Abstract

We present a design approach for diffractive phase elements (DPE’s) that generate point/ring patterns in optical systems illuminated by monochromatic and dual-wavelength light based on the general theory of phase retrieval. We carry out the numerical simulations for several diffractive patterns and appraise the performance of the designed DPE’s. The results show that the designed DPE’s can generate the desired point/ring patterns with a large signal-to-noise ratio, high average diffraction efficiency, and low color cross talk.

© 1998 Optical Society of America

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References

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  1. See feature issue on “Diffractive Optics: Design, Fabrication, and Applications,” Appl. Opt. 32, 14 (1993).
  2. See special issue on “Diffractive Optics,” J. Mod. Opt. 40, 4 (1993).
  3. F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
    [CrossRef]
  4. See special issue on Diffractive Optics Modeling, J. Opt. Soc. Am. A 12(5) (1995).
  5. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  6. H. Dammann, “Color separation gratings,” Appl. Opt. 17, 2273–2279 (1978).
    [CrossRef] [PubMed]
  7. M. Kato, K. Sakuda, “Computer-generated holograms: application to intensity variable and wavelength demultiplexing holograms,” Appl. Opt. 31, 630–635 (1992).
    [CrossRef] [PubMed]
  8. Y. Ishii, T. Kubota, “Wavelength demultiplexer in multimode fiber that uses optimized holographic optical elements,” Appl. Opt. 32, 4415–4422 (1993).
    [CrossRef] [PubMed]
  9. Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
    [CrossRef]
  10. A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 18, 534–536 (1993).
    [CrossRef] [PubMed]
  11. M. W. Farn, M. B. Stern, W. B. Weldkamp, S. S. Medeiros, “Color separation by use of binary optics,” Opt. Lett. 18, 1214–1216 (1993).
    [CrossRef] [PubMed]
  12. G. Yang, B. Gu, X. Tan, M.-P. Chang, B. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. A 11, 1632–1640 (1994).
    [CrossRef]
  13. B. Y. Gu, G. Z. Yang, B. Z. Dong, M. P. Chang, O. K. Ersoy, “Diffractive-phase-element design that implements several optical functions,” Appl. Opt. 34, 2564–2570 (1995).
    [CrossRef] [PubMed]
  14. M. P. Chang, O. K. Ersoy, B. Dong, G. Yang, B. Gu, “Iterative optimization of diffractive phase elements simultaneously implementing several optical functions,” Appl. Opt. 34, 3069–3076 (1995).
    [CrossRef] [PubMed]
  15. J. E. Ford, F. Xu, Y. Fainman, “Wavelength-selective planar hologram,” Opt. Lett. 21, 80–82 (1996).
    [CrossRef] [PubMed]
  16. G. Zhang, G. Yang, B. Gu, “Design of diffractive phase elements that produce focal annuli: a new method,” Appl. Opt. 34, 8110–8115 (1995).
    [CrossRef] [PubMed]
  17. B. Dong, G. Yang, B. Gu, G. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14, 44–48 (1997).
    [CrossRef]
  18. P. Blattner, H. P. Herzig, K. J. Weible, “Diffractive optical elements for tracking and receiving in optical space communication systems,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 132–134.
  19. B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
    [CrossRef] [PubMed]
  20. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
    [CrossRef] [PubMed]
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 4, p. 60.

1997 (1)

B. Dong, G. Yang, B. Gu, G. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14, 44–48 (1997).
[CrossRef]

1996 (1)

1995 (5)

1994 (2)

1993 (7)

1992 (1)

M. Kato, K. Sakuda, “Computer-generated holograms: application to intensity variable and wavelength demultiplexing holograms,” Appl. Opt. 31, 630–635 (1992).
[CrossRef] [PubMed]

1986 (1)

1978 (1)

Amitai, Y.

Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
[CrossRef]

Blattner, P.

P. Blattner, H. P. Herzig, K. J. Weible, “Diffractive optical elements for tracking and receiving in optical space communication systems,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 132–134.

Chang, M. P.

