Abstract

A detailed analytical study of optical scattering from a parallel ultrasonic setup is presented in which an optical beam is scattered by a surface acoustic wave that acts as a Raman–Nath-type dynamic grating and the scattering is controlled and monitored by a bulk (Bragg) acousto-optic grating placed adjacent to the surface wave. To generalize this mixed configuration, we assume the two ultrasonic cells to have a frequency ratio p:q. The intensities of the final scattered orders of light are analyzed by a multiple plane-wave scattering formalism and plotted for varying frequency ratios, Q’s, and peak phase delays ( α^). Finally, a dynamic detection method is proposed for identifying the characteristics of a surface acoustic wave with the parallel ultrasonic technique.

© 1994 Optical Society of America

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References

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  1. R. Magnusson, T. D. Black, “Enhanced detection of acoustic waves using thick and thin reference gratings,” J. Opt. Soc. Am. A 4, 498 (1987).
    [Crossref]
  2. D. A. Larson, T. D. Black, M. Green, R. G. Torti, Y. J. Wang, R. Magnusson, “Optical modulation by traveling surface acoustic wave and a holographic reference grating,” J. Opt. Soc. Am. A 7, 1745 (1990).
    [Crossref]
  3. J. S. Murty, B. R. Rao, “Diffraction of light by superposed ultrasonic waves,” Z. Phys. 157, 189 (1959).
    [Crossref]
  4. L. E. Hargrove, “Diffraction of light passing through two adjacent ultrasonic progressive waves of different frequency,”J. Opt. Soc. Am. 32, 940 (1960).
  5. R. Mertens, “On the theory of diffraction of light by two adjacent ultrasonic waves, one being the N-th harmonic of the other,”Z. Phys. 160, 291 (1960).
    [Crossref]
  6. O. Leroy, E. Blomme, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303 (1973).
  7. T.-C. Poon, P. P. Banerjee, M. R. Chatterjee, “Analysis of acoustooptic diffraction by adjacent ultrasonic beams using multiple plane-wave scattering techniques,”IEEE Trans. Sonics Ultrason. SU-32, 592 (1985).
    [Crossref]
  8. 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 (1986).
    [Crossref]
  9. A. Korpel, T.-C. Poon, “Explicit formalism for acousto-optic multiple plane wave scattering,”J. Opt. Soc. Am. 70, 817 (1980).
    [Crossref]
  10. W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction in isotropic media,”IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
    [Crossref]
  11. A. Korpel, Acousto-optics (Dekker, New York, 1988).
  12. M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81 (1990).
  13. P. P. Banerjee, C. W. Tarn, “Interaction of planar, multitone-modulated ultrasonic waves on optical beams with arbitrary profiles,” in Annual Meeting, Vol. 15 of 1990OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 134.
  14. R. Pieper, A. Korpel, “Eikonal theory of strong acousto-optic interaction with curved wave fronts of sound,” J. Opt. Soc. Am. A 2, 1435 (1985).
    [Crossref]
  15. R. Pieper, A. Korpel, W. Hereman, “Extension of the acousto-optic Bragg regime through Hamming apodization of the sound field,” J. Opt. Soc. Am. A 3, 1608 (1986).
    [Crossref]
  16. A. Korpel, C. Venzke, D. Mehrl, “Novel algorithm for strong acousto-optic interaction: application to a phase-profiled sound column,” in Proceedings of Ultrasonic International 1991 (Butterworth, London, 1991), pp. 111–114.
  17. T.-C. Poon, Department of Electrical Engineering, Virginia Polytechnic Institute, Blacksburg, Va. 24061 (personal communication, June25, 1993).

1990 (2)

D. A. Larson, T. D. Black, M. Green, R. G. Torti, Y. J. Wang, R. Magnusson, “Optical modulation by traveling surface acoustic wave and a holographic reference grating,” J. Opt. Soc. Am. A 7, 1745 (1990).
[Crossref]

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81 (1990).

