Abstract

Most of the conventional studies of acousto-optics (AO) are under the paraxial approximation; that is, the propagation directions of the scattered light are almost along the optic axis of the AO system, or, equivalently, the envelope of the light wave varies slowly with respect to the direction of propagation. This assumption should not, however, be carried too far in the cases in which the incident or the Bragg angles are large enough that the scattered light waves do not propagate closely along the optic axis of the AO system. We study the Bragg AO effect beyond the paraxial assumption for small- and large-Bragg-angle incidence. Starting from the wave equations, which are derived from Maxwell's equations, a set of coupled equations that depict the AO interaction are derived in the Bragg regime without the paraxial assumption; i.e., the second derivative terms of the scattered-light amplitudes with respect to the propagation direction are nonnegligible. Analytic solutions that describe the evolution of the scattered light that is due to the acousto-optic effect beyond the paraxial approximation can then be found from the coupled equations. Simulation results are provided to check the validity of our solutions.

© 1997 Optical Society of America

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References

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  1. See, for instance, special issue on acousto-optics: Proc. IEEE 69 (1981).
  2. A. Korpel, ed., Selected Papers on Acousto-optics, Vol. MS16 of SPIE Milestone Series (SPIE—Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1990).
  3. See, for instance, special Section on acousto-optics in Opt. Eng. 31 (1992).
  4. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  5. A. Korpel, Acousto-optics (Marcel Dekker, New York, 1989).
  6. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  7. A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989).
    [CrossRef]
  8. P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen Associates, Pacific Palisades, California, 1991).
  9. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  10. P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presenceof propagational diffraction,” Acustica 74, 181–190 (1991).

1992 (1)

See, for instance, special Section on acousto-optics in Opt. Eng. 31 (1992).

1991 (1)

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presenceof propagational diffraction,” Acustica 74, 181–190 (1991).

1981 (1)

See, for instance, special issue on acousto-optics: Proc. IEEE 69 (1981).

Banerjee, P. P.

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presenceof propagational diffraction,” Acustica 74, 181–190 (1991).

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen Associates, Pacific Palisades, California, 1991).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Ghatak, A.

A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Korpel, A.

A. Korpel, Acousto-optics (Marcel Dekker, New York, 1989).

Poon, T. C.

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen Associates, Pacific Palisades, California, 1991).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Tarn, C. W.

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presenceof propagational diffraction,” Acustica 74, 181–190 (1991).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Thyagarajan, K.

A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Acustica (1)

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presenceof propagational diffraction,” Acustica 74, 181–190 (1991).

Opt. Eng. (1)

See, for instance, special Section on acousto-optics in Opt. Eng. 31 (1992).

Proc. IEEE (1)

See, for instance, special issue on acousto-optics: Proc. IEEE 69 (1981).

Other (7)

A. Korpel, ed., Selected Papers on Acousto-optics, Vol. MS16 of SPIE Milestone Series (SPIE—Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1990).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

A. Korpel, Acousto-optics (Marcel Dekker, New York, 1989).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen Associates, Pacific Palisades, California, 1991).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

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Figures (15)

Fig. 1
Fig. 1

Configuration of the interaction of a coupled-wave sound wave with light in the Bragg regime.

Fig. 2
Fig. 2

Small-Bragg-angle AO interaction.

Fig. 3
Fig. 3

Large-Bragg-angle AO interaction.

Fig. 4
Fig. 4

Variation of the zeroth-order scattered intensities as functions of k 0 CAL / 2 for a flint glass AO cell with L = 5   cm , sound frequency = 400 MHz, Δ ϕ = 0   mrad , light wavelength λ 0 = 633   nm for paraxial and nonparaxial small-Bragg-angle AO interaction.

Fig. 5
Fig. 5

Variation of the minus-one-order scatered intensities as functions of k 0 CAL / 2 for a flint glass AO cell with L = 5   cm , sound frequency = 400 MHz, Δ ϕ = 0   mrad , and light wavelength λ 0 = 633   nm for paraxial and nonparaxial small-Bragg-angle AO interaction.

Fig. 6
Fig. 6

Difference between diffracted-light intensities with and without the paraxial approach for zeroth- and minus-one-order light under the conditions that sound frequency = 400 MHz, Δ ϕ = 0   mrad , and light wavelength λ 0 = 633   nm .

Fig. 7
Fig. 7

Variation of the zeroth-order scattered intensities as functions of k 0 CAL / 2 for a flint glass AO cell with L = 5   cm , sound frequency = 400   MHz , Δ ϕ = 0.02   mrad , and light wavelength λ 0 = 633   nm for paraxial and nonparaxial, small-Bragg-angle AO interaction.

Fig. 8
Fig. 8

Variation of the minus-one-order scattered intensities as functions of k 0 CAL / 2 for a flint glass AO cell with L = 5   cm , sound frequency = 400 MHz, Δ ϕ = 0.02   mrad , and light wavelength λ 0 = 633   nm for paraxial and nonparaxial small-Bragg-angle AO interaction.

