Abstract

In this paper, the issue of misalignment in phase retrieval by means of optical linear filtering is dis cussed. The filtering setup is based on a 4f configuration with a spatial light modulator (SLM) as an active element, located in the Fourier domain. From the analysis, crucial parameters for the alignment procedure of the setup’s optical axes and the center of the SLM are identified. Furthermore, a method to automatically as well as electronically compensate such effects by modifying the phase pattern displayed on the SLM is introduced. Experimental results are presented that validate the compensation approach.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M(2011).
    [CrossRef]

2011

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M(2011).
[CrossRef]

2010

M. Agour, C. Falldorf, and C. v. Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in optoelectronic holography,” J. Opt. 12, 055401 (2010).
[CrossRef]

C. Falldorf, M. Agour, C. von Kopylow, and R. B. Bergmann, “Design of an optical system for phase retrieval based on a spatial light modulator,” AIP Conf. Proc. 1236, 259–264(2010).
[CrossRef]

L. Waller, Y. Luo, S. Y. Yang, and G. Barbastathis, “Transport of intensity phase imaging in a volume holographic microscope,” Opt. Lett. 35, 2961–2963 (2010).
[CrossRef] [PubMed]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef] [PubMed]

2009

P. Almoro, A. Maallo, and S. Hanson, “Fast-convergent algorithm for speckle-based phase retrieval and a design for dynamic wavefront sensing,” Appl. Opt. 48, 1485–1493(2009).
[CrossRef] [PubMed]

M. Agour, C. Falldorf, E. Kolenovic, and C. v. Kopylow, “Suppression of higher diffraction orders and intensity improvement of optically reconstructed holograms from a spatial light modulator,” J. Opt. A: Pure Appl. Opt. 11, 105405(2009).
[CrossRef]

2007

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

2006

2005

1998

1992

1987

1984

1982

1972

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

1967

Agour, M.

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M(2011).
[CrossRef]

M. Agour, C. Falldorf, and C. v. Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in optoelectronic holography,” J. Opt. 12, 055401 (2010).
[CrossRef]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef] [PubMed]

C. Falldorf, M. Agour, C. von Kopylow, and R. B. Bergmann, “Design of an optical system for phase retrieval based on a spatial light modulator,” AIP Conf. Proc. 1236, 259–264(2010).
[CrossRef]

M. Agour, C. Falldorf, E. Kolenovic, and C. v. Kopylow, “Suppression of higher diffraction orders and intensity improvement of optically reconstructed holograms from a spatial light modulator,” J. Opt. A: Pure Appl. Opt. 11, 105405(2009).
[CrossRef]

Almoro, P.

Barbastathis, G.

Bergmann, R. B.

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M(2011).
[CrossRef]

C. Falldorf, M. Agour, C. von Kopylow, and R. B. Bergmann, “Design of an optical system for phase retrieval based on a spatial light modulator,” AIP Conf. Proc. 1236, 259–264(2010).
[CrossRef]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef] [PubMed]

Brady, G. R.

Falldorf, C.

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M(2011).
[CrossRef]

M. Agour, C. Falldorf, and C. v. Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in optoelectronic holography,” J. Opt. 12, 055401 (2010).
[CrossRef]

C. Falldorf, M. Agour, C. von Kopylow, and R. B. Bergmann, “Design of an optical system for phase retrieval based on a spatial light modulator,” AIP Conf. Proc. 1236, 259–264(2010).
[CrossRef]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef] [PubMed]

M. Agour, C. Falldorf, E. Kolenovic, and C. v. Kopylow, “Suppression of higher diffraction orders and intensity improvement of optically reconstructed holograms from a spatial light modulator,” J. Opt. A: Pure Appl. Opt. 11, 105405(2009).
[CrossRef]

Fienup, J. R.

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 35, 237–246 (1972).

Goodman, J.

J. Goodman, Introduction to Fourier Optics2nd ed.(McGraw-Hill, 1996).

Hanson, S.

Kolenovic, E.

M. Agour, C. Falldorf, E. Kolenovic, and C. v. Kopylow, “Suppression of higher diffraction orders and intensity improvement of optically reconstructed holograms from a spatial light modulator,” J. Opt. A: Pure Appl. Opt. 11, 105405(2009).
[CrossRef]

E. Kolenovic, “Correlation between intensity and phase in monochromatic light,” J. Opt. Soc. Am. A 22, 899–906(2005).
[CrossRef]

Kopylow, C. v.

