Abstract

A review of the principles of acoustooptical devices is given. Some very simple momentum conservation considerations indicate the optimum relationship between the optical and acoustic beam dimensions for various functions such as scanning or modulation. A calculation for the usual type of acoustic amplitude modulation is described, and serves as an example of the type of detailed considerations that are necessary and possible, as well as a verification of the validity of the simple momentum considerations. It is shown that the product of the fraction of the light that may be scattered and the bandwidth for Bragg scattering equals a materials constant times the acoustic power. This relationship is shown to be valid even to the extent of numerical constants for several configurations allowing a trade-off between these parameters. Thus, the required modulation power for any level of device performance is easily determined. The details of acoustic deflection under conditions of acoustic beam focusing or scanning are also given.

© 1966 Optical Society of America

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References

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  1. C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, “Interaction of light and microwave sound,” Proc. IEEE, vol. 53, pp. 1604–1623, October1965.
    [CrossRef]
  2. T. M. Smith, A. Korpel, “Measurement of light-sound interaction efficiencies in solids,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-1, pp. 283–284, September1965.
    [CrossRef]
  3. B. Tell, J. M. Worlock, R. J. Martin, “Enhancement of elasto-optic constants in the neighborhood of a band gap in ZnO and CdS,” Appl. Phys. Lett., vol. 6, pp. 123–124, April1, 1965.
    [CrossRef]
  4. R. W. Dixon, M. G. Cohen, “A new technique for measuring magnitudes of photoelastic tensors and its application to lithium niobate,” Appl. Phys. Lett., vol. 8, pp. 205–207, April15, 1966.E. G. Spencer, P. V. Lenzo, K. Nassau, “Optical interactions with elastic waves in lithium niobate,” IEEE J. of Quantum Electronics (Correspondence), to be published.
    [CrossRef]
  5. N. F. Foster, G. A. Rozgonyi, “Zinc oxide film transducers,” Appl. Phys. Lett., vol. 8, pp. 221–223, May1, 1966.
    [CrossRef]
  6. E. I. Gordon, M. G. Cohen, “Electro-optic gratings for light beam modulation and deflection,” presented at the 1964 IEEE WESCON.
  7. M. G. Cohen, E. I. Gordon, “Electro-optic [KTaxNb1–2O3(KTN)] gratings for light beam modulation and deflection,” Appl. Phys. Lett., vol. 5, p. 181, November1964.
    [CrossRef]
  8. A. Korpel, R. Adler, P. Desmares, T. M. Smith, “An ultrasonic light deflection system,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 60–61, April1965.
    [CrossRef]
  9. A. Korpel, R. Adler, P. Desmares, W. Watson, “A television display using acoustic deflection and modulation of coherent light,” this issue.
  10. M. G. Cohen, E. I. Gordon, “Acoustic beam probing using optical techniques,” Bell Sys. Tech. J., vol. XLIV, pp. 693–721, April1965.
  11. E. I. Gordon, M. G. Cohen, “Acoustic modulation devices,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.
  12. N. F. Foster, “Cadmium sulphide evaporated-layer transducers,” Proc. IEEE, vol. 53, pp. 1400–1405, October1965.
    [CrossRef]
  13. I. P. Kaminow, private communication.
  14. A. Korpel, R. Adler, P. Desmares, “An improved ultrasonic light deflection system,” Paper 11.5, presented at the 1965 Internat’l Electron Devices Meeting, Washington, D. C.
  15. E. I. Gordon, M. G. Cohen, “Electro-optic diffraction grating for light beam modulation and diffraction,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 191–198, August1965.
    [CrossRef]
  16. H. V. Hance, J. K. Parks, “Wide-band modulation of a laser beam using Bragg-angle diffraction by amplitude modulated ultrasonic waves,” J. Acoust. Soc. Am., vol. 38, pp. 14–23, July1965.
    [CrossRef]
  17. R. W. Dixon, E. I. Gordon, “Carrier frequency modulation using acoustic waves,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.
  18. M. G. Cohen, E. I. Gordon, “Acoustic scattering of light in a Fabry-Perot resonator,” Bell Sys. Tech. J., vol. 45, pp. 945–966, July–August 1966.
  19. E. I. Gordon, “Figure of merit for acousto-optical deflection and modulation devices,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-2, pp. 104–105, May1966.
    [CrossRef]

1966 (3)

N. F. Foster, G. A. Rozgonyi, “Zinc oxide film transducers,” Appl. Phys. Lett., vol. 8, pp. 221–223, May1, 1966.
[CrossRef]

M. G. Cohen, E. I. Gordon, “Acoustic scattering of light in a Fabry-Perot resonator,” Bell Sys. Tech. J., vol. 45, pp. 945–966, July–August 1966.

