Abstract

An analytical model for the beam spread function with time dispersion is modeled and validated against Monte Carlo calculations. The model, which is structured on simple statistical concepts and relies on only first and second moments for displacement, angle, and multipath time, is suitable for describing pulsed laser radiation propagation in nonconservative scattering media out to tens of scattering lengths. Numerical examples for marine environments are used to show its robustness and versatility.

© 1998 Optical Society of America

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References

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  1. D. Arnush, “Underwater light-beam propagation in the small-angle approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
    [CrossRef]
  2. L. S. Dolin, “Solution to the radiation transfer equation in a small-angle approximation for a stratified turbid medium with photon path dispersion taken into account,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 16, 34–39 (1980).
  3. A. Ishimaru, “Pulse propagation, scattering, and diffusion in scatterers and turbulence,” Radio Sci. 14, 269–276 (1979).
    [CrossRef]
  4. A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 21, 63–70 (1981).
  5. D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 275–281 (1987).
  6. H. C. van de Hulst, G. Kattawar, “Exact spread function for pulsed collimated beam in a medium with small-angle scattering,” Appl. Opt. 33, 5820–5829 (1994).
    [CrossRef] [PubMed]
  7. J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
    [CrossRef]
  8. A. G. Luchinin, “Some properties of backscattering signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 725–729 (1987).
  9. I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and ocean sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
    [CrossRef]
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  11. R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).
  12. R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
    [CrossRef] [PubMed]
  13. L. B. Stotts, “Closed form expression for optical pulse broadening in multiple-scattering media,” Appl. Opt. 17, 504–505 (1978).
    [CrossRef] [PubMed]
  14. J. Tessendorf, “Radiative transfer as a sum over paths,” Phys. Rev. A 35(2), 872–878 (1987).
    [CrossRef] [PubMed]
  15. R. Deutsch, Nonlinear Transformations of Random Processes (Prentice-Hall, Englewood Cliffs, N.J., 1962).
  16. C. D. Mobley, Light and Water (Academic, New York, 1994).
  17. T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).
  18. R. M. Lerner, J. D. Summers, “Monte Carlo description of time- and space-resolved multiple forward scatter in natural water,” Appl. Opt. 21, 861–869 (1982).
    [CrossRef] [PubMed]

1997 (1)

1995 (1)

1994 (1)

1987 (3)

D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 275–281 (1987).

A. G. Luchinin, “Some properties of backscattering signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 725–729 (1987).

J. Tessendorf, “Radiative transfer as a sum over paths,” Phys. Rev. A 35(2), 872–878 (1987).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 21, 63–70 (1981).

1980 (1)

L. S. Dolin, “Solution to the radiation transfer equation in a small-angle approximation for a stratified turbid medium with photon path dispersion taken into account,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 16, 34–39 (1980).

1979 (1)

A. Ishimaru, “Pulse propagation, scattering, and diffusion in scatterers and turbulence,” Radio Sci. 14, 269–276 (1979).
[CrossRef]

1978 (1)

1972 (2)

D. Arnush, “Underwater light-beam propagation in the small-angle approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
[CrossRef]

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
[CrossRef]

Arnush, D.

Ciervo, A. P.

Deutsch, R.

R. Deutsch, Nonlinear Transformations of Random Processes (Prentice-Hall, Englewood Cliffs, N.J., 1962).

Dolin, L. S.

L. S. Dolin, “Solution to the radiation transfer equation in a small-angle approximation for a stratified turbid medium with photon path dispersion taken into account,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 16, 34–39 (1980).

Hall, G. J.

Ishimaru, A.

A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 21, 63–70 (1981).

A. Ishimaru, “Pulse propagation, scattering, and diffusion in scatterers and turbulence,” Radio Sci. 14, 269–276 (1979).
[CrossRef]

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

Katsev, I. L.

Kattawar, G.

Lerner, R. M.

Luchinin, A. G.

A. G. Luchinin, “Some properties of backscattering signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 725–729 (1987).

Lutomirski, R. F.

Mobley, C. D.

C. D. Mobley, Light and Water (Academic, New York, 1994).

Petzold, T. J.

T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

Polonsky, I. N.

Prikhach, A. S.

Rogozkin, D. B.

D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 275–281 (1987).

Shipley, S. T.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
[CrossRef]

Stotts, L. B.

Summers, J. D.

Tessendorf, J.

J. Tessendorf, “Radiative transfer as a sum over paths,” Phys. Rev. A 35(2), 872–878 (1987).
[CrossRef] [PubMed]

van de Hulst, H. C.

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).

Weinman, J. A.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
[CrossRef]

Zege, E. P.

Appl. Opt. (4)

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (3)

D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 275–281 (1987).

