I ̃ α 0 + Δ α ( ν ̃ ) α 0 = 0 = A ( x ¯ , ν ¯ ) x ¯ = Δ α ν ̃ , ν ¯ = ν ̃ ,
A ( x ¯ + Δ x , ν ¯ ) A ( x ¯ , ν ¯ ) + A ( x ¯ , ν ¯ ) x ¯ Δ x .
I ̃ Δ α ( ν ̃ ) I ̃ 0 ( ν ̃ ) Δ α = A ( x ¯ , ν ¯ ) x ¯ ν ̃ x ¯ = 0 , ν ¯ = ν ̃ ,
A ( x ¯ , ν ¯ ) x ¯ x ¯ = 0 = 1 ν ¯ d I α ( x ̂ ) d α α = 0 exp ( i 2 π ν ¯ x ̂ ) d x ̂ .
ν ( x ) = 1 2 π i 1 I 0 ( x ) A ( x ¯ , ν ¯ ) x ¯ x ¯ = 0 exp ( i 2 π ν ¯ x ) d ν ¯ .
1 π i 1 u exp [ i 2 π u ( x x ̂ ) ] d u = sgn ( x ̂ x ) ,
ν ( x ) = 1 2 I 0 ( x ) d I α ( x ̂ ) d α α = 0 sgn ( x ̂ x ) d x ̂ .
ν ( x ) = 1 2 I α 0 ( x ) d I α ( x ̂ ) d α α = α 0 sgn ( x ̂ x ) d x ̂ .
ν x ( x , y ) = ϕ ( x , y ) x = 1 2 I 0 , 0 ( x , y ) I α x , α y ( x ̃ , y ̃ ) α x α x = 0 , α y = 0 sgn ( x x ̃ ) δ ( y y ̃ ) d x ̃ d y ̃
ν x ( x , y ) = ϕ ( x , y ) x = 1 2 I 0 ( x , y ) dI α ( x ̃ , y ̃ ) d α α = 0 sgn ( x x ̃ ) δ ( y y ̃ ) d x ̃ d y ̃
ϕ ( x , y ) = k x ν x ( x , y ) d x + k y ν y ( x , y ) d y ,
I α ( x ) = δ ( m 1 x m 2 ν + b 1 ) d ν ,
m 1 = 2 b 2 cos α sin α , m 2 = 2 b 2 sin α + cos α .
I α ( x ) = 1 m 2 δ ( x ν + b 1 ) d ν , x = m 1 x .
d I α ( x ) d α α = 0 = 2 b 2 δ ( x ν + b 1 ) d ν , x = 2 b 2 x ,
ν ( x ) = 1 2 I 0 ( x ) d I α ( x ) d α α = 0 sgn ( x 2 b 2 x ) d x 2 b 2 .
( Δ ν ¯ 2 ) tan Δ α < δ x ¯ ,
Δ α < 2 Δ ν ¯ 2 .
Δ ν ¯ 2 Δ ν ¯ 2 I ̃ Δ α ( ν ¯ ) I ̃ 0 ( ν ¯ ) 2 d ν ¯ > N 0 Δ ν ¯ .
Δ α < 2 b 2 2 W 2 .
I ̃ 0 ( ν ) = exp ( i π ν b 1 b 2 ) δ ( ν ) ,
I ̃ Δ α ( ν ) = 1 2 b 2 Δ α + 1 exp ( i 2 π ν b 1 2 b 2 Δ α ) δ ( ν ) .
[ 1 1 + ( 2 b 2 Δ α ) 1 ] 2 > N 0 Δ ν ¯ .
Δ α > 1 2 b 2 [ ( N 0 b 2 W ) 1 2 1 ] .
f α ( s x , s y ) = f ( s x ̃ , s y ̃ ) K α ( x , y ; x ̃ , y ̃ ) d x ̃ d y ̃ ,
K α ( x , y ; x ̃ , y ̃ ) = exp [ i π ( x 2 + y 2 tan α 2 x x ̃ + 2 y y ̃ sin α + x ̃ 2 + y ̃ 2 tan α ) ] .
d 2 = 2 f d 1 ( d 1 f ) + f 2 ( f d 1 ) 2 + f 2 ,
α = cos 1 [ d 1 ( d 1 2 f ) ( f d 1 ) 2 + f 2 ] .
d I α ( x ) d α α = 0 I α 1 ( x ) I α 2 ( x ) Δ α ,
I 0 ( x ) I α 1 ( x ) + I α 2 ( x ) 2 .

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