Abstract

We develop a numerical method to solve the coupled-wave equations for four-wave mixing phase conjugation in a photorefractive crystal with unequal complex coupling constants; by this method we can obtain both the intensities and the phases of various beams. We discuss the intensities and the phases of the interacting beams in a photorefractive crystal for the cases in which the phase shifts between the refractive-index gratings and the interference pattern are all π/2 and in which they are not all π/2.

© 1992 Optical Society of America

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References

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  1. B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981);M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
    [CrossRef] [PubMed]
  2. M. Cronin-Golomb, J. O. White, B. Fischer, A. Yariv, Opt. Lett. 7, 313 (1982).
    [CrossRef] [PubMed]
  3. A. Bledowski, W. Krolikowski, Opt. Lett. 13, 146 (1988).
    [CrossRef]
  4. M. R. Belic, Opt. Lett. 12, 105 (1987).
    [CrossRef]
  5. M. R. Belic, Phys. Rev. A 31, 3169 (1985).
    [CrossRef] [PubMed]
  6. M. R. Belic, M. Lax, Opt. Commun. 56, 197 (1985).
    [CrossRef]
  7. W. Krolikowski, Opt. Commun. 60, 319 (1986).
    [CrossRef]
  8. W. Krolikowski, M. R. Belic, Opt. Lett. 13, 149 (1988).
    [CrossRef]
  9. M. R. Belic, Phys. Rev. A 37, 1809 (1988).
    [CrossRef] [PubMed]
  10. Y. H. Ja, Appl. Opt. 25, 4306 (1986).
    [CrossRef]
  11. M. R. Belic, W. Krolikowski, J. Opt. Soc. Am. B 6, 901 (1989).
    [CrossRef]
  12. T. K. Das, K. Singh, Appl. Phys. B 49, 557 (1989).
    [CrossRef]
  13. T. K. Das, K. Singh, Opt. Commun. 76, 381 (1990).
    [CrossRef]
  14. S. K. Kwong, A. Yariv, M. Cronin-Golomb, B. Fischer, J. Opt. Soc. Am. A 3, 157 (1986).
    [CrossRef]
  15. W. Krolikowski, M. R. Belic, A. Bledowski, Phys. Rev. A 37, 2224 (1988).
    [CrossRef] [PubMed]
  16. O. N. Volkova, A. A. Zozulya, Sov. J. Quantum Electron. 19, 913 (1989).
    [CrossRef]
  17. R. W. Boyd, T. M. Habashy, A. A. Jacobs, L. Mandel, M. Nieto-Vesperinas, W. Tompkin, E. Wolf, Opt. Lett. 12, 42 (1987).
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  18. Y. Tomita, R. Yahalom, A. Yariv, Opt. Commun. 73, 413 (1989).
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  19. W. Krolikowski, K. D. Shaw, M. Cronin-Golomb, A. Bledowski, J. Opt. Soc. Am. B 6, 1828 (1989).
    [CrossRef]

1990 (1)

T. K. Das, K. Singh, Opt. Commun. 76, 381 (1990).
[CrossRef]

1989 (5)

M. R. Belic, W. Krolikowski, J. Opt. Soc. Am. B 6, 901 (1989).
[CrossRef]

T. K. Das, K. Singh, Appl. Phys. B 49, 557 (1989).
[CrossRef]

O. N. Volkova, A. A. Zozulya, Sov. J. Quantum Electron. 19, 913 (1989).
[CrossRef]

Y. Tomita, R. Yahalom, A. Yariv, Opt. Commun. 73, 413 (1989).
[CrossRef]

W. Krolikowski, K. D. Shaw, M. Cronin-Golomb, A. Bledowski, J. Opt. Soc. Am. B 6, 1828 (1989).
[CrossRef]

1988 (4)

W. Krolikowski, M. R. Belic, A. Bledowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

A. Bledowski, W. Krolikowski, Opt. Lett. 13, 146 (1988).
[CrossRef]

W. Krolikowski, M. R. Belic, Opt. Lett. 13, 149 (1988).
[CrossRef]

M. R. Belic, Phys. Rev. A 37, 1809 (1988).
[CrossRef] [PubMed]

1987 (2)

1986 (3)

1985 (2)

M. R. Belic, Phys. Rev. A 31, 3169 (1985).
[CrossRef] [PubMed]

M. R. Belic, M. Lax, Opt. Commun. 56, 197 (1985).
[CrossRef]

1982 (1)

1981 (1)

Belic, M. R.

