Abstract

The spatial coherence property of a partially coherent light during the Bragg acousto-optic interaction is investigated. Starting from the wave equation, four coupled, parabolic equations that can describe the evolution and the propagation of mutual intensity functions of the diffracted light during the acousto-optic interaction are derived. A partially coherent light beam with arbitrary spatial profile and complex degree of spatial coherence is assumed to be incident on the Bragg acousto-optic cell. With the use of a statistical theory of linear systems, a general formalism of angular-correlation functions for zero-order and minus-one-order light can be derived. The corresponding mutual intensity and complex coherence factor functions are hence implemented numerically. From the solutions one can note that, through the acousto-optic interaction, the degrees of spatial coherence of the diffracted light beams are controllable by the intensity and the frequency of the sound wave.

© 1999 Optical Society of America

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References

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  1. M. D. McNeill, T.-C. Poon, “Gaussian-beam profile shaping by acousto-optic Bragg diffraction,” Appl. Opt. 33, 4508–4515 (1994).
    [CrossRef] [PubMed]
  2. A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1988).
  3. Y. Ohtsuka, Y. Arima, Y. Imai, “Acousto-optic 2-D profile shaping of a Gaussian laser beam,” Appl. Opt. 24, 2813–2819 (1985).
    [CrossRef] [PubMed]
  4. Y. Imai, M. Imai, Y. Ohtsuka, “Optical coherence modulation by ultrasonic waves. 2: Application to speckle reduction,” Appl. Opt. 19, 3541–3544 (1980).
    [CrossRef] [PubMed]
  5. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  6. Y. Ohtsuka, “Partial coherence controlled by a progressive ultrasonic wave,” J. Opt. Soc. Am. 69, 684–689 (1979).
    [CrossRef]
  7. Y. Ohtsuka, “Modulation of optical coherence by ultrasonic waves,” J. Opt. Soc. Am. A 3, 1247–1257 (1986).
    [CrossRef]
  8. J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthetic acousto-optics holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
    [CrossRef]
  9. P. Vahimaa, J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
    [CrossRef]
  10. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  11. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  12. C.-W. Tarn, “Spatial Fourier transform approach to the study of 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]
  13. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.
  14. A. Korpel, P. Banerjee, C.-W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
    [CrossRef]
  15. C.-W. Tarn, R. S. Huang, “General formalism for Bragg acousto-optic interaction beyond the paraxial approximation,” J. Opt. Soc. Am. A 14, 3046–3056 (1997).
    [CrossRef]
  16. P. P. Banerjee, C.-W. Tarn, “A Fourier transform approach to acousto-optic interactions in the presence of propagational diffraction,” Acustica 74, 181–190 (1991).

1997 (3)

1994 (2)

1993 (1)

A. Korpel, P. Banerjee, C.-W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

1991 (1)

P. P. Banerjee, C.-W. Tarn, “A Fourier transform approach to acousto-optic interactions in the presence of propagational diffraction,” Acustica 74, 181–190 (1991).

1990 (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthetic acousto-optics holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

1986 (1)

1985 (1)

1980 (1)

1979 (1)

Arima, Y.

Banerjee, P.

A. Korpel, P. Banerjee, C.-W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

Banerjee, P. P.

P. P. Banerjee, C.-W. Tarn, “A Fourier transform approach to acousto-optic interactions in the presence of propagational diffraction,” Acustica 74, 181–190 (1991).

Friberg, A. T.

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthetic acousto-optics holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Huang, R. S.

Imai, M.

Imai, Y.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

Korpel, A.

A. Korpel, P. Banerjee, C.-W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

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

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

McNeill, M. D.

Ohtsuka, Y.

Poon, T.-C.

Tarn, C.-W.

C.-W. Tarn, “Spatial Fourier transform approach to the study of 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]

C.-W. Tarn, R. S. Huang, “General formalism for Bragg acousto-optic interaction beyond the paraxial approximation,” J. Opt. Soc. Am. A 14, 3046–3056 (1997).
[CrossRef]

A. Korpel, P. Banerjee, C.-W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

P. P. Banerjee, C.-W. Tarn, “A Fourier transform approach to acousto-optic interactions in the presence of propagational diffraction,” Acustica 74, 181–190 (1991).

