Abstract

Consideration is given to certain physical phenomena that can attenuate the intensity of a laser beam by refractively spreading or scattering the beam as it propagates. The total attenuation caused by these effects can be well above the usual geometric spreading and absorption losses. Two specific phenomena are analyzed: the nonlinear thermal blooming of the beam (thermal lens effect) and the spreading caused by random variations of the index of refraction of the medium in which the beam is propagating (turbulence effects). The intent of the investigation is to obtain reasonably simple analytic expressions that can be used to estimate or set bounds on the extent of the attenuation. Such expressions are obtained for both the transient and the steady-state effects of thermal blooming when the medium is moving relative to the laser beam, and various estimates are also obtained for the turbulence-induced attenuation of the peak intensity of a beam. For comparison, estimates are also given of the effective attenuation of the beam when it is viewed by a fixed observer, if there are random variations in its direction of propagation.

© 1972 Optical Society of America

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References

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  1. L. Landau, E. Lifshitz, Fluid Mechanics (Pergamon, London, 1959).
  2. J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 12, 1587 (1970).
    [CrossRef]
  3. A. Wood, M. Camac, E. Gerry, Appl. Opt. 10, 8, 1877 (1971).
    [CrossRef]
  4. J. Gordon et al., J. Appl. Phys. 36, 1, 3 (1965).
    [CrossRef]
  5. P. Kelley, Phys. Rev. Lett. 15, 1105 (1965); F. Gebhardt, D. Smith, Appl. Phys. Lett. 14, 52 (1969); IEEE Trans. Quant. Electron. QE-7, 63 (1971).
    [CrossRef]
  6. E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).
  7. H. Kleinman, P. Kelley, MIT Lincoln Lab, Optics Research Summary (1970).
  8. The review paper by R. Lawrence, J. Strohbehn, Proc. IEEE 58, 1523 (1970), contains an excellent bibliography on the subject.
    [CrossRef]
  9. R. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

1971 (1)

A. Wood, M. Camac, E. Gerry, Appl. Opt. 10, 8, 1877 (1971).
[CrossRef]

1970 (2)

The review paper by R. Lawrence, J. Strohbehn, Proc. IEEE 58, 1523 (1970), contains an excellent bibliography on the subject.
[CrossRef]

J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 12, 1587 (1970).
[CrossRef]

1967 (1)

R. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

1965 (2)

J. Gordon et al., J. Appl. Phys. 36, 1, 3 (1965).
[CrossRef]

P. Kelley, Phys. Rev. Lett. 15, 1105 (1965); F. Gebhardt, D. Smith, Appl. Phys. Lett. 14, 52 (1969); IEEE Trans. Quant. Electron. QE-7, 63 (1971).
[CrossRef]

Camac, M.

A. Wood, M. Camac, E. Gerry, Appl. Opt. 10, 8, 1877 (1971).
[CrossRef]

J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 12, 1587 (1970).
[CrossRef]

Emde, F.

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).

Gerry, E.

A. Wood, M. Camac, E. Gerry, Appl. Opt. 10, 8, 1877 (1971).
[CrossRef]

Gordon, J.

J. Gordon et al., J. Appl. Phys. 36, 1, 3 (1965).
[CrossRef]

Jahnke, E.

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).

Kelley, P.

P. Kelley, Phys. Rev. Lett. 15, 1105 (1965); F. Gebhardt, D. Smith, Appl. Phys. Lett. 14, 52 (1969); IEEE Trans. Quant. Electron. QE-7, 63 (1971).
[CrossRef]

H. Kleinman, P. Kelley, MIT Lincoln Lab, Optics Research Summary (1970).

Kleinman, H.

H. Kleinman, P. Kelley, MIT Lincoln Lab, Optics Research Summary (1970).

Landau, L.

L. Landau, E. Lifshitz, Fluid Mechanics (Pergamon, London, 1959).

Lawrence, R.

