Abstract

Multiple plane-wave scattering techniques are extended to analyze the diffraction of light by adjacent ultrasonics of frequency ratio 1:m to include the near-Raman–Nath regime. This is done first by modifying the existing theory, developed for the diffraction of light by sound at frequency Ω, to the general case of light–sound (frequency mΩ) interaction and thereafter deriving the coupled equations for the diffracted orders in each sound column. Results for some special cases are compared with previously published work.

© 1986 Optical Society of America

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References

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  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, 592–595 (1985).
    [CrossRef]
  2. O. Leroy, E. Blomme, “Amplitude modulation of diffracted light waves caused by adjacent ultrasonic beams of frequency ratio 1:N,” Ultrasonics 19, 173–178 (1981).
    [CrossRef]
  3. O. Leroy, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303–310 (1973).
  4. W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–132 (1967).
    [CrossRef]
  5. A. Korpel, “Two-dimensional plane wave theory of strong acousto-optic interaction in isotropic media,” J. Opt. Soc. Am. 69, 678–683 (1979).
    [CrossRef]
  6. A. Korpel, T.-C. Poon, “Explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
    [CrossRef]
  7. A. Korpel, “Acousto-optics,” in Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.
  8. R. Mertens, “On the theory of the diffraction of light by two parallel ultrasonic waves, one being the N th harmonic of the other,” Z. Phys. 160, 291–296 (1960).
    [CrossRef]
  9. L. E. Hargrove, E. A. Hiedemann, R. Mertens, “Diffraction of light by two spatially separated parallel ultrasonic waves of different frequency,” Z. Phys. 167, 326–336 (1962).
    [CrossRef]

1985 (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, 592–595 (1985).
[CrossRef]

1981 (1)

O. Leroy, E. Blomme, “Amplitude modulation of diffracted light waves caused by adjacent ultrasonic beams of frequency ratio 1:N,” Ultrasonics 19, 173–178 (1981).
[CrossRef]

1980 (1)

1979 (1)

1973 (1)

O. Leroy, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303–310 (1973).

1967 (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–132 (1967).
[CrossRef]

1962 (1)

L. E. Hargrove, E. A. Hiedemann, R. Mertens, “Diffraction of light by two spatially separated parallel ultrasonic waves of different frequency,” Z. Phys. 167, 326–336 (1962).
[CrossRef]

1960 (1)

R. Mertens, “On the theory of the diffraction of light by two parallel ultrasonic waves, one being the N th harmonic of the other,” Z. Phys. 160, 291–296 (1960).
[CrossRef]

Banerjee, P. P.

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, 592–595 (1985).
[CrossRef]

Blomme, E.

O. Leroy, E. Blomme, “Amplitude modulation of diffracted light waves caused by adjacent ultrasonic beams of frequency ratio 1:N,” Ultrasonics 19, 173–178 (1981).
[CrossRef]

Chatterjee, M. R.

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, 592–595 (1985).
[CrossRef]

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–132 (1967).
[CrossRef]

Hargrove, L. E.

L. E. Hargrove, E. A. Hiedemann, R. Mertens, “Diffraction of light by two spatially separated parallel ultrasonic waves of different frequency,” Z. Phys. 167, 326–336 (1962).
[CrossRef]

Hiedemann, E. A.

L. E. Hargrove, E. A. Hiedemann, R. Mertens, “Diffraction of light by two spatially separated parallel ultrasonic waves of different frequency,” Z. Phys. 167, 326–336 (1962).
[CrossRef]

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–132 (1967).
[CrossRef]

Korpel, A.

Leroy, O.

O. Leroy, E. Blomme, “Amplitude modulation of diffracted light waves caused by adjacent ultrasonic beams of frequency ratio 1:N,” Ultrasonics 19, 173–178 (1981).
[CrossRef]

O. Leroy, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303–310 (1973).

Mertens, R.

L. E. Hargrove, E. A. Hiedemann, R. Mertens, “Diffraction of light by two spatially separated parallel ultrasonic waves of different frequency,” Z. Phys. 167, 326–336 (1962).
[CrossRef]

R. Mertens, “On the theory of the diffraction of light by two parallel ultrasonic waves, one being the N th harmonic of the other,” Z. Phys. 160, 291–296 (1960).
[CrossRef]

Poon, T.-C.

