Abstract

A set of coupled, nonlinear differential equations is developed describing the propagation of a pulsed, quasi-monochromatic beam of electromagnetic radiation in a medium containing vaporizing aerosols. Spatiotemporal distortions in the beam are produced as a result of radiation-induced changes in the absorption and scattering coefficients as well as in the angular spread of the phase function. The irradiance distribution function of the pulse is governed by a time-dependent transfer equation in the small-angle approximation. Numerical solutions to the equations are obtained for the case of pulse propagation through a cloud of vaporizing water droplets. It is found that significant spatiotemporal distortion of the pulse accompanies enhanced aerosol vaporization.

© 1985 Optical Society of America

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References

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  1. F. G. Gebhardt, “Overview of atmospheric effects on the propagation of high energy laser radiation,” Proc. Soc. Photo-Opt. Instrum. Eng. 195, 162 (1979).
  2. T. P. Reilly, “High flux propagation through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 410, 2 (1983).
  3. R. L. Armstrong, “Aerosol heating and vaporization by pulsed light beams,” Appl. Opt. 23, 148 (1984).
    [CrossRef] [PubMed]
  4. R. L. Armstrong, “Propagation effects on pulsed light beams in absorbing media,” Appl. Opt. 23, 156 (1984).
    [CrossRef] [PubMed]
  5. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE. 68, 1424 (1980).
    [CrossRef]
  6. P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of absorption centers within irradiated spheres,” J. Opt. Soc. Am. 69, 55 (1979).
    [CrossRef]
  7. R. L. Armstrong, “Interactions of absorbing aerosols with intense light beams,” J. Appl. Phys. 56, 2142 (1984).
    [CrossRef]
  8. F. A. Williams, “On vaporization of mist by radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
    [CrossRef]
  9. B. J. Uscinski, The Elements of Wave Propagation in Random Media (McGraw-Hill, New York, 1977).
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  11. A. Zardecki, W. G. Tarn, “Pulse propagation in particulate medium,” Appl. Opt. 22, 3782 (1980).
    [CrossRef]
  12. A. Ishimaru, T. S. Hong, “Multiple scattering efects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637 (1975).
    [CrossRef]
  13. A. Ishimaru, “Diffusion of a pulse in densely distributed scattered,” J. Opt. Soc. Am. 68, 1045 (1978).
    [CrossRef]
  14. F. Furutsu, “Diffusion equation derived from space-time transport equation,” J. Opt. Soc. Am. 70, 360 (1980).
    [CrossRef]
  15. S. Ito, K. Furutsu, “Theory of light propagation through thick clouds,” J. Opt. Soc. Am. 70, 366 (1980).
    [CrossRef]
  16. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York (1979).
  17. W. G. Tarn, A. Zardecki, “Laser beam propagation in particulate media,” J. Opt. Soc. Am. 69, 68 (1979).
    [CrossRef]
  18. W. G. Tarn, A. Zardecki, “Multiple scattering corrections to the Beer-Lambert law, 1: open detector,” Appl. Opt. 21, 2405 (1982).
    [CrossRef]
  19. M. A. Box, A. Deepak, “Limiting cases of the small-angle approximation solutions for the propagation of laser beams in anisotropic scattering media,” J. Opt. Soc. Am. 71, 1534 (1981).
    [CrossRef]
  20. H. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957).
  21. G. Sageev, J. H. Seinfeld, Appl. Opt. 23, 4368 (1984).
    [CrossRef] [PubMed]

1984

1983

T. P. Reilly, “High flux propagation through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 410, 2 (1983).

1982

1981

1980

F. Furutsu, “Diffusion equation derived from space-time transport equation,” J. Opt. Soc. Am. 70, 360 (1980).
[CrossRef]

S. Ito, K. Furutsu, “Theory of light propagation through thick clouds,” J. Opt. Soc. Am. 70, 366 (1980).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE. 68, 1424 (1980).
[CrossRef]

A. Zardecki, W. G. Tarn, “Pulse propagation in particulate medium,” Appl. Opt. 22, 3782 (1980).
[CrossRef]

1979

1978

1975

A. Ishimaru, T. S. Hong, “Multiple scattering efects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637 (1975).
[CrossRef]

1965

F. A. Williams, “On vaporization of mist by radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
[CrossRef]

Armstrong, R. L.

Box, M. A.

Cooke, D. D.

Deepak, A.

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York (1979).

Dusel, P. W.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE. 68, 1424 (1980).
[CrossRef]

Furutsu, F.

