Abstract

A simple formula is derived to estimate the thermal blooming aberration remaining after correction by a modal (Zernike-ordered) adaptive-optics correction system.

© 1988 Optical Society of America

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References

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  1. J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 223–320.
    [CrossRef]
  2. C. B. Hogge, “Propagation of high energy laser beams in the atmosphere,” in High Energy Lasers and Their Applications, S. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1974), pp. 177–246.
  3. D. C. Smith, “High power laser propagation; thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
    [CrossRef]
  4. L. C. Bradley, J. Herrmann, “Phase compensation for thermal blooming,” Appl. Opt. 13, 331–334 (1974).
    [CrossRef] [PubMed]
  5. G. A. Tyler, J. F. Belsher, P. H. Roberts, “A discussion of some issues associated with the evaluation and compensation of thermal blooming,” internal rep. (Optical Sciences Company, Placentia, Calif., November1986).
  6. M. H. Lee, “Zernike decomposition of thermal-blooming-induced phase screen,” TRW internal memo. (TRW Space and Defense Sector, Redondo Beach, Calif., 1986).
  7. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976)
    [CrossRef]
  8. R. Hudgin, “Wave-front compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393–395 (1977).
    [CrossRef]
  9. H. W. Gould, Combinatorial Identities, revised ed. (University of West Virginia, Morgantown, W. Va., 1972), p. 6, identity (1.41).
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  11. D. E. Novoseller, “Adaptive optics model for thermal blooming,” Appl. Opt. 26, 4149–4150 (1987).
    [CrossRef] [PubMed]

1987 (1)

1977 (2)

D. C. Smith, “High power laser propagation; thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

R. Hudgin, “Wave-front compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393–395 (1977).
[CrossRef]

1976 (1)

1974 (1)

Belsher, J. F.

G. A. Tyler, J. F. Belsher, P. H. Roberts, “A discussion of some issues associated with the evaluation and compensation of thermal blooming,” internal rep. (Optical Sciences Company, Placentia, Calif., November1986).

Bradley, L. C.

Gould, H. W.

H. W. Gould, Combinatorial Identities, revised ed. (University of West Virginia, Morgantown, W. Va., 1972), p. 6, identity (1.41).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Herrmann, J.

Hogge, C. B.

C. B. Hogge, “Propagation of high energy laser beams in the atmosphere,” in High Energy Lasers and Their Applications, S. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1974), pp. 177–246.

Hudgin, R.

Lee, M. H.

M. H. Lee, “Zernike decomposition of thermal-blooming-induced phase screen,” TRW internal memo. (TRW Space and Defense Sector, Redondo Beach, Calif., 1986).

Noll, R. J.

Novoseller, D. E.

Roberts, P. H.

G. A. Tyler, J. F. Belsher, P. H. Roberts, “A discussion of some issues associated with the evaluation and compensation of thermal blooming,” internal rep. (Optical Sciences Company, Placentia, Calif., November1986).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Smith, D. C.

D. C. Smith, “High power laser propagation; thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

Tyler, G. A.

G. A. Tyler, J. F. Belsher, P. H. Roberts, “A discussion of some issues associated with the evaluation and compensation of thermal blooming,” internal rep. (Optical Sciences Company, Placentia, Calif., November1986).

Ulrich, P. B.

J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 223–320.
[CrossRef]

Walsh, J. L.

J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 223–320.
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

D. C. Smith, “High power laser propagation; thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

Other (6)

H. W. Gould, Combinatorial Identities, revised ed. (University of West Virginia, Morgantown, W. Va., 1972), p. 6, identity (1.41).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

G. A. Tyler, J. F. Belsher, P. H. Roberts, “A discussion of some issues associated with the evaluation and compensation of thermal blooming,” internal rep. (Optical Sciences Company, Placentia, Calif., November1986).

M. H. Lee, “Zernike decomposition of thermal-blooming-induced phase screen,” TRW internal memo. (TRW Space and Defense Sector, Redondo Beach, Calif., 1986).

J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 223–320.
[CrossRef]

C. B. Hogge, “Propagation of high energy laser beams in the atmosphere,” in High Energy Lasers and Their Applications, S. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1974), pp. 177–246.

