Abstract

We present a novel efficient gradient-based optimization algorithm for the design of diffractive optical elements (DOE’s) for synthetic spectra applications. Two design examples are given. The results demonstrate that the DOE’s obtained by the proposed algorithm can accurately produce the desired intensity spectra at a predetermined diffraction angle.

© 2003 Optical Society of America

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References

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  1. G. J. Swanson and W. B. Weldkamp, ???Diffractive optical elements for use in infrared systems,??? Opt. Eng. 28, 605???608 (1989).
  2. R. W. Gerchberg and W. O. Saxton, ???A practical algorithm for the determination of phase from image and diffraction plane pictures,??? Optik 35, 237???246 (1972).
  3. J. R. Fienup, ???Iterative method applied to image reconstruction and to computer-generated holograms,??? Opt. Eng. 19, 297???305 (1980).
  4. F. Wyrowski and O. Bryngdahl, ???Iterative Fourier transform algorithm applied to computer holography??? J. Opt. Soc. Am. A 5, 1058???1065 (1988).
    [CrossRef]
  5. J. Turunen, A. Vasara, and J. Westerholm, ???Kinoform phase relief synthesis: a stochastic method,??? Opt. Eng. 28, 1162???1176 (1989).
  6. G. Zhou, Y. Chen, Z. Wang, and H. Song, ???Genetic local search algorithm for optimization design of diffractive optical elements,??? Appl. Opt. 38, 4281???4290 (1999).
    [CrossRef]
  7. J. N. Mait, ???Understanding diffractive optic design in the scalar domain,??? J. Opt. Soc. Am. A 12, 2145???2158 (1995).
    [CrossRef]
  8. M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, ???Synthetic infrared spectra,??? Opt. Lett. 22, 1036???1038 (1997).
    [CrossRef] [PubMed]
  9. M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, ???Synthetic spectra: a tool for correlation spectroscopy,??? Appl. Opt. 36, 3342???3348 (1997).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63???90.
  11. M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, 161???167.

Appl. Opt. (2)

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

Opt. Eng. (3)

G. J. Swanson and W. B. Weldkamp, ???Diffractive optical elements for use in infrared systems,??? Opt. Eng. 28, 605???608 (1989).

J. R. Fienup, ???Iterative method applied to image reconstruction and to computer-generated holograms,??? Opt. Eng. 19, 297???305 (1980).

J. Turunen, A. Vasara, and J. Westerholm, ???Kinoform phase relief synthesis: a stochastic method,??? Opt. Eng. 28, 1162???1176 (1989).

Opt. Lett. (1)

Optik (1)

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

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63???90.

M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, 161???167.

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Figures (7)

Fig. 1.
Fig. 1.

Programmable grating-like DOE consisting of a large number of micromirror elements.

Fig. 2.
Fig. 2.

Target diffraction intensity spectrum of the first design example.

Fig. 3.
Fig. 3.

A part of the surface profile of the DOE designed for the synthesis of the spectrum shown in Fig. 2.

Fig. 4.
Fig. 4.

Diffracted intensity spectrum generated by the programmable grating-like DOE with the displacement of each micromirror set to the designed value.

Fig. 5.
Fig. 5.

Target diffraction intensity spectrum of the second design example.

Fig. 6.
Fig. 6.

A part of the surface profile of the DOE designed for the synthesis of the spectrum shown in Fig. 5.

Fig.7.
Fig.7.

Diffracted intensity spectrum generated by the programmable grating-like DOE with the displacement of each micromirror set to the designed value.

Equations (28)

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U ( x , λ ) = A m = 1 M exp ( i 4 π d m λ ) rect ( x m Δ + Δ 2 Δ ) ,
U ( λ ) = CA λ U ( x , λ ) exp [ i 2 π sin ( θ ) x λ ] dx ,
U ( λ ) = CA Δ λ sinc [ Δ sin ( θ ) λ ] exp [ i π Δ sin ( θ ) λ ]
× m = 1 M exp ( i 4 π d m λ ) exp [ i 2 π sin ( θ ) m Δ λ ] .
I n = U n 2 ,
U n = m = 1 M G nm exp ( i 4 π d m u n ) ,
G nm = u n CA Δ sinc [ Δ sin ( θ ) u n ] exp [ i 2 π Δ sin ( θ ) m u n ] .
E = n = 1 N ( I n d γ I n ) 2 ,
γ = n = 1 N I n I n d n = 1 N I n 2 .
min d 1 , d 2 , , d M { E ( d 1 , d 2 , , d M ) } .
0 d m λ max 2 , n = 1 , 2 , , M ,
E d m = k = 1 N E I k I k d m .
E I k = n = 1 N 2 ( γ I n I n d ) ( γ I n I k + I n γ I k ) ,
γ I k = I k d 2 γ I k n = 1 N I n 2 ,
I n I k = δ nk ,
δ nk = { 1 , n = k 0 n k .
E I k = 2 γ ( γ I k I k d ) .
I k d m = d m ( U k U k * ) = U k U k * d m + U k * U k d m = 2 Re { U k U k * d m } ,
U k * d m = i 4 π u k G km * exp ( i 4 π d m u k ) ,
E d m = 16 Re { i π k = 1 N γ ( γ I k I k d ) U k u k G km * exp ( i 4 π d m u k ) } .
Φ k + 1 = Φ k + s k d k ,
F ( Φ k + s k d k ) = min s { F ( Φ k + s d k ) } .
d k = H k F ( Φ k ) , H 0 = I ,
H k + 1 = H k + A k ,
y k = Φ k + 1 Φ k ,
z k = F ( Φ k + 1 ) F ( Φ k ) ,
A k = y k y k T y k T z k H k z k z k T H k T z k T H k z k .
η = I ( u p ) I 0 ( u p ) × 100 % ,

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