Abstract

A new iterative optimization method based on the Yang–Gu (Y-G) algorithm for the design of diffractive phase elements (DPE’s) that incorporate several optical functions is presented. The Y-G algorithm has been previously used to deal with various amplitude-phase retrieval problems in linear optical transform systems involving a monochromatic wave. We extend the general theory to the case of linear imaging systems with the illumination consisting of a number of components at different wavelengths. Based on a rigorous mathematical derivation, an iterative algorithm is developed to integrate several functions into one DPE. In the first set of examples numerical simulations are carried out in the design of several one-dimensional DPE’s capable of both demultiplexing different wavelength components and focusing each component wave. The second example illustrates the design of a two-dimensional DPE that performs as a demultiplexer in one direction and an array generator in the other direction.

© 1994 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, 737–745 (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, 706–711 (1992).
    [CrossRef]
  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. 31, 630–635 (1992).
    [CrossRef] [PubMed]
  10. Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
    [CrossRef]
  11. A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 18, 534–536 (1993).
    [CrossRef] [PubMed]
  12. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  13. G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).
  14. G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phases retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3153–3224 (1993).
    [CrossRef]
  15. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Er-soy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 32, 209–218 (1994).
    [CrossRef]
  16. B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981) (in Chinese).
  17. G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981) (in Chinese).
  18. E. Kreyszig, Introductory Function Analysis with Applications (Wiley, New York, 1986).
  19. B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
    [CrossRef] [PubMed]
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

1994 (1)

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Er-soy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 32, 209–218 (1994).
[CrossRef]

1993 (5)

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phases retrieval in any linear transform system and its applications,” Int. 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, 737–745 (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. 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. 18, 534–536 (1993).
[CrossRef] [PubMed]

1992 (2)

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 706–711 (1992).
[CrossRef]

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

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 non-unitary transformation,” Optik (Stuttgart) 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) (in Chinese).

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

1972 (1)

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

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, 737–745 (1993).
[CrossRef]

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 706–711 (1992).
[CrossRef]

Dong, B.

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phases retrieval in any linear transform system and its applications,” Int. 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 non-unitary transformation,” Optik (Stuttgart) 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. Er-soy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 32, 209–218 (1994).
[CrossRef]

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, 737–745 (1993).
[CrossRef]

Er-soy, O. K.

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Er-soy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 32, 209–218 (1994).
[CrossRef]

Gale, M. T.

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 706–711 (1992).
[CrossRef]

Gerchberg, R. W.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Gu, B.

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phases retrieval in any linear transform system and its applications,” Int. 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 non-unitary transformation,” Optik (Stuttgart) 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) (in Chinese).

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

Gu, B. Y.

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Er-soy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 32, 209–218 (1994).
[CrossRef]

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, 737–745 (1993).
[CrossRef]

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 706–711 (1992).
[CrossRef]

Kato, M.

Kewitsch, A.

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, 737–745 (1993).
[CrossRef]

Kreyszig, E.

E. Kreyszig, Introductory Function Analysis with Applications (Wiley, New York, 1986).

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.

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 706–711 (1992).
[CrossRef]

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, 737–745 (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 (Stuttgart) 35, 237–246 (1972).

Segev, M.

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]

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 non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Wyrowski, F.

Yang, G.

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phases retrieval in any linear transform system and its applications,” Int. 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 non-unitary transformation,” Optik (Stuttgart) 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) (in Chinese).

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

Yang, G. Z.

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Er-soy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 32, 209–218 (1994).
[CrossRef]

Yariv, A.

Zhuang, J. Y.

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Er-soy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 32, 209–218 (1994).
[CrossRef]

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) (in Chinese).

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) (in Chinese).

Appl. Opt. (6)

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

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phases retrieval in any linear transform system and its applications,” Int. 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, 737–745 (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 (1)

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)

Optik (Stuttgart) (2)

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

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

Prog. Opt. (1)

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

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

E. Kreyszig, Introductory Function Analysis with Applications (Wiley, New York, 1986).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of a diffractive optical system.

Fig. 2
Fig. 2

Flow diagram of the iterative optimization procedure.

Fig. 3
Fig. 3

Initial and final calculated phases of the DPE for the 1-D case with Nl = 8.

Fig. 4
Fig. 4

Desired output image of the 2-D DPE with wavelength-dependent zero and nonzero intensity points.

Tables (5)

Tables Icon

Table 1 Output Demultiplexed Amplitudes at 4 Wavelengths

Tables Icon

Table 2 Output Demultiplexed Amplitudes at 8 Wavelengths

Tables Icon

Table 3 Calculated Values of the Two Parameters

Tables Icon

Table 4 Output Demultiplexed Amplitudes at 4 Wavelengths for the Designed Two-Dimensional Differential Phase Element

Tables Icon

Table 5 Calculated Values of the Two Parameters in the Designed Two-Dimensional Diffraction Phase Element

Equations (36)

