Abstract

The basic problem of the diffraction of an optical plane wave by an acoustic plane wave in an anisotropic homogeneous medium is considered. The acousto-optical interaction is considered indifferently of the isotropic or of the birefringent type. Coupled-wave equations are obtained rigorously and cast into an eigenvalue value problem. A general solution is obtained for the diffraction efficiency of diffracted orders, for any interaction length and diffraction regime. The theory includes the Bragg regime, the Raman–Nath regime, and all intermediate situations in the same formulation. The method of solution is both exact and computationally efficient. It is similar in character to the rigorous coupled-wave analysis of Moharam and Gaylord but differs by the choice of basis functions adapted to propagating rather than static gratings. Examples are given for acousto-optical interaction in paratellurite, TeO2.

© 2003 Optical Society of America

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References

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  1. W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
    [CrossRef]
  2. T. C. Poon, A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
    [CrossRef]
  3. J. Xu, R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, New York, 1992).
  4. R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
    [CrossRef]
  5. A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1988).
  6. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, “Three-dimensional vec-tor coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  8. R. A. Mertens, W. Hereman, J.-P. Ottoy, “The Raman–Nath equations revisited. II. Oblique incidence of the light—Bragg reflection,” in Proceedings of Ultrasonics International (Elsevier, New York, 1987), pp. 84–89.
  9. I. C. Chang, “Acoustooptic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2–22 (1976).
    [CrossRef]
  10. V. Voloshinov, “Close to collinear acousto-optical interaction in paratellurite,” Opt. Eng. 31, 2089–2094 (1992).
    [CrossRef]
  11. F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, P. Tournois, “Arbitrary control of phase and amplitude of ultrashort pulses with an acousto-optic programmable dispersive filter: application to pulse compression and pulse shaping,” Opt. Lett. 25, 575–577 (2000).
    [CrossRef]
  12. N. Uchida, A. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40, 4692–4695 (1969).
    [CrossRef]
  13. D. L. Hecht, “Multifrequency acoustooptic diffraction,” IEEE Trans. Sonics Ultrason. SU-24, 7–18 (1977).
    [CrossRef]
  14. Y. Tao, J. Xu, “Feynman diagram analysis of intermodulation products in Bragg cells,” J. Opt. Soc. Am. A 9, 2223–2230 (1992).
    [CrossRef]
  15. F. Verluise, V. Laude, J.-P. Huignard, P. Tournois, A. Migus, “Arbitrary dispersion control of ultrashort optical pulses using acoustic waves,” J. Opt. Soc. Am. B 17, 138–145 (2000).
    [CrossRef]

2000 (2)

1992 (2)

V. Voloshinov, “Close to collinear acousto-optical interaction in paratellurite,” Opt. Eng. 31, 2089–2094 (1992).
[CrossRef]

Y. Tao, J. Xu, “Feynman diagram analysis of intermodulation products in Bragg cells,” J. Opt. Soc. Am. A 9, 2223–2230 (1992).
[CrossRef]

1983 (1)

1981 (2)

1977 (1)

D. L. Hecht, “Multifrequency acoustooptic diffraction,” IEEE Trans. Sonics Ultrason. SU-24, 7–18 (1977).
[CrossRef]

1976 (1)

I. C. Chang, “Acoustooptic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2–22 (1976).
[CrossRef]

1969 (1)

N. Uchida, A. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40, 4692–4695 (1969).
[CrossRef]

1967 (2)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

Chang, I. C.

I. C. Chang, “Acoustooptic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2–22 (1976).
[CrossRef]

Cheng, Z.

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Dixon, R. W.

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

Gaylord, T. K.

Hecht, D. L.

D. L. Hecht, “Multifrequency acoustooptic diffraction,” IEEE Trans. Sonics Ultrason. SU-24, 7–18 (1977).
[CrossRef]

Hereman, W.

R. A. Mertens, W. Hereman, J.-P. Ottoy, “The Raman–Nath equations revisited. II. Oblique incidence of the light—Bragg reflection,” in Proceedings of Ultrasonics International (Elsevier, New York, 1987), pp. 84–89.

Huignard, J.-P.

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Korpel, A.

Laude, V.

Mertens, R. A.

R. A. Mertens, W. Hereman, J.-P. Ottoy, “The Raman–Nath equations revisited. II. Oblique incidence of the light—Bragg reflection,” in Proceedings of Ultrasonics International (Elsevier, New York, 1987), pp. 84–89.

Migus, A.

