Abstract

The isoplanatic condition for imaging through the turbulent atmosphere is investigated, and theoretical expressions are derived for various applications. Explicit calculations are performed for the Pic du Midi 1 m telescope and the Hale 5 m telescope.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 9.5.
  2. P. Dumontet, Opt. Acta 2, 53 (1955).
    [Crossref]
  3. P. Laques, in Advances in Electronics and Electron Physics, 22B, edited by J. D. McGee (Academic, London, 1966).
  4. I. G. Kolchinskii, Correlation Between Image Pulsation of Stars at Small Angular Distances from Each Other (1965).
  5. D. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  6. D. Korff, J. Opt. Soc. Am. 63, 971 (1973).
    [Crossref]
  7. R. E. Hufnagel, in Restoration of Atmospherically Degraded Images, Proceedings of Woods Hole Summer Study, Vol. 2, Appendix 5, p. 29 (Defense Documentation Center, Alexandria, Va., 1966).
  8. H. T. Yura, Appl. Opt. 11, 1399 (1972).
    [Crossref] [PubMed]
  9. V. I. Tatarskii, The Effects of the Turbulent Atomsphere on Wave Propagation (U. S. Department of Commerce, NTIS, Springfield, Va., 1971).
  10. R. E. Hufnagel, in Ref. 7 Vol. 2, Appendix 3, p. 15.
  11. Merlin Miller of AERL has pointed out to us that the quantity given in Eq. (37) can be obtained directly by measuring the fringe visibility of a set of binary stars having various angular separations.
  12. R. S. Kennedy, in Ref. 7, Vol. 2, Appendix 12, p. 69.

1973 (1)

1972 (1)

1966 (1)

1955 (1)

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 9.5.

Dumontet, P.

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

Fried, D.

Hufnagel, R. E.

R. E. Hufnagel, in Restoration of Atmospherically Degraded Images, Proceedings of Woods Hole Summer Study, Vol. 2, Appendix 5, p. 29 (Defense Documentation Center, Alexandria, Va., 1966).

R. E. Hufnagel, in Ref. 7 Vol. 2, Appendix 3, p. 15.

Kennedy, R. S.

R. S. Kennedy, in Ref. 7, Vol. 2, Appendix 12, p. 69.

Kolchinskii, I. G.

I. G. Kolchinskii, Correlation Between Image Pulsation of Stars at Small Angular Distances from Each Other (1965).

Korff, D.

Laques, P.

P. Laques, in Advances in Electronics and Electron Physics, 22B, edited by J. D. McGee (Academic, London, 1966).

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atomsphere on Wave Propagation (U. S. Department of Commerce, NTIS, Springfield, Va., 1971).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 9.5.

Yura, H. T.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Acta (1)

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

Other (8)

P. Laques, in Advances in Electronics and Electron Physics, 22B, edited by J. D. McGee (Academic, London, 1966).

I. G. Kolchinskii, Correlation Between Image Pulsation of Stars at Small Angular Distances from Each Other (1965).

R. E. Hufnagel, in Restoration of Atmospherically Degraded Images, Proceedings of Woods Hole Summer Study, Vol. 2, Appendix 5, p. 29 (Defense Documentation Center, Alexandria, Va., 1966).

V. I. Tatarskii, The Effects of the Turbulent Atomsphere on Wave Propagation (U. S. Department of Commerce, NTIS, Springfield, Va., 1971).

R. E. Hufnagel, in Ref. 7 Vol. 2, Appendix 3, p. 15.

Merlin Miller of AERL has pointed out to us that the quantity given in Eq. (37) can be obtained directly by measuring the fringe visibility of a set of binary stars having various angular separations.

R. S. Kennedy, in Ref. 7, Vol. 2, Appendix 12, p. 69.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 9.5.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

FIG. 1
FIG. 1

Spatial-frequency correlation function for the 1 m telescope at the Pic du Midi observatory in France. q is the normalized frequency (λf/D) and ϕ is the angle between the q and δ ψ vectors.

FIG. 2
FIG. 2

Spatial-frequency correlation function for the 5 m Hale (Mount Palomar Observatory) telescope.