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, 592–595 (1985).
[CrossRef]

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

Acustica (1)

O. Leroy, “Diffraction of light by two adjacent parallel ultrasonic beams,” Acustica 29, 303–310 (1973).

IEEE Trans. Sonics Ultrason. (2)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–132 (1967).
[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, 592–595 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

Ultrasonics (1)

O. Leroy, E. Blomme, “Amplitude modulation of diffracted light waves caused by adjacent ultrasonic beams of frequency ratio 1:N,” Ultrasonics 19, 173–178 (1981).
[CrossRef]

Z. Phys. (2)

R. Mertens, “On the theory of the diffraction of light by two parallel ultrasonic waves, one being the N th harmonic of the other,” Z. Phys. 160, 291–296 (1960).
[CrossRef]

L. E. Hargrove, E. A. Hiedemann, R. Mertens, “Diffraction of light by two spatially separated parallel ultrasonic waves of different frequency,” Z. Phys. 167, 326–336 (1962).
[CrossRef]

Other (1)

A. Korpel, “Acousto-optics,” in Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.

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Figures (8)

Fig. 1
Fig. 1

General configuration for sound (Ω)–light interaction showing how diffracted orders n + 1 and n − 1 contribute to n through sound fields S n 1 + and S n + 1 .

Fig. 2
Fig. 2

General configuration for sound (mΩ)–light interaction showing how diffracted orders n + m and nm contribute to n through sound fields S n + m and S n m +.

Fig. 3
Fig. 3

Configuration for two adjacent sound columns showing relevant diffracted orders.

Fig. 4
Fig. 4

Variation of the diffracted intensities I0(2), I±1(2), and I±2(2) as a function of the phase difference δ between the adjacent sound fields ( α ˆ = 0.5 ).

Fig. 5
Fig. 5

Diffraction efficiencies of the orders 0 and ±1 caused by two adjacent ultrasonic beams with equal sound strength α ˆ and for ρ = 5.0.

Fig. 6
Fig. 6

Diffraction efficiencies of the orders 0, ±1, and ±2 caused by two adjacent ultrasonic beams with equal sound strength α ˆ and for Q = 0.0.

Fig. 7
Fig. 7

As in Fig. 5, but for Q = 1.0.

Fig. 8
Fig. 8

As in Fig. 6, but for Q = 10.0.

Equations (38)