Furutsu, K.

Gebhardt, F. G.

F. G. Gebhardt, “Overview of atmospheric effects on the propagation of high energy laser radiation,” Proc. Soc. Photo-Opt. Instrum. Eng. 195, 162 (1979).

Hong, T. S.

A. Ishimaru, T. S. Hong, “Multiple scattering efects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637 (1975).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Diffusion of a pulse in densely distributed scattered,” J. Opt. Soc. Am. 68, 1045 (1978).
[CrossRef]

A. Ishimaru, T. S. Hong, “Multiple scattering efects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637 (1975).
[CrossRef]

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

Ito, S.

Kerker, M.

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York (1979).

Reilly, T. P.

T. P. Reilly, “High flux propagation through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 410, 2 (1983).

Sageev, G.

Seinfeld, J. H.

Tarn, W. G.

Uscinski, B. J.

B. J. Uscinski, The Elements of Wave Propagation in Random Media (McGraw-Hill, New York, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957).

Williams, F. A.

F. A. Williams, “On vaporization of mist by radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
[CrossRef]

Zardecki, A.

Appl. Opt.

Int. J. Heat Mass Transfer

F. A. Williams, “On vaporization of mist by radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
[CrossRef]

J. Appl. Phys.

R. L. Armstrong, “Interactions of absorbing aerosols with intense light beams,” J. Appl. Phys. 56, 2142 (1984).
[CrossRef]

J. Opt. Soc. Am.

Proc. IEEE.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE. 68, 1424 (1980).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

F. G. Gebhardt, “Overview of atmospheric effects on the propagation of high energy laser radiation,” Proc. Soc. Photo-Opt. Instrum. Eng. 195, 162 (1979).

T. P. Reilly, “High flux propagation through the atmosphere,” Proc. Soc. Photo-Opt. Instrum. Eng. 410, 2 (1983).

Radio Sci.

A. Ishimaru, T. S. Hong, “Multiple scattering efects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637 (1975).
[CrossRef]

Other

B. J. Uscinski, The Elements of Wave Propagation in Random Media (McGraw-Hill, New York, 1977).

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

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York (1979).

H. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957).

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Figures (10)

Fig. 1
Fig. 1

Space–time dependence of axial-beam irradiance (W/cm2) for 10.6-μm, 1-msec, 1-J rectangular pulse propagating in monodisperse fog of 5-μm water droplets at concentration of 2 × 105 cm−3.

Fig. 2
Fig. 2

Space–time dependence of droplet temperature rise (K) for system of Fig. 1.

Fig. 3
Fig. 3

Space–time dependence of droplet radius (μm) for system of Fig. 1.

Fig. 4
Fig. 4

Space–time dependence of axial-beam irradiance (W/cm2) for system of Fig. 1, except beam energy is 15 J.

Fig. 5
Fig. 5

Space–time dependence of droplet temperature rise (K) for system of Fig. 1, except beam energy is 15 J.

Fig. 6
Fig. 6

Space–time dependence of droplet radius (μm) for system of Fig. 1, except beam energy is 15 J.

Fig. 7
Fig. 7

Space–time dependence of axial-beam irradiance (W/cm2) for 1.06-μm, 1-msec, 10-J rectangular pulse propagating in monodisperse fog of 2-μm water droplets at concentration of 5 × 105 cm−3.

Fig. 8
Fig. 8

Space–time dependence of axial-beam irradiance (W/cm2) for 1.06-μm, 0.3-msec, 2.8-J Gaussian pulse, spot size 1 cm, propagating in monodisperse fog of 2-μm water droplets of a concentration of 3×105 cm−3.

Fig. 9
Fig. 9

Space–time dependence of beam irradiance (W/cm2) 0.6 cm off beam axis for system of Fig. 8.

Fig. 10
Fig. 10

Space–time dependence of beam irradiance (W/cm2) 1.2 cm off beam axis for system of Fig. 8.