E. I. Gordon, “Figure of merit for acousto-optical deflection and modulation devices,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-2, pp. 104–105, May1966.
[CrossRef]

1965 (8)

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, “Interaction of light and microwave sound,” Proc. IEEE, vol. 53, pp. 1604–1623, October1965.
[CrossRef]

T. M. Smith, A. Korpel, “Measurement of light-sound interaction efficiencies in solids,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-1, pp. 283–284, September1965.
[CrossRef]

B. Tell, J. M. Worlock, R. J. Martin, “Enhancement of elasto-optic constants in the neighborhood of a band gap in ZnO and CdS,” Appl. Phys. Lett., vol. 6, pp. 123–124, April1, 1965.
[CrossRef]

A. Korpel, R. Adler, P. Desmares, T. M. Smith, “An ultrasonic light deflection system,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 60–61, April1965.
[CrossRef]

M. G. Cohen, E. I. Gordon, “Acoustic beam probing using optical techniques,” Bell Sys. Tech. J., vol. XLIV, pp. 693–721, April1965.

N. F. Foster, “Cadmium sulphide evaporated-layer transducers,” Proc. IEEE, vol. 53, pp. 1400–1405, October1965.
[CrossRef]

E. I. Gordon, M. G. Cohen, “Electro-optic diffraction grating for light beam modulation and diffraction,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 191–198, August1965.
[CrossRef]

H. V. Hance, J. K. Parks, “Wide-band modulation of a laser beam using Bragg-angle diffraction by amplitude modulated ultrasonic waves,” J. Acoust. Soc. Am., vol. 38, pp. 14–23, July1965.
[CrossRef]

1964 (1)

M. G. Cohen, E. I. Gordon, “Electro-optic [KTaxNb1–2O3(KTN)] gratings for light beam modulation and deflection,” Appl. Phys. Lett., vol. 5, p. 181, November1964.
[CrossRef]

Adler, R.

A. Korpel, R. Adler, P. Desmares, T. M. Smith, “An ultrasonic light deflection system,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 60–61, April1965.
[CrossRef]

A. Korpel, R. Adler, P. Desmares, W. Watson, “A television display using acoustic deflection and modulation of coherent light,” this issue.

A. Korpel, R. Adler, P. Desmares, “An improved ultrasonic light deflection system,” Paper 11.5, presented at the 1965 Internat’l Electron Devices Meeting, Washington, D. C.

Cohen, M. G.

M. G. Cohen, E. I. Gordon, “Acoustic scattering of light in a Fabry-Perot resonator,” Bell Sys. Tech. J., vol. 45, pp. 945–966, July–August 1966.

E. I. Gordon, M. G. Cohen, “Electro-optic diffraction grating for light beam modulation and diffraction,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 191–198, August1965.
[CrossRef]

M. G. Cohen, E. I. Gordon, “Acoustic beam probing using optical techniques,” Bell Sys. Tech. J., vol. XLIV, pp. 693–721, April1965.

M. G. Cohen, E. I. Gordon, “Electro-optic [KTaxNb1–2O3(KTN)] gratings for light beam modulation and deflection,” Appl. Phys. Lett., vol. 5, p. 181, November1964.
[CrossRef]

E. I. Gordon, M. G. Cohen, “Electro-optic gratings for light beam modulation and deflection,” presented at the 1964 IEEE WESCON.

R. W. Dixon, M. G. Cohen, “A new technique for measuring magnitudes of photoelastic tensors and its application to lithium niobate,” Appl. Phys. Lett., vol. 8, pp. 205–207, April15, 1966.E. G. Spencer, P. V. Lenzo, K. Nassau, “Optical interactions with elastic waves in lithium niobate,” IEEE J. of Quantum Electronics (Correspondence), to be published.
[CrossRef]

E. I. Gordon, M. G. Cohen, “Acoustic modulation devices,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.

Desmares, P.