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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 ) = Ĝ ( λ α ) U 1 ( X 1 , λ α ) ,
Ĝ + Ĝ = Â Î ,
U 1 n ( λ α ) = ρ 1 n ( λ α ) exp ( i ϕ 1 n ) , n = 1 , 2 , 3 , , N 1 ,
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 ( λ α ) , m = 1 , 2 , 3 , , N 2 s , α = 1 , 2 , 3 , , N λ .
D 2 = α U 2 α Ĝ ( λ α ) U 1 ( λ α ) 2 = α T r [ U 2 α + U 2 α U 2 α + Ĝ ( λ α ) U 1 ( λ α ) U 1 + ( λ α ) Ĝ + ( λ α ) U 2 α + U 1 + ( λ α ) Ĝ + ( λ α ) Ĝ ( λ α ) U 1 ( λ α ) ] .
δ ϕ 1 D 2 = 0 , δ ϕ 2 γ D 2 = 0 ,
δ ϕ 1 ( D 2 ) = α Tr { i δ ϕ 1 U 1 + ( λ α ) × [ Ĝ + ( λ α ) U 2 α Ĝ + ( λ α ) Ĝ ( λ α ) U 1 ( λ α ) ] i [ U 2 α + Ĝ ( λ α ) U 1 + ( λ α ) Ĝ ( λ α ) Ĝ ( λ α ) ] U 1 ( λ α ) δ ϕ 1 } = α Tr { i δ ϕ 1 U 1 + ( λ α ) [ Ĝ + ( λ α ) U 2 α Ĝ + ( λ α ) Ĝ ( λ α ) U 1 ( λ α ) + c . c } ,
δ ϕ 2 γ ( D 2 ) = Tr { i δ ϕ 2 γ U 2 γ + [ Ĝ ( λ r ) U 1 ( λ γ ) U 2 γ ] i [ U 1 + ( λ γ ) Ĝ + ( λ γ ) U 2 γ + ] U 2 γ δ ϕ 2 γ } = Tr { i δ ϕ 2 γ U 2 γ + [ Ĝ ( λ γ ) U 1 ( λ γ ) U 2 γ ] + c.c . } .
Im α { U 1 + ( λ α ) [ Ĝ + ( λ α ) U 2 α Ĝ + ( λ α ) Ĝ + ( λ α ) U 1 ( λ α ) ] } = 0 ,
Im [ U 2 γ + Ĝ + ( λ γ ) U 1 ( λ γ ) ] = 0 .
p ( X 1 ) exp [ i ϕ ( X 1 ) ] = α [ Ĝ + ( λ α ) U 2 α Ĝ + ( λ α ) Ĝ ( λ α ) U 1 ( λ α ) ρ 1 ( λ α ) = α [ Ĝ + ( λ α ) U 2 α Â ( λ α , λ α ) U 1 ( λ α ) ] ρ 1 ( λ α ) ,
q ( X 2 , λ γ ) U 2 γ ( X 2 ) = Ĝ ( λ γ ) U 1 ( λ γ ) ,
ϕ 1 ( X 1 ) = arg { α [ Ĝ + ( λ α ) U 2 α Â ND ( λ α , λ α ) U 1 ( λ α ) ] ρ 1 ( λ α ) } ,
ϕ 2 γ ( X 2 ) = arg [ Ĝ ( λ γ ) U 1 ( λ γ ) ] .
δ ϕ 1 2 ( D 2 ) = α Tr { ( δ ϕ 1 ) 2 U 1 + ( λ α ) Ĝ + ( λ α ) Ĝ ( λ α ) U 1 ( λ α ) + ( δ ϕ 1 ) 2 [ U 1 + ( λ α ) [ Ĝ ( λ α ) U 2 α Â ( λ α , λ α ) U 1 ( λ α ) ] + c.c . ] } = 2 ( δ ϕ 1 ) 2 [ α Ĝ ( λ α ) U 1 ( λ α ) + p ( X 1 ) ] 0 ,
δ ϕ 2 γ 2 ( D 2 ) = Tr { ( δ ϕ 2 γ ) 2 [ U 2 γ + ( Ĝ ( λ γ ) U 1 ( λ γ ) U 2 γ ) + U 2 γ + U 2 γ ] + c.c . } = ( δ ϕ 2 γ ) 2 2 q ( X 2 , λ γ ) U 2 γ 2 0 .
ϕ 1 k = arg [ j , α G j , k α ρ 2 j α * exp ( i ϕ 2 j α ) j k A k , j ( λ α , λ α ) ρ 1 j ( λ a ) exp ( i ϕ 1 j ) ρ 1 k ( λ α ) ] , k = 1 , 2 , 3 , , N 1 ,
ϕ 2 k γ = arg [ j G k j γ ρ 1 j ( λ γ ) exp ( i ϕ 1 j ) ] .
j | ϕ 1 j ( 0 , m ) ϕ 1 j ( 0 , m + 1 ) | | ϕ 1 j ( 0 , m ) | 1 ,
SSE = D 2 γ U 2 γ 2 = γ , k ρ 2 k γ exp [ i ϕ 2 k γ ( n ) ] j G k j γ ( λ γ ) ρ 1 j exp [ i ϕ 1 j ( n , 0 ) ] 2 y , k ρ 2 k γ 2 ,
SSE = γ , k ρ 2 k γ 2 γ , k { ρ 2 k γ j G k j γ ( λ γ ) ρ 1 j exp [ i ϕ 1 j ( n , 0 ) ] } .
G 0 ( x 2 , x 1 , l , λ α ) = 1 + cos θ i 2 λ α r exp ( i k r ) ,
G 0 ( 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 ] .
λ 0 = 1 N λ α λ α ,
λ 0 = ( λ max λ min ) / 2 .
G ( x ¯ 2 , x ¯ 1 , l , λ α ) = ( λ 0 λ α ) 1 / 2 exp ( i 2 π l / λ α ) × exp [ i π λ 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 .
ϕ 1 ( 0 ) = 2 θ [ ( λ 0 l ) 1 / 2 ] x 1 / λ 0 x 1 2 , θ = x 2 max λ 0 N λ ( λ 0 l ) 1 / 2 N 2 Δ λ l ,
ϕ 1 i = 2 π λ 0 d i ,
ϕ 1 i α = ( λ 0 λ α ) ϕ 1 i .
ω α = k i I k n ( α ) I i d ( α ) ,
β α = α α I i n ( α ) I i d ( α ) ,

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