Tervonen, E.

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthetic acousto-optics holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Turunen, J.

Vahimaa, P.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Acustica (1)

P. P. Banerjee, C.-W. Tarn, “A Fourier transform approach to acousto-optic interactions in the presence of propagational diffraction,” Acustica 74, 181–190 (1991).

Appl. Opt. (3)

J. Appl. Phys. (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthetic acousto-optics holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

A. Korpel, P. Banerjee, C.-W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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

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Figures (9)

Fig. 1
Fig. 1

Bragg acousto-optic interaction configuration.

Fig. 2
Fig. 2

Three-dimensional plots showing complex degree of spatial coherence of the zero-order light as a function of α and xd/σ for incident light that is spatially partially coherent, where σ=300λ0.

Fig. 3
Fig. 3

Same as Fig. 2, but for the minus-one-order light.

Fig. 4
Fig. 4

Comparison of the cross-section profiles of the complex degree of spatial coherence of the incident, the zero-order, and the minus-one-order light as a function of xd/σ for α=2, where the incident light is spatially partially coherent, with σ=300λ0.

Fig. 5
Fig. 5

Three-dimensional plots showing complex degree of spatial coherence of the zero-order light as a function of α and xd/σ for incident light that is spatially highly incoherent, where σ=10λ0.

Fig. 6
Fig. 6

Same as Fig. 5, but for the minus-one-order light.

Fig. 7
Fig. 7

Comparison of the cross-section profiles of the complex degree of spatial coherence of the incident, the zero-order, and the minus-one-order light as a function of xd/σ for α=2, where the incident light is spatially highly incoherent, with σ=10λ0.

Fig. 8
Fig. 8

Three-dimensional plots showing complex degree of spatial coherence of the zero-order light as a function of α and xd/σ for incident light that is spatially highly coherent, where σ=10,000λ0.

Fig. 9
Fig. 9

Same as Fig. 8, but for the minus-one-order light.

Equations (32)

