Abstract

A scheme for the design of diffractive phase elements (DPE’s) that integrates several optical functions is presented in a consistent sense based on the general theory of amplitude–phase retrieval and the Yang–Gu algorithm [Appl. Opt. 33, 209 (1994)]. We extend the original Yang–Gu algorithm to treat a system illuminated by a beam of incident light whose components are at different wavelengths, and a set of equations for determining the phase distribution of the DPE is derived. The profile of a surface-relief DPE can be designed with an iterative algorithm. Numerical simulations are carried out for the design of one-dimensional DPE’s capable of both demultiplexing different wavelength components and focusing each partial wave at predetermined positions. The influence of the extension of sampling points in the DPE’s from ideal geometric points to physical spots on design results is also investigated. The numerical simulation results show that the new algorithm can be used successfully to design the desired DPE’s. It is therefore expected to be useful in the design of DPE’s for micro-optical systems.

© 1995 Optical Society of America

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References

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  1. M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Iterative techniques to integrate different optical functions in a diffractive phase element,” Appl. Opt. 30, 4629–4635 (1991).
    [CrossRef] [PubMed]
  2. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  3. P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
    [CrossRef]
  4. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
    [CrossRef]
  5. D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
    [CrossRef] [PubMed]
  6. O. Bryngdahl, F. Wyrowski, “Digital holography-computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
    [CrossRef]
  7. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  8. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  9. M. Kato, K. Sakuda, “Computer-generated holograms: application to intensity variable and wavelength demultiplexing holograms,” Appl. Opt. 30, 630–635 (1992).
    [CrossRef]
  10. Y. Ishii, T. Kubota, “Wavelength demultiplexer in multimode fiber that uses optimized holographic optical elements,” Appl. Opt. 32, 4415–4422 (1993).
    [CrossRef] [PubMed]
  11. Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
    [CrossRef]
  12. A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
    [CrossRef]
  13. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  14. 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]
  15. G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude–phase retrieval problem in an optical system involving nonunitary transformation,” Optik 75, 68–74 (1987).
  16. G. Yang, B. Gu, B. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3153–3224 (1993).
    [CrossRef]
  17. 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]
  18. B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).
  19. G. Yang, B. Gu, “On the amplitude–phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).
  20. B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
    [CrossRef] [PubMed]
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1994 (2)

1993 (6)

G. Yang, B. Gu, B. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Y. Ishii, T. Kubota, “Wavelength demultiplexer in multimode fiber that uses optimized holographic optical elements,” Appl. Opt. 32, 4415–4422 (1993).
[CrossRef] [PubMed]

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

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

1992 (2)

1991 (1)

1990 (1)

O. Bryngdahl, F. Wyrowski, “Digital holography-computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

1989 (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1988 (1)

1987 (1)

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude–phase retrieval problem in an optical system involving nonunitary transformation,” Optik 75, 68–74 (1987).

1986 (1)

1981 (2)

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

G. Yang, B. Gu, “On the amplitude–phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

1972 (1)

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

1967 (1)

Amitai, Y.

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

Bernhardt, M.

Bryngdahl, O.

Chang, M.-P.

Dandliker, R.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
[CrossRef] [PubMed]

Dong, B.

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]

G. Yang, B. Gu, B. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude–phase retrieval problem in an optical system involving nonunitary 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.

Ehbets, P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

Ersoy, O. K.

Gale, M. T.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the 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, New York, 1968).

Gu, B.

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]

G. Yang, B. Gu, B. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude–phase retrieval problem in an optical system involving nonunitary 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]

G. Yang, B. Gu, “On the amplitude–phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

Gu, B. Y.

Herzig, H. P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
[CrossRef] [PubMed]

Ishii, Y.

Kato, M.

Kewitsch, A.

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

Kjelberg, I.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

Kubota, T.

Lohmann, A. W.

Miller, J. M.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Paris, D. P.

Prongue, D.

Regnaul, P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

Ross, N.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Sakuda, K.

Saxton, W. O.

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

Segev, M.

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Taghizadeh, M. R.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Tan, X.

Turunen, J.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Wang, L.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude–phase retrieval problem in an optical system involving nonunitary transformation,” Optik 75, 68–74 (1987).

Wyrowski, F.

Yang, G.

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]

G. Yang, B. Gu, B. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude–phase retrieval problem in an optical system involving nonunitary 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]

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

G. Yang, B. Gu, “On the amplitude–phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

Yang, G. Z.

Yariv, A.

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

Zhuang, J. Y.

