Abstract

The diffraction efficiencies by practical dielectric holograms are evaluated with the rigorous coupled-wave analysis. As discussed in Parts 1 and 2, only the first reflection light can be diffracted by general sealed volume reflection holograms. Here we discuss the possibility that complex diffraction, which occurs in multiple-grating-storage holograms, makes diffraction efficiencies of higher-order lights increase drastically, i.e., the degenerated complex diffraction. Although evident in special cases only, they appear frequently in actual holographic use.

© 1998 Optical Society of America

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References

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1998

1992

J. T. Sheridan, L. Solymar, “Spurious beams in dielectric gratings of the reflection type: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

P. Ehbets, H. P. Herzig, R. Dändliker, “TIR holography analyzed with coupled wave theory,” Opt. Commun. 89, 5–11 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

K. Curtis, D. Psaltis, “Recording of multiple holograms in photopolymer films,” Appl. Opt. 31, 7425–7428 (1992).
[CrossRef] [PubMed]

1989

1986

1983

1982

1981

1979

1978

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quant. Electron. 10, 171–178 (1978).
[CrossRef]

1975

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Alferness, R.

Case, S. K.

Curtis, K.

Dändliker, R.

P. Ehbets, H. P. Herzig, R. Dändliker, “TIR holography analyzed with coupled wave theory,” Opt. Commun. 89, 5–11 (1992).
[CrossRef]

Ehbets, P.

P. Ehbets, H. P. Herzig, R. Dändliker, “TIR holography analyzed with coupled wave theory,” Opt. Commun. 89, 5–11 (1992).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Herzig, H. P.

P. Ehbets, H. P. Herzig, R. Dändliker, “TIR holography analyzed with coupled wave theory,” Opt. Commun. 89, 5–11 (1992).
[CrossRef]

Kamiya, N.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kowarschik, R.

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quant. Electron. 10, 171–178 (1978).
[CrossRef]

Moharam, M. G.

Owen, M. P.

Psaltis, D.

Sheridan, J. T.

J. T. Sheridan, L. Solymar, “Spurious beams in dielectric gratings of the reflection type: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

Slinger, C. W.

Solymar, L.

Tomishima, K.

Tsujinishi, R.

Tsukada, N.

Ward, A. A.

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

P. Ehbets, H. P. Herzig, R. Dändliker, “TIR holography analyzed with coupled wave theory,” Opt. Commun. 89, 5–11 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in dielectric gratings of the reflection type: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

Opt. Quant. Electron.

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quant. Electron. 10, 171–178 (1978).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Cross section of a multiple-grating storage hologram.

Fig. 2
Fig. 2

Example of interactions among [j 1, j 2, j 3] order diffraction waves in a hologram in which three gratings are stored.

Fig. 3
Fig. 3

Exposure layout model used in calculations for this paper.

Fig. 4
Fig. 4

Derived gratings caused by surface reflections.

Fig. 5
Fig. 5

Vector diagram of stored gratings.

Fig. 6
Fig. 6

Classified list of main complex diffraction wave vectors.

Fig. 7
Fig. 7

Plot of Bragg matching parameter B 3,[j1,j2,j3,j4,j5]. When B 3,[j1,j2,j3,j4,j5] = 0, the diffraction wave satisfies the Bragg condition.

Fig. 8
Fig. 8

Relationship between diffraction waves and the Bragg condition. Multiple circles in region 3 represent incident light variation in the (a) incident angle and (b) wavelength. When the point of a diffraction wave vector falls within a certain circle, that wave satisfies the Bragg condition as the light that the circle represents.

Fig. 9
Fig. 9

Reflection diffraction efficiencies of a multiple-grating hologram with cover plate. The small graphs show each diffraction wave order, and the large graphs show the sum of waves with common J.

Fig. 10
Fig. 10

Transmission diffraction efficiencies of a multiple-grating hologram with cover plate. The small graphs show each diffraction wave order, and the large graphs show the sum of waves with common J.

Fig. 11
Fig. 11

Reflection diffraction efficiencies of a multiple-grating hologram without cover plate. The small graphs show each diffraction wave order, and the large graphs show the sum of waves with common J.

Fig. 12
Fig. 12

Transmission diffraction efficiencies of a multiple-grating hologram without cover plate. The small graphs show each diffraction wave order, and the large graphs show the sum of waves with common J.

