Abstract

A spatial Fourier transform approach is used to study the phenomena of polarization changing and beam profile deformation of light during the Raman–Nath, acousto-optic interaction in isotropic media. Starting from the vector version of the well-known Raman–Nath interaction equation and using a spatial Fourier transform allows analytic solutions that encompass the effects of polarization changing and beam-profile deformation for the multiple scattered light to be found in the spatial-frequency domain. Two kinds of sound wave, longitudinal and shear, are assumed to be interacted with the light, whose transverse spatial profile and state of polarization are arbitrary. It is shown that, for light with an arbitrary spatial profile after interaction with the sound wave in the Raman–Nath regime, the spatial profiles of the scattered light are almost the same shape as those of the input light. For the polarization changing part, it is found that the state of polarization and the direction of rotation can alter, depending not only on the sound amplitude but also on the propagation mode of the sound wave. Simulation results are provided to confirm the validity of this approach.

© 1998 Optical Society of America

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References

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  1. C. W. Tarn, “Spatial Fourier transform approach to the study of the polarization changing and beam profile deformation of light during Bragg acousto-optic interaction with longitudinal and shear ultrasonic waves in isotropic media,” J. Opt. Soc. Am. A 14, 2231–2242 (1997).
    [CrossRef]
  2. P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interaction in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).
  3. L. Brillouin, “La diffraction de la lumière par des ultra-sons,” Actu. Sci. Indu. 59, 1–31 (1933).
  4. C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1935).
  5. A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1989), Chap. 1, p. 1.
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, p. 77.
  7. G. S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3, p. 85.
  8. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. 2, pp. 22.1–22.36.
  9. D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
    [CrossRef]
  10. A. K. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 7, p. 370.
  12. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17, p. 663.

1997

1991

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interaction in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).

1970

D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
[CrossRef]

1935

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1935).

1933

L. Brillouin, “La diffraction de la lumière par des ultra-sons,” Actu. Sci. Indu. 59, 1–31 (1933).

Banerjee, P. P.

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interaction in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 7, p. 370.

Brillouin, L.

L. Brillouin, “La diffraction de la lumière par des ultra-sons,” Actu. Sci. Indu. 59, 1–31 (1933).

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. 2, pp. 22.1–22.36.

Ghatak, A. K.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, p. 77.

Kino, G. S.

G. S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3, p. 85.

Korpel, A.

A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1989), Chap. 1, p. 1.

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1935).

Pinnow, D. A.

D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
[CrossRef]

Raman, C. V.

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1935).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17, p. 663.

Tarn, C. W.

Thyagarajan, K.

A. K. 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), Chap. 7, p. 370.

Actu. Sci. Indu.

L. Brillouin, “La diffraction de la lumière par des ultra-sons,” Actu. Sci. Indu. 59, 1–31 (1933).

Acustica

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interaction in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).

IEEE J. Quantum Electron.

D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
[CrossRef]

J. Opt. Soc. Am. A

Proc. Indian Acad. Sci. Sect. A

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1935).

Other

A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1989), Chap. 1, p. 1.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, p. 77.

G. S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3, p. 85.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. 2, pp. 22.1–22.36.

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 7, p. 370.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17, p. 663.

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Figures (9)

Fig. 1
Fig. 1

Basic Raman–Nath acousto-optic interaction configuration.

Fig. 2
Fig. 2

Three-dimensional plots showing variations of the normalized intensity of (a) the x-polarized and (b) the y-polarized, zeroth-order light as functions of x/ D and the sound amplitude A 0 for a longitudinal sound frequency of 10 MHz.

Fig. 3
Fig. 3

Three-dimensional plots showing variations of the normalized intensity of (a) the x-polarized and (b) the y-polarized minus-one-order light as functions of x/ D and the sound amplitude A 0 for a longitudinal sound frequency of 10 MHz.

Fig. 4
Fig. 4

Three-dimensional plot showing variations of the normalized intensity of the x- and the y-polarized zeroth-order light as functions of x/ D and the sound amplitude A 0 for shear sound-wave frequency of 10 MHz.

Fig. 5
Fig. 5

Three-dimensional plot showing variation of the normalized intensity of the x- and the y-polarized minus-one-order light as functions of x/ D and the sound amplitude A 0 for a shear sound-wave frequency of 10 MHz.

Fig. 6
Fig. 6

On-axis, x = 0, variation of the degrees of the azimuthal orientation major axis of the zeroth-, minus-one-, first-, and second-order light as functions of the sound amplitude A 0 for a longitudinal sound-wave frequency of 10 MHz and for linearly polarized incident light; E x =cos 30°, E y = sin 30°.

Fig. 7
Fig. 7

On-axis, x = 0, variation of the magnitudes of the ellipticity of the zeroth-, minus-one-, first-, and second-order light as functions of the sound amplitude A 0 for a longitudinal sound-wave frequency of 10 MHz and for elliptically polarized incident light; E x = 1, E y = exp(j π/3).

Fig. 8
Fig. 8

On-axis, x = 0, variation of the degrees of the azimuthal orientation major axis of the zeroth-, minus-one-, first-, and second-order light as functions of the sound amplitude A 0 for a longitudinal sound-wave frequency of 10 MHz and for elliptically polarized incident light; E x = 1, E y = exp(j π/3).