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d E ¯ n d z = j a ( S n 1 + E ˜ n 1 + S n + I E ˜ n + 1 ) ,
E ¯ n = E ¯ inc δ n 0 at z 0 ,
S n 1 + = j | S | exp { j 1 2 [ ϕ inc ϕ B + ( 2 n 1 ) ] Q ξ }
S n + 1 = j | S | exp { j 1 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] Q ξ } ,
ξ = z / L ( L is the width of sound column ) ,
Q = K 2 L / k m ,
ϕ B = K / 2 k m ,
d E ˜ n d ξ = α ˜ 2 ( exp { j 1 2 [ ϕ inc ϕ B + ( 2 n 1 ) ] Q ξ } E ˜ n 1 exp { j 1 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] Q ξ } E ˜ n + 1 ) ,
α ˜ = k m C | S | L / 2 .
S n 1 + = S [ z , z tan ( ϕ n 1 + ϕ B ) ] = S [ z , z tan ( ϕ n ϕ B ) ] ( referring to Fig . 1 ) .
exp [ j K z ( ϕ n ϕ B ) ] = exp { j K z [ ϕ inc + ( 2 n 1 ) ϕ B ] } ( writing ϕ n = ϕ inc + 2 n ϕ B ) = exp { j 1 2 Q ξ [ ϕ inc / ϕ B + ( 2 n 1 ) ] } ,
S n m + = S [ z , z tan ( ϕ n ϕ B ) ] , ϕ B = m ϕ B ,
exp [ j K z ( ϕ n ϕ B ) ] = exp [ j m K z ( ϕ n m ϕ B ) ] = exp { j m K z [ ϕ inc + ( 2 n m ) ϕ B ] } = exp { j 1 2 m Q ξ [ ϕ inc / ϕ B + ( 2 n m ) ] }
S n m + = j | S | exp { j 1 2 m Q ξ [ ϕ inc / ϕ B + ( 2 n m ) ] }
S n + m = j | S | exp { j 1 2 m Q ξ [ ϕ inc / ϕ B + ( 2 n + m ) ] }
d E ˜ n d z = j a ( S n m + E ˜ n m + S n + m E ˜ n + m )
d E ˜ n d ξ = α ˜ 2 ( exp { j 1 2 m Q ξ [ ϕ inc ϕ B + ( 2 n m ) ] } E ˜ n m exp { j 1 2 m Q ξ [ ϕ inc ϕ B + ( 2 n + m ) ] } E ˜ n + m ) ,
d E ˜ n d ξ = α ˜ 2 ( e i × exp { j 1 2 m Q ξ [ ϕ inc ϕ B + ( 2 n m ) ] } E ˜ n m e j δ × exp { j 1 2 m Q ξ [ ϕ inc ϕ B + ( 2 n + m ) ] } ) E ˜ n + m ,
First Column d E ˜ 2 ( 2 ) d ξ = α ˆ 2 [ E ˜ 1 ( 1 ) ] ,
d E ˜ 1 ( 1 ) d ξ = α ˆ 2 [ E ˜ 2 ( 1 ) E ˜ 0 ( 1 ) ] ,
d E ˜ 0 ( 1 ) d ξ = α ˆ 2 [ E ˜ 1 ( 1 ) E ˜ 1 ( 1 ) ] ,
d E ˜ 1 ( 1 ) d ξ = α ˆ 2 [ E ˜ 0 ( 1 ) E ˜ 2 ( 1 ) ] ,
d E ˜ 2 ( 1 ) d ξ = α ˆ 2 [ E ˜ 1 ( 1 ) ] ,
Second Column d E ˜ 2 ( 2 ) d ξ = α ˆ 2 [ e j δ E ˜ 0 ( 2 ) ] ,
d E ˜ 1 ( 2 ) d ξ = α ˆ 2 [ e j δ E ˜ 1 ( 2 ) ] ,
d E ˜ 0 ( 2 ) d ξ = α ˆ 2 [ e j δ E ˜ 2 ( 2 ) e j δ E ˜ 2 ( 2 ) ] ,
d E ˜ 1 ( 2 ) d ξ = α ˆ 2 [ e i δ E ˜ 1 ( 2 ) ] ,
d E ˜ 2 ( 2 ) d ξ = α ˆ 2 [ e i δ E ˜ 0 ( 2 ) ]
d E ˜ 2 ( 1 ) d ξ = α ˆ 2 [ exp ( j 3 Q ξ / 2 ) E ˜ 1 ( 1 ) ] ,
d E ˜ 1 ( 1 ) d ξ = α ˆ 2 [ exp ( i 3 Q ξ / 2 ) E ˜ 2 ( 1 ) exp ( i Q ξ / 2 ) E ˜ 0 ( 1 ) ] ,
d E ˜ 0 ( 1 ) d ξ = α ˆ 2 [ exp ( i 3 Q ξ / 2 ) E ˜ 1 ( 1 ) exp ( i Q ξ / 2 ) E ˜ 1 ( 1 ) ] ,
d E ˜ 1 ( 1 ) d ξ = α ˆ 2 [ exp ( j Q ξ / 2 ) E ˜ 0 ( 1 ) exp ( i Q ξ / 2 ) E ˜ 2 ( 1 ) ] ,
d E ˜ 2 ( 1 ) d ξ = α ˆ 2 [ exp ( j 3 Q ξ / 2 ) E ˜ 1 ( 1 ) ] ,
Second Column d E ˜ 2 ( 2 ) d ξ = α ˆ 2 [ exp ( j 2 Q ξ ) E ˜ 0 ( 2 ) ] ,
d E ˜ 1 ( 2 ) d ξ = α ˆ 2 [ E ˜ 1 ( 2 ) ] ,
d E ˜ 0 ( 2 ) d ξ = α ˆ 2 [ exp ( j 2 Q ξ ) E ˜ 2 ( 2 ) exp ( j 2 Q ξ ) E ˜ 2 ( 2 ) ] ,
d E ˜ 1 ( 2 ) d ξ = α ˆ 2 [ E ˜ 1 ( 2 ) ] ,
d E ˜ 2 ( 2 ) d ξ = α ˆ 2 [ exp ( j 2 Q ξ ) E ˜ 0 ( 2 ) ] ,

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