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M(2011).
[CrossRef]

M. Agour, C. Falldorf, and C. v. Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in optoelectronic holography,” J. Opt. 12, 055401 (2010).
[CrossRef]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef] [PubMed]

M. Agour, C. Falldorf, E. Kolenovic, and C. v. Kopylow, “Suppression of higher diffraction orders and intensity improvement of optically reconstructed holograms from a spatial light modulator,” J. Opt. A: Pure Appl. Opt. 11, 105405(2009).
[CrossRef]

Levi, A.

Luo, Y.

Maallo, A.

Osten, W.

Pedrini, G.

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 35, 237–246 (1972).

Sherman, G. C.

Sivokon, V. P.

Stark, H.

Takahashi, T.

Takajo, H.

Taninaka, M.

Teague, M. R.

Ueda, R.

Vanov, V. Y.

von Kopylow, C.

C. Falldorf, M. Agour, C. von Kopylow, and R. B. Bergmann, “Design of an optical system for phase retrieval based on a spatial light modulator,” AIP Conf. Proc. 1236, 259–264(2010).
[CrossRef]

Vorontsov, M. A.

Waller, L.

Yang, S. Y.

Zhang, F.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Zhang, Y.

AIP Conf. Proc.

C. Falldorf, M. Agour, C. von Kopylow, and R. B. Bergmann, “Design of an optical system for phase retrieval based on a spatial light modulator,” AIP Conf. Proc. 1236, 259–264(2010).
[CrossRef]

Appl. Opt.

J. Opt.

M. Agour, C. Falldorf, and C. v. Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in optoelectronic holography,” J. Opt. 12, 055401 (2010).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

M. Agour, C. Falldorf, E. Kolenovic, and C. v. Kopylow, “Suppression of higher diffraction orders and intensity improvement of optically reconstructed holograms from a spatial light modulator,” J. Opt. A: Pure Appl. Opt. 11, 105405(2009).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Optik

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

Phys. Rev. A

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Proc. SPIE

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M(2011).
[CrossRef]

Other

J. Goodman, Introduction to Fourier Optics2nd ed.(McGraw-Hill, 1996).

Holoeye Photonics AG, PLUTO—07 in. HDTV LCOS Phase Only Kit Specification Sheet.

Supplementary Material (2)

» Media 1: AVI (1130 KB)     
» Media 2: AVI (1129 KB)     

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Figures (7)

Fig. 1
Fig. 1

Experimental configuration for phase retrieval: The intensity of a light field scattered by an object is captured using a CCD sensor across a sequence of spatially separated recording planes along the optical axis { x 1 , , x N } .

Fig. 2
Fig. 2

Sketch of the SLM-based phase retrieval setup: An object is located at plane { u } . The lenses L 1 and L 2 are arranged in a 4 f -configuration, which provides an image of the object across the camera plane { x } . A phase-only SLM is located in the corresponding Fourier plane { v } in order to enable linear filter operations. A λ / 2 plate and a linear polarizer, P, are required to adjust the polarization of the wave field in accordance with the requirements of the SLM.

Fig. 3
Fig. 3

Plane to plane propagation of a complex amplitude distribution.

Fig. 4
Fig. 4

Sketch of the scheme with a phase modulating SLM located in the Fourier plane { v } : Let A 1 and A 2 be the first and the second optical axes of the setup. Δ v 1 is the misalignment between A 1 and A 2 . Please note that the center of the SLM is shifted by the same amount of Δ v 1 with respect to A 2 . The complex amplitude across the CCD plane is u b ( x ) , and its Fourier transform across the { v } domain is u ^ b ( v ) .

Fig. 5
Fig. 5

Sketch of the scheme illustrating the second misalignment case, where Δ v 1 denotes the shift between the two optical axes, A 1 and A 2 . Please note that the transfer function is centered at different coordinates. Let Δ v 2 be the misalignment in the { v } plane between A 2 and the center of the SLM.

Fig. 6
Fig. 6

Measured results using a misaligned setup: (a) image of the investigated region on the transparent object. A movie of 20 frames of the recorded intensity of the complex amplitude generated in the CCD plane is shown in Media 1. (b) Observed real amplitude in the sensor domain corresponding to a propagation distance of z = 38 mm . (c) Corresponding phase recovered after 50 iterations. (d) Reconstructed amplitude in the front focal plane of the first lens by means of numerical propagation of the wave field obtained from (b) and (c). (All distributions: 1300 × 1300 pixels with a pixel pitch of 3.45 μm .)