Fig. 13
Fig. 13

Diffraction efficiencies of multiple-grating holograms with all the characteristics: swelling according to parabolic m [G 3 = 3, the same as the dotted curve in Figs. 6(e) and 6(f) of Part 1], nonlinear exposure-permittivity modulation characteristics [the same as in Fig. 8(b) of Part 1], and erosion in a 20-nm rectangular shape [the same as the dashed curve in Figs. 8(a) and 8(b) of Part 2, except for the eroded thickness]: (a) reflection with cover plate, (b) transmission with cover plate, (c) reflection without cover plate, (d) transmission without cover plate.

Equations (26)

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ε 3 = ε ¯ 3 + k h k = 1 Δ ε c , 3 , k , h k cos h k K 3 , k · r + Δ ε s , 3 , k , h k sin h k K 3 , k · r = ε ¯ 3 + k h k = 1 Δ ε c , 3 , k , h k + i Δ ε s , 3 , k , h k exp - ih k K 3 , k · r + Δ ε c , 3 , k , h k - i Δ ε s , 3 , k , h k exp ih k K 3 , k · r ,
Δ ε c , 3 , k , h k = 2 d 3 , k 0 d 3 , k   ε 3 K 3 , k · r cos h k K 3 , k · r d K 3 , k · r , Δ ε s , 3 , k , h k = 2 d 3 , k 0 d 3 , k   ε 3 K 3 , k · r sin h k K 3 , k · r d K 3 , k · r .
2 E + k 2 ε E = 0
E 3 = j k = -   to   at   all   k S 3 , j 1 , j 2 , j k , z × exp - i k 3 , k - k j k K 3 , k · r .
E 1 = j = -   to   at   all   k Ref j 1 , j 2 , , j k , × exp - i - ξ 1 , j 1 , j 2 , , j k , z + β j 1 , j 2 , , j k x + exp - i ξ 1 , 0,0,0,0,0 z + β 0,0,0,0,0 x , E 3 = j = -   to   at   all   k   S 3 j 1 , j 2 , , j k , z × exp - i ξ 3 , j 1 , j 2 , , j k , z + β j 1 , j 2 , , j k , x , E 4 = j = -   to   at   all   k   Tra j 1 , j 2 , , j k , × exp - i ξ 4 , j 1 , j 2 , , j k , z - T 3 + β j 1 , j 2 , j k , x + ξ 3 , j 1 , j 2 , , j k , T 3 ,
β j 1 , j 2 , , j k , = k 1 sin   θ 1 - k j k K 3 , k sin   ϕ 3 , k independent   of   g , ξ l , j 1 , j 2 , j k , = k l 2 - β j 1 , j 2 , j k , 2 1 / 2 l = 1,4 , ξ 3 , j 1 , j 2 , j k , = k 3 cos   θ 3 - k j k K 3 , k cos   ϕ 3 , k , n 1 sin   θ 1 = n ¯ 3 sin   θ 3 ,   n ¯ 3 = ε ¯ 3 , k l = 2 π n l λ l = 1,4 , k 3 = 2 π n ¯ 3 λ ,
d 2 d z 2   S 3 , j 1 , j 2 , j k , z - 2 i ξ 3 , j 1 , j 2 , j k , d d z   S 3 , j 1 , j 2 , j k , z + k 3 2 B 3 , j 1 , j 2 , j k , S 3 , j 1 , j 2 , j k , z + k 3 2 2 k h k = 1 Δ ε c , 3 , k , h k + i Δ ε s , 3 , k , h k S 3 , j 1 , j 2 , j k - h k , z + Δ ε c , 3 , k , h k - i Δ ε s , 3 , k , h k S 3 , j 1 , j 2 , j k + h k , z = 0 ,
B 3 , j 1 , j 2 , j k , k 3 2 - ξ 3 , j 1 , j 2 , j k , 2 + β j 1 , j 2 , j k , 2 / k 3 2 .