1987 (1)

R. Magnusson, T. D. Black, “Enhanced detection of acoustic waves using thick and thin reference gratings,” J. Opt. Soc. Am. A 4, 498 (1987).
[Crossref]

1986 (2)

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 (1986).
[Crossref]

R. Pieper, A. Korpel, W. Hereman, “Extension of the acousto-optic Bragg regime through Hamming apodization of the sound field,” J. Opt. Soc. Am. A 3, 1608 (1986).
[Crossref]

1985 (2)

T.-C. Poon, P. P. Banerjee, M. R. Chatterjee, “Analysis of acoustooptic diffraction by adjacent ultrasonic beams using multiple plane-wave scattering techniques,”IEEE Trans. Sonics Ultrason. SU-32, 592 (1985).
[Crossref]

R. Pieper, A. Korpel, “Eikonal theory of strong acousto-optic interaction with curved wave fronts of sound,” J. Opt. Soc. Am. A 2, 1435 (1985).
[Crossref]

1980 (1)

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

1973 (1)

O. Leroy, E. Blomme, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303 (1973).

1967 (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction in isotropic media,”IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[Crossref]

1960 (2)

L. E. Hargrove, “Diffraction of light passing through two adjacent ultrasonic progressive waves of different frequency,”J. Opt. Soc. Am. 32, 940 (1960).

R. Mertens, “On the theory of diffraction of light by two adjacent ultrasonic waves, one being the N-th harmonic of the other,”Z. Phys. 160, 291 (1960).
[Crossref]

1959 (1)

J. S. Murty, B. R. Rao, “Diffraction of light by superposed ultrasonic waves,” Z. Phys. 157, 189 (1959).
[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 (1986).
[Crossref]

T.-C. Poon, P. P. Banerjee, M. R. Chatterjee, “Analysis of acoustooptic diffraction by adjacent ultrasonic beams using multiple plane-wave scattering techniques,”IEEE Trans. Sonics Ultrason. SU-32, 592 (1985).
[Crossref]

P. P. Banerjee, C. W. Tarn, “Interaction of planar, multitone-modulated ultrasonic waves on optical beams with arbitrary profiles,” in Annual Meeting, Vol. 15 of 1990OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 134.

Black, T. D.

Blomme, E.

O. Leroy, E. Blomme, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303 (1973).

Chatterjee, M. R.

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81 (1990).

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 (1986).
[Crossref]

T.-C. Poon, P. P. Banerjee, M. R. Chatterjee, “Analysis of acoustooptic diffraction by adjacent ultrasonic beams using multiple plane-wave scattering techniques,”IEEE Trans. Sonics Ultrason. SU-32, 592 (1985).
[Crossref]

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction in isotropic media,”IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[Crossref]

Green, M.

Hargrove, L. E.

L. E. Hargrove, “Diffraction of light passing through two adjacent ultrasonic progressive waves of different frequency,”J. Opt. Soc. Am. 32, 940 (1960).

Hereman, W.

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction in isotropic media,”IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[Crossref]

Korpel, A.

R. Pieper, A. Korpel, W. Hereman, “Extension of the acousto-optic Bragg regime through Hamming apodization of the sound field,” J. Opt. Soc. Am. A 3, 1608 (1986).
[Crossref]

R. Pieper, A. Korpel, “Eikonal theory of strong acousto-optic interaction with curved wave fronts of sound,” J. Opt. Soc. Am. A 2, 1435 (1985).
[Crossref]

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

A. Korpel, Acousto-optics (Dekker, New York, 1988).

A. Korpel, C. Venzke, D. Mehrl, “Novel algorithm for strong acousto-optic interaction: application to a phase-profiled sound column,” in Proceedings of Ultrasonic International 1991 (Butterworth, London, 1991), pp. 111–114.

Larson, D. A.

Leroy, O.

O. Leroy, E. Blomme, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303 (1973).

Magnusson, R.

Mehrl, D.

A. Korpel, C. Venzke, D. Mehrl, “Novel algorithm for strong acousto-optic interaction: application to a phase-profiled sound column,” in Proceedings of Ultrasonic International 1991 (Butterworth, London, 1991), pp. 111–114.

Mertens, R.

R. Mertens, “On the theory of diffraction of light by two adjacent ultrasonic waves, one being the N-th harmonic of the other,”Z. Phys. 160, 291 (1960).
[Crossref]

Murty, J. S.

J. S. Murty, B. R. Rao, “Diffraction of light by superposed ultrasonic waves,” Z. Phys. 157, 189 (1959).
[Crossref]

Pieper, R.

Poon, T.-C.

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81 (1990).