B. Y. Gu, G. Z. Yang, B. Z. Dong, M. P. Chang, O. K. Ersoy, “Diffractive-phase-element design that implements several optical functions,” Appl. Opt. 34, 2564–2570 (1995).
[CrossRef] [PubMed]

M. P. Chang, O. K. Ersoy, B. Dong, G. Yang, B. Gu, “Iterative optimization of diffractive phase elements simultaneously implementing several optical functions,” Appl. Opt. 34, 3069–3076 (1995).
[CrossRef] [PubMed]

Chang, M.-P.

Dammann, H.

Dong, B.

B. Dong, G. Yang, B. Gu, G. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14, 44–48 (1997).
[CrossRef]

M. P. Chang, O. K. Ersoy, B. Dong, G. Yang, B. Gu, “Iterative optimization of diffractive phase elements simultaneously implementing several optical functions,” Appl. Opt. 34, 3069–3076 (1995).
[CrossRef] [PubMed]

G. Yang, B. Gu, X. Tan, M.-P. Chang, B. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. A 11, 1632–1640 (1994).
[CrossRef]

B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
[CrossRef] [PubMed]

Dong, B. Z.

Ersoy, O. K.

Fainman, Y.

Farn, M. W.

Ford, J. E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 4, p. 60.

Gu, B.

B. Dong, G. Yang, B. Gu, G. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14, 44–48 (1997).
[CrossRef]

G. Zhang, G. Yang, B. Gu, “Design of diffractive phase elements that produce focal annuli: a new method,” Appl. Opt. 34, 8110–8115 (1995).
[CrossRef] [PubMed]

M. P. Chang, O. K. Ersoy, B. Dong, G. Yang, B. Gu, “Iterative optimization of diffractive phase elements simultaneously implementing several optical functions,” Appl. Opt. 34, 3069–3076 (1995).
[CrossRef] [PubMed]

G. Yang, B. Gu, X. Tan, M.-P. Chang, B. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. A 11, 1632–1640 (1994).
[CrossRef]

B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
[CrossRef] [PubMed]

Gu, B. Y.

Herzig, H. P.

P. Blattner, H. P. Herzig, K. J. Weible, “Diffractive optical elements for tracking and receiving in optical space communication systems,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 132–134.

Ishii, Y.

Kato, M.

M. Kato, K. Sakuda, “Computer-generated holograms: application to intensity variable and wavelength demultiplexing holograms,” Appl. Opt. 31, 630–635 (1992).
[CrossRef] [PubMed]

Kewitsch, A.

Kubota, T.

Mait, J. N.

Medeiros, S. S.

Sakuda, K.

M. Kato, K. Sakuda, “Computer-generated holograms: application to intensity variable and wavelength demultiplexing holograms,” Appl. Opt. 31, 630–635 (1992).
[CrossRef] [PubMed]

Segev, M.

Stern, M. B.

Tan, X.

Weible, K. J.

P. Blattner, H. P. Herzig, K. J. Weible, “Diffractive optical elements for tracking and receiving in optical space communication systems,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 132–134.

Weldkamp, W. B.

Wyrowski, F.

Xu, F.

Yang, G.

B. Dong, G. Yang, B. Gu, G. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14, 44–48 (1997).
[CrossRef]

G. Zhang, G. Yang, B. Gu, “Design of diffractive phase elements that produce focal annuli: a new method,” Appl. Opt. 34, 8110–8115 (1995).
[CrossRef] [PubMed]

M. P. Chang, O. K. Ersoy, B. Dong, G. Yang, B. Gu, “Iterative optimization of diffractive phase elements simultaneously implementing several optical functions,” Appl. Opt. 34, 3069–3076 (1995).
[CrossRef] [PubMed]

G. Yang, B. Gu, X. Tan, M.-P. Chang, B. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. A 11, 1632–1640 (1994).
[CrossRef]

B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
[CrossRef] [PubMed]

Yang, G. Z.

Yariv, A.

Zhang, G.

B. Dong, G. Yang, B. Gu, G. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14, 44–48 (1997).
[CrossRef]

G. Zhang, G. Yang, B. Gu, “Design of diffractive phase elements that produce focal annuli: a new method,” Appl. Opt. 34, 8110–8115 (1995).
[CrossRef] [PubMed]

Zhuang, J. Y.