M. R. Belic, W. Krolikowski, J. Opt. Soc. Am. B 6, 901 (1989).
[CrossRef]

W. Krolikowski, M. R. Belic, A. Bledowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

W. Krolikowski, M. R. Belic, Opt. Lett. 13, 149 (1988).
[CrossRef]

M. R. Belic, Phys. Rev. A 37, 1809 (1988).
[CrossRef] [PubMed]

M. R. Belic, Opt. Lett. 12, 105 (1987).
[CrossRef]

M. R. Belic, Phys. Rev. A 31, 3169 (1985).
[CrossRef] [PubMed]

M. R. Belic, M. Lax, Opt. Commun. 56, 197 (1985).
[CrossRef]

Bledowski, A.

Boyd, R. W.

Cronin-Golomb, M.

Das, T. K.

T. K. Das, K. Singh, Opt. Commun. 76, 381 (1990).
[CrossRef]

T. K. Das, K. Singh, Appl. Phys. B 49, 557 (1989).
[CrossRef]

Fischer, B.

Habashy, T. M.

Ja, Y. H.

Jacobs, A. A.

Krolikowski, W.

Kwong, S. K.

Lax, M.

M. R. Belic, M. Lax, Opt. Commun. 56, 197 (1985).
[CrossRef]

Mandel, L.

Nieto-Vesperinas, M.

Shaw, K. D.

Singh, K.

T. K. Das, K. Singh, Opt. Commun. 76, 381 (1990).
[CrossRef]

T. K. Das, K. Singh, Appl. Phys. B 49, 557 (1989).
[CrossRef]

Tomita, Y.

Y. Tomita, R. Yahalom, A. Yariv, Opt. Commun. 73, 413 (1989).
[CrossRef]

Tompkin, W.

Volkova, O. N.

O. N. Volkova, A. A. Zozulya, Sov. J. Quantum Electron. 19, 913 (1989).
[CrossRef]

White, J. O.

Wolf, E.

Yahalom, R.

Y. Tomita, R. Yahalom, A. Yariv, Opt. Commun. 73, 413 (1989).
[CrossRef]

Yariv, A.

Zozulya, A. A.

O. N. Volkova, A. A. Zozulya, Sov. J. Quantum Electron. 19, 913 (1989).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

T. K. Das, K. Singh, Appl. Phys. B 49, 557 (1989).
[CrossRef]

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

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

Opt. Commun. (4)

T. K. Das, K. Singh, Opt. Commun. 76, 381 (1990).
[CrossRef]

Y. Tomita, R. Yahalom, A. Yariv, Opt. Commun. 73, 413 (1989).
[CrossRef]

M. R. Belic, M. Lax, Opt. Commun. 56, 197 (1985).
[CrossRef]

W. Krolikowski, Opt. Commun. 60, 319 (1986).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (3)

W. Krolikowski, M. R. Belic, A. Bledowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

M. R. Belic, Phys. Rev. A 31, 3169 (1985).
[CrossRef] [PubMed]

M. R. Belic, Phys. Rev. A 37, 1809 (1988).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (1)

O. N. Volkova, A. A. Zozulya, Sov. J. Quantum Electron. 19, 913 (1989).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Intensities of four waves at different zs for n1 = −560, n2 = 100, n3 = 500, and n4 = −460, in units of inverse meters. The initial wave intensities are I1(0) = I2(L) = 1 and I4(L) = 0.6. Solid and dashed curves are for θ1 = θ2 = θ3 = θ4 = π/2 and θ1 = π/8, θ2 = π/3, θ3 = θ4 = π/6, respectively. The curves a and a′, b and b′, c and c′, and d and d′ are for I1(z), I2(z), I3(z), and I4(z), respectively.

