Abstract

Thermal blooming of laser beams propagating through the atmosphere has generally been treated for cases of small wind speed transverse to the beam. The present calculation examines the associated density changes for air moving through a beam at near sonic speeds. The problem is treated for the one-dimensional case only; an exact solution to the nonlinear hydrodynamic equations for the steady state is derived. The solution shows that, at near sonic speeds, the density changes grow drastically with Mach value, while agreeing with the previous results at low wind speeds. In the vicinity of Mach I, the solution becomes invalid, probably due to the nonexistence of a steady state. The effect of the density changes on the beam are not calculated here.

© 1974 Optical Society of America

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References

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  1. c.f., J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 1587 (1970).
    [CrossRef]
  2. c.f., A. Aitken, J. Hayes, P. Ulrich, “Propagation of High Energy 10.6-μm Laser Beams Through the Atmosphere,” NRL Report 7293, May1971; see also Appl. Opt. 11, 257 (1972).
    [PubMed]
  3. Private communication, J. Wallace, Far Field, Inc., Sudbury, Mass.
  4. Private communication, C. B. Hogge, Kirtland AFB Albuquerque, N.M.
  5. Exact solutions to the time dependent linearized problem in one, two and three dimensions have been derived by the author and P. Ulrich, (and also by J. Ellinwood of Aerospace Corp.) and will be published. These show a linear growth in density change with time at Mach 1.

1970 (1)

Aitken, A.

c.f., A. Aitken, J. Hayes, P. Ulrich, “Propagation of High Energy 10.6-μm Laser Beams Through the Atmosphere,” NRL Report 7293, May1971; see also Appl. Opt. 11, 257 (1972).
[PubMed]

Camac, M.

Hayes, J.

c.f., A. Aitken, J. Hayes, P. Ulrich, “Propagation of High Energy 10.6-μm Laser Beams Through the Atmosphere,” NRL Report 7293, May1971; see also Appl. Opt. 11, 257 (1972).
[PubMed]

Hogge, C. B.

Private communication, C. B. Hogge, Kirtland AFB Albuquerque, N.M.

Ulrich, P.

c.f., A. Aitken, J. Hayes, P. Ulrich, “Propagation of High Energy 10.6-μm Laser Beams Through the Atmosphere,” NRL Report 7293, May1971; see also Appl. Opt. 11, 257 (1972).
[PubMed]

Wallace, J.

c.f., J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 1587 (1970).
[CrossRef]

Private communication, J. Wallace, Far Field, Inc., Sudbury, Mass.

J. Opt. Soc. Am. (1)

Other (4)

c.f., A. Aitken, J. Hayes, P. Ulrich, “Propagation of High Energy 10.6-μm Laser Beams Through the Atmosphere,” NRL Report 7293, May1971; see also Appl. Opt. 11, 257 (1972).
[PubMed]

Private communication, J. Wallace, Far Field, Inc., Sudbury, Mass.

Private communication, C. B. Hogge, Kirtland AFB Albuquerque, N.M.

Exact solutions to the time dependent linearized problem in one, two and three dimensions have been derived by the author and P. Ulrich, (and also by J. Ellinwood of Aerospace Corp.) and will be published. These show a linear growth in density change with time at Mach 1.

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Figures (2)

Fig. 1
Fig. 1

Downwind density change vs wind speed with αPa−1 = 106 ergs cm−2 sec−1 and 107 ergs cm−2 sec−1 for the lower and upper curves, respectively.

Fig. 2
Fig. 2

Density changes within the beam vs position; various Mach numbers are indicated. αPa−1 was chosen to be 107 ergs cm−2 sec−1 for this calculation.

Equations (18)

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d / ( d x ) ρ υ = 0.
ρ υ ( d υ ) / ( d x ) + ( d p ) / ( d x ) = 0.
υ [ ( d p ) / ( d x ) γ ( p / ρ ) ( d p ) / ( d x ) ] = ( γ 1 ) κ ρ I .
ρ ( x ) υ ( x ) = ρ 0 υ 0 ,
p ( x ) + ρ ( x ) υ ( x ) 2 = p 0 + ρ 0 υ 0 2 ,
c 0 2 = γ ( p 0 / ρ 0 ) ,
d d x [ ( ρ 0 ρ ) 3 3 ( c 0 2 υ 0 2 + γ ) 2 ( γ + 1 ) ( ρ 0 ρ ) 2 ] = 3 ( γ 1 ) ( γ + 1 ) · α I ρ 0 υ 0 3 .
ξ ( x ) ( ρ / ρ 0 ) 3 1 ζ ( ρ / ρ 0 ) + 1 = 0 ,
ζ = 2 ( γ + 1 ) 3 ( γ + c 0 2 υ 0 2 ) ,
ξ ( x ) = 1 ζ ζ [ 1 + Δ ( x , ζ ) ]
Δ ( x , ζ ) = ζ 1 ζ · 3 ( γ 1 ) ( γ + 1 ) · α x d x I ( x ) ρ 0 υ 0 3
( ρ / ρ 0 ) 3 1 / [ ζ ξ ( x ) ] ( ρ / ρ 0 ) + 1 / [ ξ ( x ) ] = 0.
ρ ( x , ζ ) ρ 0 = 2 ( 3 ¯ ) 1 / 2 cos ( 1 / 3 ) [ φ ( x , ζ ) + 2 π n ] { ( 1 ζ ) [ 1 + Δ ( x , ζ ) ] } 1 / 2 , n = 0,1,2.
φ ( x , ζ ) = arc cos [ { [ 3 ( 3 ) 1 / 2 ] / 2 } ζ { ( 1 ζ ) [ 1 + Δ ( x , ζ ) ] } 1 / 2 ] .
ρ / ρ 0 = { 1 , 1 2 { 1 ± [ 1 + 4 ζ / ( 1 ζ ) ] 1 / 2 } .
ρ ( x ) ρ 0 = 2 ( 3 ) 1 / 2 cos ( 1 / 3 ) φ ( x , ζ ) { ( 1 ζ ) [ 1 + Δ ( x , ζ ) ] } 1 / 2 ,
φ ( x , ζ ) = arc cos [ { [ 3 ( 3 ) 1 / 2 ] / 2 } ζ { ( 1 ζ ) [ 1 + Δ ( x , ζ ) ] } 1 / 2 ] ,
ρ ( x ) ρ 0 1 ( γ 1 ) α ρ 0 c 0 2 υ 0 x d x I ( x ) .

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