Abstract

A new method of digital phase contrast based on fractional-order Fourier reconstruction is proposed. We show that the diffraction patterns produced by pure phase objects exhibit linear chirp functions that can be advantageously processed using the fractional Fourier transform. The optimal fractional orders lead to the longitudinal location of the phase object, while the analysis of the reconstructed pattern leads to its diameter and to the value of the phase shift. Simulations and experimental results are given. The configuration tested in this paper is a very general Gaussian illumination.

© 2009 Optical Society of America

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References

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    [CrossRef]

2008

2007

2006

2005

2003

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

2002

2001

1999

1993

1987

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

1980

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

Allano, D.

Bevilacqua, F.

Brunel, M.

Callens, N.

Coëtmellec, S.

Cuche, E.

Depeursinge, C.

Dubois, F.

Hernández, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Illueca, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Janssen, A. J. E. M.

Kato, J.

Kemper, B.

Kerr, F. H.

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

Lebrun, D.

Lohmann, A. W.

Marquet, P.

Mas, D.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

McBride, A. C.

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

Miret, J. J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Mizuno, J.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

Nicolas, F.

Ohta, S.

Ozkul, C.

Pérez, J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Schockaert, C.

Vázquez, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Verrier, N.

von Bally, G.

Yamaguchi, I.

Yourassowsky, C.

Appl. Opt.

IMA J. Appl. Math.

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (8)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Simulated diffraction pattern produced by a transparent disk.

Fig. 3
Fig. 3

Optimal reconstruction of the simulated pattern | α x opt , α y opt [ I ] | 2 .

Fig. 4
Fig. 4

Transverse profile of the reconstructed pattern.

Fig. 5
Fig. 5

Relative amplitude discontinuity versus phase shift introduced by the plate.

Fig. 6
Fig. 6

Experimental diffraction pattern produced by a transparent disk.

Fig. 7
Fig. 7

Optimal reconstruction of the experimental pattern.

Fig. 8
Fig. 8

Transverse profile of the reconstructed pattern.

Equations (17)

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E ( ξ , η , z ) = E 0 ( z ) exp ( ξ 2 + η 2 ω ( z ) 2 ) exp ( i π λ ξ 2 + η 2 R ( z ) ) ,
E ( x , y , δ + z c ) = exp ( i 2 π z c λ ) i λ z c + + E ( ξ , η , δ ) T ( ξ , η ) exp ( i π λ z c [ ( ξ x ) 2 + ( η y ) 2 ] ) d ξ d η ,
T ( ξ , η ) = { e i φ if ξ 2 + η 2 < D / 2 1 otherwise ,
E ( x , y , δ + z c ) = E 0 ( δ ) exp ( i 2 π z c λ ) ( A 1 + ( e i φ 1 ) A 2 ) / ( i λ z c ) ,
A 1 = K 2 e π r 2 λ z c ( i M N ) ,
A 2 = π D 2 2 e i π r 2 λ z c T 0 ( r ) .
T 0 ( r ) = e i u 4 ( 2 π u ) 1 / 2 s = 0 K s ( i ) s J 2 s + 1 ( π D r λ z c ) π D r λ z c ,
u = π D 2 2 λ z c ( 1 + z c R d ) i D 2 2 ω d 2 ,
K = π ω d 2 1 + i π ω d 2 λ z c ( z c R d 1 ) ,
N = π ω d 2 λ z c 1 + π 2 ω d 4 λ 2 z c 2 ( z c R d 1 ) 2 ,
M = 1 + N π ω d 2 λ z c ( z c R d 1 ) .
α x , α y [ I ( x , y ) ] ( x a , y a ) = R 2 N α x ( x , x a ) N α y ( y , y a ) I ( x , y ) d x d y ,
N α p ( x , x a ) = C ( α p ) exp ( i π x 2 + x a 2 s p 2 tan α p ) exp ( i 2 π x a x s p 2 sin α p ) ,
C ( α p ) = exp ( i ( π 4 sign ( sin α p ) α p 2 ) ) | s p 2 sin α p | 1 / 2 .
s p 2 = N p · δ p 2 .
α x , α y [ I ] = α x , α y [ | A 1 | 2 ] F α x , α y [ 2 Re ( A 1 A 2 ¯ ( e i φ 1 ) ) ] + α x , α y [ ( 2 2 cos φ ) | A 2 | 2 ] .
π cot α x opt s x 2 = π λ z c ( M 1 ) , π cot α y opt s y 2 = π λ z c ( M 1 ) ,

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