Abstract

We present a powerful multiple-scattering approach to wave diffraction by media whose permittivity contains overlapping periodic modulations, such as those found in thick holograms or in other applications involving superposed volume gratings. For this purpose we consider sequential wave scattering in a planar model having two periodic variations that are inclined at an arbitrary angle Δϕ with respect to each other. We thus show that all the diffracted fields can be described by flow diagrams that provide physical insight into the wave-scattering process. We then examine the particularly relevant case of small angular separations Δϕ and find that the individual diffracted orders can be evaluated by applying simple flow-graph considerations to the mathematical formulation. This procedure readily provides accurate numerical results, together with an estimate of the errors incurred if simplifying approximations are introduced. Furthermore, we show that the two-grating model can be readily extended to situations having any number of superposed periodicities.

© 1990 Optical Society of America

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  1. S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
    [Crossref]
  2. R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
    [Crossref]
  3. C. W. Slinger, L. Solymar, “Grating interactions in holograms recorded with two object waves,” Appl. Opt. 25, 3283–3287 (1986).
    [Crossref] [PubMed]
  4. R. K. Kostuk, J. W. Goodman, L. Hesselink, “Volume reflection hologram with multiple gratings: an experimental and theoretical evaluation,” Appl. Opt. 25, 4362–4369 (1986).
    [Crossref] [PubMed]
  5. H. Lee, “Cross-talk effects in multiplexed volume holograms,” Opt. Lett. 13, 874–876 (1988).
    [Crossref] [PubMed]
  6. M. G. Moharam, “Crosstalk and cross coupling in multiplexed holographic gratings,” in Practical Holography III, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1051, 143–147 (1989).
    [Crossref]
  7. E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled-wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
    [Crossref] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 12, p. 593.
  9. A. Korpel, “Two-dimensional plane wave theory of strong acousto-optic interaction in isotropic media,” J. Opt. Soc. Am. 69, 678–683 (1979).
    [Crossref]
  10. A. Korpel, T.-C. Poon, “Explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
    [Crossref]
  11. T.-C. Poon, A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
    [Crossref]
  12. T.-C. Poon, M. R. Chatterjee, P. P. Banerjee, “Multiple plane-wave analysis of acousto-optic diffraction by adjacent ultrasonic beams of frequency ratio 1:m,” J. Opt. Soc. Am. A 3, 1402–1406 (1986).
    [Crossref]
  13. K. Fujiwara, “Application of higher order Born approximation to multiple elastic scattering of electrons by crystals,” J. Phys. Soc. Jpn. 14, 1513–1524 (1959).
    [Crossref]
  14. T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 41, 2461–2494 (1966).
    [Crossref]
  15. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1968).
  16. R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–500 (1970).
  17. F. Harary, Graph Theory and Theoretical Physics (Academic, New York, 1967), Chap. 2, p. 59.
  18. H. Lee, X-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
    [Crossref]
  19. K. Y. Tu, H. Lee, T. Tamir, “Crosstalk behavior of volume holographic weighted interconnections,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 159.

1989 (2)

H. Lee, X-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled-wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
[Crossref] [PubMed]

1988 (1)

1986 (3)

R. K. Kostuk, J. W. Goodman, L. Hesselink, “Volume reflection hologram with multiple gratings: an experimental and theoretical evaluation,” Appl. Opt. 25, 4362–4369 (1986).
[Crossref] [PubMed]

C. W. Slinger, L. Solymar, “Grating interactions in holograms recorded with two object waves,” Appl. Opt. 25, 3283–3287 (1986).
[Crossref] [PubMed]

T.-C. Poon, M. R. Chatterjee, P. P. Banerjee, “Multiple plane-wave analysis of acousto-optic diffraction by adjacent ultrasonic beams of frequency ratio 1:m,” J. Opt. Soc. Am. A 3, 1402–1406 (1986).
[Crossref]

1981 (1)

1980 (1)

A. Korpel, T.-C. Poon, “Explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
[Crossref]

1979 (1)

1975 (2)

S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
[Crossref]

R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[Crossref]

1970 (1)

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–500 (1970).