L. S. Dolin, “Solution to the radiation transfer equation in a small-angle approximation for a stratified turbid medium with photon path dispersion taken into account,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 16, 34–39 (1980).

A. G. Luchinin, “Some properties of backscattering signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 23, 725–729 (1987).

J. Geophys. Res. (1)

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 21, 63–70 (1981).

Phys. Rev. A (1)

J. Tessendorf, “Radiative transfer as a sum over paths,” Phys. Rev. A 35(2), 872–878 (1987).
[CrossRef] [PubMed]

Radio Sci. (1)

A. Ishimaru, “Pulse propagation, scattering, and diffusion in scatterers and turbulence,” Radio Sci. 14, 269–276 (1979).
[CrossRef]

Other (5)

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

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).

R. Deutsch, Nonlinear Transformations of Random Processes (Prentice-Hall, Englewood Cliffs, N.J., 1962).

C. D. Mobley, Light and Water (Academic, New York, 1994).

T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

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Figures (16)

Fig. 1
Fig. 1

Integrated radiance as a function of depth for clear ocean, coastal ocean, and turbid harbor waters. Symbols are calculated values. Monte Carlo results are connected by solid lines and beam spread function model results are connected by dashed lines. Downwelling attenuation coefficients are given in the text.

Fig. 2
Fig. 2

Mean multipath time versus number of scattering lengths for the three water types (legend as in Fig. 1). Conservative scattering predictions from small-angle approximation and Lutomirski et al. are displayed as solid and dashed curves, respectively.

Fig. 3
Fig. 3

Mean-square radial displacement versus number of scattering lengths for the three water types (legend as in Fig. 1). Conservative scattering predictions from small-angle approximation and Lutomirski et al. are displayed as solid and dashed curves, respectively.

Fig. 4
Fig. 4

Mean-square angular deflection versus number of scattering lengths for the three water types (legend as in Fig. 1). Conservative scattering predictions from small-angle approximation and Lutomirski et al. are displayed as solid and dashed curves, respectively.

Fig. 5
Fig. 5

Mean-square multipath time versus number of scattering lengths for the three water types (legend as in Fig. 1). Conservative scattering predictions from small-angle approximation and Lutomirski et al. are displayed as solid and dashed curves, respectively.

Fig. 6
Fig. 6

Temporal density function for coastal ocean water (depth = 15 m). Monte Carlo and beam spread function model calculations are presented as symbols and solid curve, respectively.

Fig. 7
Fig. 7

Temporal density function for coastal ocean water (depth = 60 m). Monte Carlo and beam spread function model calculations are presented as symbols and solid curve, respectively.

Fig. 8
Fig. 8

Radial density function for coastal ocean water (depth = 15 m). Monte Carlo and beam spread function model calculations are presented as symbols and solid curve, respectively.

Fig. 9
Fig. 9

Radial density function for coastal ocean water (depth = 60 m). Monte Carlo and beam spread function model calculations are presented as symbols and solid curve, respectively.

Fig. 10
Fig. 10

Temporal density function for turbid harbor water (depth = 6 m). Monte Carlo and beam spread function model calculations are presented as symbols and solid curve, respectively.

Fig. 11
Fig. 11

Radial density function for turbid harbor water (depth = 6 m). Monte Carlo and beam spread function model calculations are presented as symbols and solid curve, respectively.

Fig. 12
Fig. 12

Joint density function, log10[P(x, s x )] (coastal ocean water, depth = 30 m). Beam spread function and Monte Carlo results are provided in the upper and lower figures, respectively. Both show a ridgeline along ρ/z = (2/3)s.

Fig. 13
Fig. 13

Joint density function, log10[P(ρ, τ)] (coastal ocean water, depth = 30 m). Beam spread function and Monte Carlo results are provided in the upper and lower figures, respectively. The Monte Carlo calculations predict no photons in the region τ ≤ ρ2/(2zc), a restriction that is not rigorously satisfied by the beam spread function model.

Fig. 14
Fig. 14

Radial displacement and multipath time correlation coefficients as a function of number of scattering lengths for the three water types (legend as in Fig. 1). Dashed curve is van de Hulst and Kattawar predictions for conservative scattering.

Fig. 15
Fig. 15

Displacement and directionality correlation coefficient versus number of scattering lengths for three water types (legend as in Fig. 1). Beam spread function model calculations (short dashed curve) are same for all water types and is also that predicted by van de Hulst and Kattawar for conservative scattering.

Fig. 16
Fig. 16

Scattering phase function analytical model (solid curve) compared with values recommended by Mobley (symbols).