The review paper by R. Lawrence, J. Strohbehn, Proc. IEEE 58, 1523 (1970), contains an excellent bibliography on the subject.
[CrossRef]

Lifshitz, E.

L. Landau, E. Lifshitz, Fluid Mechanics (Pergamon, London, 1959).

Schmeltzer, R.

R. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Strohbehn, J.

The review paper by R. Lawrence, J. Strohbehn, Proc. IEEE 58, 1523 (1970), contains an excellent bibliography on the subject.
[CrossRef]

Wallace, J.

J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 12, 1587 (1970).
[CrossRef]

Wood, A.

A. Wood, M. Camac, E. Gerry, Appl. Opt. 10, 8, 1877 (1971).
[CrossRef]

Appl. Opt. (1)

A. Wood, M. Camac, E. Gerry, Appl. Opt. 10, 8, 1877 (1971).
[CrossRef]

J. Appl. Phys. (1)

J. Gordon et al., J. Appl. Phys. 36, 1, 3 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 12, 1587 (1970).
[CrossRef]

Phys. Rev. Lett. (1)

P. Kelley, Phys. Rev. Lett. 15, 1105 (1965); F. Gebhardt, D. Smith, Appl. Phys. Lett. 14, 52 (1969); IEEE Trans. Quant. Electron. QE-7, 63 (1971).
[CrossRef]

Proc. IEEE (1)

The review paper by R. Lawrence, J. Strohbehn, Proc. IEEE 58, 1523 (1970), contains an excellent bibliography on the subject.
[CrossRef]

Quart. Appl. Math. (1)

R. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Other (3)

L. Landau, E. Lifshitz, Fluid Mechanics (Pergamon, London, 1959).

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).

H. Kleinman, P. Kelley, MIT Lincoln Lab, Optics Research Summary (1970).

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Figures (3)

Fig. 1
Fig. 1

Transient attenuation of a laser beam due to thermal blooming.

Fig. 2
Fig. 2

Equal intensity contours of a thermally bloomed beam in a moving medium.

Fig. 3
Fig. 3

Attenuation due to steady-state thermal blooming effect.

Tables (1)

Tables Icon

Table I Various Attenuation Effects on a Collimated Laser Beam Having a Gaussian Profile

Equations (82)