1968 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1968).

1966 (1)

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 41, 2461–2494 (1966).
[Crossref]

1959 (1)

K. Fujiwara, “Application of higher order Born approximation to multiple elastic scattering of electrons by crystals,” J. Phys. Soc. Jpn. 14, 1513–1524 (1959).
[Crossref]

Alferness, R.

R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[Crossref]

Banerjee, P. P.

T.-C. Poon, M. R. Chatterjee, P. P. Banerjee, “Multiple plane-wave analysis of acousto-optic diffraction by adjacent ultrasonic beams of frequency ratio 1:m,” J. Opt. Soc. Am. A 3, 1402–1406 (1986).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 12, p. 593.

Case, S. K.

S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
[Crossref]

R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[Crossref]

Chatterjee, M. R.

T.-C. Poon, M. R. Chatterjee, P. P. Banerjee, “Multiple plane-wave analysis of acousto-optic diffraction by adjacent ultrasonic beams of frequency ratio 1:m,” J. Opt. Soc. Am. A 3, 1402–1406 (1986).
[Crossref]

Chu, R. S.

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–500 (1970).

Fujiwara, K.

K. Fujiwara, “Application of higher order Born approximation to multiple elastic scattering of electrons by crystals,” J. Phys. Soc. Jpn. 14, 1513–1524 (1959).
[Crossref]

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

Gu, X-G.

H. Lee, X-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

Harary, F.

F. Harary, Graph Theory and Theoretical Physics (Academic, New York, 1967), Chap. 2, p. 59.

Hesselink, L.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1968).

Korpel, A.

Kostuk, R. K.

Lee, H.

H. Lee, X-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

H. Lee, “Cross-talk effects in multiplexed volume holograms,” Opt. Lett. 13, 874–876 (1988).
[Crossref] [PubMed]

K. Y. Tu, H. Lee, T. Tamir, “Crosstalk behavior of volume holographic weighted interconnections,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 159.

Moharam, M. G.

M. G. Moharam, “Crosstalk and cross coupling in multiplexed holographic gratings,” in Practical Holography III, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1051, 143–147 (1989).
[Crossref]

Poon, T.-C.

T.-C. Poon, M. R. Chatterjee, P. P. Banerjee, “Multiple plane-wave analysis of acousto-optic diffraction by adjacent ultrasonic beams of frequency ratio 1:m,” J. Opt. Soc. Am. A 3, 1402–1406 (1986).
[Crossref]

T.-C. Poon, A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
[Crossref]

A. Korpel, T.-C. Poon, “Explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
[Crossref]

Psaltis, D.

H. Lee, X-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

Slinger, C. W.

C. W. Slinger, L. Solymar, “Grating interactions in holograms recorded with two object waves,” Appl. Opt. 25, 3283–3287 (1986).
[Crossref] [PubMed]

Solymar, L.

C. W. Slinger, L. Solymar, “Grating interactions in holograms recorded with two object waves,” Appl. Opt. 25, 3283–3287 (1986).
[Crossref] [PubMed]

Tamir, T.

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–500 (1970).

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 41, 2461–2494 (1966).
[Crossref]

K. Y. Tu, H. Lee, T. Tamir, “Crosstalk behavior of volume holographic weighted interconnections,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 159.

Tu, K. Y.

K. Y. Tu, H. Lee, T. Tamir, “Crosstalk behavior of volume holographic weighted interconnections,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 159.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 12, p. 593.

Appl. Opt. (1)

C. W. Slinger, L. Solymar, “Grating interactions in holograms recorded with two object waves,” Appl. Opt. 25, 3283–3287 (1986).
[Crossref] [PubMed]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1968).

Can. J. Phys. (1)

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 41, 2461–2494 (1966).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–500 (1970).