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ψ0(x, z)z=-j2k0 cos ϕB2ψ0(x, z)x2+2ψ0(x, z)z2-tan ϕBψ0(x, z)x-jk0CA4ψ-1(x, z),
ψ-1(x, z)z=-j2k0 cos ϕB2ψ-1(x, z)x2+2ψ-1(x, z)z2+tan ϕBψ-1(x, z)x-jk0CA4ψ0(x, z),
Γ0,0(x1, x2, z)=ψ0(x1, z)ψ0*(x2, z),
Γ-1,-1(x1, x2, z)=ψ-1(x1, z)ψ-1*(x2, z),
Γ0,-1(x1, x2, z)=ψ0(x1, z)ψ-1*(x2, z),
Γ-1,0(x1, x2, z)=ψ-1(x1, z)ψ0*(x2, z),
ψ0(x1, z)zψ0*(x2, z)=-j2k0[t12ψ0(x1, z)]ψ0*(x2, z)-ϕBψ0(x1, z)x1ψ0*(x2, z)-jk0CA4ψ-1(x1, z)ψ0*(x2, z),
ψ0*(x2, z)zψ0(x1, z)=j2k0[t22ψ0*(x2, z)]ψ0(x1, z)-ϕBψ0*(x2, z)x2ψ0(x1, z)+jk0CA4ψ-1*(x2, z)ψ0(x1, z).
Γ0,0(x1, x2, z)z=-j2k0(t12-t22)Γ0,0(x1, x2, z)-ϕBx1+x2Γ0,0(x1, x2, z)-jk0CA4[Γ-1,0(x1, x2, z)-Γ0,-1(x1, x2, z)].
Γ0,0(xc, xd, z)z=-jk02Γ0,0(xc, xd, z)xcxd-ϕBΓ0,0(xc, xd, z)xc-jk0CA4[Γ-1,0(xc, xd, z)-Γ0,-1(xc, xd, z)].
Γ-1,-1(xc, xd, z)z=-jk02Γ-1,-1(xc, xd, z)xcxd+ϕBΓ-1,-1(xc, xd, z)xc-jk0CA4[Γ0,-1(xc, xd, z)-Γ-1,0(xc, xd, z)],
Γ0,-1(xc, xd, z)z=-jk02Γ0,-1(xc, xd, z)xcxd-2ϕBΓ0,-1(xc, xd, z)xd-jk0CA4[Γ-1,-1(xc, xd, z)-Γ0,0(xc, xd, z)],
Γ-1,0(xc, xd, z)z=-jk02Γ-1,0(xc, xd, z)xcxd+2ϕBΓ-1,0(xc, xd, z)xd-jk0CA4[Γ0,0(xc, xd, z)-Γ-1,-1(xc, xd, z)].
M(k1, k2)=1(2π)2--Γ(x1, x2, z)exp(-jk1x1+jk2x2)dx1dx2,
k1x1-k2x2=kdxc+kcxd,
M(kc, kd)=1(2π)2--Γ(xc, xd, z)×exp(-jkcxd-jkdxc)dxcdxd.
M0,0(kc, kd, z)z=jkdkck0M0,0+jϕB kdM0,0-jα2L(M-1,0-M0,-1),
M-1,-1(kc, kd, z)z=jkdkck0M-1,-1-jϕB kdM-1,-1-jα2L(M0,-1-M-1,0),
M0,-1(kc, kd, z)z=jkdkck0M0,-1+2jϕB kcM0,-1-jα2L(M-1,-1-M0,0),
M-1,0(kc, kd, z)z=jkdkck0M-1,0-2jϕB kcM-1,0-jα2L(M0,0-M-1,-1),
Γ0,0(x1, x2, 0)=Γinc=exp[-(x12+x22)/w2]
×exp[-(x1-x2)2/2σ2],
Γ0,0(xc, xd, 0)=exp-2xc2w2exp-12w2+12σ2xd2.
M0,0(kc, kd, 0)=w2σ4πw2+σ2exp-w2σ22(w2+σ2)kc2×exp-w28kd2.
M-1,-1(kc, kd, 0)=M-1,0(kc, kd, 0)=M0,-1(kc, kd, 0)=0.
M0,0(kc, kd, z)=M0,0(kc, kd, 0)expjkckdk0zD2-E2×2α2L2+4(ϕBkc)2-E2cos Ez-2α2L2+4(ϕBkc)2-D2cos Dz+jE(+ϕBkdE2-4ϕB3kdkc2)sin Ez+jD(-ϕBkdD2+4ϕB3kdkc2)sin Dz,
M-1,-1(kc, kd, z)=M0,0(kc, kd, 0)α22(D2-E2)L2×expjkckdk0z(cos Ez-cos Dz),
M-1,0(kc, kd, z)=jαM0,0(kc, kd, 0)2(D2-E2)Lexpjkckdk0z×jϕB(kd-2kc)(cos Dz-cos Ez)+1D(2ϕB2kckd-D2)sin Dz-1E(2ϕB2kdkc-E2)sin Ez,
M0,-1(kc, kd, z)=jαM0,0(kc, kd, 0)2(D2-E2)Lexpjkckdk0z×jϕB(kd+2kc)(cos Ez-cos Dz)+1D(2ϕB2kckd+D2)sin Dz-1E(2ϕB2kdkc+E2)sin Ez,
D=12{[(ϕBkd)2+(2ϕBkc)2+(α/L)2+4ϕB2kckd]1/2+[(ϕBkd)2+(2ϕBkc)2+(α/L)2-4ϕB2kckd]1/2},
E=12{[(ϕBkd)2+(2ϕBkc)2+(α/L)2+4ϕBkc2kd]1/2-[(ϕBkd)2+(2ϕBkc)2+(α/L)2-4ϕBkc2kd]1/2}.
γm,n(xc, xd, z)=Γm,n(xc, xd, z)Γm,n(xc, 0, z)

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