Acta Opt. Sin. (1)

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

Acta Phys. Sin. (1)

G. Yang, B. Gu, “On the amplitude–phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

Appl. Opt. (7)

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

G. Yang, B. Gu, B. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

J. Mod. Opt. (3)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

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

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. (1)

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

Optik (2)

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

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude–phase retrieval problem in an optical system involving nonunitary transformation,” Optik 75, 68–74 (1987).

Prog. Opt. (1)

O. Bryngdahl, F. Wyrowski, “Digital holography-computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of a diffractive optical system.

Fig. 2
Fig. 2

Output pattern generated by a designed DPE that simultaneously performs wavelength demultiplexing and focusing of each wavelength. The transform kernel is G 1. The number of sampling points is N 1 = 256.

Fig. 3
Fig. 3

Output pattern generalized from the designed DPE with N 1 = 256. The transform kernel is G 2 with consideration for extension of spots: (a) for a continuous phase distribution of the DPE, (b) for a quantized phase of order k = 5.

Fig. 4
Fig. 4

Profile (depth distribution) of the designed surface-relief type DPE corresponding to the quantized phase of order k = 5. The design parameters are the same as in Fig. 3(b).

Fig. 5
Fig. 5

Output pattern obtained by the designed DPE with N 1 = 512 and the transform kernel G 2 for a quantized phase of order k = 4.

Tables (2)

Tables Icon

Table 1 Evaluation of the Performance of the Designed Diffractive Phase Element with Two Figures of Merit (N 1 = 256)

Tables Icon

Table 2 Evaluation of the Performance of the Designed Diffractive Phase Element with Two Figures of Merit (N 1 = 512)

Equations (28)

