Abstract

Arbitrary complex-valued functions can be implemented as arrays of individually specified diffusers. For any diffuser, only average step height and vertical roughness are needed to control phase and amplitude. This modulation concept suggests potentially low-cost fabrication methods in which desired topographies are patterned by exposing photoresist with partially developed speckle patterns. Analyses and experimental demonstrations that illustrate the modulation concept and aspects of the fabrication method are presented, with particular emphasis on limitations of complex recording set by various photoresist and exposure properties. Applications of diffuser array concepts to spatial light modulators and to gray-scale lithographic printing of micro-optics are also mentioned.

© 1997 Optical Society of America

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References

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  1. J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  3. R. W. Cohn, M. Liang, “Approximating fully complex spatial modulation with pseudorandom phase-only modulation,” Appl. Opt. 33, 4406–4415 (1994).
    [CrossRef] [PubMed]
  4. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  5. J. A. Davis, D. M. Cottrell, “Random mask encoding of multiplexed phase-only and binary phase-only filters,” Opt. Lett. 19, 496–498 (1994).
    [CrossRef] [PubMed]
  6. L. G. Hassebrook, M. E. Lhamon, R. C. Daley, R. W. Cohn, M. Liang, “Random phase encoding of composite fully-complex filters,” Opt. Lett. 21, 272–274 (1996).
    [CrossRef] [PubMed]
  7. R. W. Cohn, M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2498 (1996).
    [CrossRef] [PubMed]
  8. M. T. Gale, M. Rossi, J. Pedersen, H. Schutz, “Fabrication of continuous relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
    [CrossRef]
  9. W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, Chap. 3, pp. 119–232.
  10. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer, Berlin, 1980), Chap. 6, pp. 291–366.
  11. R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).
  12. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  13. F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
    [CrossRef]
  14. M. P. Dames, R. J. Dowling, P. McKee, D. Wood, “Efficient optical elements to generate intensity weighted spot arrays: design and fabrication,” Appl. Opt. 30, 2685–2691 (1991).
    [CrossRef] [PubMed]
  15. E. G. Johnson, M. A. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12, 1152–1160 (1995).
    [CrossRef]
  16. J. P. Kirk, A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971).
    [CrossRef]
  17. T. J. Suleski, D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,” Appl. Opt. 34, 7507–7517 (1995).
    [CrossRef] [PubMed]
  18. D. C. O’Shea, W. S. Rockward, “Gray-scale masks for diffractive-optics fabrication. II. Spatially filtered halftone screens,” Appl. Opt. 34, 7518–7526 (1995).
    [CrossRef]
  19. B. Wagner, H. J. Quenzer, W. Henke, W. Hoppe, W. Pilz, “Microfabrication of complex surface topographies using grey-tone lithography,” Sens. Actuators A 46–47, 89–94 (1995).
    [CrossRef]
  20. T. R. Jay, M. B. Stern, “Preshaping photoresist for refractive microlens fabrication,” Opt. Eng. 33, 3552–3555 (1994).
    [CrossRef]
  21. Shipley Corporation Microposit Products Catalog, Marlboro, Mass.
  22. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 318, Eq. (3.382.4).

1996 (2)

1995 (4)

1994 (4)

M. T. Gale, M. Rossi, J. Pedersen, H. Schutz, “Fabrication of continuous relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

T. R. Jay, M. B. Stern, “Preshaping photoresist for refractive microlens fabrication,” Opt. Eng. 33, 3552–3555 (1994).
[CrossRef]

J. A. Davis, D. M. Cottrell, “Random mask encoding of multiplexed phase-only and binary phase-only filters,” Opt. Lett. 19, 496–498 (1994).
[CrossRef] [PubMed]

R. W. Cohn, M. Liang, “Approximating fully complex spatial modulation with pseudorandom phase-only modulation,” Appl. Opt. 33, 4406–4415 (1994).
[CrossRef] [PubMed]

1991 (1)

1989 (1)

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[CrossRef]

1973 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

1971 (1)

Abushagur, M. A.

Cohn, R. W.

Cottrell, D. M.

Daley, R. C.

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer, Berlin, 1980), Chap. 6, pp. 291–366.

Dames, M. P.

Davis, J. A.

Dowling, R. J.

Gale, M. T.

M. T. Gale, M. Rossi, J. Pedersen, H. Schutz, “Fabrication of continuous relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 318, Eq. (3.382.4).

Hassebrook, L. G.

Henke, W.