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Tables (1)

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Table 1 Zernike Polynomial Indexing

Equations (56)

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ln S = 0.2944 N M 3 / 2 ( D / r 0 ) 5 / 3
ln S = μ N Z 5 / 6 ( D / r 0 ) 5 / 3 ,
ln S = 2 5 π 4 ( N D 2 / N M 2.5 ) ,
ϕ ( x , y ) = A [ y a + ( 1 x 2 a 2 ) 1 / 2 ] ,
ϕ ( x , y ) = A y + A a K Z k ( x , y ) ,
Z k ( x , y ) Z j ( x , y ) d x d y = π a 2 δ k j ,
a k = 1 π ( 1 x 2 a 2 ) 1 / 2 Z K ( x , y ) d ( x a ) d ( y a )
a k = 1 π ( 1 x 2 ) 1 / 2 Z k ( x , y ) d x d y .
Z k ( x , y ) Z n m ( x , y ) = ( n + 1 ) 1 / 2 R n m ( r ) C m ( θ ) ,
r 2 = x 2 + y 2 , C m ( θ ) = { 2 cos m θ m 0 , even solution 2 sin m θ m 0 , odd solution 1 m = 0 ,
R n m ( r ) = s = 0 ( n m ) / 2 ( ) s ( n s s ) [ n 2 s ( n 2 s ) / 2 m / 2 ] r n 2 s .
a n m = ( n + 1 ) 1 / 2 π ( 1 x 2 ) 1 / 2 R n m ( r ) C m ( θ ) d x d y = ( n + 1 ) 1 / 2 π ( 1 x 2 ) 1 / 2 s = 0 ( n m ) / 2 ( ) s ( n s s ) × [ n 2 s ( n m ) / 2 s ] r ( n m 2 s ) r m C m ( θ ) d x d y .
a n m = 8 s d m π ( n + 1 ) 1 / 2 1 ( n + 3 ) ( n + 1 ) ( n 1 ) ,
d m = { 1 m = 0 2 m 0 .
b n = m | a n m | 2 = 64 / π 2 ( n + 3 ) 2 ( n + 1 ) 2 ( n 1 ) 2 .
σ 2 = [ ϕ ( x , y ) ] 2 d x d y / d x d y ,
σ N M 2 = 2 5 π 4 ( N D 2 / N M 2.5 ) .
S exp ( σ N M 2 ) .
ln S = 0.254314 N M 5 / 6 ( D / r o ) 5 / 3 .
ϕ ( x , y ) = B y I ( x , y ) d y ,
B = 2 π λ z d z k ( z ) exp [ z d z α ( z ) ] υ ( z ) + z ω [ ( n ) n p 0 ( γ 1 ) γ ] ,
I ( x , y ) = P π a 2 { 1 x 2 + y 2 a 2 0 x 2 + y 2 > a 2 ,
ϕ ( x , y ) = B P π a 2 ( a 2 x 2 ) 1 / 2 y d y = A [ y a + ( 1 x 2 a 2 ) 1 / 2 ] ,
A = B P / π a = N D / 2 π .
( n 1 ) n T n / T n / T 2 ,
p 0 γ γ 1 = ρ c p T ,
sin m θ = sin θ [ cos ( m 1 ) θ + cos ( m 3 ) θ + ] ,
d y y 2 k + 1 = 0
a n 0 = ( n + 1 ) 1 / 2 π ( 1 x 2 ) 1 / 2 s = 0 n / 2 ( ) s ( n s s ) × [ n 2 s ( n 2 s ) / 2 ] r n 2 s d x d y = ( n + 1 ) 1 / 2 π s = 0 n / 2 ( ) s ( n s s ) ( n 2 s ) / 2 × ( 1 x 2 ) 1 / 2 ( x 2 + y 2 ) ( n / s ) s d x d y = ( n + 1 ) 1 / 2 π s = 0 n / 2 ( ) s ( n s s ) [ n 2 s ( n 2 s ) / 2 ] × p = 0 ( n / 2 ) s [ ( n 2 s ) / 2 p ] Q ,
Q = 1 1 d x ( 1 x 2 ) 1 / 2 ( 1 x 2 ) 1 / 2 d y ( 1 x 2 ) 1 / 2 x 2 ( n / 2 s p ) y 2 p = 2 2 p + 1 1 1 d x x 2 ( n / 2 s p ) ( 1 x 2 ) p + 1 = 4 2 p + 1 υ = 0 p + 1 ( p + 1 υ ) ( ) υ 1 2 ( n / 2 s p + υ ) + 1 = 4 2 p + 1 1 n 2 s 2 p + 1 1 [ ( n 2 s + 3 ) / 2 p + 1 ] ,