Appl. Opt. (2)

M. Kato, K. Sakuda, “Computer-generated holograms: application to intensity variable and wavelength demultiplexing holograms,” Appl. Opt. 31, 630–635 (1992).
[CrossRef] [PubMed]

M. P. Chang, O. K. Ersoy, B. Dong, G. Yang, B. Gu, “Iterative optimization of diffractive phase elements simultaneously implementing several optical functions,” Appl. Opt. 34, 3069–3076 (1995).
[CrossRef] [PubMed]

Appl. Opt. (7)

J. Mod. Opt. (1)

See special issue on “Diffractive Optics,” J. Mod. Opt. 40, 4 (1993).

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

B. Dong, G. Yang, B. Gu, G. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14, 44–48 (1997).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
[CrossRef]

Opt. Lett. (3)

Other (2)

P. Blattner, H. P. Herzig, K. J. Weible, “Diffractive optical elements for tracking and receiving in optical space communication systems,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 132–134.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 4, p. 60.

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Figures (5)

Fig. 1
Fig. 1

Schematic of polychromatic diffractive optical system.

Fig. 2
Fig. 2

DPE generating the pattern of one focal point and one focusing ring in a monochromatic-wave illumination system with a wavelength of λ = 632.8   nm . (a) Profile of the modulation depth of the designed DPE with eight quantized levels, (b) transverse cross section of the intensity distribution on the output plane.

Fig. 3
Fig. 3

DPE generating the pattern of one focal point and five focusing rings in a monochromatic illuminating system with a wavelength of λ = 632.8   nm . (a) Profile of the modulation depth of the designed DPE with eight quantized levels, (b) transverse cross section of the intensity distribution over the output plane.

Fig. 4
Fig. 4

DPE generating the pattern of the color point/ring pattern in a dual-wavelength illuminating optical system. (a) Profile of the surface-relief depth of the designed DPE with eight quantized levels, (b) transverse cross section of the intensity distribution for the diffractive pattern of one focal point with λ 1 = 514.5   nm and one focusing annulus with λ 2 = 632.8   nm , (c) transverse cross section of the intensity distribution for another diffractive pattern of one focal point with λ 2 = 632.8   nm and one focusing annulus with λ 1 = 514.5   nm , reversing the color of the point and ring in the diffraction pattern with respect to (b). All parameters in (c) are the same as in (b) except for the radius of the ring, r 2 = 6.25   mm .

Fig. 5
Fig. 5

Generation of the color point/ring pattern in the dual-wavelength illuminating optical system. (a) Profile of the surface-relief depth of the designed DPE with 16 quantized levels, (b) transverse cross section of the intensity distribution for the diffractive pattern of one focal point with λ 2 = 632.8   nm and one focusing annulus with λ 1 = 590.0   nm .

Equations (25)

