Abstract

The collinear anisotropic interaction of light with a standing acoustic wave generated in a lithium niobate crystal is discussed. Diffraction may be applied for harmonic modulation of laser light. The analysis of the light modulation is based on the concept of vector diagrams. A set of differential equations is derived to describe up to five interacting optic waves in the material. It is shown that solution of the equations by the method of successive approximations yields amplitudes of electric fields corresponding to diffracted and transmitted optical beams at the output of the crystal. Dependences of the amplitudes of the interacting waves on the acoustic frequency and length of the acousto-optic interaction are also discussed.

© 2009 Optical Society of America

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References

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  1. J. Xu and R. Stroud, Acousto-Optic Devices (Wiley, 1992).
  2. A. Goutzoulis and D. Pape, Design and Fabrication of Acousto-Optic Devices (Marcel Dekker, 1994).
  3. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  4. V. I. Balakshy, V. N. Parygin, and L. E. Chirkov, Optical Waves in Crystals (Radio and Communication, 1985) (in Russian).
  5. S. E. Harris, S. T. Nieh, and R. S. Feigelson, “CaMoO4 electrooptically tunable optical filter,” Appl. Phys. Lett. 17, 223-225(1970).
    [CrossRef]
  6. I. C. Chang, “Tunable acousto-optic filter utilizing acoustic beam walk-off in crystal quartz,” Appl. Phys. Lett. 25, 323-324(1974).
    [CrossRef]
  7. S. E. Harris, S. T. Nieh, and D. K. Winslow, “Electronically tunable acousto-optic filter,” Appl. Phys. Lett. 15325-326 (1969).
    [CrossRef]
  8. F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586-589 (2000).
    [CrossRef]
  9. V. N. Parygin and I. N. Zhmakin, “Collinear interaction of Gaussian acoustic and light beams,” Ultrasonics 31339-343(1993).
    [CrossRef]
  10. J. D. Feichtner, M. Gottlieb, and J. J. Conroy, “Tunable acoustooptic filters and their applications to spectroscopy,” Proc. SPIE 82, 106-118 (1976).
  11. I. C. Chang, “Tunable acoustooptic filtering: an overview,” Proc. SPIE 90, 12-22 (1976).
  12. T. J. Taylor, S. E. Harris, and S. T. Nieh, “Electronic tuning of a dye laser using the acoustooptic filter,” Appl. Phys. Lett. 19269-271 (1971).
    [CrossRef]
  13. Yu. S. Dobrolenskiy and V. B. Voloshinov, “Efficiency of collinear acousto-optic interaction in anisotropic media,” Proc. SPIE 5953, 86-94 (2005).
  14. Yu. S. Dobrolenskiy, V. B. Voloshinov, and V. N. Parygin, “Collinear acousto-optic interaction of divergent beams in paratellurite crystal,” Proc. SPIE 5828, 16-24 (2004).
  15. Yu. S. Dobrolenskiy, V. B. Voloshinov, Yu. A. Zyuryukin, and A. N. Yulaev, “Non-reciprocity of acousto-optic interaction: investigation of collinear diffraction,” in Abstracts of X International Conference for Young Researchers, Wave Electronics and Its Applications in the Information and Telecommunication Systems (2007), p. 20.
  16. Yu. S. Dobrolenskiy, V. B. Voloshinov, and Yu. A. Zyuryukin, “Influence of nonreciprocal effect on the operation of a collinear acousto-optic filter,” Quantum Electron. 3846-50 (2008).
    [CrossRef]
  17. F. Kuliasko and R. Mertens, “On the diffraction of light by standing supersonic waves: oblique incidence of the light,” Simon Stevin 34, 126-136 (1961).
  18. R. Mertens, “On the diffraction of light by supersonic waves. Part II: Standing supersonic waves,” Simon Stevin 28, 1-12(1950).
  19. R. Mertens, “Diffraction of light by standing supersonic waves: general theory,” Simon Stevin 28, 164-180 (1950).
  20. R. Mertens, “On the diffraction of light by progressive and standing supersonic waves,” Proc. Indian Acad. Sci. Sect. A 42, 195-198 (1955).
  21. Yu. A. Zyuryukin, “Coupled-wave method in the theory of light diffraction by elastic waves in crystals,” Proc. SPIE 4002, 162-174 (2000).
  22. E. Kamke, Gewöhnliche differentialgleichungen (Leipzig, 1959).