Fig. 7
Fig. 7

Measured results after the electronic alignment of the setup: A movie of 20 frames of the recorded intensity of the complex amplitude generated in the CCD plane is shown in Media 2. (a) Observed real amplitude in the sensor domain corresponding to a propagation distance of z = 38 mm . (b) Corresponding phase recovered after 50 iterations. (c) Reconstructed amplitude in the front focal plane of the first lens by means of numerical propagation of the wave field obtained from Figs. 7a, 7b. (All distributions: 1300 × 1300 pixels with a pixel pitch of 3.45 μm .)

Equations (22)

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u b ( x ) = ( u a h z ) ( x ) ,
h z ( x u ) = z i λ exp ( i 2 π | r | λ ) | r | 2 .
h ^ z ( ξ ) = exp [ i 2 π z λ 1 λ 2 | ξ | 2 ]
u ^ a ( v ) = 1 i λ f · F { u a ( u ) } ( v λ f ) .
h ^ z ( v ) = exp [ i 2 π z λ 1 1 f 2 | v | 2 ] .
t z ¯ [ m , n ] = exp [ i 2 π z λ 1 Δ p 2 f 2 ( m 2 + n 2 ) ] .
u ^ b ( v ) = u ^ a ( v + Δ v 1 ) · h ^ z ( v + Δ v 1 ) , = u ^ b ( v + Δ v 1 ) .
u b ( x ) = 1 i λ f · F { u ^ b ( v + Δ v 1 ) } ( x λ f ) , = 1 i λ f · F { u ^ b ( v ) δ ( v + Δ v 1 ) } ( x λ f ) , = exp [ i 2 π λ f ( x , Δ v 1 ) ] · u b ( x ) ,
u ^ b ( v ) = u ^ a ( v + Δ v 1 ) · h ^ z ( v + Δ v 2 ) .
h ^ z ( v + Δ v 2 ) = h ^ z ( v + Δ v 1 Δ v ) , C · h ^ z ( v + Δ v 1 ) · exp [ i 2 π z λ f 2 ( v , Δ v ) ] ,
u ^ b ( v ) = C · [ u ^ a ( v + Δ v 1 ) · h ^ z ( v + Δ v 1 ) ] · exp [ i 2 π z λ f 2 ( v , Δ v ) ] , = C · u ^ b ( v + Δ v 1 ) · exp [ i 2 π z λ f 2 ( v , Δ v ) ] .
u b ( x ) = C · 1 i λ f · F { u ^ b ( v + Δ v 1 ) · exp [ i 2 π z λ f 2 ( v , Δ v ) ] } ( x λ f ) .
u b ( x ) = C · F { u ^ b ( ξ + Δ ξ 1 ) } ( x ) F { exp [ i 2 π z λ ( ξ , Δ ξ ) ] } ( x ) .
u b ( x ) = C · { u b ( x ) · exp [ i 2 π ( Δ ξ 1 , x ) ] } δ ( x z λ Δ ξ ) , = C · exp [ i 2 π ( Δ ξ 1 , x z λ Δ ξ ) ] · u b ( x z λ Δ ξ ) , = C · exp [ i 2 π λ f ( Δ v 1 , x z f Δ v ) ] · u b ( x z f Δ v ) .
Δ x = z f Δ v .
h ^ z ( v ) exp [ i 2 π z λ ] · exp [ i π z λ f 2 | v | 2 ] .
h ^ z ( v + Δ v 1 ) = exp [ i 2 π z λ ] · exp [ i π z λ f 2 | v + Δ v 1 | 2 ] .
h ^ z ( v + Δ v 2 ) = exp [ i 2 π z λ ] · exp [ i π z λ f 2 | ( v + Δ v 1 ) Δ v | 2 ] .
h ^ z ( v + Δ v 2 ) = exp [ i 2 π z λ ] · exp [ i π z λ f 2 | v + Δ v 1 | 2 ] · exp [ i π z λ f 2 | Δ v | 2 ] · exp [ i 2 π z λ f 2 ( v , Δ v ) ] · exp [ i 2 π z λ f 2 ( Δ v 1 , Δ v ) ] .
h ^ z ( v + Δ v 2 ) = h ^ z ( v + Δ v 1 ) · exp [ i π z λ f 2 | Δ v | 2 ] · exp [ i 2 π z λ f 2 ( Δ v 1 , Δ v ) ] · exp [ i 2 π z λ f 2 ( v , Δ v ) ] .
h ^ z ( v + Δ v 2 ) C · h ^ z ( v + Δ v 1 ) · exp [ i 2 π z λ f 2 ( v , Δ v ) ] .
C = exp [ i π z λ f 2 | Δ v | 2 ] · exp [ i 2 π z λ f 2 ( Δ v 1 , Δ v ) ] .

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