Abstract

I show analytically that compensation for thermal blooming by full-field conjugation with perfect phase reproduction but with a saturable laser amplifier exhibits instability, which leads to strong scintillation on small spatial scales. I support this analysis with output from molly, a computer simulation of adaptively compensated high-energy laser-beam propagation. In the simulation, compensation for thermal blooming is effective in spite of manifest instability. I derive lower bounds on compensation performance that highlight aspects of the simulation that are responsible for good performance.

© 1992 Optical Society of America

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References

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  1. J. Herrmann, “Properties of phase conjugate adaptive optical systems,” J. Opt. Soc. Am. 67, 290–295 (1977).
    [CrossRef]
  2. T. J. Karr, “Thermal blooming compensation instabilities,” J. Opt. Soc. Am. A 6, 1038–1048 (1989).
    [CrossRef]
  3. J. F. Schonfeld, “Adaptively corrected propagation through turbulence and strong thermal blooming,” in Propagation of High-energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 118–131 (1990).
    [CrossRef]

1989 (1)

1977 (1)

Herrmann, J.

Karr, T. J.

Schonfeld, J. F.

J. F. Schonfeld, “Adaptively corrected propagation through turbulence and strong thermal blooming,” in Propagation of High-energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 118–131 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Other (1)

J. F. Schonfeld, “Adaptively corrected propagation through turbulence and strong thermal blooming,” in Propagation of High-energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 118–131 (1990).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Maximum, over all transverse wave vectors, of the real part of the instability growth rate versus the saturation parameter f for the slab atmosphere. The curve is normalized to unity at f = 0.

Fig. 2
Fig. 2

Real part of the instability growth rate with the largest real part, in the limit of large transverse wave vector, versus the saturation parameter f, for the uniform atmosphere. The curve is normalized to unity at f = 0. The largest value of f for which data are shown is 0.99. The break in the curve coincides with the point at which the left-hand side of relation (2.31) crosses one.

Fig. 3
Fig. 3

Contour plot of high-energy laser intensity at 5 km from transmitter, after one wind-clearing time, when thermal blooming is compensated for by full-field conjugation with a saturable amplifier, as defined by Eqs. (2.1) and (2.33). The wind blows toward the right. Contours linearly divide the intensity range between zero and the peak into seven equal intervals. The plot frame is 1.5 m × 1.5 m.

Fig. 4
Fig. 4

Contour plot of high-energy laser intensity at 5 km from transmitter, after one wind-clearing time, when thermal blooming is compensated for by perfect full-field conjugation. The wind blows toward the right. The contours linearly divide intensity range between zero and the peak into seven equal intervals. The plot frame is 1.5 m × 1.5 m.

Fig. 5
Fig. 5

Ratio of actual high-energy laser far-field intensity on bore sight to what the bore-sight far-field intensity would be if atmospheric aberrations could be ignored, from simulation of compensation for thermal blooming with a saturable amplifier.

Fig. 6
Fig. 6

Ratio of actual high-energy laser power leaving the transmitter to what the power would be if atmospheric aberrations could be ignored, from simulation of compensation for thermal blooming with a saturable amplifier.

Equations (49)