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U 1 α = U 1 ( X 1 , λ α ) = ρ 1 ( X 1 , λ α ) exp [ i ϕ 1 ( X 1 , λ α ) ] .
U 2 α = U 2 ( X 2 , λ α ) = ρ 2 ( X 2 , λ α ) exp [ i ϕ 2 ( X 2 , λ α ) ] .
U 2 ( X 2 , λ α ) = G ( X 2 , X 1 , λ α ) U 1 ( X 1 , λ α ) d X 1 .
U 2 ( X 2 , λ α ) = G ˆ ( λ α ) U 1 ( X 1 , λ α ) ,
G ˆ + G ˆ = A ˆ I ˆ ,
U 1 n ( λ α ) = ρ 1 n α exp ( i 2 π h 1 n / λ α ) ,
U 2 m α = ρ 2 m α exp ( i ϕ 2 m α ) ,
U 2 m α = n = 1 N 1 G m n ( λ α ) U 1 n α , n = 1 , 2 , 3 , , N 1 , m = 1 , 2 , 3 , , N 2 s , α = 1 , 2 , 3 , , N λ .
D 2 = α [ U 2 α G ˆ ( λ α ) U 1 α ] 2 = α Tr [ U 2 α + U 2 α U 2 α + G ˆ ( λ α ) U 1 α U 1 α + G ˆ + ( λ α ) U 2 α ] + U 1 α + G ˆ + ( λ α ) G ˆ ( λ α ) U 1 α ) ] = ( 1 / N ) α { i ρ 2 i α 2 + i j ρ 1 i α ρ 1 j α A i j ( λ α , λ α ) × exp [ i 2 π ( h 1 i h 1 j ) / λ α ] i j [ ρ 2 i α ρ 1 j α G i j ( λ α ) × exp [ i ( ϕ 2 i α 2 π h 1 j / λ α ) ] + c . c . ] } ,
δ h 1 D 2 = 0 , δ ϕ 2 γ D 2 = 0 ,
D 2 h 1 k = i N α ( 2 π / λ α ) [ j ( ρ 1 j α ρ 1 k α A j k ( λ α ) × exp { i [ 2 π ( h 1 j h 1 k ) / λ α ] } c . c . ) j { ρ 2 j α ρ 1 k α × G j k ( λ α ) exp [ i ( ϕ 2 j α 2 π h 1 k / λ α ) ] c . c . } ] = 0 .
Im [ Q k exp ( i 2 π h 1 k / λ 0 ) ] = 0 ,
Q k = α { j [ ρ 1 j α exp ( i 2 π h 1 j / λ α ) A j k ( λ α ) ρ 2 j α exp ( i ϕ 2 j α ) G j k ( λ α ) ] } ( 2 π / λ α ) ρ 1 k α × exp [ i ( 2 π h 1 k / λ 0 ) ( λ 0 / λ α 1 ) ] .
exp ( i 2 π h 1 k / λ 0 ) = Q ˜ k * | Q ˜ k | , k = 1 , 2 , 3 , , N 1 , N 1 = N 2 = N ,
Q ˜ k = α { j k ' ρ 1 j α exp ( i 2 π h 1 j / λ α ) A j k ( λ α ) j ρ 2 j α exp ( i ϕ 2 j α ) G j k ( λ α ) ] ( 2 π / λ α ) ρ 1 k α × exp [ i ( 2 π h 1 k / λ 0 ) ( λ 0 / λ α 1 ) ] .
D 2 ϕ 2 γ = i N [ ρ 2 k γ exp ( i ϕ 2 k γ ) j G k j ( λ γ ) ρ 1 j γ × exp ( i 2 π h 1 j / λ γ ) c . c . ] = 0 .
Im [ ρ 2 k γ exp ( i ϕ 2 k γ ) j G k j ( λ γ ) ρ 1 j γ exp ( i 2 π h 1 j / λ γ ) ] = 0 , exp ( i ϕ 2 k γ ) = j G k j ( λ γ ) ρ 1 j γ exp ( i 2 π h 1 j / λ γ ) | j G k j ( λ γ ) ρ 1 j γ exp ( i 2 π h 1 j / λ γ ) | , k = 1 , 2 , 3 , , N 2 s , γ = 1 , 2 , 3 , , N λ .
G 1 ( x 2 , x 1 ; l , λ α ) = ( 1 i λ α l ) 1 / 2 exp ( i 2 π l / λ α ) exp [ i π ( x 2 x 1 ) 2 / λ α l ] .
G 1 ( x ¯ 2 , x ¯ 1 ; l , λ α ) = ( λ 0 λ α ) 1 / 2 exp ( i 2 π l / λ α ) exp ( i πλ 0 ( x ¯ 2 x ¯ 1 ) 2 / λ α ) ,
x ¯ 1 = x 1 / ( λ 0 l ) 1 / 2 , x ¯ 2 = x 2 / ( λ 0 l ) 1 / 2 .
x 1 n = x 1 max / 2 + ( n 0 . 5 ) Δ x 1 , n = 1 , 2 , 3 , , N 1 ,
U 2 α = U 2 ( x 2 , λ α ) = ( 1 i λ α l ) 1 / 2 x 1 max / 2 x 1 max / 2 exp ( i 2 π l / λ α ) × exp [ i π ( x 2 x 1 ) 2 / λ α l ] U 1 ( x 1 , λ α ) d x 1 = ( 1 i λ α l ) 1 / 2 exp ( i 2 π l / λ α ) n = 1 N 1 U 1 ( x 1 n ) × x 1 n w / 2 x 1 n + w / 2 exp [ i π ( x 2 ξ ) 2 / λ α l ] d ξ = ( 1 2 i ) 1 / 2 exp ( i 2 π l / λ α ) n = 1 N 1 U 1 ( x 1 n ) × ( 2 / λ α l ) 1 / 2 ( x 2 x 1 n w / 2 ) ( 2 / λ α l ) 1 / 2 ( x 2 x 1 n + w / 2 ) exp ( i πη 2 / 2 ) d η = ( 1 2 i ) 1 / 2 exp ( i 2 π l / λ α ) n = 1 N 1 U 1 ( x 1 n ) × { F [ ( 2 λ α l ) 1 / 2 ( x 2 x 1 n + w / 2 ) ] F [ ( 2 λ α l ) 1 / 2 ( x 2 x 1 n w / 2 ) ] } ,
F ( w ) = 0 w exp ( i πξ 2 / 2 ) d ξ .
G 2 ( x ¯ 2 , x ¯ 1 n ; l , λ α ) = exp ( i 2 π l / λ α ) { F [ ( 2 λ 0 λ α ) 1 / 2 ( x ¯ 2 x ¯ 1 n + w ¯ / 2 ) ] F [ ( 2 λ 0 λ α ) 1 / 2 ( x ¯ 2 x ¯ 1 n w ¯ / 2 ) ] } ,
U 2 ( x ¯ 2 , λ α ) = n = 1 N 1 G 2 ( x ¯ 2 ; x ¯ 1 n ; l , λ α ) U 1 ( x ¯ 1 n , λ α ) .
ϕ 1 ( 0 ) = x 1 2 , or 0 . 5 .
A α = ( 1 N λ 1 ) k i I k Z ( α ) I i NONZ ( α ) ,
B α = ( 1 N 2 s 1 ) α α I i Z ( α ) I i NONZ ( α ) ,

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