Moharam, M. G.

Ohmachi, A.

N. Uchida, A. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40, 4692–4695 (1969).
[CrossRef]

Ottoy, J.-P.

R. A. Mertens, W. Hereman, J.-P. Ottoy, “The Raman–Nath equations revisited. II. Oblique incidence of the light—Bragg reflection,” in Proceedings of Ultrasonics International (Elsevier, New York, 1987), pp. 84–89.

Poon, T. C.

Spielmann, Ch.

Stroud, R.

J. Xu, R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, New York, 1992).

Tao, Y.

Tournois, P.

Uchida, N.

N. Uchida, A. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40, 4692–4695 (1969).
[CrossRef]

Verluise, F.

Voloshinov, V.

V. Voloshinov, “Close to collinear acousto-optical interaction in paratellurite,” Opt. Eng. 31, 2089–2094 (1992).
[CrossRef]

Xu, J.

Y. Tao, J. Xu, “Feynman diagram analysis of intermodulation products in Bragg cells,” J. Opt. Soc. Am. A 9, 2223–2230 (1992).
[CrossRef]

J. Xu, R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, New York, 1992).

IEEE J. Quantum Electron. (1)

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

IEEE Trans. Sonics Ultrason. (3)

I. C. Chang, “Acoustooptic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2–22 (1976).
[CrossRef]

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

D. L. Hecht, “Multifrequency acoustooptic diffraction,” IEEE Trans. Sonics Ultrason. SU-24, 7–18 (1977).
[CrossRef]

J. Appl. Phys. (1)

N. Uchida, A. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40, 4692–4695 (1969).
[CrossRef]

J. Opt. Soc. Am. (3)

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

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

Opt. Eng. (1)

V. Voloshinov, “Close to collinear acousto-optical interaction in paratellurite,” Opt. Eng. 31, 2089–2094 (1992).
[CrossRef]

Opt. Lett. (1)

Other (3)

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

R. A. Mertens, W. Hereman, J.-P. Ottoy, “The Raman–Nath equations revisited. II. Oblique incidence of the light—Bragg reflection,” in Proceedings of Ultrasonics International (Elsevier, New York, 1987), pp. 84–89.

J. Xu, R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, New York, 1992).

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Figures (5)

Fig. 1
Fig. 1

Schematic of the interaction region. All diffraction orders lie in the plane of incidence defined by wave vectors k0 and K.

Fig. 2
Fig. 2

Definitions for the acousto-optical interaction in TeO2 considered in Figs. 35. Acoustic waves are propagating along the [110] direction and are shear polarized along the [110¯] direction. The plane of incidence is defined by axes [001] and [110]. θ is the angle of incidence for incoming light.

Fig. 3
Fig. 3

Diffraction efficiency for isotropic (ordinary light) acousto-optical interaction in TeO2 in the intermediate regime. The interaction length is L=1 mm, the acoustic intensity is Iac=105 kg/s3, and the acoustic frequency is Ω=20 MHz. The angles of incidence are (a) 90°, (b) 89°, and (c) 88°. Figure labels are for diffraction orders.

Fig. 4
Fig. 4

Diffraction efficiency for birefringent acousto-optical interaction in TeO2 in the intermediate regime, for ordinary incident light. The acoustic frequency is Ω=60 MHz, and the angle of incidence is 89°; the Bragg condition is achieved in the first order of diffraction for λ=0.82 μm. The interaction length and the acoustic intensity are, respectively, (a) L=2 mm and Iac=119,200 kg/s3, (b) L=6 mm and Iac=13,000 kg/s3, and (c) L=10 mm and Iac=4768 kg/s3. The acoustic intensity is set to yield maximum diffraction efficiency in the first diffraction order at λ=0.82 μm. Figure labels are for diffraction orders.

Fig. 5
Fig. 5

Evolution of the diffraction efficiency for anisotropic acousto-optical interaction in TeO2 as a function of the position inside the crystal, for ordinary incident light. The acoustic frequency is Ω=60 MHz, and the angle of incidence is 89°. L=2 mm and Iac=119,200 kg/s3 as in Fig. 4(a). Figure labels are for diffraction orders.

