Abstract

Computer-generated phase-only diffractive optical elements in a cascaded setup are designed by one deterministic and one stochastic algorithm for multiplane image formation. It is hypothesized that increasing the number of elements as wavefront modulators in the longitudinal dimension would enlarge the available solution space, thus enabling enhanced image reconstruction. Numerical results show that increasing the number of holograms improves quality at the output. Design principles, computational methods, and specific conditions are discussed.

© 2013 Optical Society of America

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2011 (2)

C.-F. Ying, H. Pang, C.-J. Fan, and W.-D. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50, 055802 (2011).
[CrossRef]

A. Mazine and K. Heggarty, “Phase mapping and wavefront analysis based on multi-illumination light fields generated by a spatial light modulator,” Appl. Opt. 50, 2679–2691 (2011).
[CrossRef]

2010 (2)

2009 (1)

2008 (1)

2007 (3)

2005 (1)

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44, 125805 (2005).
[CrossRef]

2003 (1)

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

2000 (1)

1997 (1)

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (2)

1992 (1)

1984 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1972 (1)

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).

Aarts, E.

E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing (Wiley, 1989).

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Barbastathis, G.

Bartelt, H.

Borgsmüller, S.

Bryngdahl, O.

O. Bryngdahl and F. Wyrowski, in Progress in Optics, E. Wolf, ed. (North Holland, 1990), Vol. 28, pp. 3–86.

Chen, R. T.

Dannberg, P.

Deng, X.

Dietrich, C.

Dorsch, R. G.

Duda, R. O.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. (Wiley, 2000).

Fan, C.-J.

C.-F. Ying, H. Pang, C.-J. Fan, and W.-D. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50, 055802 (2011).
[CrossRef]

Fan, D.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

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).

Gerke, T. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Gulses, A. A.

A. A. Gulses and B. K. Jenkins, “Multi-plane image reconstruction using cascaded phase elements,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (Optical Society of America, 2012), paper DM4C.2.

Haist, T.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

Hart, P. E.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. (Wiley, 2000).

Heggarty, K.

Jahns, J.

S. Sinzinger and J. Jahns, Microoptics, 2nd ed. (Wiley, 2003).

Jenkins, B. K.

A. A. Gulses and B. K. Jenkins, “Multi-plane image reconstruction using cascaded phase elements,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (Optical Society of America, 2012), paper DM4C.2.

Johnson, R. V.

Kämpfe, T.

Kathman, A. D.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication and Test (SPIE, 2004).

Kley, E.-B.

Kolodziejczyk, A.

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44, 125805 (2005).
[CrossRef]

Korst, J.

E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing (Wiley, 1989).

Kresse, T.

Li, Y.

Lohmann, A. W.

Luo, Y.

Mait, J. N.

Makowski, M.

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44, 125805 (2005).
[CrossRef]

Männer, R.

Mazine, A.

Mikula, G.

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44, 125805 (2005).
[CrossRef]

Noethe, S.

Nordin, G. P.

O’Shea, D. C.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication and Test (SPIE, 2004).

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Pang, H.

C.-F. Ying, H. Pang, C.-J. Fan, and W.-D. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50, 055802 (2011).
[CrossRef]

Piestun, R.

Prather, D. W.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication and Test (SPIE, 2004).

Qiu, Y.

Roggemann, M. C.

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).

Schönleber, M.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

Shamir, J.

Shi, Y.

Sinzinger, S.

Situ, G.

Stork, D. G.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. (Wiley, 2000).

Suleski, T. J.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication and Test (SPIE, 2004).

Sypek, M.

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44, 125805 (2005).
[CrossRef]

Tanguay, A. R.

Tiziani, H. J.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

Tünnermann, A.

Voelz, D.

D. Voelz, Computational Fourier Optics (SPIE, 2010).

Voelz, D. G.

Waller, L.

Wyrowski, F.