S 3 S 3 " = O U A l , g B l , g S l , g S l , g , S 3 t [ S 3 , [ 1,1,0,0 , ] ( z ) S 3 , [ 1,0,1,0,0 , ] ( z ) S 3 , [ 0,1,1,0,0 , ] ( z ) | J = 2 S 3 , [ 1,0,0,0,0 ] ( z ) S 3 , [ 0,0,1,0,0 , ] ( z ) | J = 1 S 3 , [ 0,0,0,0,1 ] ( z ) S 3 , [ 0,0,0,1,0 , ] ( z ) S 3 , [ 0,0,0,0,0 , ] ( z ) | J = 0 S 3 , [ 1 , 1 , 1,0,0 , ] ( z ) S 3 , [ 0 , 1,0,1,0 ] ( z ) S 3 , [ 0,0 , 1,0,0 , ] ( z ) J = 1 ] .
S 3 d d z S 3 ,   S 3 d 2 d z 2 S 3 , O   :   11 × 11   zero   matrix ,   U   :   11 × 11   unit   matrix , A 3 A 3 , 2,2 A 3 , 2,1 A 3 , 2,0 A 3 , 2 , - 1 A 3 , 1,2 A 3 , 1,1 A 3 , 1,0 A 3 , 1 , - 1 A 3 , 0,2 A 3 , 0,1 A 3 , 0,0 A 3 , 0 , - 1 A 3 , - 1,2 A 3 , - 1,1 A 3 , - 1,0 A 3 , - 1 , - 1 , B 3 t c 3 , 1,1,0,0,0 c 3 , 1,0,1,0,0 c 3 , 0,1,1,0,0 | c 3 , 1,0,0,0,0 c 3 , 0,0,1,0,0 | c 3 , 0,0,0,0,1 c 3 , 0,0,0,1,0 c 3 , 0,0,0,0,0 | c 3 , 1 , - 1 , - 1,0,0 c 3 , 0 , - 1,0,1,0 c 3 , 0,0 , - 1,0,0 U .
A 3 , 2,2 b 3 , 1,1,0,0,0 0 a 3,5,1 0 b 3 , 1,0,1,0,0 a 3,4,1 a 3,5,1 * a 3,4,1 * b 3 , 0,1,1,0,0 ,   A 3 , 2,1 a 3,2,1 0 a 3,3,1 a 3,1,1 0 a 3,2,1 , A 3 , 2,0 0 a 3,2,2 0 a 3,3,2 0 0 0 0 0 ,   A 3 , 2 , - 1 0 a 3,2,3 0 0 0 0 0 0 0 , A 3 , 1,2 a 3,2,1 * a 3,3,1 * 0 0 a 3,1,1 * a 3,2,1 * ,   A 3 , 1,1 b 3 , 1,0,0,0,0 a 3,5,1 a 3,5,1 * b 3 , 0,0,1,0,0 , A 3 , 1,0 a 3,3,1 a 3,2,1 a 3,1,1 0 0 a 3,3,1 ,   A 3 , 1 , - 1 0 a 3,2,2 0 0 0 a 3,3,2 , A 3 , 0,2 0 a 3,3,2 * 0 a 3,2,2 * 0 0 0 0 0 ,   A 3 , 0,1 a 3,3,1 * 0 a 3,2,1 * 0 a 3,1,1 * a 3,3,1 * , A 3 , 0,0 b 3 , 0,0,0,0,1 0 a 3,5,1 0 b 3 , 0,0,0,1,0 a 3,4,1 a 3,5,1 * a 3,4,1 * b 3 , 0,0,0,0,0 , A 3 , 0 , - 1 a 3,2,1 0 a 3,1,1 a 3,3,1 a 3,2,1 0 0 0 a 3,3,1 , A 3 , - 1,2 0 0 0 a 3,2,3 * 0 0 0 0 0 ,   A 3 , - 1,1 0 0 a 3,2,2 * 0 0 a 3,3,2 * , A 3 , - 1,0 a 3,2,1 * a 3,3,1 * 0 0 a 3,2,1 * 0 a 3,1,1 * 0 a 3,3,1 * , A 3 , - 1 , - 1 b 3,1 , - 1 , - 1,0,0 0 a 3,4,1 0 b 3 , 0 , - 1,0,1,0 0 a 3,4,1 * 0 b 3 , 0,0 , - 1,0,0 .
a 3 , k , h k = - k 3 2 2 Δ ε c , 3 , k , h k + i Δ ε s , 3 , k , h k , a 3 , k , h k * = - k 3 2 2 Δ ε c , 3 , k , h k - i Δ ε s , 3 , k , h k , b 3 , j 1 , j 2 , j 3 , j 4 , j 5 = - k 3 2 B 3 , j 1 , j 2 , j 3 , j 4 , j 5 , c 3 , j 1 , j 2 , j 3 , j 4 , j 5 = 2 i ξ 3 , j 1 , j 2 , j 3 , j 4 , j 5 , S 3 , j 1 , j 2 , j 3 , j 4 , j 5 d d z   S 3 , j 1 , j 2 , j 3 , j 4 , j 5 z , S 3 , j 1 , j 2 , j 3 , j 4 , j 5 d 2 d z 2   S 3 , j 1 , j 2 , j 3 , j 4 , j 5 z .
S 3 , u z = m C 3 , m w 3 , u , m exp q 3 , m z     u = 1 , ,   11 ,
w 3 v 3,1 v 3,2     v 3 , m     .
m ξ 1 , j 1 , j 2 , j 3 , j 4 , j 5 + ξ 3 , j 1 , j 2 , j 3 , j 4 , j 5 + iq 3 , m C 3 , m w 3 , u , m = 0 , j 1 ,   j 2 ,   j 3 ,   j 4 ,   j 5 0,0,0,0,0 2 ξ 1 , 0,0,0,0,0 , j 1 ,   j 2 ,   j 3 ,   j 4 ,   j 5 = 0,0,0,0,0 ,
m ξ 4 , j 1 , j 2 , j 3 , j 4 , j 5 - ξ 3 , j 1 , j 2 , j 3 , j 4 , j 5 - iq 3 , m C 3 , m w 3 , u , m = 0 .
