Abstract

A new, to our knowledge, design method for diffractive optical elements (DOE’s) is described and compared with existing methods. The technique applies a nonlinear least-squares algorithm to design two-dimensional pure phase DOE’s that reconstruct a desired diffraction pattern with high uniformity, efficiency, and signal-to-noise ratio. The technique also uses a phase-shifting quantization procedure that greatly reduces the quantization error for DOE’s to a minimum level. In this paper, we compare simulated reconstruction results of DOE’s designed by use of these methods with results obtained by the commonly used two-stage iterative Fourier transform design algorithm of Wyrowski. [J. Opt. Soc. Am. A 7, 961, (1990)].

© 1997 Optical Society of America

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References

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  1. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  2. J. Jahn, S. H. Lee, Optical Computing Hardware (Academic, New York, 1995), Chap. 6, pp. 137–165.
  3. M. S. Kim, C. C. Guest, “Simulated annealing algorithm for binary phase only filters in pattern classification,” Appl. Opt. 29, 1203–1208 (1990).
    [CrossRef]
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  5. H. Akahori, “Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform,” Appl. Opt. 25, 802–811 (1986).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed., (McGraw-Hill, New York, 1995), Chap. 4, pp. 55–78.
  7. J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983), Chap. 8.
  8. J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equation in Several Variables (Academic, New York, 1970).
  9. P. Lindstrom, “A new line search algorithm for unconstrained nonlinear least squares problems,” Math. Prog. 29, 268–296 (1984).
    [CrossRef]
  10. matlab software (Math Works, Inc., Natick, Mass., 1994).
  11. Å. Björck, P. G. Ciarlet, J. L. Lions, Handbook of Numerical Analysis (Elsevier, North Holland, Amsterdam, The Netherlands, 1987), Chap. 6, pp. 152–169.
  12. K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).
  13. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef]
  14. C. B. Kuznia, A. A. Sawchuk, “Time multiplexing and control for optical cellular hypercube arrays,” Appl. Opt. 35, 1836–1847 (1996).
    [CrossRef]
  15. M. J. B. Duff, “CLIP4: a large scale integrated circuit array parallel processor,” in Proceedings of the International Joint Conference of Pattern Recognition (IJCPR) (IEEE Computer Society, Long Beach, Calif., 1976), pp. 728–733.
  16. H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
    [CrossRef]
  17. A. Goldstein, B. K. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–25 October 1996, paper ILS–XII.

1996 (1)

1994 (1)

H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
[CrossRef]

1990 (2)

1989 (1)

1986 (1)

1984 (1)

P. Lindstrom, “A new line search algorithm for unconstrained nonlinear least squares problems,” Math. Prog. 29, 268–296 (1984).
[CrossRef]

1972 (1)

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

1944 (1)

K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Akahori, H.

Björck, Å.

Å. Björck, P. G. Ciarlet, J. L. Lions, Handbook of Numerical Analysis (Elsevier, North Holland, Amsterdam, The Netherlands, 1987), Chap. 6, pp. 152–169.

Ciarlet, P. G.

Å. Björck, P. G. Ciarlet, J. L. Lions, Handbook of Numerical Analysis (Elsevier, North Holland, Amsterdam, The Netherlands, 1987), Chap. 6, pp. 152–169.

Cloonan, T. J.

H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
[CrossRef]

Dennis, J. E.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983), Chap. 8.

Duff, M. J. B.

M. J. B. Duff, “CLIP4: a large scale integrated circuit array parallel processor,” in Proceedings of the International Joint Conference of Pattern Recognition (IJCPR) (IEEE Computer Society, Long Beach, Calif., 1976), pp. 728–733.

Gerchberg, R. W.

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

Goldstein, A.

A. Goldstein, B. K. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–25 October 1996, paper ILS–XII.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed., (McGraw-Hill, New York, 1995), Chap. 4, pp. 55–78.

Guest, C. C.

Hinton, H. S.

H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
[CrossRef]

Jahn, J.

J. Jahn, S. H. Lee, Optical Computing Hardware (Academic, New York, 1995), Chap. 6, pp. 137–165.

Jenkins, B. K.

A. Goldstein, B. K. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–25 October 1996, paper ILS–XII.

Kim, M. S.

Kuznia, C. B.

Lee, S. H.

J. Jahn, S. H. Lee, Optical Computing Hardware (Academic, New York, 1995), Chap. 6, pp. 137–165.

Lentine, A. L.

H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
[CrossRef]

Levenberg, K.

K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Lindstrom, P.

P. Lindstrom, “A new line search algorithm for unconstrained nonlinear least squares problems,” Math. Prog. 29, 268–296 (1984).
[CrossRef]

Lions, J. L.

Å. Björck, P. G. Ciarlet, J. L. Lions, Handbook of Numerical Analysis (Elsevier, North Holland, Amsterdam, The Netherlands, 1987), Chap. 6, pp. 152–169.

McCormic, F. B.

H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
[CrossRef]

Ortega, J. M.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equation in Several Variables (Academic, New York, 1970).

Rheinboldt, W. C.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equation in Several Variables (Academic, New York, 1970).

Sawchuk, A. A.

Saxton, W. O.

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

Schnabel, R. B.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983), Chap. 8.

Tooley, F. A. P.

H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
[CrossRef]

Wyrowski, F.

