Abstract

We propose a method of designing two-dimensional random surfaces that scatter light uniformly within a specified range of angles and produce no scattering outside that range. The method is first tested by means of computer simulations. Then a procedure for fabricating such structures on photoresist is described, and light-scattering measurements with the fabricated samples are presented. The results validate the design procedure and show that the fabrication method is feasible.

© 2001 Optical Society of America

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References

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  1. C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am. 62, 929–989 (1972).
  2. C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am. 63, 1080–1092 (1973).
    [CrossRef]
  3. Y. Nakayama, M. Kato, “Linear recording of Fourier transform holograms using a pseudorandom diffuser,” Appl. Opt. 21, 1410–1418 (1982).
    [CrossRef] [PubMed]
  4. M. Kowalczyk, “Spectral and imaging properties of uniform diffusers,” J. Opt. Soc. Am. A 1, 192–200 (1984).
    [CrossRef]
  5. See, e.g., Thorlabs Inc.Newton, New Jersey 07860-0366.
  6. Physical Optics Corp., Torrance, Calif. 90501-1821.
  7. T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
    [CrossRef]
  8. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 20.
  9. A. A. Maradudin, I. Simonsen, T. A. Leskova, E. R. Méndez are preparing a manuscript to be called “Design of one-dimensional Lambertian diffusers of light.”
  10. W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–387 (1977).
    [CrossRef]
  11. Z. H. Gu, H. M. Escamilla, E. R. Méndez, A. A. Maradudin, J. Q. Lu, T. Michel, M. Nieto-Vesperinas, “Interaction of two optical beams at a symmetric random surface,” Appl. Opt. 31, 5878–5889 (1992).
    [CrossRef] [PubMed]
  12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 13.
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 281–282.
  14. K. A. O’Donnell, E. R. Méndez, “An experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  15. R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
    [CrossRef]
  16. E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
    [CrossRef]

1998 (1)

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

1995 (1)

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

1992 (1)

1987 (1)

1984 (1)

1982 (1)

1981 (1)

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

1977 (1)

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–387 (1977).
[CrossRef]

1973 (1)

1972 (1)

C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am. 62, 929–989 (1972).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 20.

DePalma, J. J.

Escamilla, H. M.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 281–282.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 13.

Gu, Z. H.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Z. H. Gu, H. M. Escamilla, E. R. Méndez, A. A. Maradudin, J. Q. Lu, T. Michel, M. Nieto-Vesperinas, “Interaction of two optical beams at a symmetric random surface,” Appl. Opt. 31, 5878–5889 (1992).
[CrossRef] [PubMed]

Hoadley, H. O.

Jakeman, E.

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

Kato, M.

Kowalczyk, M.

Kurtz, C. N.

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am. 63, 1080–1092 (1973).
[CrossRef]

C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am. 62, 929–989 (1972).

Leskova, T. A.

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

A. A. Maradudin, I. Simonsen, T. A. Leskova, E. R. Méndez are preparing a manuscript to be called “Design of one-dimensional Lambertian diffusers of light.”

Lu, J. Q.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Z. H. Gu, H. M. Escamilla, E. R. Méndez, A. A. Maradudin, J. Q. Lu, T. Michel, M. Nieto-Vesperinas, “Interaction of two optical beams at a symmetric random surface,” Appl. Opt. 31, 5878–5889 (1992).
[CrossRef] [PubMed]

Luna, R. E.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Maradudin, A. A.

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

Z. H. Gu, H. M. Escamilla, E. R. Méndez, A. A. Maradudin, J. Q. Lu, T. Michel, M. Nieto-Vesperinas, “Interaction of two optical beams at a symmetric random surface,” Appl. Opt. 31, 5878–5889 (1992).
[CrossRef] [PubMed]

A. A. Maradudin, I. Simonsen, T. A. Leskova, E. R. Méndez are preparing a manuscript to be called “Design of one-dimensional Lambertian diffusers of light.”

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

Méndez, E. R.

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Z. H. Gu, H. M. Escamilla, E. R. Méndez, A. A. Maradudin, J. Q. Lu, T. Michel, M. Nieto-Vesperinas, “Interaction of two optical beams at a symmetric random surface,” Appl. Opt. 31, 5878–5889 (1992).
[CrossRef] [PubMed]

K. A. O’Donnell, E. R. Méndez, “An experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

A. A. Maradudin, I. Simonsen, T. A. Leskova, E. R. Méndez are preparing a manuscript to be called “Design of one-dimensional Lambertian diffusers of light.”