FIG. 3
FIG. 3

Spatial correlation function for the 1 m Pic du Midi telescope.

FIG. 4
FIG. 4

Improved OTF of the Pic du Midi 1 m telescope for several reference-point spacings, compared to Fried’s short-exposure OTF.

FIG. 5
FIG. 5

Ratio of speckle-correction terms to zeroth-order terms for several object sizes. D = 5 m, λ = 0.5 μm.

FIG. 6
FIG. 6

Finest angular detail observable by speckle interferometry as a function of object size for various tolerances. D = 5 m, λ = 0.5 μm.

FIG. 7
FIG. 7

Correlation function for image wander, Gao Telescope (Soviet Ukraine). ○ = 17° elevation; ● = 60° elevation. D = 0.4 m.

FIG. 8
FIG. 8

Effective value of 〈|τ|2〉 as a function of angle traversed for a moving point object.

Equations (76)

Equations on this page are rendered with MathJax. Learn more.

K ( u , u ) = K A ( u - u ) ,
K ( u , u ) = K A ( u - u ) .
I ( u ) = O ( u ) G ( u , u ) d u ,
I ( u ) O ( u ) G ( u - u ) d u ,
Ĩ ( f ) - O ˜ ( f ) τ ( f ) ,
τ ( f ; r 1 ) = B Σ U ( v , r 1 ) U * ( v - λ f ; r 1 ) d v ,
τ ( f ; r 2 ) = I ( r - r 2 ) exp ( 2 π i f · r / R ) d r = I ( r - r 1 ) exp ( 2 π i f · ( r 2 - r 1 ) / R ) exp ( 2 π i f · r / R ) d r = τ ( f , r 1 ) exp ( 2 π i f · ( r 2 - r 1 ) / R ) ,
G ( r ; r + r 1 ) = τ ( f ; r + r 1 ) exp ( - 2 π i f · r / R ) d f = exp [ 2 π i f · ( r 1 - r ) / R ] τ ( f ; r ) d f = G ( r - r 1 ; r ) ,
Q = [ τ ( f ; r 1 ) τ * ( f ; r 2 ) - τ ( f , r 1 ) τ * ( f , r 2 ) ] exp [ - 2 π i f · ( r 2 - r 1 ) / R ] [ τ ( f , r 1 ) 2 - τ ( f , r 1 ) 2 ] 1 / 2 [ τ ( f ; r 2 ) 2 - τ ( f ; r 2 ) 2 ] 1 / 2 ,
τ ( f ; r 1 ) 2 τ ( f ; r 2 ) 2
τ ( f ; r 1 ) = exp ( 2 π i f · ( r 1 - r 2 ) / R ) τ ( f ; r 2 ) ,
Q τ ( f ; r 2 ) τ * ( f ; r 1 ) exp [ - 2 π i f · ( r 2 - r 1 ) / R ] - τ ( f , r 1 ) 2 τ ( f ; r 1 ) 2 - τ ( f ; r 1 ) 2 .
Q s = d r ( I ( r ; 0 ) - I ) ( I ( r - R δ ψ ; δ ψ ) - I ) d r ( I ( r ; 0 ) - I ) 2 ,
I ( r ; 0 ) = τ ( f ; 0 ) exp ( - 2 π i f · r / R ) d f
I ( r - R δ ψ ; δ ψ ) = τ * ( f ; z δ ψ ) exp ( 2 π i f · r / R ) d f ,