J. Appl. Phys. (1)

H. Lee, X-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

J. Opt. Soc. Am. (2)

A. Korpel, T.-C. Poon, “Explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
[Crossref]

S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
[Crossref]

J. Opt. Soc. Am. (1)

R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[Crossref]

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

T.-C. Poon, M. R. Chatterjee, P. P. Banerjee, “Multiple plane-wave analysis of acousto-optic diffraction by adjacent ultrasonic beams of frequency ratio 1:m,” J. Opt. Soc. Am. A 3, 1402–1406 (1986).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. Soc. Jpn. (1)

K. Fujiwara, “Application of higher order Born approximation to multiple elastic scattering of electrons by crystals,” J. Phys. Soc. Jpn. 14, 1513–1524 (1959).
[Crossref]

Opt. Lett. (1)

Other (4)

M. G. Moharam, “Crosstalk and cross coupling in multiplexed holographic gratings,” in Practical Holography III, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1051, 143–147 (1989).
[Crossref]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 12, p. 593.

F. Harary, Graph Theory and Theoretical Physics (Academic, New York, 1967), Chap. 2, p. 59.

K. Y. Tu, H. Lee, T. Tamir, “Crosstalk behavior of volume holographic weighted interconnections,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 159.

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Figures (11)

Fig. 1
Fig. 1

Geometry of a volume hologram consisting of two superposed gratings.

Fig. 2
Fig. 2

Wave-number diagram illustrating the relationship between diffraction vectors k n 1 , n 2 and propagation vectors k ¯ n 1 , n 2 .

Fig. 3
Fig. 3

Flow chart for scattering by two gratings to the m = 3 level.

Fig. 4
Fig. 4

Wave-number diagram showing higher-order diffracted waves: (a) diffraction vectors k n 1 , n 2 , (b) actual propagation vectors k ¯ n 1 , n 2 . The situation is chosen so that Bragg conditions are satisfied for the (1, 0) and (1, −1) orders.

Fig. 5
Fig. 5

Partial flow chart showing only the propagating (nonevanescent) scattered components that generate the (1, 0) order to the m = 5 scattering level. Bragg and off-Bragg transitions are indicated by solid and dashed branches, respectively.

Fig. 6
Fig. 6

Variation of diffraction efficiency η n 1 , n 2 versus angular separation Δϕ for the situation shown in Fig. 4, with K1/k0 = 1.33, θ0 = 41.68 deg, z0/d1 = 6.5 × 104: (a) M1 = M2 = 6 × 10−6; (b) M1 = 6 × 10−6, M2 = 0; (c) M1 = M2 = 3 × 10−6; (d) M1 = 3 × 10−6, M2 = 0.

Fig. 7
Fig. 7

Partial flow chart showing only the principal scattered components that generate the (0, 1) diffracted order up to the m = 5 scattering level.

Fig. 8
Fig. 8

Variation of phase-correlation-function magnitude |ΨOj| versus angular separation Δϕ.

Fig. 9
Fig. 9

Variation of diffraction efficiency η n 1 , n 2 versus angular separation Δϕ for the situation shown in Fig. 4 and with the parameters of case (a) in Fig. 6.

Fig. 10
Fig. 10

Variation of diffraction efficiency η n 1 , n 2 versus angular separation Δϕ for the situation shown in Fig. 4, with K1/k0 = 1.33, θ0 = 41.68 deg, Mz0/d1 = 0.39: (a) M1 = M2 = M = 6 × 10−6, (b) M1 = M2 = M = 1.2 × 10−5.

Fig. 11
Fig. 11

Variation of diffraction efficiency η n 1 , n 2 versus angular separation Δϕ for the situation shown in Fig. 4, with K1/k0 = 1.33, θ0 = 41.68 deg, z0/d1 = 6.5 × 104: (a) M2 < M1 = 6 × 10−6, (b) M2 > M1 = 3 × 10−6.