Equations (32)

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4 π a 3 3 ρ C T t + 4 π a 2 [ m L K ( T r ) r = a + m 3 2 ρ 2 ] = π a 2 Q a F ,
m = D a ln [ 1 Y o 1 Y 0 exp ( L M R T 0 L M R T ) ] ,
K ( T r ) r = a = m C p ( T T 0 ) [ exp ( m C p a / K ) 1 ] ,
T t = 3 Q a F 4 a ρ C 3 m a ρ C { L + C p ( T T 0 ) [ exp ( m C p a / K ) 1 ] + m 2 2 ρ 2 } ,
a t = m ρ .
( 1 c t + Ω + σ t ) I ( Ω , r , t ) = σ s p ( Ω , Ω ) I ( Ω , r , t ) d 2 Ω ,
Ω = x ̂ sin θ cos ψ + ŷ sin θ sin ψ + cos θ , Ω = x ̂ sin θ cos ψ + ŷ sin θ sin ψ + cos θ .
ϕ = x ̂ θ cos ψ + ŷ θ sin ψ , ϕ = x ̂ θ cos ψ + ŷ θ sin ψ .
τ = t z / c z = z ,
( z + ϕ ρ + σ t ) I ( ϕ , ρ , z , τ ) = σ s p ( ϕ ϕ ) I ( ϕ , ρ , z , τ ) d 2 ϕ ,
I ( ϕ , ρ , z = 0 , τ ) = S ( ϕ , ρ ) T ( τ ) ,
p ( ϕ ) = ( α 2 / π ) exp ( α 2 ϕ 2 ) ,
S ( ϕ , ρ ) = ( β 2 γ 2 / π 2 ) exp ( β 2 ϕ 2 γ 2 ρ 2 ) ,
Î ( ξ , η , z = 0 , τ ) = Ŝ ( ξ , η ) T ( τ ) ,
Ŝ ( ξ η ) = S ( ϕ ρ ) exp [ i ( ξ ϕ + η ρ ) ] d 2 ϕ d 2 ρ .
a ( ρ ) = a ( ) exp [ const . × t × exp ( γ 2 ρ 2 ) ] ,
1 ( 2 π ) 2 σ ̂ t ( η , z ) Î ( ξ , η η , z ) d 2 η .
Î ( ξ , η , z , τ ) = Ŝ ( ξ + z η , η ) T ( τ ) × exp [ Ω ( z , τ ) 0 z σ t ( z , τ ) d z ] ,
Ω ( z , τ ) = 0 z σ s ( z , τ ) P [ ξ + η ( z z ) , z , τ ] d z .
P ( ξ , z , τ ) = p ( ϕ , z , τ ) exp ( i ξ ϕ ) d 2 ϕ .
F ( ρ , z , τ ) = I ( ϕ , ρ , z , τ ) d 2 ϕ
F ( ρ , z , τ ) = 1 ( 2 π ) 2 T ( τ ) Ŝ ( z η , η ) exp [ Ω 0 ( z , τ ) 0 z σ t ( z , τ ) d z ] exp ( i η ρ ) d 2 η ,
Ω 0 ( z , τ ) = 0 z σ s ( z , τ ) P [ η ( z z ) , z , τ ] d z .
F ( ρ , z , τ ) = 1 ( 2 π ) 2 T ( τ ) F ̂ ( η , z , τ ) exp ( i η ρ ) d 2 η ,
F ̂ ( ρ , z , τ ) = exp [ 1 4 ( z 2 η 2 β 2 + η 2 γ 2 ) ] exp { 0 z σ s ( z , τ ) × exp [ η 2 ( z z ) 2 4 α 2 ( z , τ ) ] d z 0 z σ t ( z , τ ) d z } .
F ( ρ , z + Δ , τ ) = 1 ( 2 π ) 2 T ( τ ) exp [ σ t ( z , τ ) Δ ] F ̂ ( η , z , τ ) k × F 1 ( η , z , τ ) exp ( i η ρ ) d 2 η ,
F ̂ 1 ( η , z , τ ) = exp ( z Δ η 2 2 β 2 ) { 1 + σ s ( z , τ ) Δ exp [ Δ 2 η 2 4 α 2 ( z , τ ) ] } .
F ( ρ , z + Δ , τ ) = 1 ( 2 π ) 2 exp [ σ t ( z , τ ) Δ ] × F ( ρ , z , τ ) F 1 ( ρ ρ , z , τ ) d 2 ρ ,
F 1 ( ρ , z , τ ) = 1 π [ K 1 exp ( ρ 2 / K ) + L 1 exp ( ρ 2 / L ) ] .
K ( z ) = 2 z Δ / β 2
L ( z , τ ) = 2 z Δ / β 2 + Δ 2 α 2 ( z , τ ) .
F ( ρ , z + Δ , τ ) = i = 1 2 2 f i exp ( ρ 2 / f i ) × 0 exp ( ρ 2 / f i ) F ( ρ , z , τ ) × I 0 ( 2 ρ ρ / f i ) ρ d ρ ,

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