η 1 , j 1 , j 2 , j 3 , j 4 , j 5 = ξ 1 , j 1 , j 2 , j 3 , j 4 , j 5 ξ 1 , 0,0,0,0,0 Ref j 1 , j 2 , j 3 , j 4 , j 5   Ref j 1 , j 2 , j 3 , j 4 , j 5 * = ξ 1 , j 1 , j 2 , j 3 , j 4 , j 5 ξ 1 , 0,0,0,0,0   S 3 , j 1 , j 2 , j 3 , j 4 , j 5 0 × S 3 , j 1 , j 2 , j 3 , j 4 , j 5 * 0 , η 4 , j 1 , j 2 , j 3 , j 4 , j 5 = ξ 4 , j 1 , j 2 , j 3 , j 4 , j 5 ξ 1 , 0,0,0,0,0 Tra j 1 , j 2 , j 3 , j 4 , j 5 Tra j 1 , j 2 , j 3 , j 4 , j 5 * = ξ 4 , j 1 , j 2 , j 3 , j 4 , j 5 ξ 1 , 0,0,0,0,0   S 3 , j 1 , j 2 , j 3 , j 4 , j 5 T 3 × S 3 , j 1 , j 2 , j 3 , j 4 , j 5 * T 3 .
E 1 = j k = -   to   at   all   k Ref j 1 , j 2 , j k , exp - i - ξ 1 , j 1 , j 2 , j k , z + β j 1 , j 2 , j k , x + exp - i ξ 1 , 0,0 , z + β 0,0 , x , E 2 , g = j k = -   to   at   all   k   S 2 , g , j 1 , j 2 , j k , z × exp - i ξ 2 , g , j 1 , j 2 , j k , z - g = 1 g - 1   t 2 , g + β j 1 , j 2 , j k , x + g = 1 g - 1   ξ 2 , g , j 1 , j 2 , j k , t 2 , g , E 3 , g = j k = -   to   at   all   k   S 3 , g , j 1 , j 2 , j k , z × exp - i ξ 3 , g , j 1 , j 2 , j k , z - T 2 - g = 1 g - 1   t 3 , g + β j 1 , j 2 , j k , x + g = 1 G 2   ξ 2 , g , j 1 , j 2 , j k , t 2 , g + g = 1 g - 1   ξ 3 , g , j 1 , j 2 , j k , t 3 , g , E 4 = j k = -   to   at   all   k Tra j 1 , j 2 , j k , × exp - i ξ 4 , j 1 , j 2 , j k , z - T 2 - T 3 + β j 1 , j 2 , j k , x + g = 1 G 2   ξ 2 , g , j 1 , j 2 , j k , t 2 , g + g = 1 G 3   ξ 3 , g , j 1 , j 2 , j k , t 3 , g ,
A 2 , g , 2,2 b 2 , g , 1,1,0,0,0 0 0 0 b 2 , g , 1,0,1,0,0 0 0 0 b 2 , g , 0,1,1,0,0 , A 2 , g , 2,1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 , A 2 , g , 2,0 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 , A 2 , g , 2 , - 1 a 2 , g , 3 a 2 , g , 3 a 2 , g , 3 a 2 , g , 3 a 2 , g , 3 a 2 , g , 3 a 2 , g , 3 a 2 , g , 3 a 2 , g , 3 , A 2 , g , 1,2 a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * , A 2 , g , 1,1 b 2 , g , 1,0,0,0,0 0 0 b 2 , g , 0,0,1,0,0 , A 2 , g , 1,0 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 , A 2 , g , 1 , - 1 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 a 2 , g , 2 , A 2 , g , 0,2 a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * , A 2 , g , 0,1 a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * , A 2 , g , 0,0 b 2 , g , 0,0,0,0,1 0 0 0 b 2 , g , 0,0,0,1,0 0 0 0 b 2 , g , 0,0,0,0,0 , A 2 , g , 0 , - 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 a 2 , g , 1 , A 2 , g , - 1,2 a 2 , g , 3 * a 2 , g , 3 * a 2 , g , 3 * a 2 , g , 3 * a 2 , g , 3 * a 2 , g , 3 * a 2 , g , 3 * a 2 , g , 3 * a 2 , g , 3 * , A 2 , g , - 1,1 a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * a 2 , g , 2 * , A 2 , g , - 1,0 a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * a 2 , g , 1 * , A 2 , g , - 1 , - 1 b 2 , g , 1 , - 1 , - 1,0,0 0 0 0 b 2 , g , 0 , - 1,0,1,0 0 0 0 b 2 , g , 0,0 , - 1,0,0 .