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D T / D t ( T / t ) + V · T = ( Q / ρ C p ) + ( k / ρ C p ) 2 T ,
Q = 0.24 [ α t I ( D E / D t ) ]
D E / D t ( E / t ) + V · E = α I ( E / τ ) ,
T t + υ T x = α t 024 ρ C p { ( 1 δ ) I + δ τ 0 t exp [ ( t ζ ) τ ] × I [ x υ ( t ζ ) , y , z , ζ ] d ζ } + k ρ C p 2 T ,
T = α t 0.24 ρ C p 0 t { ( 1 δ ) I [ x υ ( t ζ ) , y , z , ζ ] + δ τ 0 t × exp ( ( t θ ) τ ) × I [ x υ ( t ζ ) υ ( t θ ) , y , z , θ ] d θ } d ζ .
T = 0.24 α t ρ C p υ x { ( 1 δ ) I ( x , y , z , ) + δ exp [ ( x x ) / υ τ ] υ τ x I ( q , y , z , ) d q } d x ,
T / t = ( α t 0.24 / ρ C p ) I + ( k / ρ C p ) 2 T .
I ( r ) = ( 2 / π ) ( W / ω 0 2 ) exp [ 2 ( r / ω 0 ) 2 ] ,
T = 0.24 α t 4 π k W 1 / ω 0 2 1 / ( ω 0 2 + 8 D t ) exp ( 2 r 2 y ) y d y ,
T ( 0.24 α t / 4 π k ) W { [ 1 / ( ω 0 2 + 8 D t ) ] ( 1 / ω 0 2 ) } { exp [ ( 2 r 2 / ω 0 2 ) ] / ( 1 / ω 0 2 ) } ( 0.24 α t / ρ C p ) I t ,
T α t 0.24 ( 1 δ ) ρ C p 0 t I [ x υ ( t ζ ) , y , z , ζ ] d ζ α t 0.24 ( 1 δ ) ρ C p 0 t [ I ( x , y , z , ζ ) + υ ( ζ t ) I x ] d ζ ,
t | I / υ ( I / x ) | .
t 1 / | υ | ( 4 r / ω 0 2 ) .
T = 0.24 ( 1 δ ) α t ρ C p 0 t I ( x , y , z , η ) d η ,
d 2 r / d z 2 = ( 1 / n ) n ,
d 2 r / d z 2 = [ ( 1 / n ) ( d n / d T ) ] T .
d 2 r d z 2 = 1 n d n d T 0.24 ( 1 δ ) α t ρ C p 0 t I ( x , y , z , n ) d n .
d 2 r d z 2 = 1 n d n d T 0.24 ( 1 δ ) α t ρ C p 0 t I ( x , y , z , n ) r d n .
I ( r , 0 , t ) Δ r ( 0 ) r ( 0 ) = I ( r , z , t ) Δ r ( z ) r ( z ) ,
d 2 r / d z 2 = a t exp [ 2 ( r / ω 0 ) 2 ] r ,
a = [ ( d n / d T ) / n ] [ 0.24 α t ( 1 δ ) / ρ C p ] ( 8 / π ) ( W / ω 0 4 ) .
I ( r , z , t ) / I ( r , 0 , t ) = 1 / cosh 2 ( a t ) 1 2 z ,
t c = ( ω 0 / z ) 2 { 1 / [ ( d n / d T ) / n ] [ 0.96 α t ( 1 δ ) / ρ C p ] I ( 0,0 ) } ,
E ( t c ) t c I ( 0,0 ) = ( ω 0 / z ) 2 { 1 / [ ( d n / d T ) / n ] [ 0.96 α t ( 1 δ ) / ρ C p ] } .
I ( r , z , t ) / I ( r , 0 , t ) = 1 / cos 2 ( | a | t ) 1 2 z ,
I ( r , z , t ) = ( 2 / π ) [ W / ω 0 2 ( z , t ) ] exp { 2 [ r / ω 0 ( z , t ) ] 2 } ,
d 2 ω 0 d z 2 = 8 π W ( d n d T / n ) 0.24 α t ( 1 δ ) ρ C p e 2 0 t d n ω 0 3 ( z , n ) ,
ω 0 ( 0 , t ) = ω 0 , ( d ω 0 / d z ) ( 0 ) = 0 ,
