Abstract

We describe a new approach to suppress undesired diffraction orders in the signal area of a Fourier plane diffractive optical element (DOE). We implement this new approach for the DOE design by a two-stage iterative Fourier transform algorithm that incorporates an adaptive optimization of the signal-to-noise ratio and does not require the introduction of a dummy output area outside the field of view. A comparison among this approach and three other approaches are presented on the basis of numerical results from several sample diffraction patterns.

© 1997 Optical Society of America

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References

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  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  2. H. Akahori, “Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform,” Appl. Opt. 25, 802–811 (1986).
    [CrossRef] [PubMed]
  3. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  4. C. B. Kuznia, “Cellular hypercube interconnections for optoelectronic smart pixel cellular arrays,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1994).
  5. K.–S. Huang, C. B. Kuznia, B. K. Jenkins, and A. A. Sawchuk, “Parallel Architectures for digital optical cellular image processing,” Proc. IEEE 82, 1711–1723 (1994).
    [CrossRef]
  6. C. B. Kuznia and A. A. Sawchuk, “Time multiplexing and control for optical cellular–hypercube arrays,” Appl. Opt. 35, 1836–1847 (1996).
    [CrossRef] [PubMed]
  7. M. R. Taghizadeh, J. M. Miller, P. Blair, and F. A. P. Tooley, “Developing diffractive optics for optical computing,” IEEE Micro 14, 10–18 (Dec. 1994).
    [CrossRef]
  8. M. R. Taghizaden and J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).
  9. M. W. Farn, M. B. Stern, and W. Veldkamp, “The making of binary optics,” Opt. Photon. News 2 (5), 20–22 (1991).
    [CrossRef]
  10. Y. A. Carts, “Microelectronic methods push binary optics frontiers,” Laser Focus World 28, (2), 87–95 (Feb. 1992).

1996

1994

K.–S. Huang, C. B. Kuznia, B. K. Jenkins, and A. A. Sawchuk, “Parallel Architectures for digital optical cellular image processing,” Proc. IEEE 82, 1711–1723 (1994).
[CrossRef]

1992

M. R. Taghizaden and J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

1990

1986

1972

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

Akahori, H.

Blair, P.

M. R. Taghizadeh, J. M. Miller, P. Blair, and F. A. P. Tooley, “Developing diffractive optics for optical computing,” IEEE Micro 14, 10–18 (Dec. 1994).
[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 (Stuttgart) 35, 237–246 (1972).

Huang, K.–S.

K.–S. Huang, C. B. Kuznia, B. K. Jenkins, and A. A. Sawchuk, “Parallel Architectures for digital optical cellular image processing,” Proc. IEEE 82, 1711–1723 (1994).
[CrossRef]

Jenkins, B. K.

K.–S. Huang, C. B. Kuznia, B. K. Jenkins, and A. A. Sawchuk, “Parallel Architectures for digital optical cellular image processing,” Proc. IEEE 82, 1711–1723 (1994).
[CrossRef]

Kuznia, C. B.

C. B. Kuznia and A. A. Sawchuk, “Time multiplexing and control for optical cellular–hypercube arrays,” Appl. Opt. 35, 1836–1847 (1996).
[CrossRef] [PubMed]

K.–S. Huang, C. B. Kuznia, B. K. Jenkins, and A. A. Sawchuk, “Parallel Architectures for digital optical cellular image processing,” Proc. IEEE 82, 1711–1723 (1994).
[CrossRef]

Miller, J. M.

M. R. Taghizadeh, J. M. Miller, P. Blair, and F. A. P. Tooley, “Developing diffractive optics for optical computing,” IEEE Micro 14, 10–18 (Dec. 1994).
[CrossRef]

Sawchuk, A. A.

C. B. Kuznia and A. A. Sawchuk, “Time multiplexing and control for optical cellular–hypercube arrays,” Appl. Opt. 35, 1836–1847 (1996).
[CrossRef] [PubMed]

K.–S. Huang, C. B. Kuznia, B. K. Jenkins, and A. A. Sawchuk, “Parallel Architectures for digital optical cellular image processing,” Proc. IEEE 82, 1711–1723 (1994).
[CrossRef]

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 (Stuttgart) 35, 237–246 (1972).

Taghizadeh, M. R.

M. R. Taghizadeh, J. M. Miller, P. Blair, and F. A. P. Tooley, “Developing diffractive optics for optical computing,” IEEE Micro 14, 10–18 (Dec. 1994).
[CrossRef]

Taghizaden, M. R.

M. R. Taghizaden and J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

Tooley, F. A. P.

M. R. Taghizadeh, J. M. Miller, P. Blair, and F. A. P. Tooley, “Developing diffractive optics for optical computing,” IEEE Micro 14, 10–18 (Dec. 1994).
[CrossRef]

Turunen, J.

M. R. Taghizaden and J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

Wyrowski, F.

Appl. Opt.

IEEE Micro

M. R. Taghizadeh, J. M. Miller, P. Blair, and F. A. P. Tooley, “Developing diffractive optics for optical computing,” IEEE Micro 14, 10–18 (Dec. 1994).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Comput. Process.

