Abstract

Special-purpose diffractive optical devices are often designed by iterative methods without consideration of convergence properties and related factors that affect performance. We examine the properties of iterative algorithms in a vector-space setting and illustrate, with examples, differences in convergence performance based on starting point, sequential versus parallel projections, and intersecting versus nonintersecting sets.

© 1999 Optical Society of America

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References

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  1. D. Wood, P. McKee, M. Dames, “Multiple-imaging and multiple-focussing Fresnel lenses with high numerical aperture,” in Holographics International ’92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 307–316 (1993).
    [CrossRef]
  2. G. Hatakoshi, M. Nakamura, “Grating lenses for optical branching,” Appl. Opt. 32, 3661–3667 (1993).
    [CrossRef] [PubMed]
  3. M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital holography,” Opt. Commun. 77, 4–8 (1990).
    [CrossRef]
  4. M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Iterative techniques to integrate different optical functions in a diffractive phase element,” Appl. Opt. 30, 4629–4635 (1991).
    [CrossRef] [PubMed]
  5. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 235–246 (1972).
  6. B. C. Kress, S. H. Lee, “Iterative design of computer generated Fresnel holograms for free-space optical interconnections,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 22–25.
  7. F. Wyrowski, G. M. Morris, guest eds., “Diffractive optics,” feature issue of J. Mod. Opt. (April1993).
  8. M. G. Moharam, J. Leger, guest eds., “Diffractive optics modeling,” feature issue of J. Opt. Soc. Am. A 12, 1025–1169 (1995).
  9. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  10. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  11. H. Stark, W. C. Catino, J. L. LoCicero, “Design of phase gratings by generalized projections,” J. Opt. Soc. Am. A 8, 566–571 (1991).
    [CrossRef]
  12. W. C. Catino, J. L. LoCicero, H. Stark, “Design of continuous and quantized phase holograms by generalized projections,” J. Opt. Soc. Am. A 14, 2715–2725 (1997).
    [CrossRef]
  13. W. C. Catino, J. L. LoCicero, H. Stark, “Design of continuous and quantized amplitude holograms by generalized projections,” J. Opt. Soc. Am. A 15, 68–76 (1998).
    [CrossRef]
  14. K. S. Urquhart, P. Marchand, Y. Fainman, S. H. Lee, “Diffractive optics applied to free-space optical interconnects,” Appl. Opt. 33, 3670–3682 (1994).
    [CrossRef] [PubMed]
  15. J. Leger, M. G. Moharam, guest eds., “Diffractive optics,” feature issue of Appl. Opt. 36, 2389–2604 (1995).
  16. I. Cindrich, S. H. Lee, eds., Diffractive and Holographic Optics Technology III, Proc. SPIE2689 (1996).
  17. Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996).
  18. H. P. Herzig, Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1970).
  19. A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
    [CrossRef]
  20. G. M. Morris, “Diffractive optics technology and its applications,” short course notes 104, in Annual Meeting (Optical Society of America, Washington, D.C., 1997).
  21. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  22. R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distribution,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–327 (1995).
    [CrossRef]
  23. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  24. H. Stark, Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, New York, 1998).
  25. T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel-projection methods to correlation-filter design,” Appl. Opt. 34, 3883–3894 (1995).
    [CrossRef] [PubMed]
  26. G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 28, 96–115 (1984).
    [CrossRef]
  27. P. L. Combettes, “Inconsistent signal feasibility problem: least-squares solution in a product space, IEEE Trans. Signal Process. 42, 2955–2966 (1994).
    [CrossRef]
  28. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  29. A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).
  30. H. Stark, “Theory and measurements of the optical Fourier transform,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 1–40.
  31. R. V. Churchill, J. W. Brown, R. F. Verhey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974), p. 167.