A. Korpel, R. Adler, P. Desmares, T. M. Smith, “An ultrasonic light deflection system,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 60–61, April1965.
[CrossRef]

A. Korpel, R. Adler, P. Desmares, W. Watson, “A television display using acoustic deflection and modulation of coherent light,” this issue.

A. Korpel, R. Adler, P. Desmares, “An improved ultrasonic light deflection system,” Paper 11.5, presented at the 1965 Internat’l Electron Devices Meeting, Washington, D. C.

Dixon, R. W.

R. W. Dixon, E. I. Gordon, “Carrier frequency modulation using acoustic waves,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.

R. W. Dixon, M. G. Cohen, “A new technique for measuring magnitudes of photoelastic tensors and its application to lithium niobate,” Appl. Phys. Lett., vol. 8, pp. 205–207, April15, 1966.E. G. Spencer, P. V. Lenzo, K. Nassau, “Optical interactions with elastic waves in lithium niobate,” IEEE J. of Quantum Electronics (Correspondence), to be published.
[CrossRef]

Foster, N. F.

N. F. Foster, G. A. Rozgonyi, “Zinc oxide film transducers,” Appl. Phys. Lett., vol. 8, pp. 221–223, May1, 1966.
[CrossRef]

N. F. Foster, “Cadmium sulphide evaporated-layer transducers,” Proc. IEEE, vol. 53, pp. 1400–1405, October1965.
[CrossRef]

Gordon, E. I.

M. G. Cohen, E. I. Gordon, “Acoustic scattering of light in a Fabry-Perot resonator,” Bell Sys. Tech. J., vol. 45, pp. 945–966, July–August 1966.

E. I. Gordon, “Figure of merit for acousto-optical deflection and modulation devices,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-2, pp. 104–105, May1966.
[CrossRef]

E. I. Gordon, M. G. Cohen, “Electro-optic diffraction grating for light beam modulation and diffraction,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 191–198, August1965.
[CrossRef]

M. G. Cohen, E. I. Gordon, “Acoustic beam probing using optical techniques,” Bell Sys. Tech. J., vol. XLIV, pp. 693–721, April1965.

M. G. Cohen, E. I. Gordon, “Electro-optic [KTaxNb1–2O3(KTN)] gratings for light beam modulation and deflection,” Appl. Phys. Lett., vol. 5, p. 181, November1964.
[CrossRef]

E. I. Gordon, M. G. Cohen, “Electro-optic gratings for light beam modulation and deflection,” presented at the 1964 IEEE WESCON.

E. I. Gordon, M. G. Cohen, “Acoustic modulation devices,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.

R. W. Dixon, E. I. Gordon, “Carrier frequency modulation using acoustic waves,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.

Hance, H. V.

H. V. Hance, J. K. Parks, “Wide-band modulation of a laser beam using Bragg-angle diffraction by amplitude modulated ultrasonic waves,” J. Acoust. Soc. Am., vol. 38, pp. 14–23, July1965.
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, private communication.

Korpel, A.

A. Korpel, R. Adler, P. Desmares, T. M. Smith, “An ultrasonic light deflection system,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 60–61, April1965.
[CrossRef]

T. M. Smith, A. Korpel, “Measurement of light-sound interaction efficiencies in solids,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-1, pp. 283–284, September1965.
[CrossRef]

A. Korpel, R. Adler, P. Desmares, W. Watson, “A television display using acoustic deflection and modulation of coherent light,” this issue.

A. Korpel, R. Adler, P. Desmares, “An improved ultrasonic light deflection system,” Paper 11.5, presented at the 1965 Internat’l Electron Devices Meeting, Washington, D. C.

Martin, R. J.

B. Tell, J. M. Worlock, R. J. Martin, “Enhancement of elasto-optic constants in the neighborhood of a band gap in ZnO and CdS,” Appl. Phys. Lett., vol. 6, pp. 123–124, April1, 1965.
[CrossRef]

Parks, J. K.

H. V. Hance, J. K. Parks, “Wide-band modulation of a laser beam using Bragg-angle diffraction by amplitude modulated ultrasonic waves,” J. Acoust. Soc. Am., vol. 38, pp. 14–23, July1965.
[CrossRef]

Quate, C. F.