a 2 , g , h = - k 2 , g 2 2 Δ ε c , 2 , g , h + i Δ ε s , 2 , g , h , a 2 , g , h * = - k 2 , g 2 2 Δ ε c , 2 , g , h - i Δ ε s , 2 , g , h , b 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 = - k 2 , g 2 B 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 , c 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 = 2 i ξ 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 , S 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 d d z   S 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 z , S 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 d 2 d z 2   S 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 z .
ε 2 , g x = n 2 , g x 2 = ε ¯ 2 , g + h = 1 Δ ε c , 2 , g , h cos 2 π hx / p + Δ ε s , 2 , g , h sin 2 π hx / p = ε ¯ 2 , g + h = 1 Δ ε c , 2 , g , h + i Δ ε s , 2 , g , h × exp - 2 π ihx / p + Δ ε c , 2 , g , h - i Δ ε s , 2 , g , h × exp 2 π ihx / p , ε ¯ 2 , g = 1 p 0 p   ε 2 , g x d x , Δ ε c , 2 , g , h = 2 p 0 p   ε 2 , g x cos 2 π hx / p d x , Δ ε s , 2 , g , h = 2 p 0 p   ε 2 , g x sin 2 π hx / p d x .
m ξ 1 , j 1 , j 2 , j 3 , j 4 , j 5 + ξ 2,1 , j 1 , j 2 , j 3 , j 4 , j 5 + iq 2,1 , m C 2,1 , m w 2,1 , u , m = 0 , j 1 ,   j 2 ,   j 3 ,   j 4 ,   j 5 0,0,0,0,0 2 ξ 1 , 0,0,0,0,0 , j 1 ,   j 2 ,   j 3 ,   j 4 ,   j 5 = 0,0,0,0,0 .
m C 2 , g - 1 , m w 2 , g - 1 , u , m exp q 2 , g - 1 , m t 2 , g - 1 = m C 2 , g , m w 2 , g , u , m , m q 2 , g - 1 , m - i ξ 2 , g - 1 , j 1 , j 2 , j 3 , j 4 , j 5 C 2 , g - 1 , m w 2 , g - 1 , u , m × exp q 2 , g - 1 , m t 2 , g - 1 = m q 2 , g , m - i ξ 2 , g , j 1 , j 2 , j 3 , j 4 , j 5 C 2 , g , m w 2 , g , u , m .
m C 2 , G 2 , m w 2 , G 2 , u , m exp q 2 , G 2 , m t 2 , G 2 = m C 3,1 , m w 3,1 , u , m , m q 2 , G 2 , m - i ξ 2 , G 2 , j 1 , j 2 , j 3 , j 4 , j 5 C 2 , G 2 , m w 2 , G 2 , u , m × exp q 2 , G 2 , m t 2 , G 2 = m q 3,1 , m - i ξ 3,1 , j 1 , j 2 , j 3 , j 4 , j 5 C 3,1 , m w 3,1 , u , m .
m C 3 , g - 1 , m w 3 , g - 1 , u , m exp q 3 , g - 1 , m t 3 , g - 1 = m C 3 , g , m w 3 , g , u , m , m q 3 , g - 1 , m - i ξ 3 , g - 1 , j 1 , j 2 , j 3 , j 4 , j 5 C 3 , g - 1 , m w 3 , g - 1 , u , m × exp q 3 , g - 1 , m t 3 , g - 1 = m q 3 , g , m - i ξ 3 , g , j 1 , j 2 , j 3 , j 4 , j 5 C 3 , g , m w 3 , g , u , m .
m ξ 4 , j 1 , j 2 , j 3 , j 4 , j 5 - ξ 3 , G 3 , j 1 , j 2 , j 3 , j 4 , j 5 - iq 3 , G 3 , m C 3 , G 3 , m w 3 , G 3 , u , m = 0 .

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