Q s = d f [ τ ( f ; 0 ) τ * ( f ; z δ ψ ) - τ ( f ) 2 ] d f [ τ ( f ; 0 ) 2 - τ ( f ) 2 ] .
P τ ( f ; r 1 ) τ * ( f ; r 2 ) exp ( - 2 π i f · ( r 2 - r 1 ) / R ) .
P = B 2 Σ d v d v U ( v ; r 1 ) U * ( v - λ f ; r 1 ) U * ( v ; r 2 ) × U ( v - λ f ; r 2 ) exp ( - 2 π i f · ( r 2 - r 1 ) / R ) ,
P = B 2 d v d v W ( v ) W ( v - λ f ) W ( v ) W ( v - λ f ) exp [ i k r 1 - v - i k r 1 - v + λ f - i k r 2 - v + i k r 2 - v + λ f ] × exp [ ψ ( r 1 , v ) + ψ * ( r 1 , v - λ f ) + ψ * ( r 2 , v ) + ψ ( r 2 , v - λ f ) ] 1 r 1 2 r 2 2 exp ( - 2 π i f · ( r 2 - r 1 ) / R ) ,
W ( v ) = { 1 , v D / 2 , 0 , otherwise .
exp [ - 1 2 U 2 ] = exp [ - 2 ψ 1 2 + ψ 1 ( r 1 , v ) ψ 1 * ( r 1 , v - λ f ) + ψ 1 ( r 1 , v ) ψ 1 * ( r 2 , v ) + ψ 1 ( r 1 , v ) ψ 1 ( r 2 , v - λ f ) + ψ 1 * ( r 1 , v - λ f ) ψ 1 * ( r 2 , v ) + ψ 1 * ( r 2 , v ) ψ 1 ( r 2 , v - λ f ) + ψ 1 * ( r 1 , v - f ) ψ 1 ( r 2 , v - λ f ) ] ,
ψ 1 ( r , v ) = U 0 - 1 ( r , v ) ( k 2 2 π ) d 3 x n 1 ( x ) U 0 ( r , x ) U 0 ( x , v ) ,
i k { r 1 - v - r 1 - v + λ f - r 2 - v + r 2 - v + λ f } × 2 π i ( r 2 - r 1 ) · f / z + 2 π i ( v - v ) · f / z ,
P = B 2 z 4 d v d v exp ( 2 π i ( v - v ) · f / z ) × W ( v ) W ( v - λ f ) W ( v ) W ( v - λ f ) exp [ - 1 2 U 2 ] .
1 2 U 2 = 6.88 α 5 / 3 0 d z 1 C n 2 ( z 1 ) R ( z 1 ) 0 d z 1 C n 2 ( z 1 ) ,
R = q 5 / 3 + | y + z 1 δ ψ D | 5 / 3 - 1 2 | y + z 1 δ ψ D + q | 5 / 3 - 1 2 | y + z 1 δ ψ D - q | 5 / 3 .
q = λ f / D ,             α = D / r 0 , y = ( v - v ) / D ,             r 0 = [ 0.42 C n 2 ( z 1 ) d z 1 ] - 3 / 5 , δ ψ = ( r 1 - r 2 ) / z .
P = B 2 4 z 4 d v - d v + W ( v + + v - - 2 λ f 2 ) W ( v + + v - 2 ) × W ( v + - v - - 2 λ f 2 ) W ( v + - v - 2 ) exp ( - 1 2 U 2 ) .
v + 1 = 2 λ f - v - , v + 2 = - v - , v + 3 = 2 λ f + v - , v + 4 = + v - .
P = D 4 B 2 4 z 4 d y S ( q , y , θ ) exp ( - 1 2 U 2 ) ,
τ ( 0 ) O ˜ ( 0 ) = Ĩ ( 0 ) ,
τ ( 0 ) O ( x ) d x = I ( x ) d x = I T .
τ ( 0 ) = 1.
τ ( 0 ) = B z 2 d v W 2 ( v ) = π B D 2 4 z 2
P = 4 π 2 d y S ( q , y , θ ) exp ( - 1 2 U 2 ) .
P = 4 π 2 d y S ( q , y , θ ) = 16 π 2 D 4 d v d v W 2 ( v ) W 2 ( v ) = 1.