B. Wagner, H. J. Quenzer, W. Henke, W. Hoppe, W. Pilz, “Microfabrication of complex surface topographies using grey-tone lithography,” Sens. Actuators A 46–47, 89–94 (1995).
[CrossRef]

Hoppe, W.

B. Wagner, H. J. Quenzer, W. Henke, W. Hoppe, W. Pilz, “Microfabrication of complex surface topographies using grey-tone lithography,” Sens. Actuators A 46–47, 89–94 (1995).
[CrossRef]

Jay, T. R.

T. R. Jay, M. B. Stern, “Preshaping photoresist for refractive microlens fabrication,” Opt. Eng. 33, 3552–3555 (1994).
[CrossRef]

Johnson, E. G.

Jones, A. L.

Kirk, J. P.

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, Chap. 3, pp. 119–232.

Lhamon, M. E.

Liang, M.

Liu, B.

McCormick, F. B.

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[CrossRef]

McKee, P.

O’Shea, D. C.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Pedersen, J.

M. T. Gale, M. Rossi, J. Pedersen, H. Schutz, “Fabrication of continuous relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Pilz, W.

B. Wagner, H. J. Quenzer, W. Henke, W. Hoppe, W. Pilz, “Microfabrication of complex surface topographies using grey-tone lithography,” Sens. Actuators A 46–47, 89–94 (1995).
[CrossRef]

Quenzer, H. J.

B. Wagner, H. J. Quenzer, W. Henke, W. Hoppe, W. Pilz, “Microfabrication of complex surface topographies using grey-tone lithography,” Sens. Actuators A 46–47, 89–94 (1995).
[CrossRef]

Rockward, W. S.

Rossi, M.

M. T. Gale, M. Rossi, J. Pedersen, H. Schutz, “Fabrication of continuous relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 318, Eq. (3.382.4).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

Schutz, H.

M. T. Gale, M. Rossi, J. Pedersen, H. Schutz, “Fabrication of continuous relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Stern, M. B.

T. R. Jay, M. B. Stern, “Preshaping photoresist for refractive microlens fabrication,” Opt. Eng. 33, 3552–3555 (1994).
[CrossRef]

Suleski, T. J.

Wagner, B.

B. Wagner, H. J. Quenzer, W. Henke, W. Hoppe, W. Pilz, “Microfabrication of complex surface topographies using grey-tone lithography,” Sens. Actuators A 46–47, 89–94 (1995).
[CrossRef]

Wood, D.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

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

Opt. Eng. (3)

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[CrossRef]

M. T. Gale, M. Rossi, J. Pedersen, H. Schutz, “Fabrication of continuous relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

T. R. Jay, M. B. Stern, “Preshaping photoresist for refractive microlens fabrication,” Opt. Eng. 33, 3552–3555 (1994).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

Sens. Actuators A (1)

B. Wagner, H. J. Quenzer, W. Henke, W. Hoppe, W. Pilz, “Microfabrication of complex surface topographies using grey-tone lithography,” Sens. Actuators A 46–47, 89–94 (1995).
[CrossRef]

Other (7)

Shipley Corporation Microposit Products Catalog, Marlboro, Mass.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 318, Eq. (3.382.4).

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, Chap. 3, pp. 119–232.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer, Berlin, 1980), Chap. 6, pp. 291–366.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (11)

Fig. 1
Fig. 1

Controlling specular intensity by varying surface roughness.

Fig. 2
Fig. 2

Array of diffusers that produces a custom complex-valued modulation.

Fig. 3
Fig. 3

Comparison of diffraction patterns from random encoding a 100×100 array of desired complex values to a 100×100 array of phase-only pixels and to a 100×100 array of diffuser pixels. Each diffuser pixel is a 3×3 array of phases that are randomly encoded to produce the same effective value of amplitude ap. Shown are gray-scale images of diffraction pattern intensity for arrays of (a) phase-only pixels, theory; (b) diffuser pixels, theory; (c) phase-only pixels, experiment; and (d) diffuser pixels, experiment. The on-axis or dc component [upper left of (c) and (d)] is primarily due to Fresnel reflection from the cover glass, which has not been antireflection coated for this liquid-crystal light valve.

Fig. 4
Fig. 4

Proximity exposure systems for producing complex-valued pixels. Phase offsets produced by (a) time-averaged recording of speckle patterns from a spinning diffuser and (b) adding a spatially uniform exposure, which, as shown, is derived from a single-mode optical fiber used as a point source.