2008 (1)

Yu. S. Dobrolenskiy, V. B. Voloshinov, and Yu. A. Zyuryukin, “Influence of nonreciprocal effect on the operation of a collinear acousto-optic filter,” Quantum Electron. 3846-50 (2008).
[CrossRef]

2005 (1)

Yu. S. Dobrolenskiy and V. B. Voloshinov, “Efficiency of collinear acousto-optic interaction in anisotropic media,” Proc. SPIE 5953, 86-94 (2005).

2004 (1)

Yu. S. Dobrolenskiy, V. B. Voloshinov, and V. N. Parygin, “Collinear acousto-optic interaction of divergent beams in paratellurite crystal,” Proc. SPIE 5828, 16-24 (2004).

2000 (2)

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586-589 (2000).
[CrossRef]

Yu. A. Zyuryukin, “Coupled-wave method in the theory of light diffraction by elastic waves in crystals,” Proc. SPIE 4002, 162-174 (2000).

1993 (1)

V. N. Parygin and I. N. Zhmakin, “Collinear interaction of Gaussian acoustic and light beams,” Ultrasonics 31339-343(1993).
[CrossRef]

1976 (2)

J. D. Feichtner, M. Gottlieb, and J. J. Conroy, “Tunable acoustooptic filters and their applications to spectroscopy,” Proc. SPIE 82, 106-118 (1976).

I. C. Chang, “Tunable acoustooptic filtering: an overview,” Proc. SPIE 90, 12-22 (1976).

1974 (1)

I. C. Chang, “Tunable acousto-optic filter utilizing acoustic beam walk-off in crystal quartz,” Appl. Phys. Lett. 25, 323-324(1974).
[CrossRef]

1971 (1)

T. J. Taylor, S. E. Harris, and S. T. Nieh, “Electronic tuning of a dye laser using the acoustooptic filter,” Appl. Phys. Lett. 19269-271 (1971).
[CrossRef]

1970 (1)

S. E. Harris, S. T. Nieh, and R. S. Feigelson, “CaMoO4 electrooptically tunable optical filter,” Appl. Phys. Lett. 17, 223-225(1970).
[CrossRef]

1969 (1)

S. E. Harris, S. T. Nieh, and D. K. Winslow, “Electronically tunable acousto-optic filter,” Appl. Phys. Lett. 15325-326 (1969).
[CrossRef]

1961 (1)

F. Kuliasko and R. Mertens, “On the diffraction of light by standing supersonic waves: oblique incidence of the light,” Simon Stevin 34, 126-136 (1961).

1955 (1)

R. Mertens, “On the diffraction of light by progressive and standing supersonic waves,” Proc. Indian Acad. Sci. Sect. A 42, 195-198 (1955).

1950 (2)

R. Mertens, “On the diffraction of light by supersonic waves. Part II: Standing supersonic waves,” Simon Stevin 28, 1-12(1950).

R. Mertens, “Diffraction of light by standing supersonic waves: general theory,” Simon Stevin 28, 164-180 (1950).

Balakshy, V. I.

V. I. Balakshy, V. N. Parygin, and L. E. Chirkov, Optical Waves in Crystals (Radio and Communication, 1985) (in Russian).

Chang, I. C.

I. C. Chang, “Tunable acoustooptic filtering: an overview,” Proc. SPIE 90, 12-22 (1976).

I. C. Chang, “Tunable acousto-optic filter utilizing acoustic beam walk-off in crystal quartz,” Appl. Phys. Lett. 25, 323-324(1974).
[CrossRef]

Chirkov, L. E.

V. I. Balakshy, V. N. Parygin, and L. E. Chirkov, Optical Waves in Crystals (Radio and Communication, 1985) (in Russian).

Conroy, J. J.

J. D. Feichtner, M. Gottlieb, and J. J. Conroy, “Tunable acoustooptic filters and their applications to spectroscopy,” Proc. SPIE 82, 106-118 (1976).

