Abstract

We present an experimental configuration for phase retrieval from a set of intensity measurements. The key component is a spatial light modulator located in the Fourier domain of an imaging system. It performs a linear filter operation that is associated to the process of propagation in the image plane. In contrast to the state of the art, no mechanical adjustment is required during the recording process, thus reducing the measurement time considerably. The method is experimentally demonstrated by investigating a wave field scattered by a diffuser, and the results are verified by comparing them to those obtained from standard interferometry.

© 2010 Optical Society of America

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References

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2009 (2)

2006 (2)

2005 (1)

1997 (1)

1994 (2)

1993 (1)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247-264 (1993).
[CrossRef]

1992 (1)

1987 (1)

1986 (3)

1984 (1)

1982 (2)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

1967 (2)

J. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

G. C. Sherman, “Application of the convolution theorem to Rayleigh's integral formulas,” J. Opt. Soc. Am. 57, 546-547(1967).
[CrossRef] [PubMed]

AG, Holoeye Photonics

Holoeye Photonics AG, PLUTO--0. 7 in. HDTV LCOS Phase Only Kit Specification Sheet.

Almoro, P.

Almoro, P. F.

Anand, A.

Chhaniwal, V. K.

Dong, B.

Dong, B.-Z.

Ersoy, O. K.

Fienup, J.

Fienup, J. R.

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247-264 (1993).
[CrossRef]

Freund, I.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247-264 (1993).
[CrossRef]

George, N.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Goodman, J.

J. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gu, B.

Gu, B.-Y.

Hanson, S. G.

Ina, H.

Ivanov, V. Y.

Jüptner, W.

Kobayashi, S.

Kohler, C.

Kreis, T.

T. Kreis, Holographic Interferometry, 1st ed. (Akademie Verlag, 1996).

Lawrence, R.

J. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Levi, A.

Maallo, A. M. S.

Merzlyakov, N.

L. P. Yaroslavskii and N. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).

Osten, W.

Pedrini, G.

Rolleston, R.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Schnars, U.

Schwab, X.

Sherman, G. C.

Shvartsman, N.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247-264 (1993).
[CrossRef]

Sivokon, V. P.

Stark, H.

Takeda, M.

Vorontsov, M. A.

Wackerman, C.

Yang, G.

Yang, G.-Z.

Yaroslavskii, L. P.

L. P. Yaroslavskii and N. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).

Zhang, Y.

Zhuang, J.

Appl. Opt. (7)

Appl. Phys. Lett. (1)

J. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247-264 (1993).
[CrossRef]

Opt. Lett. (2)

Optik (Jena) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Other (4)

L. P. Yaroslavskii and N. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Holoeye Photonics AG, PLUTO--0. 7 in. HDTV LCOS Phase Only Kit Specification Sheet.

T. Kreis, Holographic Interferometry, 1st ed. (Akademie Verlag, 1996).

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. The lenses L 1 and L 2 are arranged in a 4 f configuration that provides an image of the object (D) across the camera plane { x } . A phase-modulating SLM is located in the corresponding Fourier domain { v } in order to enable linear filter operations. Optionally, a beam splitter (BS) allows for the superposition of the wave field across the camera target with a known reference wave in order to compare the results of the phase retrieval to those obtained from conventional interferometry.

Fig. 2
Fig. 2

Results obtained from phase retrieval. (a) Example of captured speckle intensity across the image plane of the 4 f configuration. (b) Phase distribution obtained using the captured intensity distributions associated with 10 different propagation states of the wave field and after k = 100 iterations. (All distributions are 504 × 504 pixel in size with a pixel pitch of Δ p ccd = 3.45 μm .)

Fig. 3
Fig. 3

Results obtained from standard interferometry. (a) Intensity of the interference pattern arising from the superposition of the speckle field in the image plane with a tilted reference wave. (b) Phase distribution obtained by means of the spatial carrier method. (All distributions are 504 × 504 pixel in size with a pixel pitch of Δ p ccd = 3.45 μm .)

Fig. 4
Fig. 4

Difference map of the obtained phase distributions. The standard deviation is σ ϕ = 0.14 rad , where differences of up to Δ ϕ max = 0.25 rad are mainly observed in the direct vicinity of phase dislocations.

Equations (10)

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u b ( x ) = ( u a h z ) ( x ) .
u ^ b ( ξ ) = u ^ a ( ξ ) · h ^ z ( ξ ) ,
h ^ z ( ξ ) = exp [ i 2 π z λ 1 λ 2 | ξ | 2 ] .
u ( v ) = 1 i λ f · F { u ( u ) } ( v λ f ) .
t z ( v ) = exp [ i 2 π z λ 1 1 f 2 | v | 2 ] .
t ¯ z ( v ) = t z ( v ) · m , n δ ( v i m Δ p , v j n Δ p ) .
t ¯ z [ m , n ] = exp [ i 2 π z λ 1 Δ p 2 f 2 ( m 2 + n 2 ) ] .
u ˜ m ( k ) ( x ) = F 1 { u ^ n ( k ) ( ξ ) · h ^ z ( ξ ) } .
u m ( k ) ( x ) = I m ( x ) · u ˜ m ( k ) ( x ) · | u ˜ m ( k ) ( x ) | 1 .
A S ( x ) A R ( x ) exp [ i ( ϕ S ( x ) ϕ R ( x ) + 2 π g · x ) ] = F 1 { F { I C ( x ) } · P W ( ξ ) } .

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