1998 (1)

1997 (1)

1995 (3)

J. Leger, M. G. Moharam, guest eds., “Diffractive optics,” feature issue of Appl. Opt. 36, 2389–2604 (1995).

T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel-projection methods to correlation-filter design,” Appl. Opt. 34, 3883–3894 (1995).
[CrossRef] [PubMed]

M. G. Moharam, J. Leger, guest eds., “Diffractive optics modeling,” feature issue of J. Opt. Soc. Am. A 12, 1025–1169 (1995).

1994 (3)

1993 (2)

F. Wyrowski, G. M. Morris, guest eds., “Diffractive optics,” feature issue of J. Mod. Opt. (April1993).

G. Hatakoshi, M. Nakamura, “Grating lenses for optical branching,” Appl. Opt. 32, 3661–3667 (1993).
[CrossRef] [PubMed]

1991 (2)

1990 (2)

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
[CrossRef]

M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital holography,” Opt. Commun. 77, 4–8 (1990).
[CrossRef]

1989 (1)

1984 (3)

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 28, 96–115 (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 (Stuttgart) 35, 235–246 (1972).

Bernhardt, M.

Brown, J. W.

R. V. Churchill, J. W. Brown, R. F. Verhey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974), p. 167.

Bryngdahl, O.

Catino, W. C.

Churchill, R. V.

R. V. Churchill, J. W. Brown, R. F. Verhey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974), p. 167.

Combettes, P. L.

P. L. Combettes, “Inconsistent signal feasibility problem: least-squares solution in a product space, IEEE Trans. Signal Process. 42, 2955–2966 (1994).
[CrossRef]

Dames, M.

D. Wood, P. McKee, M. Dames, “Multiple-imaging and multiple-focussing Fresnel lenses with high numerical aperture,” in Holographics International ’92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 307–316 (1993).
[CrossRef]

Fainman, Y.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 235–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hatakoshi, G.

Herzig, H. P.

H. P. Herzig, Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1970).

Kotzer, T.

Kress, B. C.

B. C. Kress, S. H. Lee, “Iterative design of computer generated Fresnel holograms for free-space optical interconnections,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 22–25.

Lee, S. H.

K. S. Urquhart, P. Marchand, Y. Fainman, S. H. Lee, “Diffractive optics applied to free-space optical interconnects,” Appl. Opt. 33, 3670–3682 (1994).
[CrossRef] [PubMed]

B. C. Kress, S. H. Lee, “Iterative design of computer generated Fresnel holograms for free-space optical interconnections,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 22–25.

Levi, A.

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

LoCicero, J. L.

Marchand, P.

McKee, P.

D. Wood, P. McKee, M. Dames, “Multiple-imaging and multiple-focussing Fresnel lenses with high numerical aperture,” in Holographics International ’92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 307–316 (1993).
[CrossRef]

Morris, G. M.

G. M. Morris, “Diffractive optics technology and its applications,” short course notes 104, in Annual Meeting (Optical Society of America, Washington, D.C., 1997).

Nakamura, M.

Pierra, G.

G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 28, 96–115 (1984).
[CrossRef]

Piestun, R.

R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
[CrossRef] [PubMed]

R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distribution,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–327 (1995).
[CrossRef]

Rosen, J.

Sawchuk, A. A.

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 235–246 (1972).

Shamir, J.

Spektor, B.

R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distribution,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–327 (1995).
[CrossRef]

Stark, H.

Strand, T. C.

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

Urquhart, K. S.

Verhey, R. F.

R. V. Churchill, J. W. Brown, R. F. Verhey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974), p. 167.

Wood, D.

D. Wood, P. McKee, M. Dames, “Multiple-imaging and multiple-focussing Fresnel lenses with high numerical aperture,” in Holographics International ’92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 307–316 (1993).
[CrossRef]

Wyrowski, F.

Yang, Y.

H. Stark, Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, New York, 1998).

Appl. Opt. (6)

IEEE Trans. Signal Process. (1)

P. L. Combettes, “Inconsistent signal feasibility problem: least-squares solution in a product space, IEEE Trans. Signal Process. 42, 2955–2966 (1994).
[CrossRef]

J. Mod. Opt. (1)

F. Wyrowski, G. M. Morris, guest eds., “Diffractive optics,” feature issue of J. Mod. Opt. (April1993).

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

Math. Program. (1)

G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 28, 96–115 (1984).
[CrossRef]

Opt. Commun. (1)

M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital holography,” Opt. Commun. 77, 4–8 (1990).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 235–246 (1972).