Dobrolenskiy, Yu. S.

Yu. S. Dobrolenskiy, V. B. Voloshinov, and Yu. A. Zyuryukin, “Influence of nonreciprocal effect on the operation of a collinear acousto-optic filter,” Quantum Electron. 3846-50 (2008).
[CrossRef]

Yu. S. Dobrolenskiy and V. B. Voloshinov, “Efficiency of collinear acousto-optic interaction in anisotropic media,” Proc. SPIE 5953, 86-94 (2005).

Yu. S. Dobrolenskiy, V. B. Voloshinov, and V. N. Parygin, “Collinear acousto-optic interaction of divergent beams in paratellurite crystal,” Proc. SPIE 5828, 16-24 (2004).

Yu. S. Dobrolenskiy, V. B. Voloshinov, Yu. A. Zyuryukin, and A. N. Yulaev, “Non-reciprocity of acousto-optic interaction: investigation of collinear diffraction,” in Abstracts of X International Conference for Young Researchers, Wave Electronics and Its Applications in the Information and Telecommunication Systems (2007), p. 20.

Feichtner, J. D.

J. D. Feichtner, M. Gottlieb, and J. J. Conroy, “Tunable acoustooptic filters and their applications to spectroscopy,” Proc. SPIE 82, 106-118 (1976).

Feigelson, R. S.

S. E. Harris, S. T. Nieh, and R. S. Feigelson, “CaMoO4 electrooptically tunable optical filter,” Appl. Phys. Lett. 17, 223-225(1970).
[CrossRef]

Gottlieb, M.

J. D. Feichtner, M. Gottlieb, and J. J. Conroy, “Tunable acoustooptic filters and their applications to spectroscopy,” Proc. SPIE 82, 106-118 (1976).

Goutzoulis, A.

A. Goutzoulis and D. Pape, Design and Fabrication of Acousto-Optic Devices (Marcel Dekker, 1994).

Harris, S. E.

T. J. Taylor, S. E. Harris, and S. T. Nieh, “Electronic tuning of a dye laser using the acoustooptic filter,” Appl. Phys. Lett. 19269-271 (1971).
[CrossRef]

S. E. Harris, S. T. Nieh, and R. S. Feigelson, “CaMoO4 electrooptically tunable optical filter,” Appl. Phys. Lett. 17, 223-225(1970).
[CrossRef]

S. E. Harris, S. T. Nieh, and D. K. Winslow, “Electronically tunable acousto-optic filter,” Appl. Phys. Lett. 15325-326 (1969).
[CrossRef]

Kamke, E.

E. Kamke, Gewöhnliche differentialgleichungen (Leipzig, 1959).

Kuliasko, F.

F. Kuliasko and R. Mertens, “On the diffraction of light by standing supersonic waves: oblique incidence of the light,” Simon Stevin 34, 126-136 (1961).

Leroy, O.

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586-589 (2000).
[CrossRef]

Mertens, R.

F. Kuliasko and R. Mertens, “On the diffraction of light by standing supersonic waves: oblique incidence of the light,” Simon Stevin 34, 126-136 (1961).

R. Mertens, “On the diffraction of light by progressive and standing supersonic waves,” Proc. Indian Acad. Sci. Sect. A 42, 195-198 (1955).

R. Mertens, “On the diffraction of light by supersonic waves. Part II: Standing supersonic waves,” Simon Stevin 28, 1-12(1950).

R. Mertens, “Diffraction of light by standing supersonic waves: general theory,” Simon Stevin 28, 164-180 (1950).

Nieh, S. T.

T. J. Taylor, S. E. Harris, and S. T. Nieh, “Electronic tuning of a dye laser using the acoustooptic filter,” Appl. Phys. Lett. 19269-271 (1971).
[CrossRef]

S. E. Harris, S. T. Nieh, and R. S. Feigelson, “CaMoO4 electrooptically tunable optical filter,” Appl. Phys. Lett. 17, 223-225(1970).
[CrossRef]

S. E. Harris, S. T. Nieh, and D. K. Winslow, “Electronically tunable acousto-optic filter,” Appl. Phys. Lett. 15325-326 (1969).
[CrossRef]

Pape, D.