Fig. 2
Fig. 2

Output phase-conjugate reflectivity R versus log r with n’s and θ ’s as parameters. (a) n1 = 800, n2 = 600, n3 = 500, n4 = 460. Curves 1, θ1 = θ2 = θ3 = θ4 = π/2; 2, θ1 = 0, θ2 = θ3 = θ4 = π/2; 3, θ2 = 0, θ1 = θ3 = θ4 = π/2. (b) n1 = −800, n2 = −600, n3 = −500, n4 = −460. Curves 1, θ1 = θ2 = θ3 =θ4 = π/2; 2, θ1 = 0, θ2 = θ3 = θ4 = π/2; 3, θ2 = 0, θ1 = θ3 = θ4 = π/2; 4, θ1 = θ2 = θ3 = θ4 = π/2. (c) n1 = 800, n2 = −600, n3 = −500, n4 = −460. Curves 1, θ1 = θ2 = θ3 = θ4 = π/2; 2, θ1 = π/6, θ2 = θ3 = θ4 = π/2; 3, θ2 = π/6, θ1 = θ3 = θ4 = π/2. (d) n1 = 800, n2 = −600, n3 = 500, n4 = 460. Curves 1, θ1 = θ2 = θ3 = θ4 = π/2; 2, θ1 = π/6, θ2 = θ3 = θ4 = π/2; 3, θ2 = π/6, θ1 = θ3 = θ4 = π/2. (e) n1 = −800, n2 = 600, n3 = −500, n4 = −460. Curves 1, θ1 = θ2 = θ3 = θ4 = π/2; 2, θ1 = π/6, θ2 = θ3 = θ4 = π2; 3, θ2 = π/6, θ1 = θ3 = θ4 = π/2.

Fig. 3
Fig. 3

Phases of four waves at different z’s for various n’s and θ ’s with r = 1, q = 0.3, and φ1(0) = φ4(0) = φ2(L) = 0. Curves a and a′, b and b′, c and c′, and d and d′ are for φ1(z), φ2(z), φ3(z), and φ4(z), respectively, (a) n1 = −560, n2 = −500, n3 = −500, n4 = −460. Solid and dashed curves are for θ1 = θ2 = θ3 = θ4 = π/3 and for θ1 = θ2 = θ3 = θ4 = π/6, respectively, (b) θ1 = π/6, θ2 = π/3, θ3 = π/4, θ4 = π/2. Solid and dashed curves are for n1 = −560, n2 = −600, n3 = −500, n4 = −460 and for n1 = −1120,n2 = −600,n3 = −500, and n4 = −500, respectively.

Fig. 4
Fig. 4

φ3(0) as a function of log r and log q for n1 = −800, n2 = −300, n3= −500, n4= −460 and θ1 = π/6, θ2 = π/4, θ3 = θ4 = π/3. φ1(0) = φ4(0) = φ2(L) = 0.

Fig. 5
Fig. 5

Absolute phase φ(z) as a function of z for different n’s and various θ ’s with r = 1 and q = 0.3. (a) n1 = −1000, n2 = −600, n3 = −500, n4 = −460. Curves 1, θ1 = θ2 = θ3 = θ4 = π/2; 2, θ1 = π/8, θ2 = θ3 = θ4 = π/2; 3, θ1 = 0, θ2 = θ3 = θ4 = π/2; 4, θ2 = π/6, θ1 = θ3 = θ4 = π/2; 5, θ2 = 0, θ1 = θ3 = θ4 = π/2; 6, θ1 = θ2 = π/2, θ3 = θ4 = 0; 7, θ1 = θ2 = θ3 = θ4 = 0. (b) n1 = 1000, n2 = −600, n3 = −500, n4 = −460. Curves 1, θ1 = θ2 = θ3 = θ4 = π/2; 2, θ1 = π/3, θ2 = θ3 = θ4 = π/2; 3, θ1 = 0, θ2 = θ3 = θ4 = π/2; 4, θ2 = π/6, θ1 = θ3 = θ4 = π/2; 5, θ2 = 0, θ1 = θ3 = θ4 = π/2; 6, θ1 = θ2 = π/2, θ3 = θ4 = 0; 7, θ1 = θ2 = θ3 = θ4 = 0.

