Abstract

In this paper we present the theoretical background required for evaluating phase compensation as a technique for reducing the effects of thermal blooming. By using a time-dependent approach for thermal blooming, we can define and evaluate the instantaneous Green’s function for the heated atmosphere. The phase at the mirror is determined by propagating a point source from the focal point to the laser. The effectiveness of this technique is determined by propagating point sources within a diffraction-limited spot to the laser and comparing the resulting phase to the phase of a point source at the focal point. The phase difference will be small and the increase in irradiance large if the heating occurs where geometric optics is valid. The phase difference is large and the increase in irradiance small if atmospheric heating occurs where significant diffractive spreading is also occurring. The resulting gradients lie in the geometric shadow of the aperture and are not accessible to correction by contouring the mirror. Numerical results illustrating the analysis are presented for both cw and pulsed lasers.

© 1977 Optical Society of America

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References

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  1. L. C. Bradley and J. Herrmann, “Phase compensation for thermal blooming,” Appl. Opt. 13, 331 (1974).
    [Crossref] [PubMed]
  2. J. Wallace and J. Pasciak, “Compensating for thermal blooming of repetitively pulsed lasers,” J. Opt. Soc. Am. 65, 1257 (1975).
    [Crossref]
  3. J. R. Dunphy and D. C. Smith, “Multiple-pulse thermal blooming and phase compensation,” J. Opt. Soc. Am. 67, 295 (1977).
    [Crossref]
  4. W. P. Brown, “Computer simulation of adaptive optical systems,” Hughes Research Laboratory Report N60921-74-C-0249 (September1975).
  5. F. G. Gebhardt, “Experimental demonstration of the use of phase correction to reduce thermal blooming” (private communication).
  6. G. S. Avila and J. B. Keller, “High-frequency asymptolic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363 (1963).
    [Crossref]
  7. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).
  8. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II (Interscience, New York, 1962).
  9. J. Hermann, “Properties of phase conjugate adaptive optical systems,” J. Opt. Soc. Am. 67, 290 (1977).
    [Crossref]
  10. P. T. Berger, P. B. Ulrich, J. J. Ulrich, and F. G. Gebhardt, “Transient thermal blooming of a slewed laser beam containing a region of stagnant absorber,” Appl. Opt. 16, 345 (1977).
    [Crossref] [PubMed]
  11. P. B. Ulrich, Naval Research Laboratory, Washington D. C. (private communication).
  12. C. B. Hogge, “Propagation of high-energy laser beams in the atmosphere,” Air Force Weapons Lab TR-74-74 (1974).

1977 (3)

1975 (1)

1974 (1)

1963 (1)

G. S. Avila and J. B. Keller, “High-frequency asymptolic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

Avila, G. S.

G. S. Avila and J. B. Keller, “High-frequency asymptolic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

Berger, P. T.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Bradley, L. C.

Brown, W. P.

W. P. Brown, “Computer simulation of adaptive optical systems,” Hughes Research Laboratory Report N60921-74-C-0249 (September1975).

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II (Interscience, New York, 1962).

Dunphy, J. R.

Gebhardt, F. G.

Hermann, J.

Herrmann, J.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II (Interscience, New York, 1962).

Hogge, C. B.

C. B. Hogge, “Propagation of high-energy laser beams in the atmosphere,” Air Force Weapons Lab TR-74-74 (1974).

Keller, J. B.

G. S. Avila and J. B. Keller, “High-frequency asymptolic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

Pasciak, J.

Smith, D. C.

Ulrich, J. J.

Ulrich, P. B.

Wallace, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Appl. Opt. (2)

Commun. Pure Appl. Math. (1)

G. S. Avila and J. B. Keller, “High-frequency asymptolic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

J. Opt. Soc. Am. (3)

Other (6)

W. P. Brown, “Computer simulation of adaptive optical systems,” Hughes Research Laboratory Report N60921-74-C-0249 (September1975).

F. G. Gebhardt, “Experimental demonstration of the use of phase correction to reduce thermal blooming” (private communication).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II (Interscience, New York, 1962).

P. B. Ulrich, Naval Research Laboratory, Washington D. C. (private communication).

C. B. Hogge, “Propagation of high-energy laser beams in the atmosphere,” Air Force Weapons Lab TR-74-74 (1974).

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Figures (7)

FIG. 1
FIG. 1

Point-source geometry. Compensating phase is determined from the on-axis calculation. Effectiveness is determined by the off-axis calculation.

FIG. 2
FIG. 2

Dependence of the irradiance on the distortion parameter for two linear, distributed, parabolic lenses. The solid lines represent the irradiance with phase compensation and the dashed lines without compensation.

FIG. 3
FIG. 3

Flow diagram for determining the phase contours and focal plane irradiance in the nonlinear problem.

FIG. 4
FIG. 4

Time dependence of the isoirradiance and the isophase contours for a cw beam. The uncorrected steady-state results at the focal plane are also presented.

FIG. 5
FIG. 5

Time dependence of the isoirradiance and the isophase contours in the presence of a stagnation zone located midway to the focal plane. Also shown for comparison, are the uncorrected irradiance distributions.

FIG. 6
FIG. 6

Comparison of the steady-state irradiance distribution with phase compensation for 2 pulses per flow time.

FIG. 7
FIG. 7

Time dependence of the isoirradiance and the isophase contours for a repetitively pulsed laser in the presence of a stagnation zone located midway to the focal plane.