Appl. Opt. (4)

J. Opt. Soc. Am. A (1)

Math. Prog. (1)

P. Lindstrom, “A new line search algorithm for unconstrained nonlinear least squares problems,” Math. Prog. 29, 268–296 (1984).
[CrossRef]

Optik (1)

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

Proc. IEEE (1)

H. S. Hinton, T. J. Cloonan, F. B. McCormic, A. L. Lentine, F. A. P. Tooley, “Free-space digital optical systems,” Proc. IEEE 82, 1632–1648 (1994).
[CrossRef]

Q. Appl. Math. (1)

K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Other (8)

A. Goldstein, B. K. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–25 October 1996, paper ILS–XII.

matlab software (Math Works, Inc., Natick, Mass., 1994).

Å. Björck, P. G. Ciarlet, J. L. Lions, Handbook of Numerical Analysis (Elsevier, North Holland, Amsterdam, The Netherlands, 1987), Chap. 6, pp. 152–169.

M. J. B. Duff, “CLIP4: a large scale integrated circuit array parallel processor,” in Proceedings of the International Joint Conference of Pattern Recognition (IJCPR) (IEEE Computer Society, Long Beach, Calif., 1976), pp. 728–733.

J. Jahn, S. H. Lee, Optical Computing Hardware (Academic, New York, 1995), Chap. 6, pp. 137–165.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed., (McGraw-Hill, New York, 1995), Chap. 4, pp. 55–78.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983), Chap. 8.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equation in Several Variables (Academic, New York, 1970).

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Figures (6)

Fig. 1
Fig. 1

Diagram of the first stage of the two-stage IFT method.

Fig. 2
Fig. 2

Diagram of the second stage of the two-stage IFT method.

Fig. 3
Fig. 3

Phase-shifting effect in quantization (three phase levels): (a) The constant phase shift δϕ results in different quantization pattern. (b) The effect of a constant phase shift in the phase coordinate.

Fig. 4
Fig. 4

Improvement of uniformity by use of phase-shifting quantization with 16 phase levels.

Fig. 5
Fig. 5

Flowchart of DOE design by use of the NLS method and the phase-shifting quantization scheme.

Fig. 6
Fig. 6

MSE versus computation cost (in FLOPS) for a 16-phase-level cellular hypercube interconnection pattern [1, 2, 4].

Tables (3)

Tables Icon

Table 1 Simulation Results of Different Methods for Cellular Hypercube Pattern [1,2,4]

Tables Icon

Table 2 Simulation Results of Different Methods for the 3 × 3 Uniform Spot Array

Tables Icon

Table 3 Simulation Results of Different Methods for a 4:2:1 Weighted Fan-Out DOE

Equations (34)

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

Gj, k=expiθj, k,-πθj, kπ.
g=W×FT-1G,
MSE=1MNm,ngm, n2-Im, n21/2.
F=Fm, n|m=0,1,,M-1,n=0, 1,, N-1, Fm, n=gm, n2-Im, n,
Im, n=Im, n/sincm/Msincn/N2,
gm, n=gm, n/sincm/M/sincn/N,
gm, n=j=0M-1k=0N-1 Gj, kexp-i2π jmM+knN.
Fm, n=j=0M-1k=0N-1p=0M-1q=0N-1expi2πp-jmM+q-knN+θj, k-θp, q-Im, n.
Θl+1=Θl+αlHl,
hl=-JHJ-1JHf,
hv=hjN+k=Hj, k, v=0, 1, 2,,MN-1, j=0, 1, 2,,M-1, k=0, 1,2,,N-1,
fu=fmN+n=Fm, n, u=0, 1, 2,,MN-1, m=0, 1, 2,,M-1, n=0, 1, 2,,N-1.
Ju, v=JmN+n,jN+k=Fm, n/θj, k.
Ju, v=JmN+n, jN+k=p=0pjM-1q=0qkN-1-2 sinθ j, k-θp, q+2πmp-jM+nq-kN.
fΘl2-fΘl+αlH212αlJΘlhl2.
minαlfΘl+αlHl2.
g0m, n=Im, n1/2,
G¯lj, k=expiGlj, k,
gl+1m, n=Im, n1/2 expig¯lm, n for m, n R=0  for m, nR,
gl+1m, n=clIm, n1/2 expig¯lm, n for m, n  R=g¯lm, n for m, nR,
U=Ŝmax-Ŝmin/Ŝmax+Ŝmin,
ηw=m,nSrm, nm,nIrm, n,
ηg=m,n Srm, n,
SNR=10 log10Smin/Nmax,
QF=w11-U+w2ηw+w3ηg+w4SNR, i-14wi=1.
Ir=W×FT-1G expiδϕ2=W×FT-1Gexpiδϕ2=W×FT-1G2,
ηo=m,nRIm, nm,nIm, n,
I=00000000.20000.20.20.20000.20000000,
I=0.0050.0050.0050.0050.0050.0050.0050.180.0050.0050.0050.180.180.180.0050.0050.0050.180.0050.0050.0050.0050.0050.0050.005.
gl+1m, n=ηoIm, n1/2 expig¯lm, n for m, n R=c for m, nR,
I=0000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000010000000000000001000000000001011011010000000000010000000000000001000000000000000000000000000000010000000000000000000000000000000000000000000000000000000×112.
QF=57100-U+17ηg+17SNR,
I=0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011100000000000001110000000000000111000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000×19.
I=0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012100000000000002420000000000000121000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000×116.

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