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, “Interaction of light and microwave sound,” Proc. IEEE, vol. 53, pp. 1604–1623, October1965.
[CrossRef]

Rozgonyi, G. A.

N. F. Foster, G. A. Rozgonyi, “Zinc oxide film transducers,” Appl. Phys. Lett., vol. 8, pp. 221–223, May1, 1966.
[CrossRef]

Smith, T. M.

T. M. Smith, A. Korpel, “Measurement of light-sound interaction efficiencies in solids,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-1, pp. 283–284, September1965.
[CrossRef]

A. Korpel, R. Adler, P. Desmares, T. M. Smith, “An ultrasonic light deflection system,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 60–61, April1965.
[CrossRef]

Tell, B.

B. Tell, J. M. Worlock, R. J. Martin, “Enhancement of elasto-optic constants in the neighborhood of a band gap in ZnO and CdS,” Appl. Phys. Lett., vol. 6, pp. 123–124, April1, 1965.
[CrossRef]

Watson, W.

A. Korpel, R. Adler, P. Desmares, W. Watson, “A television display using acoustic deflection and modulation of coherent light,” this issue.

Wilkinson, C. D. W.

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, “Interaction of light and microwave sound,” Proc. IEEE, vol. 53, pp. 1604–1623, October1965.
[CrossRef]

Winslow, D. K.

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, “Interaction of light and microwave sound,” Proc. IEEE, vol. 53, pp. 1604–1623, October1965.
[CrossRef]

Worlock, J. M.

B. Tell, J. M. Worlock, R. J. Martin, “Enhancement of elasto-optic constants in the neighborhood of a band gap in ZnO and CdS,” Appl. Phys. Lett., vol. 6, pp. 123–124, April1, 1965.
[CrossRef]

Appl. Phys. Lett. (3)

B. Tell, J. M. Worlock, R. J. Martin, “Enhancement of elasto-optic constants in the neighborhood of a band gap in ZnO and CdS,” Appl. Phys. Lett., vol. 6, pp. 123–124, April1, 1965.
[CrossRef]

N. F. Foster, G. A. Rozgonyi, “Zinc oxide film transducers,” Appl. Phys. Lett., vol. 8, pp. 221–223, May1, 1966.
[CrossRef]

M. G. Cohen, E. I. Gordon, “Electro-optic [KTaxNb1–2O3(KTN)] gratings for light beam modulation and deflection,” Appl. Phys. Lett., vol. 5, p. 181, November1964.
[CrossRef]

Bell Sys. Tech. J. (2)

M. G. Cohen, E. I. Gordon, “Acoustic beam probing using optical techniques,” Bell Sys. Tech. J., vol. XLIV, pp. 693–721, April1965.

M. G. Cohen, E. I. Gordon, “Acoustic scattering of light in a Fabry-Perot resonator,” Bell Sys. Tech. J., vol. 45, pp. 945–966, July–August 1966.

IEEE J. of Quantum Electronics (2)

E. I. Gordon, M. G. Cohen, “Electro-optic diffraction grating for light beam modulation and diffraction,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 191–198, August1965.
[CrossRef]

A. Korpel, R. Adler, P. Desmares, T. M. Smith, “An ultrasonic light deflection system,” IEEE J. of Quantum Electronics, vol. QE-1, pp. 60–61, April1965.
[CrossRef]

IEEE J. of Quantum Electronics (Correspondence) (2)

T. M. Smith, A. Korpel, “Measurement of light-sound interaction efficiencies in solids,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-1, pp. 283–284, September1965.
[CrossRef]

E. I. Gordon, “Figure of merit for acousto-optical deflection and modulation devices,” IEEE J. of Quantum Electronics (Correspondence), vol. QE-2, pp. 104–105, May1966.
[CrossRef]

J. Acoust. Soc. Am. (1)

H. V. Hance, J. K. Parks, “Wide-band modulation of a laser beam using Bragg-angle diffraction by amplitude modulated ultrasonic waves,” J. Acoust. Soc. Am., vol. 38, pp. 14–23, July1965.
[CrossRef]

Proc. IEEE (2)

