Abstract

In the previous paper, the diffusion equation was derived from the ordinary space-time transport equation; corresponding boundary conditions on a surface of medium discontinuity were also obtained. In this paper, the theory is applied to evaluate the reflected and transmitted wave intensities of a plane wave pulse incident upon a scattering slab and the pulse shapes are displayed for a wide range of parameters. In both cases, the pulses are found to decrease exponentially with time t in the asymptotic domain, even when the scatterers have no absorption. This forms a marked contrast to the case of semi-infinite or infinite scattering media where the intensities decrease by the factor <i>t</i><sup>3/2</sup>. An expression for average pulse width is obtained analytically for the pulses transmitted through the scattering slab and is given as a function of the mean cosine of the scattering angle, the optical thickness, and the absorption cross section. The expression obtained for pulse width, with a reasonable value of absorption cross section, seems to show good agreement with the experimental results of Bucher and Lerner concerning optical pulse propagation through thick clouds.

© 1980 Optical Society of America

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  1. A. Ishimaru and S. T. Hong, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).
  2. C. H. Liu and K. C. Yeh, "Propagation of pulsed beam waves through turbulence, cloud, rain, or fog," J. Opt. Soc. Am. 67, 1261–1266 (1977).
  3. S. T. Hong, I. Sreenivasiah, and A. Ishimaru, "Plane wave pulse propagation through random media," IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).
  4. E. A. Bucher, "Computer simulation of light pulse propagation for communication through thick clouds," Appl. Opt. 12, 2391–2400 (1973).
  5. E. A. Bucher and R. M. Lerner, "Experiments on light pulse communication and propagation through atmospheric clouds," Appl. Opt. 12, 2401–2414 (1973).
  6. A. Ishimaru, "Diffusion of a pulse in densely distributed scatterers," J. Opt. Soc. Am. 68, 1045–1050 (1978).
  7. K. Furutsu, "On the diffusion equation derived from the space-time transport equation," J. Opt. Soc. Am. 70, 360–366 (1980).
  8. When asymmetrical scatterers are likely to be oriented in a particular direction in space, the scattering cross section becomes anisotropic. The diffusion equation can be derived also for this case, and the details will be treated elsewhere [K. Furutsu, J. Math. Phys. (to be published)].
  9. A different expression for pulse time spread was proposed by Stotts who also asserted a good agreement with Bucher and Lerner's experimental results: L. B. Stotts, "Closed form expression for optical pulse broadening in multiple-scattering media," Appl. Opt. 17, 504–505 (1978). But, the theory was developed by using ray tracing methods in a statistical sense and was essentially based on the forward scattering approximation. Therefore the theory should not be valid for large time spread in the range of at least 〈Δt2½t0 (refer to Fig. 10).
  10. R. E. Danielson, D. R. Moore, and H. C. van de Hulst, "The transfer of visible radiation through clouds," J. Atmos. Sci. 26, 1078–1087 (1969).
  11. The albedo of clouds is defined by the ratio of the reflected flux to the incident flux and therefore is different from ω0.
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.

1980

1978

1977

C. H. Liu and K. C. Yeh, "Propagation of pulsed beam waves through turbulence, cloud, rain, or fog," J. Opt. Soc. Am. 67, 1261–1266 (1977).

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, "Plane wave pulse propagation through random media," IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).

1975

A. Ishimaru and S. T. Hong, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

1973

1969

R. E. Danielson, D. R. Moore, and H. C. van de Hulst, "The transfer of visible radiation through clouds," J. Atmos. Sci. 26, 1078–1087 (1969).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.

Bucher, E. A.

Danielson, R. E.

R. E. Danielson, D. R. Moore, and H. C. van de Hulst, "The transfer of visible radiation through clouds," J. Atmos. Sci. 26, 1078–1087 (1969).

Furutsu, K.

Hong, S. T.

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, "Plane wave pulse propagation through random media," IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).

A. Ishimaru and S. T. Hong, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

Ishimaru, A.

A. Ishimaru, "Diffusion of a pulse in densely distributed scatterers," J. Opt. Soc. Am. 68, 1045–1050 (1978).

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, "Plane wave pulse propagation through random media," IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).

A. Ishimaru and S. T. Hong, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

Lerner, R. M.

Liu, C. H.

Moore, D. R.

R. E. Danielson, D. R. Moore, and H. C. van de Hulst, "The transfer of visible radiation through clouds," J. Atmos. Sci. 26, 1078–1087 (1969).

Sreenivasiah, I.

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, "Plane wave pulse propagation through random media," IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.

Stotts, L. B.

van de Hulst, H. C.

R. E. Danielson, D. R. Moore, and H. C. van de Hulst, "The transfer of visible radiation through clouds," J. Atmos. Sci. 26, 1078–1087 (1969).

Yeh, K. C.

Appl. Opt.

IEEE Trans. Antennas Propag.

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, "Plane wave pulse propagation through random media," IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).

J. Atmos. Sci.

R. E. Danielson, D. R. Moore, and H. C. van de Hulst, "The transfer of visible radiation through clouds," J. Atmos. Sci. 26, 1078–1087 (1969).

J. Opt. Soc. Am.

Radio Sci.

A. Ishimaru and S. T. Hong, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

Other

When asymmetrical scatterers are likely to be oriented in a particular direction in space, the scattering cross section becomes anisotropic. The diffusion equation can be derived also for this case, and the details will be treated elsewhere [K. Furutsu, J. Math. Phys. (to be published)].

The albedo of clouds is defined by the ratio of the reflected flux to the incident flux and therefore is different from ω0.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.

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