Abstract

The effect of thermal blooming in the air is formulated within the framework of the radiative transfer equation. This is accomplished on the basis of the ray optics approach and the notion of an effective time-dependent absorption coefficient that describes the changes in the phase of the electric field caused by thermal blooming. In this approach, the beam spreading effects are entirely due to scattering from the aerosol cloud; the effect of vaporization of aerosols leads to a nonlinear coupling of the aerosols to radiation. The general setup is used to model numerically the propagation of a high energy laser beam through a cloud of aerosols distributed in the atmosphere. The limitations of the approach are outlined.

© 1988 Optical Society of America

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References

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  1. R. L. Armstrong, “Aerosol Heating and Vaporization by Pulsed Laser Beams,” Appl. Opt. 23, 148 (1984).
    [CrossRef] [PubMed]
  2. R. L. Armstrong, S. A. W. Gerstl, A. Zardecki, “Nonlinear Pulse Propagation in the Presence of Evaporating Aerosols,” J. Opt. Soc. Am. A 2, 1739 (1985).
    [CrossRef]
  3. S. M. Chitanvis, S. A. W. Gerstl, “Aerosol Clearing Model for a High Energy Laser Beam Propagating Through Vaporizing Media,” J. Appl. Phys. 62, 3091 (1987).
    [CrossRef]
  4. S. C. Davies, J. R. Brock, “Laser Evaporation of Droplets,” Appl. Opt. 26, 786 (1987).
    [CrossRef] [PubMed]
  5. R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Involuted Droplets,” Phys. Fluids 29, 3573 (1986).
    [CrossRef]
  6. J. L. Walsh, P. B. Ulrich, “Thermal Blooming in the Atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohben, Ed. (Springer-Verlag, Berlin, 1978), p. 223.
    [CrossRef]
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  8. R. L. Fante, “Relationship Between Radiative-Transport Theory and Maxwell’s Equations in Dielectric Media,” J. Opt. Soc. Am. 71, 460 (1981).
    [CrossRef]

1987

S. M. Chitanvis, S. A. W. Gerstl, “Aerosol Clearing Model for a High Energy Laser Beam Propagating Through Vaporizing Media,” J. Appl. Phys. 62, 3091 (1987).
[CrossRef]

S. C. Davies, J. R. Brock, “Laser Evaporation of Droplets,” Appl. Opt. 26, 786 (1987).
[CrossRef] [PubMed]

1986

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Involuted Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

1985

1984

1981

Armstrong, R. L.

Brock, J. R.

Chitanvis, S. M.

S. M. Chitanvis, S. A. W. Gerstl, “Aerosol Clearing Model for a High Energy Laser Beam Propagating Through Vaporizing Media,” J. Appl. Phys. 62, 3091 (1987).
[CrossRef]

Davies, S. C.

Fante, R. L.

Gerstl, S. A. W.

S. M. Chitanvis, S. A. W. Gerstl, “Aerosol Clearing Model for a High Energy Laser Beam Propagating Through Vaporizing Media,” J. Appl. Phys. 62, 3091 (1987).
[CrossRef]

R. L. Armstrong, S. A. W. Gerstl, A. Zardecki, “Nonlinear Pulse Propagation in the Presence of Evaporating Aerosols,” J. Opt. Soc. Am. A 2, 1739 (1985).
[CrossRef]

Ishimaru, A.

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

O’Rourke, P. J.

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Involuted Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

Ulrich, P. B.

J. L. Walsh, P. B. Ulrich, “Thermal Blooming in the Atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohben, Ed. (Springer-Verlag, Berlin, 1978), p. 223.
[CrossRef]

Walsh, J. L.

J. L. Walsh, P. B. Ulrich, “Thermal Blooming in the Atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohben, Ed. (Springer-Verlag, Berlin, 1978), p. 223.
[CrossRef]

Zardecki, A.

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Involuted Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

R. L. Armstrong, S. A. W. Gerstl, A. Zardecki, “Nonlinear Pulse Propagation in the Presence of Evaporating Aerosols,” J. Opt. Soc. Am. A 2, 1739 (1985).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

S. M. Chitanvis, S. A. W. Gerstl, “Aerosol Clearing Model for a High Energy Laser Beam Propagating Through Vaporizing Media,” J. Appl. Phys. 62, 3091 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Fluids

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Involuted Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

Other

J. L. Walsh, P. B. Ulrich, “Thermal Blooming in the Atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohben, Ed. (Springer-Verlag, Berlin, 1978), p. 223.
[CrossRef]

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

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Figures (10)

Fig. 1
Fig. 1

Functional shape of the laser pulse at the input plane, z = 0. The independent variables are time t and radial distance r from the beam axis; the latter is denoted ρ in the text. Peak power Fmax = 103 W/cm2.

Fig. 2
Fig. 2

Functional shape of the pulse shown in Fig. 1 at a distance z = 200 m.

Fig. 3
Fig. 3

Functional shape of the pulse shown in Fig. 1 at a distance z = 400 m.

Fig. 4
Fig. 4

Functional shape of the pulse at z = 0. Fmax = 5 × 103 W/cm2.

Fig. 5
Fig. 5

Functional shape of the pulse shown in Fig. 4 at a distance z = 200 m.

Fig. 6
Fig. 6

Functional shape of the pulse shown in Fig. 4 at a distance z = 400 m.