M. R. Taghizaden and J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

Optik (Stuttgart)

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

Proc. IEEE

K.–S. Huang, C. B. Kuznia, B. K. Jenkins, and A. A. Sawchuk, “Parallel Architectures for digital optical cellular image processing,” Proc. IEEE 82, 1711–1723 (1994).
[CrossRef]

Other

C. B. Kuznia, “Cellular hypercube interconnections for optoelectronic smart pixel cellular arrays,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1994).

M. W. Farn, M. B. Stern, and W. Veldkamp, “The making of binary optics,” Opt. Photon. News 2 (5), 20–22 (1991).
[CrossRef]

Y. A. Carts, “Microelectronic methods push binary optics frontiers,” Laser Focus World 28, (2), 87–95 (Feb. 1992).

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Figures (14)

Fig. 1
Fig. 1

Schematic diagrams for three diffraction patterns: an 8 × 8 spot array, a CH, and the letters USC. The signal area is shown and includes 32 × 32 spots or sample points. Dark and empty spots represent signal and noise orders, respectively.

Fig. 2
Fig. 2

Schematic diagram of the phase profile of a 1-D DOE with phase levels of Z = 4 and Q = 4.

Fig. 3
Fig. 3

Normalized (norm.) intensity distribution function along the K x axis of a 2-D CH DOE with N = 4, Z = 8, and Q = 8, and P = 5: (a) normalized intensity resulting from the factors of Eq. (7), i.e., sinc factor, combination of the constant and sine-over-sine factors, and the summation factor, and (b) resultant normalized intensity function.

Fig. 4
Fig. 4

Main procedures of the basic IFT algorithm. IDFT denotes the inverse DFT.

Fig. 5
Fig. 5

Two operators inside the iterative loop.

Fig. 6
Fig. 6

Proposed two-stage algorithm for the DOE design.

Fig. 7
Fig. 7

Flowchart for the P stage.

Fig. 8
Fig. 8

Flowchart for the AP L stage.

Fig. 9
Fig. 9

Flowchart for the AP H stage. Qntz, quantize.

Fig. 10
Fig. 10

Normalized intensity distribution of the CH along the K x axis in the observation area for the direct, Darea, and new Pnoise approaches.

Fig. 11
Fig. 11

Arrangement of the dummy area for the Darea approach when the signal area has 32 × 32 sample points.

Fig. 12
Fig. 12

SNRmin, U, and ηcgg of the DOE’s versus the maximum SNRmin for the 8 × 8 spot array, the CH with N = 16, and the letter pattern USC. All DOE’s are with Q = 32 and Z = 8.

Fig. 13
Fig. 13

Relation among three performance indicators (SNRmin, U, and ηcgg) and the number of phase levels (Z) for CH DOE’s: (a) SNRmin versus Z, (b) U versus Z, and (c) ηcgg versus Z. Note that the AP H algorithm is applied in the AP stage when Z ≥ 16.

Fig. 14
Fig. 14

Statistical relation of the uniformity U and the efficiency ηcgg to SNRmin for five values of the maximum SNRmin on the basis of the same experiment that generated Fig. 12. For each maximum SNRmin, 10 runs were performed. (a) Statistical relation between the uniformity U and SNRmin. (b) Statistical relation between the efficiency ηcgg and SNRmin.

Tables (3)

Tables Icon

Table 1 Numerical Results of Four Approaches for the CH with N = 16 and Z = 8

Tables Icon

Table 2 Comparison between the Darea and the New Pnoise Approaches on the Basis of Results from Three Patternsa

Tables Icon

Table 3 Comparison between the Old and New Pnoise Approaches on the Basis of Results for the CH with N = 16

Equations (15)

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ϕqΔx=2πΔ1-uqΔxλ Δn,
gx=Px=0Px-1qx=0Qx-1gqxΔxrectx-qxΔx-pxQxΔxΔx,
Gfx=1PxLxΔx sincΔxfxexp-jπΔxfxQxPx-1×sinπΔxfxQxPxsinπΔxfxQxqx=0Qx-1gqxΔxexp-j2πΔxfxqx.
Kx=LxxfλFlens
GKx=1PxQxsincKxQxsinπPxKxsinπKxqx=0Qx-1gqxΔx×exp-j2πqxQxKx.
DFTgqxΔx=qx=0Qx-1gqxΔxexp-j2πqxQxKx.
Gm=1Qx-1mPx-1sincmQxDFTgqxΔx.
GKx, Ky=1PxPyQxQxsincKxQxsincKyQysinπPxKxsinπKx×sinπPyKysinπKy×qx=0Qx-1qy=0Qy-1gqxΔx, qyΔy×exp-j2πqxQx Kx+qyQyKy,
Gm, n=1QxQy-1mPx-1+nPy-1sincmQxsincnQy×DFTgqxΔx, qyΔy.
G¯m, n=Gm, n×QxQy-1mPx-1+nPy-1sincmQxsincnQy.
G¯m, n=DFTgqxΔx,qyΔy.
SNRmin=10 log10ηsminηnmax,
U=ηsmax-ηsminηsmax+ηsmin,
ηcgg=m,nSsηmn,
ηcggw=m,nSsηmnm,nSsηmn+m,nSnηmn.

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