Fig. 9
Fig. 9

Difference between diffracted-light intensities with and without the paraxial approach for zeroth- and minus-one-order light under the conditions that sound frequency = 400 MHz, Δ ϕ = 0.02   mrad , and light wavelength λ 0 = 633   nm .

Fig. 10
Fig. 10

Variation of the zeroth-order scattered intensities as functions of x / D for a TeO 2 AO cell with L = 5   cm , sound frequency = 4 GHz, Δ ϕ = 0.12   mrad , light wavelength λ 0 = 633   nm , and k 0 CAD / 4   sin   ϕ B = 2.3 for paraxial and nonparaxial large-Bragg-angle AO interaction.

Fig. 11
Fig. 11

Variation of the minus-one-order scattered intensities as functions of x / D for a TeO 2 AO cell with L = 5   cm , sound frequency = 4 GHz, Δ ϕ = 0.12   mrad , light wavelength λ 0 = 633   nm , and k 0 CAD / 4   sin   ϕ B = 2.3 for paraxial and nonparaxial large-Bragg-angle AO interaction.

Fig. 12
Fig. 12

Difference between the diffracted-light intensities with and without the paraxial approach for zeroth- and minus-one-order light during large-Bragg-angle AO diffraction under the conditions that sound frequency = 4 GHz, Δ ϕ = 0.12   mrad , light wavelength λ 0 = 633   nm , and k 0 CAD / 4   sin   ϕ B = 2.3 .

Fig. 13
Fig. 13

Variation of the zeroth-order scattered intensities as functions of x / D for a TeO 2 AO cell with L = 5   cm , sound frequency = 4 GHz, Δ ϕ = 2.1   mrad , light wavelength λ 0 = 633   nm , and k 0 CAD / 4   sin   ϕ B = 40.8 for paraxial and nonparaxial large-Bragg-angle AO interaction.

Fig. 14
Fig. 14

Variation of the minus-one-order scattered intensities as functions of x / D for a TeO 2 AO cell with L = 5   cm , sound frequency = 4 GHz, Δ ϕ = 2.1   mrad , light wavelength λ 0 = 633   nm , and k 0 CAD / 4   sin   ϕ B = 40.8 for paraxial and nonparaxial large-Bragg-angle AO interaction.

Fig. 15
Fig. 15

Differences between the diffracted-light intensities with and without the paraxial approach for the zeroth- and minus-one-order light during large-Bragg-angle AO diffraction under the conditions that sound frequency = 4 GHz, Δ ϕ = 2.1   mrad , light wavelength λ 0 = 633   nm , and k 0 CAD / 4   sin   ϕ B = 40.8 .

Equations (44)