Fig. 9
Fig. 9

On-axis, x = 0, variation of the magnitudes of the ellipticity of the zeroth-, minus-one-, first-, and second-order light as functions of the sound amplitude A 0 for a longitudinal sound-wave frequency of 10 MHz and for circularly polarized incident light; E x = 1, E y = exp(-j π/2).

Equations (18)

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E inc r ,   t = 1 2 E x a x + E y a y | E x | 2 + | E y | 2 1 / 2   ψ inc r × exp j ω 0 t - jk 0 x   sin   ϕ inc - jk 0 z   cos   ϕ inc + c.c. , = 1 2 E x a x + E y a y | E x | 2 + | E y | 2 1 / 2 ψ inc r × exp j ω 0 t - jk 0 z + c.c. ,
S = 1 / 2   A 0   exp j Ω t - Kx + c.c. ,
E r ,   t = 1 2 m = - e m ψ m r exp j ω 0 t + m Ω t - jnk 0 x   sin   ϕ m - jnk 0 z   cos   ϕ m + c.c. ,
sin   ϕ m = m K / nk 0 .
2 ψ m x 2 + 2 ψ m y 2 + 2 ψ m z 2 - 2 jnk 0   sin   ϕ m ψ m x - 2 jnk 0   cos   ϕ m ψ m z e m = 1 2   k 0 2 n 4 A 0 ψ m - 1   exp - jnk 0 z cos   ϕ m - 1 - cos   ϕ m P ¯ ¯ e m - 1 + 1 2   k 0 2 n 4 A 0 * ψ m + 1   exp - jnk 0 z cos   ϕ m + 1 - cos   ϕ m P ¯ ¯ e m + 1 .
P ¯ ¯ = P ¯ ¯ L = p 11 0 0 0 p 12 0 0 0 p 12
P ¯ ¯ = P ¯ ¯ S = 0 p 11 - p 12 0 p 11 - p 12 0 0 0 0 0
ψ m z e m = - tan   ϕ m ψ m x e m - j nk 0   cos   ϕ m × 2 ψ m x 2 + 2 ψ m y 2 e m + j   k 0 n 3 A 0 4   cos   ϕ m   P ¯ ¯ ψ m - 1 e m - 1 + j   k 0 n 3 A 0 * 4   cos   ϕ m   P ¯ ¯ ψ m + 1 e m + 1 .
Ψ k x ,   k y ;   z = ψ x ,   y ,   z = -   ψ x ,   y ,   z exp jk x x + jk y y d x d y ,
ψ x ,   y ,   z = - 1 Ψ k x ,   k y ;   z = 1 4 π 2 -   Ψ k x ,   k y ;   z exp - jk x x - jk y y d k x d k y ,
Ψ m k x ,   k y ;   z z e m = j k x 2 + k y 2 - 2 nk 0 k x   sin   ϕ m 2 nk 0   cos   ϕ m × Ψ m k x ,   k y ;   z e m +   j   k 0 n 3 A 0 4   cos   ϕ m   P ¯ ¯ Ψ m - 1 k x ,   k y ;   z e m - 1 +   j   k 0 n 3 A 0 * 4   cos   ϕ m   P ¯ ¯ Ψ m + 1 k x ,   k y ;   z e m + 1 ,
Ψ m = Ψ inc δ m 0   at   z 0 ,
E m r ,   t = 1 4 π 2 -   Ψ inc k x ,   k y ;   0 H ¯ ¯ m k x ,   k y ;   z × exp j - k x x - k y y + ω 0 t + m Ω t + nk 0 x   sin   ϕ m + nk 0 z   cos   ϕ m d k x d k y ,
H ¯ ¯ m , L k x ,   k y ;   z = j m | E x | 2 + | E y | 2 1 / 2   exp j k x 2 + k y 2 z 2 nk 0   cos   ϕ m × exp - jmk x K 2 k 0   z × E x a x | p 11 | p 11   J m k 0 n 3 A 0 | p 11 | 2   cos   ϕ m sin k x K 2 k 0   z k x K 2 k 0 + E y a y | p 12 | p 12   J m k 0 n 3 A 0 | p 12 | 2   cos   ϕ m sin k x K 2 k 0   z k x K 2 k 0 .
H ¯ ¯ 2 m , S k x ,   k y ;   z = j 2 m   exp j k x 2 + k y 2 z 2 nk 0   cos   ϕ 2 m × exp - j 2 mk x K 2 k 0   z × E x a x + E y a y | E x | 2 + | E y | 2 1 / 2 | p 11 - p 12 | p 11 - p 12 × J 2 m k 0 n 3 A 0 | p 11 - p 12 | 2   cos   ϕ 2 m sin k x K 2 k 0   z k x K 2 k 0 .
H ¯ ¯ 2 m - 1 , S k x ,   k y ;   z = j 2 m - 1   exp j k x 2 + k y 2 z 2 nk 0   cos   ϕ 2 m - 1 × exp - j 2 m - 1 k x K 2 k 0   z × E y a x + E x a y | E x | 2 + | E y | 2 1 / 2 | p 11 - p 12 | p 11 - p 12 × J 2 m - 1 k 0 n 3 A 0 | p 11 - p 12 | 2   cos   ϕ 2 m - 1 sin k x K 2 k 0   z k x K 2 k 0 .
ψ inc x ,   y ,   0 = exp - x 2 + y 2 D 2 ,
Ψ inc k x ,   k y ;   0 = π D 2   exp - k x 2 + k y 2 4   D 2 ,

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