N. F. Foster, “Cadmium sulphide evaporated-layer transducers,” Proc. IEEE, vol. 53, pp. 1400–1405, October1965.
[CrossRef]

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, “Interaction of light and microwave sound,” Proc. IEEE, vol. 53, pp. 1604–1623, October1965.
[CrossRef]

Other (7)

R. W. Dixon, M. G. Cohen, “A new technique for measuring magnitudes of photoelastic tensors and its application to lithium niobate,” Appl. Phys. Lett., vol. 8, pp. 205–207, April15, 1966.E. G. Spencer, P. V. Lenzo, K. Nassau, “Optical interactions with elastic waves in lithium niobate,” IEEE J. of Quantum Electronics (Correspondence), to be published.
[CrossRef]

A. Korpel, R. Adler, P. Desmares, W. Watson, “A television display using acoustic deflection and modulation of coherent light,” this issue.

E. I. Gordon, M. G. Cohen, “Electro-optic gratings for light beam modulation and deflection,” presented at the 1964 IEEE WESCON.

I. P. Kaminow, private communication.

A. Korpel, R. Adler, P. Desmares, “An improved ultrasonic light deflection system,” Paper 11.5, presented at the 1965 Internat’l Electron Devices Meeting, Washington, D. C.

E. I. Gordon, M. G. Cohen, “Acoustic modulation devices,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.

R. W. Dixon, E. I. Gordon, “Carrier frequency modulation using acoustic waves,” presented at the 1966 Conference on Electron Device Research, California Institute of Technology, Pasadena, Calif.

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Figures (11)

Fig. 1
Fig. 1

Momentum scattering for plane, monochromatic optical, and acoustic waves. If the direction of k or K is changed, thereby changing the angle of incidence denoted Θ, the vector sum k+K no longer falls on the circle and no scattering can occur.

Fig. 2
Fig. 2

Scattering with acoustic waves of finite width corresponding to nonzero diffraction angle. The angular distribution of acoustic energy for a rectangular, flat transducer is shown above the cone. Only one plane wave component of the acoustic beam is effective for scattering. The amplitude of the scattered light as a function of the direction of the acoustic beam measures its far-field pattern.

Fig. 3
Fig. 3

Scattering with acoustic and light beams of finite width. The diffraction angle of the light is δϕ and that of the sound is δθ. For scanning devices there are two possible configurations: (a) δθδϕ, or (b) δϕδθ. For (a) the scattered light beam has a diffraction angle δϕ, for (b) the diffraction angle is δθ.

Fig. 4
Fig. 4

Geometry of the interacting beams illustrating that the coherence width of the light beam is unchanged from that of incident light for δθδϕ, but is increased for δθδϕ. The diffraction angle of the scattered light always corresponds to the smaller of the two angles.

Fig. 5
Fig. 5

Arrangement for the usual amplitude modulation device. Two configurations are shown: (a) the optimum situation for which δθδϕ, and (b) the case δϕδθ.

Fig. 6
Fig. 6

Arrangement for carrier frequency modulation of the light using a Kösters prism for producing symmetrical, coherent beams.

Fig. 7
Fig. 7

Momentum diagram for the arrangement of Fig. 6.

Fig. 8
Fig. 8

Geometry of scattering calculation. The rectangular acoustic beam of height H and width W0 moves along the x0-axis and the incident Gaussian light beam moves at an angle θ0. The scattering to the far-field point X=R sin θ, Y=R cos θ cos δ, Z= R cos θ sin δ is calculated.

Fig. 9
Fig. 9

Focused acoustic beam geometry.

Fig. 10
Fig. 10

Scattering arrangement for a multielement transducer. As the acoustic frequency increases, ψ decreases and the angle of incidence of the light relative to the acoustic lobe on the right increases. Thus, the angle of incidence can approximately track the Bragg angle which increases with frequency.