Michel, T.

Nakayama, Y.

Nieto-Vesperinas, M.

Novikov, I. V.

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

O’Donnell, K. A.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 281–282.

Shchegrov, A. V.

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

Simonsen, I.

A. A. Maradudin, I. Simonsen, T. A. Leskova, E. R. Méndez are preparing a manuscript to be called “Design of one-dimensional Lambertian diffusers of light.”

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 20.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 281–282.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 281–282.

Welford, W. T.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–387 (1977).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

Appl. Phys. Lett. (1)

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

J. Mod. Opt. (1)

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am. 63, 1080–1092 (1973).
[CrossRef]

C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am. 62, 929–989 (1972).

J. Opt. Soc. Am. A (2)

Opt. Quantum Electron. (1)

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–387 (1977).
[CrossRef]

Other (6)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 13.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 281–282.

See, e.g., Thorlabs Inc.Newton, New Jersey 07860-0366.

Physical Optics Corp., Torrance, Calif. 90501-1821.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 20.

A. A. Maradudin, I. Simonsen, T. A. Leskova, E. R. Méndez are preparing a manuscript to be called “Design of one-dimensional Lambertian diffusers of light.”

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Figures (13)

Fig. 1
Fig. 1

Schematic diagram of the scattering geometry for (a) reflection by a surface and (b) transmission through a dielectric diffuser.

Fig. 2
Fig. 2

Functions s(x) and d(x) for the case n = 2.

Fig. 3
Fig. 3

Illustration of the sum represented by Eq. (28) with n = 1. At any one of the segments of length b shown in the figure there is only one nonzero contribution to ζ′(x).

Fig. 4
Fig. 4

Numerical generation of a surface profile and its derivative. The parameters used are b = 60 µm, n = 1, and θ m = 2h = 8°.

Fig. 5
Fig. 5

Mean differential reflection coefficient estimated from N p = 6000 realizations of the surface-profile function for the case of normal incidence and a wavelength λ = 0.6328 µm. The parameters used are b = 60 µm, n = 1, and θ m = 2m = 5°. The sampling on the surface was Δx = λ/5, and the size of the surface was L = 2000λ.

Fig. 6
Fig. 6

Same as Fig. 5 but with random deviates c j for the generation of the surface along x 1 drawn from the distribution given by Eq. (41) with ∊ = 0.01. The illumination wavelengths are (a) λ = 0.6328 µm, (b) λ = 0.532 µm, and (c) λ = 0.442 µm.

Fig. 7
Fig. 7

Realization of a two-dimensional surface profile, ζ(x ) = ζ1(x 1) + ζ2(x 2), with b 1 = b 2 = 60 µm, n 1 = n 2 = 1, and θ1m = θ2m = 5°.

Fig. 8
Fig. 8

Mean differential reflection coefficient estimated from N p = 2000 realizations of the surface-profile function for the case of normal incidence and a wavelength λ = 0.6328 µm. The parameters used are b 1 = b 2 = 60 µm, n 1 = n 2 = 1, and θ1m = θ2m = 5°. The sampling on the surface was Δx 1 = Δx 2 = λ/5, and the size of the surface was L 1 = L 2 = 2000λ.

Fig. 9
Fig. 9

Schematic diagram of the experimental arrangement used for the fabrication of the diffusers.

Fig. 10
Fig. 10

Measured profile that illustrates the experimental realization of the function s(x). The profile was estimated by means of a Dektakst mechanical profilometer and has parameters H ≈ 7 µm, b ≈ 15 µm, and n = 1. The ideal profile is shown by the dashed curve.

Fig. 11
Fig. 11

Measured segment of the surface profile of a fabricated sample. The parameters are b ≈ 60 µm and n = 0.

Fig. 12
Fig. 12

Mean intensity (unnormalized differential reflection coefficient) produced by a sample with b ≈ 6 µm and n = 1. The experiment was done in transmission, and the sample was illuminated at normal incidence.

Fig. 13
Fig. 13

Photograph of the transmission scattering pattern produced by a two-dimensional diffuser with b 1 = b 2 ≈ 6 µm and n 1 = n 2 = 1.