A. Goutzoulis and D. Pape, Design and Fabrication of Acousto-Optic Devices (Marcel Dekker, 1994).

Parygin, V. N.

Yu. S. Dobrolenskiy, V. B. Voloshinov, and V. N. Parygin, “Collinear acousto-optic interaction of divergent beams in paratellurite crystal,” Proc. SPIE 5828, 16-24 (2004).

V. N. Parygin and I. N. Zhmakin, “Collinear interaction of Gaussian acoustic and light beams,” Ultrasonics 31339-343(1993).
[CrossRef]

V. I. Balakshy, V. N. Parygin, and L. E. Chirkov, Optical Waves in Crystals (Radio and Communication, 1985) (in Russian).

Pustovoit, V. I.

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586-589 (2000).
[CrossRef]

Stroud, R.

J. Xu and R. Stroud, Acousto-Optic Devices (Wiley, 1992).

Taylor, T. J.

T. J. Taylor, S. E. Harris, and S. T. Nieh, “Electronic tuning of a dye laser using the acoustooptic filter,” Appl. Phys. Lett. 19269-271 (1971).
[CrossRef]

Voloshinov, V. B.

Yu. S. Dobrolenskiy, V. B. Voloshinov, and Yu. A. Zyuryukin, “Influence of nonreciprocal effect on the operation of a collinear acousto-optic filter,” Quantum Electron. 3846-50 (2008).
[CrossRef]

Yu. S. Dobrolenskiy and V. B. Voloshinov, “Efficiency of collinear acousto-optic interaction in anisotropic media,” Proc. SPIE 5953, 86-94 (2005).

Yu. S. Dobrolenskiy, V. B. Voloshinov, and V. N. Parygin, “Collinear acousto-optic interaction of divergent beams in paratellurite crystal,” Proc. SPIE 5828, 16-24 (2004).

Yu. S. Dobrolenskiy, V. B. Voloshinov, Yu. A. Zyuryukin, and A. N. Yulaev, “Non-reciprocity of acousto-optic interaction: investigation of collinear diffraction,” in Abstracts of X International Conference for Young Researchers, Wave Electronics and Its Applications in the Information and Telecommunication Systems (2007), p. 20.

Windels, F. W.

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586-589 (2000).
[CrossRef]

Winslow, D. K.

S. E. Harris, S. T. Nieh, and D. K. Winslow, “Electronically tunable acousto-optic filter,” Appl. Phys. Lett. 15325-326 (1969).
[CrossRef]

Xu, J.

J. Xu and R. Stroud, Acousto-Optic Devices (Wiley, 1992).

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yulaev, A. N.

Yu. S. Dobrolenskiy, V. B. Voloshinov, Yu. A. Zyuryukin, and A. N. Yulaev, “Non-reciprocity of acousto-optic interaction: investigation of collinear diffraction,” in Abstracts of X International Conference for Young Researchers, Wave Electronics and Its Applications in the Information and Telecommunication Systems (2007), p. 20.

Zhmakin, I. N.

V. N. Parygin and I. N. Zhmakin, “Collinear interaction of Gaussian acoustic and light beams,” Ultrasonics 31339-343(1993).
[CrossRef]

Zyuryukin, Yu. A.

Yu. S. Dobrolenskiy, V. B. Voloshinov, and Yu. A. Zyuryukin, “Influence of nonreciprocal effect on the operation of a collinear acousto-optic filter,” Quantum Electron. 3846-50 (2008).
[CrossRef]

Yu. A. Zyuryukin, “Coupled-wave method in the theory of light diffraction by elastic waves in crystals,” Proc. SPIE 4002, 162-174 (2000).

Yu. S. Dobrolenskiy, V. B. Voloshinov, Yu. A. Zyuryukin, and A. N. Yulaev, “Non-reciprocity of acousto-optic interaction: investigation of collinear diffraction,” in Abstracts of X International Conference for Young Researchers, Wave Electronics and Its Applications in the Information and Telecommunication Systems (2007), p. 20.