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U 1 α = U 1 ( r 1 ,   λ α ) = ρ 1 ( r 1 ,   λ α ) exp [ i ϕ 1 ( r 1 ,   λ α ) ] ,
U 2 α = U 2 ( r 2 ,   λ α ) = ρ 2 ( r 2 ,   λ α ) exp [ i ϕ 2 ( r 2 ,   λ α ) ] .
U 2 ( r 2 ,   λ α ) = G ˆ ( r 2 ,   r 1 ,   λ α ) U 1 ( r 1 ,   λ α ) ,
U 1 j ( λ α ) = ρ 1 j α   exp [ i 2 π h 1 j ( n s α - 1 ) / λ α ] ,
j = 1 ,   2 ,   3 N 1 ,
U 2 k α = ρ 2 k α   exp ( i ϕ 2 k α ) , k = 1 ,   2 ,   3 N 2 r ,
U 2 k α = j = 1 N 1 G kj ( λ α ) U 1 j ( λ α ) .
D 2 = α = 1 1   or   2 | U 2 α - G ˆ ( λ α ) U 1 α | 2 = 1 N 2 α = 1 1   or   2 i ρ 2 i α 2 + i , j ρ 1 i α ρ 1 j α A ij ( λ α ,   λ α ) × exp [ - i 2 π ( h 1 i - h 1 j ) ( n s α - 1 ) / λ α ] - i , j { ρ 2 i α ρ 1 j α G ij ( λ α ) exp [ - i ( ϕ 2 i α - 2 π h 1 j ( n s α - 1 ) / λ α ) ] + c . c . } ,
δ h 1 D 2 = 0 , δ ϕ 2 γ D 2 = 0 ,
D 2 h 1 k = i N 2 α ( 2 π ( n s α - 1 ) / λ α ) j { ρ 1 j α ρ 1 k α A jk ( λ α ) × exp [ - i 2 π ( h 1 j - h 1 k ) ( n s α - 1 ) / λ α ] - c . c . } - j { ρ 2 j α ρ 1 k α G jk ( λ α ) exp [ - i ( ϕ 2 j α - 2 π h 1 k ( n s α - 1 ) / λ α ) ] - c . c . } = 0 .
Im [ Q k   exp ( i 2 π h 1 k ( n 0 - 1 ) / λ 0 ) ] = 0 .
exp [ i 2 π h 1 k ( n 0 - 1 ) / λ 0 ] = Q ˜ k * | Q ˜ k | , k = 1 ,   2 ,   3 N 1 ,
Q ˜ k = α [ 2 π ( n s α - 1 ) / λ α ] × j k ρ 1 j α   exp [ - i 2 π h 1 j ( n s α - 1 ) / λ α ] × A jk ( λ α ) - j ρ 2 j α   exp ( - i ϕ 2 j α ) G jk ( λ α ) ρ 1 k α × exp i [ 2 π h 1 k ( n 0 - 1 ) / λ 0 ] × λ 0 λ α ( n s α - 1 ) ( n 0 - 1 ) - 1 ,
D 2 ϕ 2 γ = i N 2 ρ 2 k γ   exp ( - i ϕ 2 k γ ) j G kj ( λ γ ) ρ 1 j γ × exp [ i 2 π h 1 j ( n s γ - 1 ) / λ γ ] - c . c . = 0 ,
Im ρ 2 k γ   exp ( - i ϕ 2 k γ ) j G kj ( λ γ ) ρ 1 j γ × exp [ i 2 π h 1 j ( n s γ - 1 ) / λ γ ] = 0 .
exp ( i ϕ 2 k γ )
= j G kj ( λ γ ) ρ 1 j γ   exp [ i 2 π h 1 j ( n s γ - 1 ) / λ γ ] j G kj ( λ γ ) ρ 1 j γ   exp [ i 2 π h 1 j ( n s γ - 1 ) / λ γ ] ,
k = 1 ,   2 ,   3 N 2 r .
j | h ˜ 1 j ( 0 ,   m ) - h ˜ 1 j ( 0 ,   m + 1 ) | 1 ,
SSE = γ , k | ρ 2 k γ   exp ( i ϕ ˜ 2 k γ ( n ) ) - j G kj ( λ γ ) ρ 1 j γ × exp [ i 2 π ( n s γ - 1 ) h ˜ 1 j ( n ,   0 ) ] | 2 γ , k ρ 2 k γ 2 ,
G ( r 2 ,   r 1 ; l ,   λ α ) = 2 π i λ α l exp ( i 2 π l / λ α ) × exp i π ( r 2 2 + r 1 2 ) λ α l × J 0 2 π r 2 r 1 λ α l r 1 ,
I ( r 2 ,   λ α ) = 0 R 1 m G ( r 2 ,   r 1 ; l ,   λ α ) × ρ 1 α   exp [ i 2 π h 1 ( r 1 ) ( n s α - 1 ) / λ α ] d r 1 2 .
SNR = 1 N λ α   s I ( r 2 s ,   λ α ) n s I ( r 2 n ,   λ α ) / ( N 2 r - 1 ) ,
η = 1 N λ α   s signal I ( r 2 s ,   λ α ) n I ( r 2 n ,   λ α ) .
CCT = 1 N λ α   β α I ( r 2 s ,   λ β ) I ( r 2 s ,   λ α ) .

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