Equations (48)

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1c22Dt2=2(η:D)-[·(η:D)],
ijηjk=δik,
Di=D0di exp[j(ωt-kjxj)]
1n2=ω2k2c2=d:η:d=diηijdj.
|S|=12c0n3D02,
Sij(t, r)=Ssˆij sin(Ωt-K·r),
Pac=IacSt,
Iac=12ρVa3|S|2
Δηij(t, r)=pijklSkl(t, r)=S(pijklsˆkl)sin(Ωt-K·r)=Δηij sin(Ωt-K·r).
D(r, t)=m=-Dm(zm)dm exp[j(ωmt-km·r)],
ωm=ω+mΩ,
km=k+mK.
zm=um·r,
f(zm)xi=(um)i df(zm)dzm=(um)if(zm).
um·dmDm(zm)-jkm·dmDm(zm)=0,
um·dm=0,
km·dm=0.
1nm2=dm:η:dm.
Dˆm(Z)=1nm3LmDm(zm).
Dˆ(Z)=jMDˆ(Z),
Mm,m=12kmLm1-ωm2nm2c2km2,
Mm,m-1=-jω4c(LmLm-1nm3nm-13)1/2Δn|mm-1,
Mm,m+1=jω4c(LmLm+1nm3nm+13)1/2Δn|mm+1,
Δn|mm±1=dm:Δη:dm±1.
Dˆ(Z)=XΔ(Z)a,
Δm,m(Z)=exp(jvmZ)
Dˆ(0)=Xm,nan=δm,
XX=I,
m|am|2=1
m|Dˆm(Z)|2=1.
an=X0,n*,
Dˆm(Z)=nXm,nX0,n* exp(jvnZ),
Q=(K2L)/k.
1c22Dt2=-m ωm2c2Dm(zm)dm exp(jϕm),
ϕm=ωmt-km·r.
η(t, r)=η+12jΔη{exp[j(Ωt-K·r)]-exp[-j(Ωt-K·r)]}.
η:D(r, t)=mDm(zm)η:dm exp(jϕm)+12jmDm(zm)Δη:dm exp(jϕm+1)-12jmDm(zm)Δη:dm exp(jϕm-1).
2[η:D(r, t)]=-m[km2Dm(zm)+2j(km·um)Dm(zm)]η:dm exp(jϕm)-12jmDm(zm)km+12Δη:dm×exp(jϕm+1)+12jmDm(zm)km-12×Δη:dm exp(jϕm-1).
2[η:D(r, t)]=-m[km2Dm(zm)+2jkmDm(zm)]η:dm exp(jϕm)-12jmkm2Dm-1(zm-1)Δη:dm-1×exp(jϕm)+12jmkm2Dm+1(zm+1)×Δη:dm+1 exp(jϕm).
·[η:D(r, t)]=m[-jDm(zm)(km:η:dm)+Dm(zm)(um:η:dm)]exp(jϕm)-12mDm(zm)(km+1:Δη:dm)exp(jϕm+1)+12mDm(zm)(km-1:Δη:dm)exp(jϕm-1)
{·[η:D(r, t)]}=-mkm[Dm(zm)(km:η:dm)+2jDm(zm)(lm:η:dm)]×exp(jϕm)+j2mDm(zm)×km+1(km+1:Δη:dm)×exp(jϕm+1)-j2mDm(zm)×km-1(km-1:Δη:dm)×exp(jϕm-1)
{·[η:D(r, t)]}=-mkm[Dm(zm)(km:η:dm)+2jDm(zm)(lm:η:dm)]exp(jϕm)+j2mkmDm-1(zm-1)(km:Δη:dm-1)exp(jϕm)-j2mkmDm+1(zm+1)(km:Δη:dm+1)exp(jϕm).
ωm2c2-km2nm2Dm(zm)-2jkmnm2Dm(zm)=km22j[Δn|mm-1Dm-1(zm-1)-Δn|mm+1Dm+1(zm+1)].
Dm(zm)=j2km1-ωm2nm2c2km2Dm(zm)+14nm2km×[Δn|mm-1Dm-1(zm-1)-Δn|mm+1Dm+1(zm+1)].
Dm(Z)=Lm/nm3Dm(zm)
Dm(Z)=j2kmLm1-ωm2nm2c2km2Dˆm(Z)+14nmLmkmnm-13L-1Δn|m-1Dˆ-1(Z)-14nmLmkmnm+13Lm+1Δn|m+1Dˆm+1(Z).
kmnmωc,
Dˆm(Z)=j2kmLm1-ωm2nm2c2km2Dˆm(Z)+14ωcnm3Lmnm-13Lm-1Δn|mm-1Dˆ-1(Z)-14ωcnm3Lmnm+13Lm+1Δn|mm+1Dˆm+1(Z).

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