Appl. Phys. Lett. (4)

S. E. Harris, S. T. Nieh, and R. S. Feigelson, “CaMoO4 electrooptically tunable optical filter,” Appl. Phys. Lett. 17, 223-225(1970).
[CrossRef]

I. C. Chang, “Tunable acousto-optic filter utilizing acoustic beam walk-off in crystal quartz,” Appl. Phys. Lett. 25, 323-324(1974).
[CrossRef]

S. E. Harris, S. T. Nieh, and D. K. Winslow, “Electronically tunable acousto-optic filter,” Appl. Phys. Lett. 15325-326 (1969).
[CrossRef]

T. J. Taylor, S. E. Harris, and S. T. Nieh, “Electronic tuning of a dye laser using the acoustooptic filter,” Appl. Phys. Lett. 19269-271 (1971).
[CrossRef]

Proc. Indian Acad. Sci. Sect. A (1)

R. Mertens, “On the diffraction of light by progressive and standing supersonic waves,” Proc. Indian Acad. Sci. Sect. A 42, 195-198 (1955).

Proc. SPIE (5)

Yu. A. Zyuryukin, “Coupled-wave method in the theory of light diffraction by elastic waves in crystals,” Proc. SPIE 4002, 162-174 (2000).

J. D. Feichtner, M. Gottlieb, and J. J. Conroy, “Tunable acoustooptic filters and their applications to spectroscopy,” Proc. SPIE 82, 106-118 (1976).

I. C. Chang, “Tunable acoustooptic filtering: an overview,” Proc. SPIE 90, 12-22 (1976).

Yu. S. Dobrolenskiy and V. B. Voloshinov, “Efficiency of collinear acousto-optic interaction in anisotropic media,” Proc. SPIE 5953, 86-94 (2005).

Yu. S. Dobrolenskiy, V. B. Voloshinov, and V. N. Parygin, “Collinear acousto-optic interaction of divergent beams in paratellurite crystal,” Proc. SPIE 5828, 16-24 (2004).

Quantum Electron. (1)

Yu. S. Dobrolenskiy, V. B. Voloshinov, and Yu. A. Zyuryukin, “Influence of nonreciprocal effect on the operation of a collinear acousto-optic filter,” Quantum Electron. 3846-50 (2008).
[CrossRef]

Simon Stevin (3)

F. Kuliasko and R. Mertens, “On the diffraction of light by standing supersonic waves: oblique incidence of the light,” Simon Stevin 34, 126-136 (1961).

R. Mertens, “On the diffraction of light by supersonic waves. Part II: Standing supersonic waves,” Simon Stevin 28, 1-12(1950).

R. Mertens, “Diffraction of light by standing supersonic waves: general theory,” Simon Stevin 28, 164-180 (1950).

Ultrasonics (2)

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586-589 (2000).
[CrossRef]

V. N. Parygin and I. N. Zhmakin, “Collinear interaction of Gaussian acoustic and light beams,” Ultrasonics 31339-343(1993).
[CrossRef]

Other (6)

J. Xu and R. Stroud, Acousto-Optic Devices (Wiley, 1992).

A. Goutzoulis and D. Pape, Design and Fabrication of Acousto-Optic Devices (Marcel Dekker, 1994).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

V. I. Balakshy, V. N. Parygin, and L. E. Chirkov, Optical Waves in Crystals (Radio and Communication, 1985) (in Russian).

Yu. S. Dobrolenskiy, V. B. Voloshinov, Yu. A. Zyuryukin, and A. N. Yulaev, “Non-reciprocity of acousto-optic interaction: investigation of collinear diffraction,” in Abstracts of X International Conference for Young Researchers, Wave Electronics and Its Applications in the Information and Telecommunication Systems (2007), p. 20.

E. Kamke, Gewöhnliche differentialgleichungen (Leipzig, 1959).

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Figures (4)

Fig. 1
Fig. 1

Vector diagrams as the explanation for the nonreciprocity effects of collinear anisotropic diffraction by a standing acoustic wave. (a) Diffraction that corresponds to the plus (+) in Eqs. (1, 2); the solid arc relates to the wave vector surface at ω 1 , the dashed arc relates to the wave vector surface at frequency ω 2 = ω 1 + Ω . (b) Diffraction that corresponds to the minus (−) in Eqs. (1, 2); the dotted–dashed arc relates to the wave vector surface at frequency ω 2 = ω 1 Ω .