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 (1986).
[Crossref]

T.-C. Poon, P. P. Banerjee, M. R. Chatterjee, “Analysis of acoustooptic diffraction by adjacent ultrasonic beams using multiple plane-wave scattering techniques,”IEEE Trans. Sonics Ultrason. SU-32, 592 (1985).
[Crossref]

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

T.-C. Poon, Department of Electrical Engineering, Virginia Polytechnic Institute, Blacksburg, Va. 24061 (personal communication, June25, 1993).

Rao, B. R.

J. S. Murty, B. R. Rao, “Diffraction of light by superposed ultrasonic waves,” Z. Phys. 157, 189 (1959).
[Crossref]

Sitter, D. N.

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81 (1990).

Tarn, C. W.

P. P. Banerjee, C. W. Tarn, “Interaction of planar, multitone-modulated ultrasonic waves on optical beams with arbitrary profiles,” in Annual Meeting, Vol. 15 of 1990OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 134.

Torti, R. G.

Venzke, C.

A. Korpel, C. Venzke, D. Mehrl, “Novel algorithm for strong acousto-optic interaction: application to a phase-profiled sound column,” in Proceedings of Ultrasonic International 1991 (Butterworth, London, 1991), pp. 111–114.

Wang, Y. J.

Acustica (2)

O. Leroy, E. Blomme, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303 (1973).

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81 (1990).

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction in isotropic media,”IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[Crossref]

IEEE Trans. Sonics Ultrason. (1)

T.-C. Poon, P. P. Banerjee, M. R. Chatterjee, “Analysis of acoustooptic diffraction by adjacent ultrasonic beams using multiple plane-wave scattering techniques,”IEEE Trans. Sonics Ultrason. SU-32, 592 (1985).
[Crossref]

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

R. Magnusson, T. D. Black, “Enhanced detection of acoustic waves using thick and thin reference gratings,” J. Opt. Soc. Am. A 4, 498 (1987).
[Crossref]

J. Opt. Soc. Am. (1)

A. Korpel, T.-C. Poon, “Explicit formalism for acousto-optic multiple plane wave scattering,”J. Opt. Soc. Am. 70, 817 (1980).
[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 (1986).
[Crossref]

J. Opt. Soc. Am. (1)

L. E. Hargrove, “Diffraction of light passing through two adjacent ultrasonic progressive waves of different frequency,”J. Opt. Soc. Am. 32, 940 (1960).

J. Opt. Soc. Am. A (3)

Z. Phys. (2)

J. S. Murty, B. R. Rao, “Diffraction of light by superposed ultrasonic waves,” Z. Phys. 157, 189 (1959).
[Crossref]

R. Mertens, “On the theory of diffraction of light by two adjacent ultrasonic waves, one being the N-th harmonic of the other,”Z. Phys. 160, 291 (1960).
[Crossref]

Other (4)

A. Korpel, Acousto-optics (Dekker, New York, 1988).

A. Korpel, C. Venzke, D. Mehrl, “Novel algorithm for strong acousto-optic interaction: application to a phase-profiled sound column,” in Proceedings of Ultrasonic International 1991 (Butterworth, London, 1991), pp. 111–114.

T.-C. Poon, Department of Electrical Engineering, Virginia Polytechnic Institute, Blacksburg, Va. 24061 (personal communication, June25, 1993).

P. P. Banerjee, C. W. Tarn, “Interaction of planar, multitone-modulated ultrasonic waves on optical beams with arbitrary profiles,” in Annual Meeting, Vol. 15 of 1990OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 134.

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Figures (15)

Fig. 1
Fig. 1

R–N-to-Bragg parallel ultrasonic system.

Fig. 2
Fig. 2

Acousto-optic diffraction geometry for sound frequencies Ω and mΩ.

Fig. 3
Fig. 3

Output orders leaving the R–N cell at viewing angle 8B.

Fig. 4
Fig. 4

Order labeling for R–N and Bragg cells.

Fig. 5
Fig. 5

Relative angles of output beams exiting the Bragg cell. The vertical axis is in units of B. The horizontal axis is the p:q ratio.

Fig. 6
Fig. 6

Ray tracing for the R–N-to-Bragg configuration with a p:q ratio of 1:4.

Fig. 7
Fig. 7

Field intensity distribution in the R–N and Bragg cells, where ξ = ξ1 + ξ2 and Q = 10π. Only four orders are used in R–N computation.

Fig. 8
Fig. 8

Overlapped field intensities for frequency ratio 1:2. αRN = π/2. αBRAGG is varied from 0 to 4π. ϕinc = −ϕB.