A = [ ( d n / d T ) / n ] [ 0.24 α t ( 1 δ ) / ρ C p e 2 ] ( 8 / π ) ( W / ω 0 4 ) ,
( d / d y ) [ y 2 ( d 2 f / d y 2 ) ] = y / f 3
I ( 0 , z , t ) / I ( 0,0 , t ) 1 / f 2 ( y ) = 1 / f 2 [ ( 2 A t ) 1 2 z ]
E = ( ω 0 / z ) 2 { 1 / [ ( d n / d T ) / n ] [ 0.052 α t ( 1 δ ) / ρ C p ] }
t ˆ c = E / ( 2 / π ) ( W / ω 0 2 ) ;
d 2 x d z 2 = 1 n d n d x = 1 n d n d T d T d x = 1 n d n d T 0.24 α t ρ C p υ [ ( 1 δ ) I + δ υ τ x I ( q , y , z , ) d q ]
d 2 y d z 2 = 1 n d n d T 0.24 α t ρ C p υ y x { ( 1 δ ) I ( x , y , z , ) + δ exp { [ ( x x ) / υ τ ] } υ τ x I d q } d x ,
d 2 x / d z 2 = C , d 2 y / d z 2 = D y ,
C = 1 n d n d T 0.24 α t ρ C p υ [ ( 1 δ ) I 0 ( 0 ) + δ υ τ 0 I 0 ( q , y , z , ) d q ] , D = 1 n d n d T 0.96 α t ρ C p υ ω 0 2 { 0 [ ( 1 δ ) I 0 ( x , y , z , ) + δ υ τ e ( x / υ τ ) x I 0 d q ] d x } ,
I 0 ( x , y , z , ) = ( 2 / π ) ( W / ω 0 2 ) exp { [ 2 ( x 2 + y 2 ) / ω 0 2 ] } .
x = 1 2 C z 2 + x ( 0 ) , y = y 0 cosh ( D ) 1 2 z ,
Δ x = Δ x ( 0 ) , Δ y = Δ y ( 0 ) cosh ( D ) 1 2 z .
I ( 0 , z , ) / I ( 0,0 , ) = 1 / cosh ( D ) 1 2 z .
z C 2 = υ / ( 1 n d n d T ) 0.96 α t ρ C p ω 0 2 { 0 [ ( 1 δ ) I 0 + δ exp ( x / υ τ ) υ τ x I 0 ( q ) d q ] d x } ,
z C 2 = ρ C p υ ( 2 π ) 1 2 ω 0 3 / [ ( d n / d T ) / n ] 0.96 α t W .
I / x ~ ( I / y ) , y > 0
I / x ~ I / y , y < 0 ,
d 2 x / d z 2 = [ ( 1 / n ) ( d n / d T ) ( 0.24 α t / ρ C p υ ) ] I
d 2 y / d z 2 = ( 1 / n ) ( d n / d T ) ( 0.24 α t / ρ C p υ ) { I , y > 0 I , y < 0 } .
d 2 Δ y / d z 2 = ( 2 / n ) ( d n / d T ) ( 0.24 α t / ρ C p υ ) I ,
I ( z ) = I ( 0 ) [ Δ x ( 0 ) Δ y ( 0 ) / Δ x Δ y ] = I ( 0 ) [ Δ y ( 0 ) / Δ y ] ,
d 2 Δ y / d z 2 = ( 2 / n ) ( d n / d T ) ( 0.24 α t / ρ C p υ ) I ( 0 ) [ Δ y ( 0 ) / Δ y ] .
[ ( d n d T / n ) I ( 0 ) 2 ω 0 0.24 α t ρ C p υ ] 1 2 z = 0 { ln [ Δ y / Δ y ( 0 ) ] } 1 2 e t 2 d t 0 { ln [ I ( 0 ) / I ( z ) ] } 1 2 e t 2 d t ,
0 x e t 2 d t e x 2 / 2 x
I ( z ) / I ( 0 ) ( 1 / 2 { ln [ I ( 0 ) / I ( z ) ] } 1 2 ) ( 1 / { [ ( d n / d T ) / n ] × [ I ( 0 ) / 2 ω 0 ] ( 0.24 α t / ρ C p υ ) } 1 2 z ) ,
D = ( 2 z f λ / D ) + [ ( z f z ) / z f ] [ D ( 2 z f λ / D ) ] ,
( d Δ y / d z ) ( 0 ) = ( D / z f ) [ 1 ( 2 z f λ / D 2 ) ] ,