O. Bryngdahl and F. Wyrowski, in Progress in Optics, E. Wolf, ed. (North Holland, 1990), Vol. 28, pp. 3–86.

Yang, S. Young

Ying, C.-F.

C.-F. Ying, H. Pang, C.-J. Fan, and W.-D. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50, 055802 (2011).
[CrossRef]

Zhang, J.

Zhou, W.-D.

C.-F. Ying, H. Pang, C.-J. Fan, and W.-D. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50, 055802 (2011).
[CrossRef]

Appl. Opt. (6)

J. Opt. Soc. Am. A (3)

Nat. Photonics (1)

T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4, 188–193 (2010).
[CrossRef]

Opt. Commun. (2)

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Opt. Eng. (3)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikuła, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44, 125805 (2005).
[CrossRef]

C.-F. Ying, H. Pang, C.-J. Fan, and W.-D. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50, 055802 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Optik (1)

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).

Other (8)

A. A. Gulses and B. K. Jenkins, “Multi-plane image reconstruction using cascaded phase elements,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (Optical Society of America, 2012), paper DM4C.2.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

D. Voelz, Computational Fourier Optics (SPIE, 2010).

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication and Test (SPIE, 2004).

O. Bryngdahl and F. Wyrowski, in Progress in Optics, E. Wolf, ed. (North Holland, 1990), Vol. 28, pp. 3–86.

S. Sinzinger and J. Jahns, Microoptics, 2nd ed. (Wiley, 2003).

E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing (Wiley, 1989).

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. (Wiley, 2000).

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Figures (5)

Fig. 1.
Fig. 1.

System configuration. N thin PEs, each L by L, with a separation “d”, are concatenated in order to create output, namely target planes 1 and 2 with separation “”. The physical field propagation is to the +z direction. The symbol I denotes Fourier transform. A lens is in use for Fourier transform from PEN to target plane 1.

Fig. 2.
Fig. 2.

Flow chart for the extended IFTA in optimization of cascaded DOEs for multiplane imaging. One subiteration that is employed in the computation of one PE is shown. At the end, the element index is updated and another subiteration starts for the subsequent PE. When every component in the stack is finished, iteration is completed and the next one starts. This is controlled by mod(N) factor.

Fig. 3.
Fig. 3.

Mean squared error (MSE) versus number of PEs used in two-output plane imaging, when extended IFTA is used with 64 allowed phase values.

Fig. 4.
Fig. 4.

Some computational results of extended IFTA are compared (referring to Fig. 1). (a) Desired target images. (b) Final reconstructions at target planes using just one PE. (c) Final reconstructions at target planes using five PEs. Note that reconstructed image intensities are unnormalized here and a little bit lower than the original due to other diffracted orders.

Fig. 5.
Fig. 5.

Flow chart for the extended SA algorithm in optimization of cascaded DOEs for multiplane imaging. “k” shows the loop number. MSE is denoted as “E” in the diagram for brevity. (Eave)new represents the average error of two targets for the new trial, and (Eave)old is the current average error. The new attempt will be accepted if the conditional statement holds.

Tables (2)

Tables Icon

Table 1. Performance Merits for One and Five PEs According to Allowable Phase Levels on Device for Extended IFTA Case

Tables Icon

Table 2. Performance Merits for One and Five PEs When Allowed Number of Phase Levels is Four, for Extended SA Case

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

H(fx,fy)=exp(i2πd1λ2fx2fy2)
ψi=I1[I(ψi1)×H]×PEi.
MSE(%)=All pixels(|ψtarget|C|ψreconstructed|)2Number of pixels×100.
C=All Pixels|ψtarget||ψreconstructed|All Pixels|ψreconstructed|2.
φ=2πz1λ2fx2.
1Δfx2|x|max.
|x|max=12π|φfx|max.
Δfx|φfx|maxπ.
|φfx|=2πzfx1λ2fx2.
Δfx1λ2fx,max22zfx,max.
Δxλ21+(2zL)2.

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