Fig. 11
Fig. 11

A plot of the element-to-element shift ϕ=κL [as defined by (40)] required for optimum scattering from an acoustic beam generated by a phased array of transducer elements. The construction pertains to the case ϕ=π

Equations (62)

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ω s = ω ± Ω k s = k ± K
sin Θ = 1 2 K / k .
N = 2 δ θ / δ ϕ
N = ( Δ Ω / 2 π ) w 0 / v cos θ 0
N = 1 2 ( Δ Ω / 2 π ) w s / v cos θ 0
π Δ f k v = δ θ cos Θ
δ θ Δ ϕ 2 π / k w 0
1 2 Δ f ( v cos Θ ) / w 0 .
ψ s , Ω ( r ) = ρ G k s ( r r 0 ) d v 0
G k ( r r 0 ) = exp - i k r - r 0 r - r 0
ψ i ( x 0 , y 0 , z 0 ) ψ 0 exp - [ 2 ln 2 w 0 2 [ ( - x 0 cos θ 0 + y 0 sin θ 0 ) 2 + z 0 2 ] + i k ( y 0 cos θ 0 + x 0 sin θ 0 ) ]
S = S 0 exp - i K x 0
ρ ψ i S .
= ψ 0 S 0 R ( exp - i ( ω + Ω ) R / c ) ( π W 0 / β ) Erf [ 1 2 β 1 / 2 H ] × exp - [ ( cos θ sin ϕ ) 2 / 4 β + η 2 / 4 β cos 2 θ 0 ] × sin 1 2 ( ξ + η tan θ 0 ) W 0 / 1 2 ( ξ + η tan θ 0 ) W 0
η = - [ k s sin θ + k sin θ 0 - K ] ξ = [ k s cos θ cos ϕ - k cos θ 0 ] β = ( 2 ln 2 ) / w 0 2 .
P s = R 2 0 2 π d ϕ 0 π / 2 cos θ d θ | Ω ψ s , Ω exp i Ω t | 2 .
P s [ S 0 2 W 0 H ] [ ψ 0 2 / β ] · [ ( Erf [ 1 2 β 1 / 2 H ] ) 2 / 1 2 β 1 / 2 H ] Ɨ
Ɨ = π - 1 - + d x e - x 2 ( sin a x ) 2 a x 2 + π - 1 - + d x e - x 2 sin 2 a ( x + b ) a ( x + b ) 2 + 2 π - 1 e - b 2 - + d x e - x 2 sin a x sin a ( x + b ) a x ( x + b ) × cos 2 π f ( t - R / c ) a = ( 2 β ) 1 / 2 W 0 sin θ 0 b = π f / v ( 2 β ) 1 / 2 cos θ 0 .
1 2 β 1 / 2 H = ( 1 2 ln 2 ) 1 / 2 H / w 0 = 0.99
a - 1 0 a Erf ( a ) d a
a = ( ln 2 ) 1 / 2 K W 0 / k w 0 1
2 sin a b a b ( exp - b 2 ) cos 2 π f ( t - R / c )
Δ f ( 1.8 n v 2 cos θ 0 ) λ 0 f 0 W 0
η = 1 2 π 2 ( n 6 p 2 ρ v 3 ) ( λ 0 2 H cos 2 θ 0 ) - 1 W 0 P a
2 f 0 Δ f · η 1.8 π 2 ( n 7 p 2 ρ v ) ( λ 0 3 H cos θ 0 ) - 1 P a
2 f 0 Δ f ( Δ f ) 2 } · η 6 × 10 17 P a ( watts ) .
η 10 P a ( watts )
S ( x , y , t ) = S c ( x , y ) cos ( Ω t - K x ) + S s ( x , y ) sin ( Ω t - K x ) = Re S ¯ * ( x , y ) exp i ( Ω t - K x )
S ¯ = S c + i S s
S ¯ * ( x , y ) exp - i K x = 1 4 π A d A [ n ( S ¯ * exp - i K x ) e - i K r r - ( S ¯ * exp - i K x ) n e - i K r r ]
r = [ ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 ] 1 / 2 .