Fig. 5
Fig. 5

Gray-scale intensity images of speckle patterns recorded at (a) 0 µm, (b) 100 µm, and (c) 500 µm past a 100-µm slit. The diameter of the speckle is approximately 2.5 µm. Patterns were imaged onto a 1/3-in. (0.85-cm) CCD camera by using a 40× microscope objective approximately 160 mm from the CCD. The images were then recorded with a video frame grabber.

Fig. 6
Fig. 6

Effective amplitude ap and phase ϕp for log and linear resists. For linear resist, which depends on absolute intensity, the x axis is defined to be γ=ψs/π.

Fig. 7
Fig. 7

Probability density functions for depths of speckle recorded into log and linear resists. Each distribution produces effective amplitude ap=0.025.

Fig. 8
Fig. 8

Effective complex amplitude for time-averaged recording of speckle in linear resist. Average recorded depth is proportional to average exposure energy Es. The amplitude and phase curves use the same style for a given value of M. For M of 80, 180, and 602, the effective phase curves are nearly identical and, for this reason, are plotted with a single style. The dots (●) indicate where effective phase is 2.5π.

Fig. 9
Fig. 9

Probability density functions (pdf's) for time-averaged recording in linear resist. Each pdf produces identical effective amplitude aP=0.025.

Fig. 10
Fig. 10

Effective amplitude for time-averaged recording in linear resist for a constant value of effective phase. The dots (●) indicate identical points from Fig. 8. The diamonds (♦) indicate points identical in amplitude but differing in phase by an integer multiple of 2π.

Fig. 11
Fig. 11

Experimental demonstration of speckle recording using a phase-only liquid-crystal light valve to represent a photoresist. The plots show how the effective amplitude curves change for (a) different levels of uniform bias Eb and (b) different speckle diameters. Specific values used in the experiment and the theory are given in Tables 2 and 3.

Tables (3)

Tables Icon

Table 1 Measures of Improvement of the Spot Array Design by Using Diffuser Pixels in Place of Single Phase-Only Pixels

Tables Icon

Table 2 Parameters Used for the Measured and Theoretical Curves in Fig. 11(a)

Tables Icon

Table 3 Parameters Used for the Measured and Theoretical Curvesa in Fig. 11(b)

Equations (30)

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A=F [a],
I=|A|2=|A|2+Is,
a= p(ψ)exp(jψ)dψ=ap exp(jϕp),
ap=sinc(ν/2π).
a(x, y)=exp[jψ(x, y)]=exp{j[αh(2πf0x)+ψα(x, y)]}.
acac exp(jψc)=J0(α)exp(jψα),
p(Is)=1Isexp-IsIs,
a=exp[j(ψb+arctanψs)]1+ψs2.
a=exp(jψb)0ψms p(ψ)exp(jψ)dψ+exp(jψms) ψms p(ψ)dψ,
a=exp[j(ψb+arctanψs)]1+ψs2×1+ψsexp[j(ψms-π/2)]exp-ψmsψs,
(ac)exp(-acψms),
ψtψmb+ψms=2π-(ln min)/amin,
(ac)=(min)ac/amin.
ψt=2π+2J0-1(amin),
t(E)=m ln(E/Eb),
ψt=ψb+ψs=α ln(Es+Eb),
ψs=α ln(1+Es/Eb).
a=exp(jψb) 0 exp(-x)exp[jα ln(1+γx)]dx=exp[j(ψb+α ln γ)]exp(1/γ)Γ(1+jα, 1/γ),
a=a+=a+exp(jψb)exp-γmsγ+jψms-γms/γ exp[-x+jα ln(1+γx)]dx,
p(ψs)=1αγexpψsα+1γ1-expψsγ.
aexp[j(ψb+α ln γ)]Γ(1+jα),
p(ψs)=ψsM-1Γ(M)MψsM exp-Mψsψs,
a=1+ψsM2-M/2 expjM arctan ψsM.
Esψs=Map-2/M-1.
ap=|cos(ϕp/M)|M.
p(ψs)12πMψsexp12Mψs-ψsψs2.
aexp(jψs)exp-ψs22M.
a=exp(jψb) 0 xM-1Γ(M)exp(-x)×expjα ln1+γxMdx.
aexp(jψb)2π- exp(-x2/2)×exp[jα ln(1+γ+γx/M)]dx.
aexp[jψb+jα ln(1+γ)]exp-12Mαγ1+γ2.

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