z / z 0 = exp [ ( z 0 / 2 z f ) 2 ] { ( z 0 / 2 z f ) 2 + ln [ Δ y / Δ y ( 0 ) ] } 1 2 z 0 / 2 z f e t 2 d t ,
1 / z f = ( 1 / z f ) [ 1 ( 2 z f λ / D 2 ) ]
z 0 = 1 / { [ ( d n / d T ) / n ] ( I 0 / D ) ( 0.24 α t / ρ C p υ ) } 1 2 .
1 2 z f λ D 2 = ln [ I ( 0 ) / I ( z f ) ] 0 e η [ 1 + η ( 2 z f / z 0 ) 2 ] 1 2 d η ,
| ln [ I ( 0 ) / I ( z f ) ] ( 2 z f / z 0 ) 2 | 1 ,
1 2 z f λ D 2 = ln [ I ( 0 ) / I ] 0 e η d η = 1 I ( 0 ) I
I ( 0 ) / I ( z f ) = ( 2 z f λ / D ) / D ,
| ln ( 2 z f λ / D 2 ) | ( 2 z f / z 0 ) 2 1 ,
| ln ( 2 z f λ / D 2 ) | ( z 0 / 2 z f ) 2 = ( 1 / 4 z f 2 ) { 1 / [ ( d n / d T ) / n ] ( I 0 / D ) ( 0.24 α t / ρ C p υ ) } .
ε { ln I I ˆ } = k 4 π 0 z d s 0 s d q 0 σ Φ ( σ 1 2 ; q ) exp [ Re γ ( s , q ) σ / 2 ] sin [ Im γ ( s , q ) σ 2 ] d σ ,
γ ( s , q ) = [ 2 ( s q ) / i k ] × { [ q + ( q z f ) ( i k α 0 2 / z f ) ] / [ s + ( s z f ) ( i k α 0 2 / z f ) ] } ,
Φ ( K , z ) = 8.16 C N 2 ( z ) exp [ ( K / K m ) 2 ] / ( K 0 2 + K 2 ) 1 1 6 ,
| ε { ln I I ˆ } | k 4 π 0 z d s 0 s d q 0 σ Φ ( σ 1 2 ; q ) d σ .
| ε { ln I I ˆ } | k 8 π C N 2 z 2 0 σ 8.16 exp ( σ / K m 2 ) ( K 0 2 + σ ) 1 1 6 d σ ,
0 σ exp ( σ / K m 2 ) ( K 0 2 + σ ) 1 1 6 d σ 0 σ 5 6 exp ( σ / K m 2 ) d σ = ( K m 2 ) 1 6 ( 5 / 6 ) ! ,
| ε { ln ( I / I ˆ ) } | 11.2 C N 2 z 2 K m 1 3 / λ ( 11.2 C N 2 z 2 / λ ) ( 2 π / l 0 ) 1 3 ,
γ ( s , q ) [ 2 ( s q ) / i k ] [ ( q z f ) / ( s z f ) ] ,
ε { ln I I ˆ } = k 4 π 0 z d s 0 s d q 0 σ Φ ( σ 1 2 ; q ) sin ( s q ) σ k d σ .
ε { ln I I ˆ } = k 2 4 π 8.16 C N 2 0 e ( σ / K m 2 ) ( K 0 2 + σ ) 1 1 6 [ z k σ sin σ z k ] d σ .
0 [ z ( k / σ ) sin ( σ z / k ) ] σ 1 1 6 d σ ,
ε { ln ( I / I ˆ ) } 0.312 k 7 6 z 1 1 6 C N 2 .
I ( x , y ) = ( 2 / π ) ( W / ω z 2 ) exp { 2 [ ( x 2 + y 2 ) / ω z 2 ] } ,
I ( x , y ) = ( 2 / π ) ( W / ω z 2 ) exp ( 2 { [ ( x + x ) 2 + ( y + y ) 2 ] / ω z 2 } ) ,
ε [ I ( 0,0 ) ] = 2 π W ω z 2 exp [ 2 ( x 2 + y 2 ω z 2 ) ] × p ( x , y ) d x d y ,
p ( x , y ) = ( 1 / 2 π σ 2 ) exp [ ( x 2 + y 2 ) / 2 σ 2 ] ,
ε [ I ( 0,0 ) / I ] = { ( 2 / π ) [ W / ( ω z 2 + 2 σ 2 z 2 ) ] } / [ ( 2 / π ) ( W / ω z 2 ) ] = 1 / [ 1 + ( 2 σ θ 2 z 2 / ω z 2 ] ,
ε [ I ( 0,0 ) ] W / π σ θ 2 z 2 ,

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