Abstract

A new optimization method based on the general theory of amplitude-phase retrieval is proposed for designing the diffractive phase elements (DPE's) that produce focal annular patterns. A set of equations for determining the phase distribution of the DPE is given. The profile of a surface-relief DPE can be designed with an iterative algorithm. Numerical calculations are carried out for several examples. A comparison of the performance of the DPE's designed with the Gerchberg–Saxton algorithm and the new algorithm is presented. The effect of quantization of the phase distribution of the DPE's on the results is also investigated. The results show that the new algorithm can successfully achieve the design of the DPE's that convert the uniform incident beam into the focal annular patterns.

© 1995 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
  13. G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitudephase retrieval problem in an optical system involved non-unitary transformation,” Optik 75, 68–74 (1987).
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    [Crossref]
  15. 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]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968). Chap. 4, p. 58, [Eq. (4-3)]; p. 60 [Eq. (4-9)].
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    [Crossref]

1994 (3)

1993 (4)

J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2542 (1993).
[Crossref] [PubMed]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[Crossref] [PubMed]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Modern Opt. 40, 761–769 (1993).
[Crossref]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitudephase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3152–3224 (1993).
[Crossref]

1992 (1)

1991 (2)

1987 (1)

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitudephase retrieval problem in an optical system involved non-unitary transformation,” Optik 75, 68–74 (1987).

1986 (1)

1982 (2)

1974 (1)

1972 (1)

R. W. Gerchburg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Bernhardt, M.

Bryngdahl, O.

Cordingley, J.

Dandliker, R.

Dong, B.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitudephase retrieval problem in an optical system involved non-unitary transformation,” Optik 75, 68–74 (1987).

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

Dong, B. Z.

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]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitudephase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3152–3224 (1993).
[Crossref]

Ersoy, O. K.

Fienup, J. R.

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[Crossref] [PubMed]

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).

Gale, M. T.

Gerchburg, R. W.

R. W. Gerchburg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968). Chap. 4, p. 58, [Eq. (4-3)]; p. 60 [Eq. (4-9)].

Gu, B.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitudephase retrieval problem in an optical system involved non-unitary transformation,” Optik 75, 68–74 (1987).

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

Gu, B. Y.

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]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitudephase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3152–3224 (1993).
[Crossref]

Herzig, H. P.

Kastner, C. J.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Modern Opt. 40, 761–769 (1993).
[Crossref]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Modern Opt. 40, 761–769 (1993).
[Crossref]

Piestun, R.

Prongue, D.

Rosen, J.

Roux, F. S.

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[Crossref]

Saxton, W. O.

R. W. Gerchburg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shamir, J.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Modern Opt. 40, 761–769 (1993).
[Crossref]

Veldkamp, W. B.

Wang, L.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitudephase retrieval problem in an optical system involved non-unitary transformation,” Optik 75, 68–74 (1987).

Wyrowski, F.

Yang, G.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitudephase retrieval problem in an optical system involved non-unitary transformation,” Optik 75, 68–74 (1987).

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

Yang, G. Z.

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]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitudephase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3152–3224 (1993).
[Crossref]

Yariv, A.

Zhuang, J. Y.

Appl. Opt. (8)

Int. J. Mod. Phys. B (1)

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitudephase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3152–3224 (1993).
[Crossref]

J. Modern Opt. (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Modern Opt. 40, 761–769 (1993).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[Crossref]

Opt. Lett. (2)

Optik (2)

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitudephase retrieval problem in an optical system involved non-unitary transformation,” Optik 75, 68–74 (1987).

R. W. Gerchburg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968). Chap. 4, p. 58, [Eq. (4-3)]; p. 60 [Eq. (4-9)].

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a diffractive optical system for converting a uniform incident beam into focal annular patterns.

Fig. 2
Fig. 2

Amplitude profile of a single focal annular with the central radius r 0 = 0.712 mm and width dr 0 = 0.057 mm: (a) the amplitude profile of the desired focal annular, (b) the calculated amplitude distribution of the focal annular generated by the designed DPE.

Fig. 3
Fig. 3

Phase distribution of the designed DPE. The relevant parameters are the same as those of Fig. 2.

Fig. 4
Fig. 4

Output amplitude distribution generated by the designed DPE with the quantized phase with order k = 5.

Fig. 5
Fig. 5

Amplitude profile of a single focal annular with central radius r 0 = 0.712 mm and width dr 0 = 0.114 mm: (a) the amplitude profile of the desired focal annular, (b) the calculated amplitude of the focal annular generated by the designed DPE.