Equations (49)

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= 0 [ 1 + g ( r ) ] , g ( r ) = M 1 cos ( K 1 · r ) + M 2 cos ( K 2 · r ) ,
( 2 + k 0 2 ) E = k 0 2 g ( r ) E ,
E ( r ) = E ( 0 ) ( r ) + k 0 2 G ( r , r i ) g ( r i ) E ( r i ) d r i ,
( 2 + k 0 2 ) G ( r , r i ) = δ ( r , r i ) = δ ( r r i )
E ( r ) = m = 0 E ( m ) ( r ) ,
E ( m ) ( r ) = k 0 2 G ( r , r i ) g ( r i ) E ( m 1 ) ( r i ) d r i ,
E ( 0 ) = exp ( i k 0 · r ) ,
G ( r , r i ) = 1 ( 2 π ) 2 exp [ i k · ( r r i ) ] k 2 k 0 2 d k ,
E ( 1 ) ( r ) = ( k 0 2 π ) 2 exp [ i k · ( r r i ) ] k 2 k 0 2 { M 1 2 [ exp ( i k 1 · r i ) + exp ( i K 1 · r i ) ] + M 2 2 [ exp ( i K 2 · r i ) + exp ( i K 2 · r i ) ] } exp ( i k 0 · r i ) d k d r i .
E 1,0 ( 1 ) ( r ) = ( k 0 2 π ) 2 M 1 2 exp [ i k · ( r r i ) ] k 2 k 0 2 × exp [ i ( K 1 + k 0 ) · r i ] d k d r i = k 0 2 M 1 2 I ( r ) exp [ i ( k 0 + K 1 ) · r ] ,
I ( r ) = 1 ( 2 π ) 2 exp [ i ( k k 0 K 1 ) · r ] k 2 k 0 2 d k d r = 1 ( 2 π ) 2 d υ d u u 2 + υ 2 k 0 2 exp [ i ( u u 0 U 1 ) x ] × d x 0 z exp [ i ( υ υ 0 V 1 ) z ] d z ,
I = 1 2 π i 1 υ 2 + ( u 0 + U 1 ) 2 k 0 2 × exp [ i ( υ υ 0 V 1 ) z ] 1 υ υ 0 V 1 d υ .
I = 1 2 w 1,0 exp [ i ( w 1,0 υ 0 V 1 ) z ] 1 w 1,0 υ 0 V 1 ,
u n 1 , n 2 = u 0 + n 1 U 1 + n 2 U 2 ,
υ n 1 , n 2 = υ 0 + n 1 V 1 + n 2 V 2 ,
w n 1 , n 2 = ( k 0 2 u n 1 , n 2 2 ) 1 / 2 ,
p n 1 , n 2 = w n 1 , n 2 υ n 1 , n 2 ,
E 1,0 ( 1 ) ( r ) = k 0 2 M 1 4 w 1,0 exp ( i p 1,0 z ) 1 p 1,0 exp [ i ( k 0 + K 1 ) · r ] = k 0 2 M 1 4 w 1,0 1 exp ( i p 1,0 z ) p 1,0 exp ( i k ¯ 1,0 · r ) ,
k ¯ n 1 , n 2 = k n 1 , n 2 + Δ k n 1 , n 2 = x ˆ u n 1 , n 2 + z ˆ w n 1 , n 2 .
E ( 1 ) = ν ( 1 ) k 0 2 M μ ( 1 ) 4 w n ( 1 ) 1 exp [ i p n ( 1 ) z ] p n ( 1 ) exp [ i k ¯ n ( 1 ) · r ] ,
n ( m + 1 ) = [ n 1 ( m + 1 ) , n 2 ( m + 1 ) ] = { [ n 1 ( m ) + 1 , n 2 ( m ) ] [ n 1 ( m ) 1 , n 2 ( m ) ] } with μ ( m + 1 ) = 1 [ n 1 ( m ) , n 2 ( m ) + 1 ] [ n 1 ( m ) , n 2 ( m ) 1 ] } with μ ( m + 1 ) = 2 ,