Proc. IEEE (1)

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

Other (12)

G. M. Morris, “Diffractive optics technology and its applications,” short course notes 104, in Annual Meeting (Optical Society of America, Washington, D.C., 1997).

I. Cindrich, S. H. Lee, eds., Diffractive and Holographic Optics Technology III, Proc. SPIE2689 (1996).

Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996).

H. P. Herzig, Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1970).

B. C. Kress, S. H. Lee, “Iterative design of computer generated Fresnel holograms for free-space optical interconnections,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 22–25.

R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distribution,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–327 (1995).
[CrossRef]

H. Stark, Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, New York, 1998).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

H. Stark, “Theory and measurements of the optical Fourier transform,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 1–40.

R. V. Churchill, J. W. Brown, R. F. Verhey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974), p. 167.

D. Wood, P. McKee, M. Dames, “Multiple-imaging and multiple-focussing Fresnel lenses with high numerical aperture,” in Holographics International ’92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 307–316 (1993).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Block diagram of various possibilities in solving a problem by vector-space projections. POCS stands for projections onto convex sets.

Fig. 2
Fig. 2

Progress of the iterates toward a solution for the sequential and parallel cases.

Fig. 3
Fig. 3

Point y˜*, which minimizes the distance from h˜ to C˜1, permits the computation of y*, the point that minimizes the distance from h to C1, according to y*=y˜*Ω(x, z)¯.

Fig. 4
Fig. 4

Diffractive element that produces a specified optical field magnitude when illuminated by a point source a distance x0 above the axis.

Fig. 5
Fig. 5

Evolution of a lens that meets the design constraints when the starting point is a lens of insufficient size. On the left is the image; on the right is the phase of the DOE. The top row is the initial image formed from the initial phase. The middle row shows the same after 5 iterations. The bottom row is a feasible solution obtained after 15 iterations.

Fig. 6
Fig. 6

Evolution of a lens that meets the design constraints when the starting point is a uniform section of glass. On the left is the image; on the right is the phase of the DOE. The top row is the initial image formed from the initial phase. The middle row shows the same after 20 iterations. The bottom row is a feasible solution obtained after 100 iterations. The extra iterations compared with Fig. 5 are due to the poorer starting point.

Fig. 7
Fig. 7

Results of applying the SGPA to the low-light-level problem at dc after 350 iterations. (a) Diffraction peaks moved away from the origin, (b) a small portion of the highly fluctuating phase of the DOE, (c) SDE behavior with application of the SGPA.

Fig. 8
Fig. 8

Results of applying the PGPA to the low-light-level problem at dc after 350 iterations. (a) Diffraction peaks moved away from the origin, (b) SDE behavior with application of the PGPA.

Fig. 9
Fig. 9

Failure of both the SGPA and the PGPA to converge strongly when a more stringent constraint is imposed. (a) SDE behavior computed with the SGPA formula, (b) SDE behavior computed with the PGPA formula, (c) resulting magnitude of DOE transmittance. Note the inability of the PGPA to furnish a pure-phase solution.

Fig. 10
Fig. 10

Image magnitude due to a point source at infinity: 10% misfocusing error (a) in front of image (focal) plane, (b) in focal plane, (c) in back of focal plane.

Fig. 11
Fig. 11

Designing increased depth-of-focus optics with the SGPA. Magnitude diffraction pattern due to a point source at infinity: 10% misfocusing error (a) in front of image (focal) plane, (b) in focal plane, (c) in back of focal plane.

Fig. 12
Fig. 12

Designing increased depth-of-focus optics with the PGPA. Magnitude diffraction pattern due to a point source at infinity: 10% misfocusing error (a) in front of image (focal) plane, (b) in focal plane, (c) in back of focal plane.