a n o = ( n + 1 ) 1 / 2 π s = 0 n / 2 ( ) s ( n s s ) [ n 2 s ( n 2 s ) / 2 ] × p = 0 ( n / 2 ) s [ ( n 2 s ) / 2 p ] 4 2 p + 1 1 n 2 s 2 p + 1 × 1 [ ( n 2 s + 3 ) / 2 p + 1 ] = ( n + 1 ) 1 / 2 π 1 2 ! 3 2 ! s = 0 n / 2 ( ) s ( n s s ) [ n 2 s ( n 2 s ) / 2 ] [ ( n 2 s + 3 ) / 2 3 / 2 ] × p = 0 ( n 2 s ) / 2 ( 2 p + 2 2 p + 1 ) 1 [ ( n 2 s 2 p ) / 2 1 / 2 ] .
a n m = ± 8 π 1 ( n + 1 ) 1 / 2 ( n + 3 ) ( n + 1 ) ( n 1 )
a n m 2 1 / 2 = ( n + 1 ) 1 / 2 π ( 1 x 2 ) 1 / 2 s = 0 ( n m ) / 2 ( ) s ( n s s ) × [ n 2 s ( n m ) / 2 s ] r n m 2 s r m cos m θ d x d y ,
cos m θ = k = 0 m / 2 ( m 2 k ) ( ) k cos θ m 2 k sin θ 2 k ,
a n m 2 1 / 2 = ( n + 1 ) 1 / 2 π s = 0 ( n m ) / 2 ( ) s ( n s s ) × [ n 2 s ( n m ) / 2 s ] k = 0 m / 2 ( m 2 k ) ( 1 ) k F ,
F = ( 1 x 2 ) 1 / 2 r n m 2 s r m cos θ m 2 k sin θ 2 k d x d y .
2 N = n m 2 s , 2 M = m ,
F = d x d y ( 1 x 2 ) 1 / 2 ( x 2 + y 2 ) N x 2 M 2 k y 2 k = p ( N P ) d x d y ( 1 x 2 ) 1 / 2 x 2 N 2 p + 2 M 2 k y 2 p + 2 k = p ( N P ) 2 2 p + 2 k + 1 d x x 2 N 2 p + 2 M 2 k ( 1 x 2 ) p + k + 1 = p ( N P ) 2 2 p + 2 k + 1 ν ( ) ν ( p + k + 1 ν ) 1 1 d x x 2 N + 2 M 2 p 2 k + 2 ν = p ( N P ) 2 2 p + 2 k + 1 ν ( ) ν ( p + k + 1 ν ) 1 2 N + 2 M 2 p 2 k + 2 ν + 1 = p ( N P ) 4 2 p + 2 k + 1 1 2 N + 2 M 2 p 2 k + 1 1 [ ( 2 N + 2 M + 3 ) / 2 p + k + 1 ] ,
a n m 2 1 / 2 = ( n + 1 ) 1 / 2 π s = 0 ( n m ) / 2 ( ) s ( n s s ) [ n 2 s ( n m ) / 2 s ] × k = 0 ( m 2 k ) ( ) k p [ ( n m ) / 2 s p ] × 4 2 p + 2 k + 1 1 n 2 s 2 p 2 k + 1 1 [ ( n 2 s + 3 ) / 2 p + k + 1 ] .
σ 2 [ ϕ ( x , y ) ] 2 d x d y / d x d y = A 2 π 2 ( 1 x 2 ) d x d y = 3 4 A 2 .
σ 2 = π A 2 ( n , m a n m 2 ) / π = A 2 n ( even ) b n ,
b n = 1 π 2 [ 1 ( n + 3 ) 2 + 4 ( n + 1 ) 2 + 1 ( n 1 ) 2 6 ( n + 3 ) ( n 1 ) ]
σ 2 = A 2 π 2 k [ 1 ( 2 k + 3 ) 2 + 4 ( 2 k + 1 ) 2 + 1 ( 2 k 1 ) 2 6 ( 2 k + 3 ) ( 2 k 1 ) ] .
k = 0 1 ( 2 k + 1 ) 2 = π 2 8 ,
k = 0 1 ( 2 k + 3 ) 2 = π 2 8 1 ,
k = 0 1 ( 2 k 1 ) 2 = π 2 8 + 1.
k = 1 1 ( 2 k 1 ) 2 x 2 = π 4 x tan π x 2 ,
0 = k = 1 1 ( 2 k 3 ) ( 2 k + 1 )
= k = 0 1 ( 2 k + 3 ) ( 2 k 1 ) .
σ 2 = 3 4 A 2 ,
N M = ( N + 1 ) ( N + 2 ) / 2 N 2 / 2
σ N 2 = A 2 n = N + 2 | a n m | 2 = A 2 n = N + 2 b n 64 A 2 π 2 Δ n = 2 1 n 6 A 2 π 2 Δ k = 1 1 k 6 A 2 π 2 N / 2 d k k 6 A 2 5 π 2 ( 1 N / 2 ) 5 32 5 π 2 A 2 N 5 ,
σ N 2 = 2 5 π 4 N D 2 N M 2.5
σ 0 2 = 3 4 A 2 A 2 b 0 = ( 3 4 64 9 π 2 ) N D 2 4 π 2 ,
S exp ( σ 0 2 ) ,
S 16 / N D

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