E ( 2 ) = ν ( 1 ) k 0 2 M μ ( 1 ) 4 w n ( 1 ) ν ( 2 ) k 0 2 M μ ( 2 ) 4 w n ( 2 ) { 1 exp ( i p n ( 2 ) z ) p n ( 1 ) p n ( 2 ) + 1 exp [ i ( p n ( 1 ) p n ( 2 ) ) z ] p n ( 1 ) ( p n ( 1 ) p n ( 2 ) ) } exp [ i k ¯ n ( 2 ) · r ] .
E 1 , 1 ( 2 ) = k 0 2 M 1 4 w 1,0 k 0 2 M 2 4 w 1 , 1 { 1 exp ( i p 1 , 1 z ) p 1,0 p 1 , 1 + 1 exp [ i ( p 1,0 p 1 , 1 ) z ] p 1,0 ( p 1,0 p 1 , 1 ) } exp [ i k ¯ 1 , 1 · r ] .
E ( 3 ) = ν ( 1 ) k 0 2 M μ ( 1 ) 4 w n ( 1 ) ν ( 2 ) k 0 2 M μ ( 2 ) 4 w n ( 2 ) ν ( 3 ) k 0 2 M μ ( 3 ) 4 w n ( 3 ) × { 1 exp ( i p n ( 3 ) z ) p n ( 1 ) p n ( 2 ) p n ( 3 ) 1 exp [ i ( p n ( 1 ) p n ( 3 ) ) z ] p n ( 1 ) ( p n ( 1 ) p n ( 2 ) ) ( p n ( 1 ) p n ( 3 ) ) 1 exp [ i ( p n ( 2 ) p n ( 3 ) ) z ] p n ( 2 ) ( p n ( 2 ) p n ( 1 ) ) ( p n ( 2 ) p n ( 3 ) ) } exp [ i k ¯ n ( 3 ) · r ] .
α n ( q ) = k 0 2 M μ ( q ) z 4 w n ( q )
Δ n ( q ) = q n ( q ) z .
E ( m ) = S m Ψ n ( m ) exp [ i k ¯ n ( m ) · r ] ,
S m = ν ( 1 ) α n ( 1 ) ν ( 2 ) α n ( 2 ) ν ( 3 ) α n ( 3 ) ν ( m ) α n ( m ) ,
Ψ n ( m ) = ( ) m + 1 1 exp ( i Δ n ( m ) ) q = 1 m Δ n ( q ) q = 1 m 1 1 exp [ i ( Δ n ( q ) Δ n ( m ) ) ] Δ n ( q ) h = 1 h q m ( Δ n ( q ) Δ n ( h ) ) .
E n 1 , n 2 = m = 0 E n 1 , n 2 ( m ) .
Ψ n ( m ) = q = m ( i ) q q h 1 = 0 q m ( ) h 1 ( q h 1 1 ) ! h 1 ! h 2 = 0 q m h 1 h 3 = 0 q m ( h 1 + h 2 ) h m 1 = 0 q m ( h 1 + h 2 + + h m 2 ) { [ Δ n ( m ) ] h 1 [ Δ n ( m 1 ) ] h 2 [ Δ n ( 1 ) ] q m ( h 1 + h 2 + + h m 1 ) } .
E 1,0 ( r ) = i α 1,0 + i 3 α 1,0 α 0,0 α 1,0 3 ! + i 5 α 1,0 α 0,0 α 1,0 α 0,0 α 1,0 5 ! + = i ( α 1,0 α 0,0 ) 1 / 2 [ ( α 1,0 α 0,0 ) 1 / 2 1 ! ( α 1,0 α 0,0 ) 3 / 2 3 ! + ( α 1,0 α 0,0 ) 5 / 2 5 ! + ] = i ( α 1,0 α 0,0 ) 1 / 2 sin [ ( α 1,0 α 0,0 ) 1 / 2 ] = i ( w 0,0 w 1,0 ) 1 / 2 sin [ k 0 2 M 1 z 4 ( w 1,0 w 0,0 ) 1 / 2 ] ,