Fig. 9
Fig. 9

Output intensities versus phase mismatch δ showing identical transmissions for δ = 0 and δ = 2π.

Fig. 10
Fig. 10

SAW detection schematic. PD’s, photodetectors.

Fig. 11
Fig. 11

Effective overlapped intensities versus α ^ BRAGG for α ^ RN = π/4.

Fig. 12
Fig. 12

Effective overlapped intensities versus α ^ BRAGG for α ^ RN = π/2.

Fig. 13
Fig. 13

Effective overlapped intensities versus α ^ BRAGG for α ^ RN = 3π/4.

Fig. 14
Fig. 14

Effective overlapped intensities versus α ^ BRAGG for α ^ RN = π.

Fig. 15
Fig. 15

ac Peak-to-peak separation and dc level of intensities I0 and I2 for α ^ RN varying from 0 to π.

Tables (2)

Tables Icon

Table 1 Number of Bragg-Scattered Orders, Nature of Overlap, and Angular Range and Separation for Four R–N Scattered Orders at Various p:q Ratios

Tables Icon

Table 2 Number of Bragg-Scattered Orders, Nature of Overlap, and Angular Range and Separation for Eight R–N Scattered Orders of Various p:q Ratios

Equations (60)

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π n s λ m L Λ 2 1 ,
Q = K 2 L k m = 2 π λ m L Λ 2 ,
d E n d ξ = - j α ^ 2 [ E n - 1 - E n + 1 ] ,
d E n d ξ = - j α ^ 2 [ E n - m - E n + m ] .
d E n d ξ = - j α ^ 2 [ E n - 1 - E n + 1 + E n - m - E n + m ] ,
d E n d ξ = - j α ^ 2 [ E n - 1 - E n + 1 + exp ( j δ ) E n - m - exp ( - j δ ) E n + m ] ,
d E - 2 d ξ = - j α ^ 2 [ - E - 1 - exp ( - j δ ) E 0 ] ,
d E - 1 d ξ = - j α ^ 2 [ E - 2 - E 0 - exp ( - j δ ) E 1 ] ,
d E 0 d ξ = - j α ^ 2 [ exp ( j δ ) E - 2 + E - 1 - E 1 + exp ( - j δ ) E 2 ] ,
d E 1 d ξ = - j α ^ 2 [ E 0 - E 2 + exp ( j δ ) E - 1 ] ,
d E 2 d ξ = - j α ^ 2 [ exp ( j δ ) E 0 + E 1 ] .
d E n ( 1 ) d ξ 1 = - j α ^ 2 ( exp { - j 1 2 [ ϕ inc ϕ B + ( 2 n - 1 ) ] Q ξ 1 } E n - 1 ( 1 ) - exp { j 1 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] Q ξ 1 } E n + 1 ( 1 ) ) ,
d E n ( 2 ) d ξ 2 = - j α ^ 2 ( exp { - j m Q ξ 2 2 [ ϕ inc ϕ B + ( 2 n - m ) ] } E n - m ( 2 ) - exp { j m Q ξ 2 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] } E n + m ( 2 ) ) ,
d E - 2 ( 1 ) d ξ 1 = - j α ^ 2 [ - exp ( - j 3 Q ξ 1 2 ) E - 1 ( 1 ) ] ,
d E - 1 ( 1 ) d ξ 1 = - j α ^ 2 [ exp ( j 3 Q ξ 1 2 ) E - 2 ( 1 ) - exp ( - j Q ξ 1 2 ) E 0 ( 1 ) ] ,
d E 0 ( 1 ) d ξ 1 = - j α ^ 2 [ exp ( j Q ξ 1 2 ) E - 1 ( 1 ) - exp ( j Q ξ 1 2 ) E 1 ( 1 ) ] ,