Equations (52)

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tn+1(x)=P1P2  PMtn(x),
J(y)=i=1My-Piy.
J(y)=y-P1y+y-P2y.
J(tn+1)J(tn)forn=1, 2, .
xn+1=i=1mwiPi xn
J(xn+1)J(xn),
J(xn)=i=1mwiPixn-xn21/2.
gz(ξ, η)=1λzRt(x, y)×expj πλz[(x-ξ)2+(y-η)2]dxdy,
gz(ξ)=1λz-a/2a/2t(x)expj πλz(x-ξ)2dx,
1λz-a/2a/2t(x)expjπλz(x-ξ)2dx=M(ξ).
C1={y(x)L2:|Fr[y]|=M(ξ)},
C2={y(x)L2:|y(x)|=1forx[-a/2, a/2]and0 elsewhere}.
C1={y(x)L2:|Fr[y]|c(ξ)},
-a/2a/2t˜(x, z)exp{-jωx}dx=Mz(ω),
t˜(x, z)t(x)Ω(x, z),
Ω(x, z)expj πλ zx2,
ω2πξ/λ z,
Mz(ω)=λzM(λzω/2π).
y*=argminyC1h-y,
C˜1={y˜(x)L2:|Y˜(ω)|=Mz(ω)}.
y˜*(x)=P˜1h˜Mz(ω)exp[j θH˜(ω)],
y*(x)=y˜*Ω(x, z)¯.
tk+1(x)=P1 P2tk(x),t0arbitrary,
P1h=Ω¯F-1{Mz(ω)exp[jθH˜(ω)]},
P2h=0x[-a/2, a/2]exp[jϕh(x)]x[-a/2, a/2],
gz1(ξ)=1λz1-δ(x-x0)expj πλz1(x-ξ)2dx
=1λz1expj πλz1(x0-ξ)2,
gz2(ζ)=1λz2-gz1(ξ)t(ξ)expj πλz2(ζ-ξ)2dξ.
-a/2a/2t(ξ)expj πλzξ2exp(-jωξ)dξ=Mz(ω),
z=z1z2z1+z2,
ω=x0λz1+ζλz2,
Mz(ω)=λz1z2Mλω z2-z2z1x0.
M(ζ)=4sin 2πζ2πζ.
-a/2a/2t(x)exp-j 2πλ fx2expj πλf(x-ζ)2dx,
=uζ/λz-a/2a/2t(x)exp(-j2πux)dxϵ(u)
foruD1,
C1={y(x)L2:|Y(u)|ϵ(u)foruD1}
C2=y(x)L2:|y(x)|=1forx-a2, a2y(x)=0otherwise,
C3={y(x)L2:Y(u)=Y(-u)foruD3}.
P1h(x)ϵ(u)exp[jθH(u)]if|H(u)|>ϵ(u)anduD1H(u)otherwise.
P3 h(x)12[H(u)+H(-u)]ifH(u)H(-u)anduD3H(u)otherwise.
tn+1=P2 P1 P3tn(x),t0(x)=1
Uz(ζ)=KzTz(ζ),
Tz(ζ)=-0.50.5 exp[-jπb(1-r)y2]exp(-j2πrρζy)dy,
Kz=aλz,r=f/z,
ρ=a/λf,b=a2/λf=aρ,
U˜z(ζ)=KzT˜z(ζ),
T˜z(ζ)=-0.50.5t(y)exp[-jπb(1-r)y2]exp(-j2πrρζy)dy.
C1={t(x)L2:|U˜f-ϵ(ζ)|c(ζ), |U˜f-ϵ(0)|=Kf},
C2={t(x)L2:|U˜f(ζ)|c(ζ),U˜f (0)=Kf},
C3={t(x)L2:|U˜f+ϵ(ζ)|c(ζ),U˜f+ϵ(0)=Kf},
C4={t(x)L2:|t(x)|=rect(x/a)},

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