E 1,0 = i α 1,0 [ 1 ( α 0,0 α 1,0 + α 1 , 1 α 1,0 ) 1 3 ! + α 0,0 α 1,0 ( α 0,0 α 1,0 + α 1 , 1 α 1,0 ) 1 5 ! + α 1 , 1 α 1,0 ( α 0,0 α 1,0 + α 1 , 1 α 1,0 ) 1 5 ! + ] = i ( α 1,0 α 0,0 ) 1 / 2 ( α 0,0 α 1,0 ) 1 / 2 ( α 1,0 α 0,0 + α 1 , 1 α 1,0 ) 1 / 2 × sin ( α 1,0 α 0,0 + α 1 , 1 α 1,0 ) 1 / 2 ,
E 1,0 = m = 1 n 1 + n 2 = 1 E n 1 , n 2 ( m ) = i { α 1,0 [ 1 ( α 0,0 α 1,0 + α 0,0 α 0,1 + α 1 , 1 α 2 , 1 + α 1 , 1 α 1,0 ) 1 3 ! + ] + α 0,1 [ 1 ( α 1,1 α 0,1 + α 1,1 α 1,2 + α 0,0 α 1,0 + α 0,0 α 0,1 ) 1 3 ! + ] } = i [ k 0 2 ( M 1 + M 2 ) z 4 w 1 k 0 6 ( M 1 + M 2 ) 3 z 3 4 3 w 0 w 1 2 1 3 ! + ] = i ( w 0 w 1 ) 1 / 2 sin [ k 0 2 ( M 1 + M 2 ) z 4 ( w 0 w 1 ) 1 / 2 ] ,
E B 6 = α 1,0 α 1 , 1 α 1,0 α 0,0 α 1,0 Ψ B 6 exp ( i k ¯ 1,0 · r ) ,
Ψ n ( m ) = 1 2 π i z m × C i C + i { ( i ) m s ( s + i p n ( m ) ) q = 1 m 1 [ s + i ( p n ( m ) p n ( q ) ) ] } exp ( s z ) d s .
Ψ O 1 = 1 ( Δ 0,1 ) 3 [ exp ( i Δ 0,1 ) q = 0 2 ( i Δ 0,1 ) q q ! ] , Ψ O 2 = 1 ( Δ 0,1 ) 5 [ exp ( i Δ 0,1 ) q = 0 4 ( i Δ 0,1 ) q q ! ] , Ψ O 3 = 1 ( Δ 2 , 1 ) 5 [ exp ( i Δ 2 , 1 ) q = 0 4 ( i Δ 2 , 1 ) q q ! ] , Ψ O 4 = 1 ( Δ 0,1 ) 5 ( 1 γ ) 2 { 4 3 γ + ( 3 2 γ ) ( i Δ 0,1 ) + ( 2 γ ) ( i Δ 0,1 ) 2 2 + [ q = 3 ( i Δ 1,1 ) q 3 q ! ] ( i Δ 0,1 ) 3 + [ 4 + 3 γ + ( 1 γ ) ( i Δ 0,1 ) ] exp ( i Δ 0,1 ) } Ψ O 5 = Ψ O 6 = Ψ O 2 , Ψ O 7 = 1 ( Δ 0,1 ) 5 [ ( 4 + i Δ 0,1 ) exp ( i Δ 0,1 ) + q = 0 3 4 q q ! ( i Δ 0,1 ) q ] ,
Ψ O 1 = q = 3 i q q ! ( Δ 0,1 ) q 3 , Ψ O 2 = q = 5 i q q ! ( Δ 0,1 ) q 5 , Ψ O 3 = q = 5 i q q ! ( Δ 2 , 1 ) q 5 , Ψ O 4 = q = 5 i q q ! ( Δ 0,1 ) q 5 h 2 = 0 q 5 h 3 = 0 q 5 h 2 γ h 3 , Ψ O 5 = Ψ O 6 = Ψ O 2 , Ψ O 7 = q = 5 i q q ! ( q 4 ) ( Δ 0,1 ) q 5 .
η n 1 , n 2 = ( w n 1 , n 2 / w 0,0 ) | E n 1 , n 2 | 2 .
Γ n ( m ) = ( 1,0 ) m > m 0 | S m Ψ n ( m ) | < n ( m ) = ( 1,0 ) m > m 0 ( α 1,0 ) m + 1 2 ( α 0,0 ) m 1 2 m ! P m ,