Fig. 6
Fig. 6

Absolute phase φ(z) as a function of z for various r’s and q’s. θ1 = θ2 = θ3 = θ4 = π/6. (a) n1 = −1000, n2= −600, n3 = −500, n4= −460. Curves 1, r = 1, q = 0.01; 2, r = 1, q = 0.1; 3, r = 1, q = 1; 4, r = 1, q = 100; 5, r = 0.01, q = 1; 6, r = 100, q = 1. (b) n1 = 1000, n2 = −600, n3 = −500, n4 = 460. Curves 1, r = 1, q = 0.01; 2, r = 1, q = 1; 3, r = 1, q = 100; 4, r = 0.01, q = 1; 5, r = 100, q = 1.

Fig. 7
Fig. 7

Curves of φ(L) as functions of θi. (a) n1 = −1000, n2 = −600, n3 = −500, n4 = −460. Curves 1, θ1 is a variable and θ2 = θ3 = θ4 = π/2; 2, θ2 is a variable and θ1 = θ3 = θ4 = π/2; 3, θ3 or θ4 is a variable and other θi = π/2. (b) n1 = 1000, n2 = −600, n3 = −500, n4 = −460. Curves 1, θ1 is a variable and θ2 = θ3 = θ4 = π/2; 2, θ2 is a variable and θ1 = θ3 = θ4 = π/2; 3, θ3 or θ4 is a variable and other θi = π/2.

Fig. 8
Fig. 8

(a) φ3(0) and (b) R versus log q for r = 1, n1 = 3000, n2 = 2000, n3 = n4 = 1500 and θ1 = π/18, θ2 = π/6, θ3 = θ4 = π/4, φ1(0) = φ4(0) = φ2(L) = 0.

Fig. 9
Fig. 9

φ3 (0) versus log q for r = 1, n1 = 3000, n2 = n3 = n4 = 0, φ1 (0) = φ4 (0) = φ2 (L) = 0.(a) θ1 = 0, (b) θ1 = π/24, (c) θ1 = π/6.

Fig. 10
Fig. 10

R versus log q for r = 1, n1 = 3000, n2 = n3 = n4 = 0. (a) θ1 = 0, (b) θ1 = π/24, (c) θ1 = π/6.

Fig. 11
Fig. 11

(a) φ3(0) and (b) R versus log q for r = 1, n1 = n3 = n4 = 0, n2 = −3000, θ2 = 0, φ1(0) = φ4(0) = φ2 (L) = 0.

Equations (41)