Fig. 6
Fig. 6

Amplitude profile for multiple focal annuli: (a) the calculated amplitude for two focal annuli, (b) the calculated amplitude for three focal annuli.

Fig. 7
Fig. 7

Calculated amplitude for a single focal annular with the central radius r 0 = 0.712 mm and width dr 0 = 0.057 mm: (a) the designed result with the GS algorithm, starting with a random initial-phase function; (b) the designed result with the YG algorithm with a random initial-phase function.

Fig. 8
Fig. 8

Amplitude profile for two focal annuli: (a) the designed result with the GS algorithm, starting with a random initial-phase function; (b) the designed result with the YG algorithm with a random initial-phase function.

Equations (17)

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U 1 ( X 1 ) = ρ 1 ( X 1 ) exp [ i ϕ 1 ( X 1 ) ] ,
U 2 ( X 2 ) = ρ 2 ( X 2 ) exp [ i ϕ 2 ( X 2 ) ] ,
U 2 ( X 2 ) = G ( X 2 , X 1 ) U 1 ( X 1 ) d X 1 .
U 2 ( X 2 ) = G U 1 ( X 1 ) ,
U 1 n = ρ 1 n exp ( i ϕ 1 n ) , U 2 m = ρ 2 m exp ( i ϕ 2 m ) , U 2 m = n = 1 N 1 G m n U 1 n , n = 1 , 2 , 3 , , N 1 , m = 1 , 2 , 3 , , N 2 .
D 2 ( ρ 1 , ϕ 1 ; ρ 2 , ϕ 2 ) = U 2 G U 1 2 = Tr ( U 2 + U 2 U 2 + G U 1 U 1 + G + U 2 + U 2 + G + G U 1 ) = 1 N 2 k = 1 N 2 | U 2 k ( G U 1 ) k | 2 = ( 1 / N 2 ) ( k ρ 2 k 2 + k j ρ 1 k ρ 1 j A k j exp [ i ( ϕ 1 k ϕ 1 j ) ] k j { ρ 2 k ρ 1 j G k j exp [ i ( ϕ 2 k ϕ 1 j ) ] + c . c . } )
D 2 ϕ 1 k = i N 2 ( j { ρ 1 j ρ 1 k A j k exp [ i ( ϕ 1 j ϕ 1 k ) ] c . c . } j { ρ 2 j ρ 1 k G j k exp [ i ( ϕ 2 j ϕ 1 k ) ] c . c . } ) = 0 .
Im [ Q k exp ( i ϕ 1 k ) ] = 0 ,
Q k = [ j k ρ 1 j exp ( i ϕ 1 j ) A j k j ρ 2 j exp ( i ϕ 2 j ) G j k ] ρ 1 k .
ϕ 1 k = arg [ j G j k * ρ 2 j exp ( i ϕ 2 j ) j k A k j ρ 1 j exp ( i ϕ 1 j ) ] , k = 1 , 2 , 3 , , N 1 .
D 2 ϕ 2 k = i N 2 [ ρ 2 k exp ( i ϕ 2 k ) j G k j ρ 1 j exp ( i ϕ 1 j ) c . c . ] = 0 .
Im [ ρ 2 k exp ( i ϕ 2 k ) j G k j ρ 1 j exp ( i ϕ 1 , j ) ] = 0 , ϕ 2 k = arg [ j G k j ρ 1 j exp ( i ϕ 1 j ) ] , k = 1 , 2 , 3 , , N 2 .
1 N 1 k = 1 N 1 | ϕ 1 k ( 0 , m 1 ) ϕ 1 k ( 0 , m 1 + 1 ) | 1 ,
SE = k = 1 N 2 | ρ 2 k exp ( i ϕ 2 k ) j = 1 N 1 G k j ρ 1 j exp ( i ϕ 1 j ) | 2 k = 1 N 2 ρ 2 k 2 ,
G ( X 2 , X 1 ) = exp ( i 2 π l / λ ) i λ l exp { i π λ l [ ( x 2 = x 1 ) 2 + ( y 2 = y 1 ) 2 ] } .
U 2 ( r 2 ) = 0 R G ( r 2 , r 1 ) U 1 ( r 1 ) d r 1 ,
G ( r 2 , r 1 ) = 2 π i λ l exp ( i 2 π l / λ ) × exp [ i π ( r 1 2 + r 2 2 ) / λ l ] J 0 ( 2 π r 2 r 1 / λ l ) r 1 ,

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