Fig. 2
Fig. 2

Two families of wave vectors as the result of the extraordinary optic beam diffraction by the standing acoustic wave. (a) Family of extraordinary wave vectors and (b) family of ordinary wave vectors.

Fig. 3
Fig. 3

Dependences of amplitudes of optic fields on inter action length x under the condition of, a,  Δ k 1 = 0 m 1 and, b,  Δ k 1 = 83 m 1 .

Fig. 4
Fig. 4

Dependences of amplitudes of optic fields on acoustic frequency Ω under the condition x = 20 mm .

Equations (37)

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ω 2 = ω 1 ± Ω ,
k 2 = k 1 ± K .
Δ k 1 = k 2 k 1 K ,
Δ k 2 = k 4 k 1 K ,
Δ k 3 = k 2 k 3 K ,
Δ k 4 = k 4 k 5 K .
Δ k 1 = ω 1 + Ω c o ω 1 c e Ω v .
Δ k 2 = Δ k 1 2 n o Ω c ,
Δ k 3 = Δ k 1 2 n e Ω c ,
Δ k 4 = Δ k 1 2 Δ n Ω c .
Δ E 1 c 2 2 t 2 [ ε 0 × E ] = 1 c 2 2 t 2 [ ε 1 × E ] ,
E = E 1 + E 2 + E 3 + E 4 + E 5 = e 1 E 1 + e 2 E 2 + e 3 E 3 + e 4 E 4 + e 5 E 5 ,
E i ( x , t ) = 1 2 { E i exp [ j ( ω i t k i x ) ] + E i * exp [ j ( ω i t k i x ) ] } , i = 1 , 2 5 ,
e 1 [ ε 1 × e 2 ] = e 2 [ ε 1 × e 1 ] = χ ( x , t ) .
χ ( x , t ) = 1 2 { χ 1 exp [ j ( Ω t K x ) ] + χ 2 exp [ j ( Ω t + K x ) ] + χ 1 * exp [ j ( Ω t K x ) ] + χ 2 * exp [ j ( Ω t + K x ) ] } ,
E 1 x = j a 12 exp ( j Δ k 1 x ) E 2 j a 14 exp ( j Δ k 2 x ) E 4 ,
E 2 x = j a 21 exp ( j Δ k 1 x ) E 1 j a 23 exp ( j Δ k 3 x ) E 3 ,
E 3 x = j a 32 exp ( j Δ k 3 x ) E 2 ,
E 4 x = j a 45 exp ( j Δ k 4 x ) E 5 j a 41 exp ( j Δ k 2 x ) E 1 ,
E 5 x = j a 54 exp ( j Δ k 4 x ) E 4 .
E 1 = E 0 , E 2 = E 3 = E 4 = E 5 = 0.
E i = C i exp ( j λ i x ) , i = 1 , 2 5 ,
Δ k 2 λ 4 + λ 1 = 0 ,
Δ k 4 λ 4 + λ 5 = 0 ,
Δ k 3 λ 2 + λ 3 = 0 ,
Δ k 1 + λ 1 λ 2 = 0 ,
λ 1 = 0 ,
C 1 x = j a 12 C 2 j a 14 C 4 ,
C 2 x = j Δ k 1 C 2 j a 21 C 1 j a 23 C 3 ,
C 3 x = j Δ k 13 C 3 j a 32 C 2 ,
C 4 x = j Δ k 2 C 4 j a 45 C 5 j a 41 C 1 ,
C 5 x = j Δ k 24 C 5 j a 54 C 4 .
C 1 = E 0 , C 2 = C 3 = C 4 = C 5 = 0.
C ( x ) = exp ( A x ) C 0 ,
exp ( A x ) = 1 + A x + A 2 x 2 2 ! + + A n x n n ! + ,
χ 2 = χ 1 exp ( j φ ) ,
E i ( Ω , x , t ) = E i ( Ω , x ) exp j ( ω i t k i x ) , i = 1 , 2 5 .

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