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I 0 d A 1 / d z = γ 1 * ( A 1 A 4 * + A 2 * A 3 ) A 4 γ 2 ( A 1 A 3 * + A 2 * A 4 ) A 3 γ 3 A 1 | A 2 | 2 α I 0 A 1 ,
I 0 d A 2 * / d z = γ 1 * ( A 1 A 4 * + A 2 * A 3 ) A 3 * γ 2 ( A 1 A 3 * + A 2 * A 4 ) A 4 * γ 3 | A 1 | 2 A 2 * + α I 0 A 2 * ,
I 0 d A 3 / d z = γ 1 * ( A 1 A 4 * + A 2 * A 3 ) A 2 γ 2 * ( A 1 * A 3 + A 2 A 4 * ) A 1 γ 4 * A 3 | A 4 | 2 + α I 0 A 3 ,
I 0 d A 4 * / d z = γ 1 * ( A 1 A 4 * + A 2 * A 3 ) A 1 * γ 2 ( A 1 * A 3 + A 2 A 4 * ) A 2 * γ 4 * | A 3 | 2 A 4 * α I 0 A 4 * .
γ j = i n j exp ( i θ j ) ( j = 1 , 2 , 3 , 4 ) ,
n j = [ ω n 0 3 / ( 4 C cos Θ ) ] r eff E p × { ( E 0 2 + E d 2 ) / [ E 0 2 + ( E d + E p ) 2 ] } 1 / 2 ,
tan θ j = [ E d ( E d + E p ) E 0 2 ] / ( E 0 E p ) ,
A ( z ) j = [ I j ( z ) ] 1 / 2 exp [ i φ j ( z ) ] .
I 0 d I 1 / d z = 2 n 1 [ I 1 I 4 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 2 [ I 1 I 3 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 3 I 1 I 2 2 α I 0 I 1 ,
I 0 d I 2 / d z = 2 n 1 [ I 2 I 3 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 2 [ I 2 I 4 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 3 I 2 I 1 + 2 α I 0 I 2 ,
I 0 d I 3 / d z = 2 n 1 [ I 3 I 2 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 2 [ I 3 I 1 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 4 I 3 I 4 + 2 α I 0 I 3 ,
I 0 d I 4 / d z = 2 n 1 [ I 4 I 1 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 2 [ I 4 I 2 + ( I 1 I 2 I 3 I 4 ) 1 / 2 cos φ ] + 2 n 4 I 4 I 3 2 α I 0 I 4 ,
I 0 d φ 1 / d z = ( n 2 n 1 ) ( I 2 I 3 I 4 / I 1 ) 1 / 2 sin φ ,
I 0 d φ 2 / d z = ( n 2 n 1 ) ( I 1 I 3 I 4 / I 2 ) 1 / 2 sin φ ,
I 0 d φ 3 / d z = ( n 1 + n 2 ) ( I 1 I 2 I 4 / I 3 ) 1 / 2 sin φ ,
I 0 d φ 4 / d z = ( n 1 + n 2 ) ( I 1 I 2 I 3 / I 4 ) 1 / 2 sin φ ,
φ = φ 3 + φ 4 φ 1 φ 2 .
A j ( z ) = ρ j ( z ) exp [ i φ j ( z ) ] ( j = 1 , 2 , 4 ) ,
A 3 ( z ) = a ( z ) + i b ( z ) .
I 0 d ρ 1 / d z = n 1 { ρ 1 ρ 4 2 sin θ 1 + ρ 2 ρ 4 [ a sin ( δ θ 1 ) ] + b cos ( δ θ 1 ) } + n 2 { ρ 1 ( a 2 + b 2 ) sin θ 2 + ρ 2 ρ 4 [ a sin ( δ + θ 2 ) + b cos ( δ + θ 2 ) } + n 3 ρ 1 ρ 2 2 sin θ 3 α I 0 ρ 1 ,
I 0 d ρ 2 / d z = n 1 { ρ 2 ( a 2 + b 2 ) sin θ 1 ρ 1 ρ 4 [ a sin ( δ + θ 1 ) ] + b cos ( δ + θ 1 ) } + n 2 { ρ 2 ρ 4 sin θ 2 ρ 1 ρ 4 [ a sin ( δ θ 2 ) + b cos ( δ θ 2 ) ] } + n 3 ρ 2 ρ 1 2 sin θ 3 + α I 0 p 2 ,