S ¯ * ( x , y ) = e i K x 4 π - + d y - + d y S ¯ * ( 0 , y ) · [ i K e i K r r - x r r e - i K r r ] r = [ x 2 + ( y - y ) 2 + ( z - z ) 2 ] 1 / 2 .
V 1 ( θ 0 ) x = V 0 [ - 1 2 i exp ( i K y sin θ 0 - sin Θ cos θ 0 ) ] × - + d y ξ ( x , y ) * · exp - ( i K y sin θ 0 - sin Θ cos θ 0 ) .
ξ ( x , y ) = k 0 Δ n ( x , y ) cos θ 0
Δ n = 1 2 n 3 p ( S c + i S s ) = 1 2 n 3 p S ¯
sin Θ = 1 2 K n k 0 .
V 1 ( θ 0 ) x = 1 2 k 0 n 3 p V 0 cos θ 0 · [ - 1 2 i exp ( i K y sin θ 0 - sin Θ cos θ 0 ) ] × - + d y S ¯ * ( x , y ) · exp - ( i K y sin θ 0 - sin Θ cos θ 0 ) .
V 1 ( θ 0 ) x = V 1 ( θ 0 ) 0 e i K x 4 π - + d z - + d y · exp - ( i K y sin θ 0 - sin Θ cos θ 0 ) × [ i K e - i K ρ ρ - x ρ ρ e - i K ρ ρ ] ρ = [ x 2 + y 2 + ( z - z ) 2 ] 1 / 2 .
e - i ( K x x + K y y ) 1 4 π - + d z - + d y [ exp - i ( K x x + K y y ) ] x = 0 × [ i K x e - i K ρ ρ - x ρ ρ e - i K ρ ρ ] K x 2 + K y 2 = K 2 ρ = [ x 2 + ( y - y ) 2 + ( z - z ) ] 1 / 2 .
V 1 ( θ 0 ) x = V 1 ( θ 0 ) 0 · exp i K x [ 1 - [ 1 - ( sin θ 0 - sin Θ cos θ 0 ) 2 ] 1 / 2 ] .
V 1 ( θ 0 ) x / V 0 2 = 1 16 k 0 2 p 2 n 6 cos 2 θ 0 | - + d y S ¯ ( x , y ) * · exp - i K ( sin θ 0 - sin Θ cos θ 0 ) y |
η = V 1 / V 0 2 = 1 16 k 0 2 p 2 n 6 W 0 6 cos 2 θ 0 [ sin     1 2 K W 0 Δ θ 1 2 K W 0 Δ θ ] 2 S ¯ 2
1 2 K W 0 Δ θ 1 / 2 ± 0.45 π
2 Δ θ 1 / 2 1.8 π / K W 0
δ θ π Δ f / n k 0 v cos θ 0
Δ f 1.8 n k 0 v 2 cos θ 0 / 2 π f 0 W 0 .
η = ( 1 4 n 3 k 0 p W 0 / cos θ 0 ) 2 S ¯ 2 .
P a = 1 2 ρ v 3 S ¯ 2 W 0 H
η = 1 16 k 0 2 p 2 n 6 cos 2 θ 0 | l = 0 2 m - 1 S ¯ * exp i l ϕ - ( m - l ) L - ( m - l ) L + D exp - i k y d y | 2
κ K [ sin θ 0 - 1 2 K / k 0 n ] / cos θ 0 .
η = 1 16 ( k 0 2 p 2 n 6 S ¯ 2 cos 2 θ 0 ) ( sin 1 2 κ D 1 2 κ ) · ( sin ( κ L - ϕ ) m sin 1 2 ( κ L - ϕ ) ) 2 .
κ L = ϕ ± 0.45 π / m .
f = f 0 ± f 0 [ 1 - ( κ L cos θ 0 / π ) λ 0 / L n sin 2 θ 0 ] 1 / 2
f 0 = ( v n / λ 0 ) sin θ 0
( κ L ) max = ( π L n λ 0 ) sin 2 θ 0 / cos θ 0 .
L = v 2 ( n / λ 0 f 0 2 ) ( 1 + 0.45 / m ) cos θ 0
f 1 / 2 = f 0 ± f 0 ( 0.9 / m 1 + 0.45 / m ) 1 / 2
Δ f = 2 f 0 ( 0.9 / m 1 + 0.45 / m ) 1 / 2
= v ( 3.6 n cos θ 0 / λ 0 L m ) 1 / 2 .
P a = 1 2 ρ v 3 S ¯ 2 2 m D H
( Δ f ) 2 η = 14.4 ( n 7 p 2 / ρ v ) ( λ 0 3 H cos θ 0 ) - 1 P a · ( L / D ) sin 2 ( 1 2 π D / L ) .
( Δ f ) 2 η = 16.4 ( n 7 p 2 / ρ v ) ( λ 0 3 H cos θ 0 ) - 1 P a .

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