d E 1 ( 1 ) d ξ 1 = - j α ^ 2 [ exp ( - j Q ξ 1 2 ) E 0 ( 1 ) - exp ( j 3 Q ξ 1 2 ) E 2 ( 1 ) ] ,
d E 2 ( 1 ) d ξ 1 = - j α ^ 2 [ exp ( - j 3 Q ξ 1 2 ) E 1 ( 1 ) ] ,
d E - 2 ( 2 ) d ξ 2 = - j α ^ 2 [ - exp ( - j 2 Q ξ 2 ) E 0 ( 2 ) ] ,
d E - 1 ( 2 ) d ξ 2 = - j α ^ 2 [ - E 1 ( 2 ) ] ,
d E 0 ( 2 ) d ξ 2 = - j α ^ 2 [ exp ( j 2 Q ξ 2 ) E - 2 ( 2 ) - exp ( j 2 Q ξ 2 ) E 2 ( 2 ) ] ,
d E 1 ( 2 ) d ξ 2 = - j α ^ 2 [ E - 1 ( 2 ) ] ,
d E 2 ( 2 ) d ξ 2 = - j α ^ 2 [ exp ( - j 2 Q ξ 2 ) E 0 ( 2 ) ] ,
ϕ B sin - 1 λ 2 Λ ,
ϕ B 1 = sin - 1 p λ 2 Λ p ϕ B ,
ϕ B 2 = sin - 1 q λ 2 Λ q ϕ B ,
E 2 = ϕ inc + 4 p ϕ B ,
E 1 = ϕ inc + 2 p ϕ B ,
E 0 = ϕ i n c ,
E - 1 = ϕ inc - 2 p ϕ B .
E m = ϕ inc + 2 m p ϕ B ,             m = 0 , ± 1 , ± 2 ,
E m , n = E m + n 2 q ϕ B = ϕ inc + m 2 p ϕ B + n 2 q ϕ B = ϕ inc + 2 ϕ B ( p m + q n ) ,
( m , n ) { ( 2 , - 1 ) , ( 2 , 0 ) , ( 1 , - 1 ) , ( 1 , 0 ) , ( 0 , 0 ) , ( 0 , 1 ) , ( - 1 , 1 ) , ( - 1 , 0 ) } .
ϕ inc = 0 ,             q ϕ B = 1 ,
E m , n = 2 ( m p q + n ) q ϕ B = 2 ( m p q + n ) .
d E n d ξ = - j α ^ 2 [ E n - 1 + E n + 1 ] ,
d E - 1 d ξ 1 = - j α ^ 2 E 0 ,
d E 0 d ξ 1 = - j α ^ 2 [ E - 1 + E 1 ] ,
d E 1 d ξ 1 = - j α ^ 2 [ E 0 + E 2 ] ,
d E 2 d ξ 1 = - j α ^ 2 E 1 .
ϕ inc , 2 = E 2 = ϕ inc + 4 p ϕ B ,
ϕ inc , 1 = E 1 = ϕ inc + 2 p ϕ B ,
ϕ inc , 0 = E 0 = ϕ inc ,
ϕ inc , - 1 = E - 1 = ϕ inc - 2 p ϕ B ,
ϕ inc , m = E m = ϕ inc + 2 m p ϕ B ,
d E n d ξ 2 = - j α ^ 2 ( exp { - j 1 2 [ ϕ inc ϕ B + ( 2 n - 1 ) ] Q ξ 2 } E n - 1 - exp { j 1 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] Q ξ 2 } E n + 1 ) .
ϕ inc , m = ϕ inc + 2 m p ϕ B = - ( 1 + δ m ) q ϕ B ,
δ m = - ϕ inc q ϕ B - 2 m p q - 1.
ϕ inc , m = ϕ inc + 2 m p ϕ B = ( 1 + δ m ) q ϕ B ,
δ m = ϕ inc q ϕ B + 2 m p q - 1.
d E m , 0 d ξ = - j α ^ 2 exp ( - j Q ξ δ m ) E m , ± 1 ,
d E ^ m , ± 1 d ξ = - j α ^ 2 exp ( j Q ξ δ m ) E m , 0 ,
I - 1 = E - 1 , 0 + E 1 , - 1 2 ,
I 0 = E 0 , 0 + E 2 , - 1 2 ,
I 1 = E 1 , 0 + E - 1 , 1 2 ,
I 2 = E 2 , 0 + E 0 , 1 2 .
d E n d ξ 2 = - j α ^ 2 ( exp { - j 1 2 [ ϕ inc ϕ B + ( 2 n - 1 ) ] Q ξ 2 } × exp ( j δ ) E n - 1 + exp { j 1 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] Q ξ 2 } × exp ( - j δ ) E n + 1 ) .
d E m , 0 d ξ 2 = - j α ^ 2 exp [ - j ( Q ξ 2 δ m - δ ) ] E m , ± 1 ,
d E m , ± 1 d ξ 2 = - j α ^ 2 exp [ j ( Q ξ 2 δ m - δ ) ] E m , 0 .
α = C k m S L 2 ,

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