Γ < Γ max = m > m 0 ( α 1,0 ) m + 1 2 ( α 0,0 ) m 1 2 [ ( m 1 ) / 2 ] ! [ ( m + 1 ) / 2 ] ! .
Γ max = ( 0.6167 ) 7 3 ! 4 ! + ( 0.6167 ) 9 4 ! 5 ! + 2.4 × 10 4 .
Q m = 1 ( Δ 1,0 ) m [ exp ( i Δ 1,0 ) q = 0 m 1 ( i Δ 1,0 ) q q ! ] ( i ) m + 1 Δ 1,0 ( m 1 ) ! ,
Γ n ( m ) = ( 1,0 ) m 3 | S m Q m | n ( m ) = ( 1,0 ) m 3 | α 1,0 | ( α 1,0 ) m 1 | Δ 1,0 | ( m 1 ) ! P m ,
Γ max = [ 6 ( α 1,0 ) 2 2 ! + 90 ( α 1,0 ) 2 4 ! + 1190 ( α 1,0 ) 6 6 ! + 15,750 ( α 1,0 ) 8 8 ! + ] | α 1,0 Δ 1,0 | .
n ( m + 1 ) = [ n 1 ( m + 1 ) , n 2 ( m + 1 ) , n 3 ( m + 1 ) ] = { [ n 1 ( m ) + 1 , n 2 ( m ) , n 3 ( m ) ] [ n 1 ( m ) 1 , n 2 ( m ) , n 3 ( m ) ] } with μ ( m + 1 ) = 1 [ n 1 ( m ) , n 2 ( m ) + 1 , n 3 ( m ) ] [ n 1 ( m ) , n 2 ( m ) 1 , n 3 ( m ) ] } with μ ( m + 1 ) = 2. [ n 1 ( m ) , n 2 ( m ) , n 3 ( m ) + 1 ] [ n 1 ( m ) , n 2 ( m ) , n 3 ( m ) 1 ] } with μ ( m + 1 ) = 3
Ψ n ( 1 ) = q = 1 ( i ) q q ! [ Δ n ( 1 ) ] q 1 .
Ψ n ( 2 ) = i 2 2 ! + i 3 3 ! [ Δ n ( 1 ) 2 Δ n ( 2 ) ] + i 4 4 ! { [ Δ n ( 1 ) ] 2 3 Δ n ( 1 ) Δ n ( 2 ) + 3 [ Δ n ( 2 ) ] 2 } + + i n n ! { [ Δ n ( 1 ) ] n 2 ( n 1 1 ) [ Δ n ( 1 ) ] n 3 × Δ n ( 2 ) + + ( ) n 2 ( n 1 n 2 ) [ Δ n ( 2 ) ] n 2 } + = q = 2 i q q h 1 = 0 q 2 [ Δ n ( 2 ) ] h 1 Δ n ( 1 ) q 2 h 1 ( q h 1 1 ) ! h 1 ! .
Ψ n ( 3 ) = 1 Δ n ( 1 ) Δ n ( 2 ) q = 1 ( i ) q q ! [ Δ n ( 3 ) ] q 1 + 1 Δ n ( 1 ) [ Δ n ( 1 ) Δ n ( 2 ) ] × q = 1 i q q ! [ Δ n ( 1 ) Δ n ( 3 ) ] q 1 + 1 Δ n ( 2 ) [ Δ n ( 2 ) Δ n ( 1 ) ] q = 1 i q q ! × [ Δ n ( 2 ) Δ n ( 3 ) ] q 1 = q = 1 i q q ! { ( Δ n ( 3 ) ) q 1 Δ n ( 1 ) Δ n ( 2 ) + [ Δ n ( 1 ) Δ n ( 3 ) ] q 1 Δ n ( 1 ) [ Δ n ( 1 ) Δ n ( 2 ) ] + [ Δ n ( 2 ) Δ n ( 3 ) ] q 1 Δ n ( 2 ) [ Δ n ( 2 ) Δ n ( 1 ) ] } = q = 3 i q q ! h 1 q 3 ( ) h 1 ( q h 1 1 ) ! h 1 ! × h 2 = 0 q 3 h 1 [ Δ n ( 3 ) ] h 1 [ Δ n ( 2 ) ] h 2 [ Δ n ( 1 ) ] q 3 h 1 h 2 .

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