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2 E ( r ,   t ) - μ 0 0   2 E ( r ,   t ) t 2 μ 0 Δ   2 E ( r ,   t ) t 2 .
Δ ( r ,   t ) = 0 CS ( r ,   t ) ,
E inc ( r ,   t ) = 1 2 ψ inc ( x ,   z ) exp ( j ω 0 t - jk 0 x   sin   ϕ inc - jk 0 z   cos   ϕ inc ) + c . c . ,
S ( r ,   t ) = 1 2 A   exp ( j Ω t - Kx ) + c . c . ,
E ( r ,   t ) = 1 2   m = - ψ m ( x ,   z ) exp [ j ( ω 0 t + m Ω t - k 0 x   sin   ϕ m - k 0 z   cos   ϕ m ) ] + c . c . ,
sin   ϕ m = sin   ϕ inc + m   K k 0 .
2 ψ m x 2 + 2 ψ m z 2 - 2 jk 0   sin   ϕ m   ψ m x - 2 jk 0   cos   ϕ m   ψ m z - 1 2 k 0 2 CA ψ m - 1   exp [ - jk 0 ( cos   ϕ m - 1 - cos   ϕ m ) z - jk 0 ( sin   ϕ m - 1 - sin   ϕ m ) x ] - 1 2 k 0 2 CA * ψ m + 1 × exp [ - jk 0 ( cos   ϕ m + 1 - cos   ϕ m ) z - jk 0 ( sin   ϕ m + 1 - sin   ϕ m ) x ] = 0 .
ψ 0 ( z ) z = - j   k 0 CA 4   cos   ϕ B   ψ - 1 ( z ) ,
ψ - 1 ( z ) z = - j   k 0 CA 4   cos   ϕ B   ψ 0 ( z ) ,
ψ 0 ( x ) x = + j   k 0 CA 4   sin   ϕ B   ψ - 1 ( x ) ,
ψ - 1 ( x ) x = - j   k 0 CA 4   sin   ϕ B   ψ 0 ( x ) ,
2 ψ 0 z 2 - 2 jk 0   cos   ϕ 0   ψ 0 z + 1 2 k 0 2 CA ψ - 1
× exp ( 2 jk 0 γ c z ) = 0 ,
2 ψ - 1 z 2 - 2 jk 0   cos   ϕ - 1   ψ - 1 z + 1 2 k 0 2 CA * ψ 0
× exp ( - 2 jk 0 γ c z ) = 0 ,
2 γ c = cos   ϕ 0 - cos   ϕ - 1 .
2 ψ 0 x 2 - 2 jk 0   sin   ϕ 0   ψ 0 x + 1 2 k 0 2 CA ψ - 1
× exp ( 2 jk 0 rx ) = 0 ,
2 ψ - 1 x 2 - 2 jk 0   sin   ϕ - 1   ψ - 1 x + 1 2 k 0 2 CA * ψ 0
× exp ( - 2 jk 0 rx ) = 0 ,
ψ 0 = C 1   exp ( λ 1 z ) + C 2   exp ( λ 2 z ) + C 3   exp ( λ 3 z ) + C 4   exp ( λ 4 z ) ,
ψ - 1 = C 5   exp ( λ 1 z ) + C 6   exp ( λ 2 z ) + C 7   exp ( λ 3 z ) + C 8   exp ( λ 4 z ) ,
ψ 0 ( z = 0 ) = ψ inc ,
ψ - 1 ( z = 0 ) = 0 .
ψ 0 ( z ) = ( cos   ϕ 0 2 - g 2 2 ) exp [ j ( k 0   cos   ϕ 0 - k 0 g 1 ) z ] - ( cos   ϕ 0 2 - g 1 2 ) exp [ j ( k 0   cos   ϕ 0 - k 0 g 2 ) z ] 2 ( 4 γ + 2 γ c 2 + 1 4 C 2 A 2 ) 1 / 2   ψ inc ,
ψ - 1 ( z ) = 1 2 CA { exp [ j ( k 0   cos   ϕ - 1 - k 0 g 1 ) z ] - exp [ j ( k 0   cos   ϕ - 1 - k 0 g 2 ) z ] } 2 ( 4 γ + 2 γ c 2 + 1 4 C 2 A 2 ) 1 / 2   ψ inc ,
g 1 = ( γ + 2 + γ c 2 + 4 γ + 2 + γ c 2 + 1 4 C 2 A 2 ) 1 / 2 ,
g 2 = ( γ + 2 + γ c 2 - 4 γ + 2 + γ c 2 + 1 4 C 2 A 2 ) 1 / 2 ,
2 γ + = cos   ϕ 0 + cos   ϕ - 1 .
ψ 0 , p = ( cos   s 1 x - j   k 0 γ c 2 s 1   sin   s 1 x ) exp ( jk 0 γ c z ) ψ inc ,
ψ - 1 , p = - 4 jk 0 CA 4 s 1   cos   ϕ B   sin   s 1 z   exp ( - jk 0 γ c z ) ψ inc ,
s 1 = 1 4   k 0 2 γ c 2 + k 0 2 C 2 A 2 16   cos 2   ϕ B 1 / 2 .
| ψ 0 ( z ) | 2 + | ψ - 1 ( z ) | 2 = | ψ inc | 2 .
ψ 0 ( x = 0 ) = ψ inc ,
ψ - 1 ( x = D ) = 0 ,
ψ 0 ( x ) = g   cosh [ k 0 g ( D - x ) ] + j   g 2 + 2 r   sin   ϕ 0 - r 2 2 ( sin   ϕ 0 - r )   sinh [ k 0 g ( D - x ) ] g   cosh ( k 0 gD ) + j   g 2 + 2 r   sin   ϕ 0 - r 2 2 ( sin   ϕ 0 - r )   sinh ( k 0 gD )   exp ( jk 0 rx ) ψ inc ,
ψ - 1 ( x )
= - j   CA 4 ( sin   ϕ 0 - r )   sinh [ k 0 g ( D - x ) ] g   cosh ( k 0 gD ) + j   g 2 + 2 r   sin   ϕ 0 - r 2 2 ( sin   ϕ 0 - r )   sinh ( k 0 gD ) × exp ( - jk 0 rx ) ψ inc ,
g = { 2   sin   ϕ 0 ( r - sin   ϕ 0 ) + [ 2 ( 2   sin 2   ϕ 0 - r 2 ) × ( sin   ϕ 0 - r ) 2 + 1 4 C 2 A 2 ] 1 / 2 } 1 / 2 .
ψ 0 , p ( x ) = s   cosh [ k 0 s ( D - x ) ] + jr   sinh [ k 0 s ( D - x ) ] s   cosh ( k 0 sD ) + jr   sinh ( k 0 sD ) × exp ( jk 0 rx ) ψ inc ,
ψ - 1 , p ( x ) = - j ( CA ) / ( 4   sin   ϕ 0 ) sinh [ k 0 s ( D - x ) ] s   cosh ( k 0 sD ) + jr   sinh ( k 0 sD ) × exp ( - jk 0 rx ) ψ inc ,
s = CA 4   sin   ϕ 0 2 - r 2 1 / 2 .
d d x   [ | ψ 0 ( x ) | 2 - | ψ - 1 ( x ) | 2 ]
= 1 2 jk 0 ( sin   ϕ 0 - r )   ( | ψ 0 | 2 + | ψ - 1 | 2 ) ( λ 2 - λ * 2 ) ,

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