Fig. 7
Fig. 7

Space–time dependence of beam irradiance on-axis for Fmax = 103 W/cm2. Multiple scattering in the absence of blooming.

Fig. 8
Fig. 8

Same as Fig. 7, but blooming is present. Notice that blooming has little affect on pulse propagation.

Fig. 9
Fig. 9

Space–time dependence of beam irradiance on-axis for Fmax = 104 W/cm2. Multiple scattering in the absence of blooming.

Fig. 10
Fig. 10

Same as Fig. 9, but blooming is present. Notice the lack of punch-through compared with Fig. 9.

Equations (45)

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ρ t ( C T L + 1 2 v 2 ) + ρ · [ ( H + 1 2 v 2 ) v ] + · ( - κ T L ) = W ,
ρ C d d t ( 4 π a 3 T D 3 ) + 4 π a 2 [ ρ v H - ρ C T L ( d a d t ) r = a - ( K T r ) r = a + ρ 3 v 3 2 ρ 2 ] = π a 2 Q a F .
H = L + C ( T L - T 0 ) + C g ( T g - T 0 ) ,
T D = V T L d V V d V ,
4 π a 3 3 ρ C d T d t + 4 π a 2 m [ L + C ( T - T 0 ) ] - 4 π a 2 K ( T r ) r = a + 4 π a 2 m 3 2 ρ 2 = π a 2 Q a F .
m = D a ln [ 1 - Y 0 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 ( T - T 0 ) + C p ( T - T 0 ) exp ( m C p a / K ) - 1 + m 2 2 ρ 2 ] ,
a t = - m ρ .
[ 2 + k 0 2 ( r , t ) ] E ( r , t ) = 0 ,
E ( r , t ) = A ( r , t ) exp [ i k 0 S ( r , t ) ] .
S ( r , t ) · A ( r , t ) + ½ [ 2 S ( r , t ) + k 0 2 ( r , t ) ] A ( r , t ) = 0 ,
[ S ( r , t ) ] 2 = 1 ( r , t ) .
S ( r , t ) = S 0 ( r , t ) + S 1 ( r , t ) ,
= 0 + δ ( r , t ) ,
S 0 ( r , t ) = z ^ 1 0 .
1 0 A ( r , t ) z + 1 2 [ 2 S 1 ( r , t ) + k 0 2 ( r , t ) ] A ( r , t ) = 0 ,
S 1 ( r , t ) z = δ 1 ( r , t ) 2 1 0 .
Γ ( r , r , t ) = E ( r , t ) E * ( r , t ) ,
I ( ρ c , z , ϕ , t ) = c 4 π Γ ( ρ c , k ρ d , z , t ) exp ( - i k ϕ · ρ d ) d 2 ( k ρ d ) .
I ( ρ c , z , ϕ , t ) z + 1 1 0 [ 1 4 ρ c 2 S 1 ( ρ c , z , t ) + 2 z 2 S 1 ( ρ c , z , t ) + k 0 2 ] I ( ρ c , z , ϕ , t ) = 0.
σ air = 1 1 0 [ 1 4 ρ c 2 S 1 ( ρ c , z , t ) + 2 z 2 S 1 ( ρ c , z , t + k 0 2 ( ρ c , z , t ) ]
( z + ϕ · ρ + σ t ) I ( ρ , z , ϕ , τ ) = σ s p ( ϕ - ϕ ) I ( ρ , z , ϕ , τ ) d 2 ϕ .
σ t = σ s + σ a + σ air ,
p ( ϕ ) = ( α 2 / π ) exp ( - a 2 ϕ 2 ) ,
I ( ρ , z = 0 , ϕ , τ ) = S ( ρ , ϕ ) T ( τ ) .
F ( ρ , z , τ ) = I ( ρ , z , ϕ , τ ) d 2 ϕ
F ( ρ , z + Δ , τ ) = 1 ( 2 π ) 2 exp [ - σ t ( z , τ ) Δ ] × G ( ρ - ρ , z , τ ) F ( ρ , z , τ ) d 2 ρ .
G ( ρ , z , τ ) = 1 π [ K - 1 exp ( - ρ 2 / K ) + σ s Δ L - 1 exp ( - ρ 2 / L ) ] ;
K ( z ) = 2 z Δ / β 2 ,
L ( z ) = 2 z Δ / β 2 + Δ 2 / α 2 .
ρ t = - · ( ρ v ) ,
ρ [ v t + ( v · ) v ] = - p ,
ρ C v [ T t + ( v · ) T ] = - p · v + σ air F ,
ρ t - ρ 0 · v ,
ρ 0 v t - p 0 = 0 ,
ρ 0 C v T t - p 0 ( · v ) + σ air F .
1 ρ 0 ρ t - γ - 1 γ p 0 σ air F .
t 0 t d ρ = δ ρ ,
δ ρ ρ 0 = - γ - 1 γ p 0 σ air t 0 t F ( r , t ) d t .
( n - 1 ) = K ρ ,
δ n n 0 - 1 = δ ρ ρ 0 .
δ n ( r , t ) = - ( n 0 - 1 ) γ - 1 γ p 0 σ air t 0 t F ( r , t ) d t .
δ 1 ( r , t ) = - 2 n 0 ( n 0 - 1 ) γ - 1 γ p 0 σ air t 0 t F ( r , t ) d t .
δ 2 ( r , t ) = - 2 k ( n 0 - 1 ) γ - 1 γ p 0 σ air t 0 t F ( r , t ) d t .

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