Tables (2)

Tables Icon

Table 1 Conservative Scattering Means and Variances in the Small-Angle Approximation

Tables Icon

Table 2 Water Optical Propertiesa

Equations (29)

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k z ,   ρ ,   s ,   τ = 1 2 π 5         d κ d q d ω K z ,   κ ,   q ,   ω × exp - i κ · ρ + q · s + ω τ , K z ,   κ ,   q ,   ω =       d ρ d s d τ k z ,   ρ ,   s ,   τ × exp i κ · ρ + q · s + ω τ .
k z ,   ρ ,   s ,   τ = exp - act g z ,   τ h z ,   ρ ,   s ,   τ , = exp - az exp - ac τ g z ,   τ × h z ,   ρ ,   s ,   τ .
g z ,   τ = μ σ 2 Γ μ 2 / σ 2 μ τ σ 2 μ 2 / σ 2 - 1 exp - μ τ σ 2 ,
μ τ z / c = 1 - 1 - exp - bzv bzv 1 4   bz Θ 2 - 1 24 bz 2 Θ 2 2 +   ,
σ τ 2 z / c 2 = 2 3 w 2 - 3 wv exp - bzv - 1 + bzv + 2 v 2 exp - bzw - 1 + bzw b 2 z 2 wv 2 w - v - 1 - exp - bzv bzv 2 1 12   bz Θ 4 + 1 24 bz 2 Θ 2 2 +   ,
h z ,   ρ ,   s ,   τ = 3 4 π 2 c τ 2 exp - s 2 - 3 s · ρ / z + 3 ρ 2 / z 2 τ c / z
  d s h z ,   ρ ,   s ,   τ = 3 4 π τ cz exp - 3 4 ρ 2 τ cz
  d ρ h z ,   ρ ,   s ,   τ = 1 4 π τ c / z exp - 1 4 s 2 τ c / z
σ ρ 2 z 2 =       d τ d ρ d s ρ / z 2 g z ,   τ h z ,   ρ ,   s ,   τ = 4 3 μ τ z / c 1 3   bz Θ 2 ,
σ s 2 =       d τ d ρ d ss 2 g z ,   τ h z ,   ρ ,   s ,   τ = 4   μ τ z / c bz Θ 2 .
      k z ,   ρ ,   s ,   τ d τ d ρ d s = exp - az 1 + ac σ τ 2 / μ τ μ τ 2 / σ τ 2 .
M mnk =       ρ / z m s n τ c / z k k z ,   ρ ,   s ,   τ d ρ d s d τ       k z ,   ρ ,   s ,   τ d ρ d s d τ
M mnk = M mnk 0 1 + ac σ τ 2 / μ τ 1 / 2 m + n + k ,
k z ,   ρ ,   s ,   τ = exp - bz k unsc z ,   ρ ,   s ,   τ + 1 - exp - bz k sc z ,   ρ ,   s ,   τ .
k unsc z ,   ρ ,   s ,   τ = δ τ δ ρ δ s exp - az ,
k z ,   ρ ,   s ,   τ = δ ρ δ s δ τ exp - a + b z + 1 - exp - bz exp - a z + c τ × g z ,   τ h z ,   ρ ,   s ,   τ ,
P z ,   τ =       k z ,   ρ ,   s ,   τ d ρ d s       k z ,   ρ ,   s ,   τ d τ d ρ d s ,
P z ,   ρ =       k z ,   ρ ,   s ,   τ d τ d s       k z ,   ρ ,   s ,   τ d τ d ρ d s .
P z ,   ρ ,   τ =       k z ,   ρ ,   s ,   τ d s       k z ,   ρ ,   s ,   τ d τ d ρ d s ,
P z ,   x ,   s x =       k z ,   ρ ,   s ,   τ d τ d s y d y       k z ,   ρ ,   s ,   τ d τ d ρ d s
r ρ 2 τ = ρ 2 τ - ρ 2 τ ρ 4 - ρ 2 2 τ 2 - τ 2 1 / 2 ,
r ρ · s = ρ · s ρ 2 s 2 1 / 2 .
K z ,   κ ,   q = exp - bz + b   0 z P q + ( z - z κ ] 4 π d z ,
P q 4 π =   p s 4 π exp - i s · q d s =   p s 4 π   J 0 qs 2 π s d s
P q 4 π 1 - 1 4   θ 2 q 2 + 1 32   θ 4 q 4 +   .
k z ,   ρ ,   s = 3 4 π 2 z 2 ¼ bz θ 2 2 × exp - s 2 - 3 s · ρ / z + 3 ρ 2 / z 2 ¼ bz θ 2 ,
K z ,   κ ,   q exp - ¼ bz θ 2 q 2 + z q · κ + z 2 κ 2 .
p θ 4 π = A   exp - Θ / Θ 0 1 / 2 4 π Θ / Θ 0 3 / 2 Θ 0 2 ,
p θ 4 π = exp - θ / θ 0 1 / 2 4 π θ / θ 0 3 / 2 θ 0 2 .

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