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ψ out ( x ) = ψ in * ( x ) [ F ( | ψ in ( x ) | 2 ) ] 1 / 2
I p = lim I I F ( I ) ,
d ( I ) = I F ( I ) / I p .
k E z p 2 Θ = 0 , 2 k Θ z + 1 2 p 2 E 2 k 2 μ = 0 ,
k E B z p 2 Θ B = 0 , 2 k Θ B z + 1 2 p 2 E B 2 k 2 μ = 0 ,
μ t i v · p μ = Γ E ,
E B z = L = Θ B z = L = 0 ,
Θ ( p ) z = 0 = Θ B ( p ) z = 0 , E ( p ) z = 0 = f E B ( p ) z = 0 .
Γ = n ρ c p n T I H α ( z ) exp { 0 z [ α ( z ) + α s ( z ) ] d z } ,
f = I I log [ I F ( I ) ] I = I b .
E , E B , Θ , Θ B , μ exp ( Ω t )
2 k 2 p 2 2 E z 2 + 1 2 p 2 E + 2 k 2 Γ Ω i v · p E = 0 ,
2 k 2 p 2 2 E B z 2 + 1 2 p 2 E B + 2 k 2 Γ Ω i v · p E = 0
E z ( p ) z = 0 = E B z ( p ) z = 0 .
E B = E E z = L cos ω ( z L 1 ) E z z = L L ω sin ω ( z L 1 ) ,
ω = p 2 L 2 k .
[ E B E B z ] z = 0 = ( 1 [ cos ω L ω sin ω ω L sin ω cos ω ] V ) [ E E z ] z = 0 U [ E E z ] z = 0 ,
[ E E z ] z = L = V [ E E z ] z = L .
det V = 1 .
0 = det ( 1 U [ f 0 0 1 ] ) = 1 + ( f 1 ) ( 1 V 12 ω L sin ω V 22 cos ω ) ,
Γ ( z ) = γ δ ( z L ) .
V 12 ( p ) = [ L / ω ( p ) ] sin ω ( p ) ,
V 22 ( p ) = cos ω ( p ) 2 k γ sin ω ( p ) Ω i v · p ,
1 V 12 ( p ) ω ( p ) L sin ω ( p ) V 22 ( p ) cos ω ( p ) = k γ sin 2 ω ( p ) Ω i v · p ,
Ω = i p · v + ( 1 f ) k γ sin 2 ω .
Re Ω max = | 1 f | k γ > 0 ,
V 12 ( p ) = L β ( p ) sin β ( p ) ,
V 22 ( p ) = cos β ( p ) ,
β ( p ) = [ ω 2 ( p ) + ω ( p ) G Ω i v · p ] 1 / 2 ,
Ω i p · v = O ( G ) .
1 1 f 1 cos ( G / 2 Ω i p · v ) .
π 4 ln [ 2 / ( 1 f ) ] .
F ( I ) = A 1 + ( I / I 0 ) ,
f ( I b ) = 1 d ( I b ) = F ( I b ) / A .
P = π ( 1.0 m 2 ) 2 A I 0 2 .
N D 0 ( 4 2 k P ρ c p D ) ( d n d T ) [ α ( z ) υ ( z ) ] × exp { 0 z [ α ( z ) + α s ( z ) ] } d z ,
Re Ω max = G / ( 2 π )
F ( I ) A ,
S = | ψ out ( x ) g ( x ) d 2 x [ I b F ( I b ) ] 1 / 2 g 0 ( x ) d 2 x | 2 ,
R = | ψ out ( x ) | 2 d 2 x I b F ( I b ) a = | ψ in ( x ) | 2 F ( | ψ in ( x ) | 2 ) d 2 x I b F ( I b ) a ,
S = | ψ out ( x ) ψ in ( x ) d 2 x [ I b F ( I b ) ] 1 / 2 ( I b ) 1 / 2 d 2 x | 2 = { | ψ in | 2 [ F ( | ψ in | 2 ) ] 1 / 2 } 2 I b 2 F ( I b ) a 2 .
S { c 1 [ F ( c 1 ) ] 1 / 2 + c 2 [ F ( c 2 ) ] 1 / 2 } 2 4 I b 2 F ( I b ) ,
R c 1 F ( c 1 ) + c 2 F ( c 2 ) 2 I b F ( I b ) .
a 2 ( c 1 + c 2 ) | ψ in ( x ) | 2 d 2 x beacon power in aperture projection at top of atmosphere a I b ,
c 1 + c 2 2 I b .
S F ( 2 I b ) / F ( I b ) ,
R F ( 2 I b ) / F ( I b ) .
H ( 0 ) H ( I b ) = [ F ( 2 I b ) F ( I b ) ] 1 / 2 or F ( 2 I b ) F ( I b ) ,
S , R 2 / 3 ,

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