I 0 d a / d z = n 1 [ ρ 1 ρ 2 ρ 4 sin ( δ + θ 1 ) + ρ 2 2 [ a sin θ 1 b cos θ 1 ) ] + n 2 [ ρ 1 ρ 2 ρ 4 sin ( δ + θ 2 ) + ρ 1 2 ( a sin θ 2 b cos θ 2 ) ] + n 4 ρ 4 2 ( a sin θ 4 b cos θ 4 ) + α I 0 a ,
I 0 d ρ 4 / d z = n 1 { ρ 4 ρ 1 2 sin θ 1 ρ 1 ρ 2 [ a sin ( δ θ 1 ) + b cos ( δ θ 1 ) ] } + n 2 { ρ 4 ρ 2 2 sin θ 2 ρ 1 ρ 2 [ a sin ( δ θ 2 ) + b cos ( δ θ 2 ) ] } + n 4 ρ 4 ( a 2 + b 2 ) sin θ 4 α I 0 ρ 4 ,
I 0 d φ 1 / d z = n 1 { ρ 4 2 cos θ 1 ρ 2 ρ 4 / ρ 1 [ a cos ( δ θ 1 ) b sin ( δ θ 1 ) ] } + n 2 { ( a 2 + b 2 ) cos θ 2 ρ 2 ρ 4 / ρ 1 [ a cos ( δ + θ 2 ) b sin ( δ + θ 2 ) ] } n 3 ρ 2 2 cos θ 3 ,
I 0 d φ 2 / d z = n 1 { ( a 2 + b 2 ) cos θ 1 + ρ 1 ρ 4 / ρ 2 [ a cos ( δ + θ 1 ) + b sin ( δ + θ 1 ) ] } n 2 { ρ 4 2 cos θ 2 + ρ 1 ρ 4 / ρ 2 [ a cos ( δ θ 2 ) + b sin ( δ θ 2 ) ] } + n 3 ρ 1 2 cos θ 3 ,
I 0 d b / d z = n 1 [ ρ 1 ρ 2 ρ 4 cos ( δ + θ 1 ) + ρ 2 2 ( a cos θ 1 + b sin θ 1 ) ] + n 2 [ ρ 1 ρ 2 ρ 4 cos ( δ + θ 2 ) + ρ 1 2 ( a cos θ 2 + b sin θ 2 ) ] + n 4 ρ 4 2 ( a cos θ 4 + b sin θ 4 ) + α I 0 b ,
I 0 d φ 4 / d z = n 1 { ρ 1 2 cos θ 1 + ρ 1 ρ 2 / ρ 4 [ a cos ( δ θ 1 ) b sin ( δ θ 1 ) ] } n 2 { ρ 2 2 cos θ 2 + ρ 1 ρ 2 / ρ 4 [ a cos ( δ θ 2 ) b sin ( δ θ 2 ) ] } n 4 ( a 2 + b 2 ) cos θ 4 ,
δ = φ 4 φ 1 φ 2 .
Y i ( L ) = Y i ( x 1 , x 2 , x 3 , x 4 ) ( i = 1 , 2 , 3 , 4 ) .
Y i ( L ) = Y i ( x 1 , x 2 , x 3 , x 4 ) .
Y i ( x 1 , x 2 , x 3 , x 4 ) = Y i ( x 1 , x 2 , x 3 , x 4 ) + Y i / x 1 ( x 1 x 1 ) + Y i / x 2 ( x 2 x 2 ) + Y i / x 3 ( x 3 x 3 ) + Y i / x 4 ( x 4 x 4 ) .
Y i / x 1 = [ Y i ( x 1 + Δ x 1 , x 2 , x 3 , x 4 ) Y i ( x 1 , x 2 , x 3 , x 4 ) ] / Δ x 1 ,
Y i / x 2 = [ Y i ( x 1 , x 2 + Δ x 2 , x 3 , x 4 ) Y i ( x 1 , x 2 , x 3 , x 4 ) ] / Δ x 2 ,
Y i / x 3 = [ Y i ( x 1 , x 2 , x 3 + Δ x 3 , x 4 ) Y i ( x 1 , x 2 , x 3 , x 4 ) ] / Δ x 3 ,
Y i / x 4 = [ Y i ( x 1 , x 2 , x 3 , x 4 + Δ x 4 ) Y i ( x 1 , x 2 , x 3 , x 4 ) ] / Δ x 4 .
Y 1 ( x 1 , x 2 , x 3 , x 4 ) = ρ 2 ( L ) ,
Y 2 ( x 1 , x 2 , x 3 , x 4 ) = φ 2 ( L ) ,
Y 3 ( x 1 , x 2 , x 3 , x 4 ) = 0 ,
Y 4 ( x 1 , x 2 , x 3 , x 4 ) = 0 .
φ ( z ) φ 3 ( z ) + φ 4 ( z ) φ 1 ( z ) φ 2 ( z ) = 0
φ ( z ) φ 3 ( z ) + φ 4 ( z ) φ 1 ( z ) φ 2 ( z ) = π .

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