Equations (46)

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ψxinc=ψ0 expik·x-iα0kx3,
ψxsc=-d2q2π2 Rq|kexpiq·x+iα0qx3,
Rq|k=12iα0q-d2x exp-iq·x-iα0qζxFx.
Fx=2-ζx1x1-ζx2x2+x3ψxincx3=ζx.
Rq|k=-ψ0α0q-d2xζx1 k1+ζx2 k2+α0k×exp-iq-k·x-iα0q+α0kζx.
Rq|k=ψ0F3q, k-d2x exp-iq-k·x-ikaq, kζx,
F3q, k=-ω/c2+α0qα0k-q·kα0qα0q+α0k
ϕx=kaζx=a ωc ζx,
a=2in reflectionn0-ngin transmission,
Rq|kψ0κ0-d2x exp-iq-k·x-ikaζx,
Rq|kψ0κ0R1q1|k1R2q2|k2,
Rjqj|kj=-dxj exp-iqj-kjxj-ikaζjxj,
RΩs=1|ψ0|21L1L2ω/c2π2α02qα0k Rq|k2,
RΩs|κ0|2I1q1|k1λL1I2q2|k2λL2,
Ijqj|kj=|Rjqj|kj|2.
Ijqj|kj=-dxj-dxj exp-iqj-kjxj-xj×exp-ikaζjxj-ζjxj.
Iq|k=-dx -du expiq-ku×exp-ikaζx-ζx+u.
Iq|k=L -du expiq-kugu,
gu=exp-ikaζx-ζx+u.
Iq|k=K 12θmrectθs-θ02θmK 12qmrectq-k2qm,
gu=exp-ikauζx,
Iq|k=L -duexp-ikauζxexpiq-ku=2πLka Pζq-kka,
Iq|k2πLka Pζθs-θ0a.
gu=exp-δ2ka21-cu,
Iq|k-dx -du expiq-ku×exp-ikauζx.
ζx=j=- cjsx-2jb,
sx=0for x-n+1b-n+1H-mxfor -n+1b<x<-nb-Hfor -nbxnb-n+1H+mxfor nb<x<n+1b0for n+1bx,
ζx=j=- cjdx-2jb,
dx=0for x-n+1b-mfor -n+1b<x<-nb0for -nbxnbmfor nb<x<n+1b0for n+1bx,
expikauζx=expikau j=- cjdx-2jb=j=-expikaucjdx-2jb=j=- expikaucjdx-2jb,
expikauζx=expikaumcj=- fγexpikaumγdγ,
expikauζx=exp-ikaumcj=- fγexp-ikaumγdγ.
Iq|k=j2jb2j+1bdx -du expiq-ku×- fγexpikaumγdγ+j2j-1b2jbdx -du expiq-ku×- fγexp-ikaumγdγ.
Iq|k=L2-du expiq-ku- fγexpikaγmu+exp-ikaγmudγ.
Iq|k=πL -dγfγδq-k+kaγm+δq-k-kaγm=πLkamfk-qkam+fq-kkam.
RΩs=|κ0|212am1f1k1-q1kam1+f1q1-k1kam1×12am2f2k2-q2kam2+f2q2-k2kam2,
RΩs=|κ0|212am1f1θ10-θ1sam1+f1θ1s-θ10am1×12am2f2θ20-θ2sam2+f2θ2s-θ20am2,
fγ=rectγ-12,
RΩs=|κ0|212am1rectθ10-θ1sam1-12+rectθ1s-θ10am1-12×12am2rectθ20-θ2sam2-12+rectθ2s-θ20am2-12
=|κ0|212θ1mrectθ10-θ1s2θ1m×12θ2mrectθ20-θ2s2θ2m,
fγ=rectγ-12+,
Rθ1s=12θ1mrectθ10-θ1sθ1m-12++rectθ1s-θ10θ1m-12+.
Isx-x0I0rectx-x0l*,
Ex=-T/2T/2 Isx-vtdt,
Ex=E0b-rectsbrectx-s2n+1bds,
Ex=0for x-n+1b-n+1E0-E0/bxfor -n+1b<x<-nb-E0for -nbxnb-n+1E0+E0/bxfor nb<x<n+1b0for n+1bx.

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