Equations (28)

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E = A ( x , y , z ) exp ( i k n z α z / 2 ) ,
2 i F n A z + 2 A + 2 k 2 ( n 1 ) R m 2 ρ A = 0 ,
A = A 0 ( x , y , 0 ) exp [ i F ( ψ c ( x , y ) 1 2 ( x 2 + y 2 ) ) ] .
2 i F n A z s + 2 A s + 2 k 2 R m 2 ( n 1 ) ρ A s = δ ( x x 1 , y y 1 , z z f ) .
A s = [ a s ( ξ , η , z ) / 4 π ( 1 z ) ] × exp [ i F ( ψ s ( ξ , η , z ) + 1 2 ( ξ 2 + η 2 ) ( 1 z ) ) ] ,
ξ = ( x x 1 ) / ( 1 z ) , η = ( y y 1 ) / ( 1 z ) ,
ψ z s + 1 2 [ ( ψ ξ s ) 2 + ( ψ η s ) 2 ] / ( 1 z ) 2 = z f 2 ( n 1 ) [ ρ ( x 1 + ξ ( 1 z ) , y 1 + η ( 1 z ) , z ) ] / R m 2 ,
( 1 z ) 2 a z s + ( a s ) · ( ψ s ) + 1 2 a s ( 2 ψ s ) i ( 2 a s ) / 2 F = 0 ,
ψ = 0 , a s = 1. 0 at z = z f ,
A = F 2 π i A 0 ( x , y ) a s ( ξ , η ; x 1 , y 1 ) × exp [ i F ( ψ s ( ξ , η ; x 1 , y 1 ) + ψ c ( x , y ) + 1 2 ( x 1 2 + y 1 2 ) x 1 x y 1 y ) ] dxdy ,
ξ = x x 1 , η = y y 1 .
Δ ψ = ψ s ( ξ , η ; x 1 , y 1 ) + ψ c ( ξ , η ) = ψ s ( ξ , η ; x 1 , y 1 ) ψ s ( ξ , η ; 0 , 0 ) .
A 1 = F 2 π i A 0 a s ( x , y ; 0 , 0 ) × exp [ i F ( 1 2 ( x 1 2 + y 1 2 ) x 1 x y 1 y ) ] dxdy
A 2 = F 2 π i A 0 [ a s ( ξ , η ; x 1 , y 1 ) a s ( x , y ; 0 , 0 ) exp ( i Δ ψ F ) ] × exp [ i F ( Δ ψ + 1 2 ( x 1 2 + y 1 2 ) x 1 x y 1 y ) ] dxdy .
A 2 = F 2 π i A 0 a s ( x , y ; 0 , 0 ) × ( 0 1 2 ( Δ ψ ) d z / 2 + i F Δ ψ + O ( Δ ψ ) 2 ) × exp [ i F ( 1 2 ( x 1 2 + y 1 2 ) x 1 x y 1 y ) ] dxdy ,
[ ξ ( 1 z ) ] = z f 2 ( n 1 ) ρ ξ / R m 2 ( 1 z ) ; ξ ( 1 ) = ξ i = x x 1 , ξ ( 1 ) = 0 ; [ η ( 1 z ) ] = z f 2 ( n 1 ) ρ η / R m 2 ( 1 z ) ; η ( 1 ) = η i = y y 1 , η ( 1 ) = 0 .
ψ c = z f 2 ( n 1 ) R m 2 0 1 ρ ( ξ ( 1 z ) , η ( 1 z ) , z ) d z ,
F Δ ψ = F z f 2 ( n 1 ) R m 2 0 1 [ ρ ( x 1 + ξ ( 1 z ) , y 1 + η ( 1 z ) , z ) ρ ( ξ ( 1 z ) , η ( 1 z ) , z ] d z ,
ψ s = ( N 1 / 2 ) ψ s ( ξ / N 1 / 2 , η / N 1 / 2 , ( 1 z ) N 1 / 2 ) , N = z f 2 ( n 1 ) / R m 2 ,
ψ z s + [ ( ψ ξ s ) 2 + ( ψ η s ) 2 ] / 2 z 2 = ρ ( x 1 + ξ z , y 1 + η z , z ) ,
n = n ( n 1 ) ( 1 ( x 2 + y 2 ) / 2 ( 1 z ) 2 ) / ( 1 z ) 2 ; 0 < z < z h ,
A = exp [ ( x 2 + y 2 ) / R 2 ( z ) + i F ( x 2 + y 2 ) R ( z ) / 2 R ( z ) + g ( z ) ] / R 2 ( z ) .
R ( z ) = ( 1 z ) [ 1 + ( 2 / F N 1 / 2 ) 2 tanh 2 ( N 1 / 2 ζ ) ] 1 / 2 cosh ( N 1 / 2 ζ ) , ζ = z / ( 1 z ) , F = k R m 2 / z f .
ψ c = N 1 / 2 ( x 2 + y 2 ) tanh ( N 1 / 2 ζ h ) / 2 ,
R ( z ) = ( 1 z ) { 1 + ( 2 / F N 1 / 2 ) [ tanh ( N 1 / 2 ζ h ) tanh ( N 1 / 2 ( ζ h ζ ) ) ] 2 cosh 4 ( N 1 / 2 ζ h ) } 1 / 2 × cosh ( N 1 / 2 ( ζ h ζ ) ) / cosh ( N 1 / 2 ζ h ) .
R ( z ) = { [ R ( z h ) + ( z z h ) R ( z h ) ] 2 + [ 2 ( z z h ) / F R ( z h ) ] 2 } 1 / 2 , z h < z
ρ t + U ρ x = exp ( α z ) I ( x , y , z , t ) , = ( γ 1 ) α I 0 / γ ρ .
ρ ( x , y , z , t ) = ρ ( x U ( t t 0 ) / R , y , z , t 0 ) exp ( α z ) t 0 t I ( x U ( t t ) / R , y , z , t ) d t .