P = τ ( 0 ) 2 = 1 + σ p 2 ,
Q ( 4 / π 2 ) d y S ( q , y , θ ) exp [ - 1 2 U 2 ( δ ψ ) ] - τ ( f ; r 1 ) 2 ( 4 / π 2 ) d y S ( q , y , θ ) exp [ - 1 2 U 2 ( 0 ) ] - τ ( f ; r 1 ) 2 ,
0 ˜ ( f , r 1 ) = Ĩ ( f ; r 1 ) τ ( f ; r 1 ) .
O ˜ rec ( f ) = O ˜ ( f ) τ ( f , 0 ) τ ( f , z δ ψ ) .
O rec ( r ) = exp [ - 2 π i r · f / z ] O ˜ ( f ) τ ( f , 0 ) τ * ( f , z δ ψ ) τ ( f , z δ ψ ) 2 d f .
τ rec ( f ˜ ) = τ ( f , 0 ) τ * ( f , z δ ψ ) ( f , z δ ψ ) 2 ,
Ĩ ( f ) 2 = τ ( f ) 2 O ˜ ( f ) 2 .
G ( r ; r ) G ( 0 ; r - r ) + r · G ξ ( ξ ; r - r ) | ξ = 0 + 1 2 r i r j 2 G ξ i ξ j ( ξ ; r - r ) | ξ = 0
Ĩ ( f ) = 0 ˜ ( f ) τ ( f ; 0 ) + 1 2 π i f 0 ˜ ( f ) · τ ( f ; ξ ) | ξ = 0 + ( 1 2 π i ) 2 · 1 2 · f i f j 0 ˜ ( f ) ξ i ξ j τ ( f ; ξ ) | ξ = 0 .
Ĩ ( f ) 2 0 ˜ ( f ) 2 τ ( f ; 0 ) 2 + 1 2 π i O ˜ ( f ) f 0 ˜ * ( f ) τ ( f ; 0 ) ξ τ * ( f ; ξ ) | ξ = 0 - 1 2 π i 0 ˜ * ( f ) f 0 ˜ ( f ) τ * ( f ; 0 ) ξ τ ( f ; ξ ) | ξ = 0 + 1 4 π 2 f i 0 ˜ * ( f ) f j 0 ˜ ( f ) ξ i ξ j τ * ( f ; ξ ) τ ( f ; ξ ) | ξ = ξ = 0 - 1 2 ( 4 π 2 ) 0 ˜ ( f ) 2 f i f j 0 ˜ * ( f ) τ ( f ; 0 ) 2 ξ i ξ j τ * ( f ; ξ ) | ξ = 0 - 1 2 ( 4 π 2 ) 0 ˜ * ( f ) 2 f i f j 0 ˜ ( f ) τ * ( f ; 0 ) 2 ξ i ξ j τ ( f ; ξ ) | ξ = 0 .
τ ( f ; 0 ) ξ τ * ( f ; ξ ) ξ = 0 = ξ τ ( f ; 0 ) τ * ( f ; ξ ) ξ = 0 ,
Ĩ ( f ) 2 = 0 ˜ ( f ) 2 τ ( f ; 0 ) 2 + 1 4 π 2 [ f i 0 ˜ * ( f ) ] [ f i 0 ˜ ( f ) ] 2 ξ i ξ i τ ( f ; ξ ) τ * ( f ; ξ ) | ξ = ξ = 0 - 1 8 π 2 [ 0 ˜ ( f ) 2 f i 2 0 ˜ * ( f ) + 0 ˜ * ( f ) 2 f i 2 0 ˜ ( f ) ] τ ( f ; 0 ) 2 ξ i 2 τ * ( f ; ξ ) | ξ = 0 = O ˜ 2 τ 2 - 1 4 π 2 2 ξ i 2 τ ( f ; 0 ) τ * ( f ; ξ ) | ξ = 0 [ O f i O * f i + 1 2 ( O * 2 f i 2 O + O 2 f i 2 0 * ) ] .
0 ( x , y ) = 1 2 π σ x σ y exp [ - 1 2 ( x 2 σ x 2 + y 2 σ y 2 ) ] ;
O ( x ) = α δ ( x ) + 1 2 π σ x σ y exp [ - ( x 2 2 σ x 2 + y 2 2 σ y 2 ) ] ,
n ~ ( θ 0 ) 2 z 1 2 / r 0 2
z 1 2 = 0 d z 1 C n 2 ( z 1 ) z 1 2 / 0 d z 1 C n 2 ( z 1 )
I ( r ; 0 )             and             I ( r - r 1 ; r 1 )
δ r A = r I ( r ; 0 ) d r I ( r ; 0 ) d r
δ r B = ( r - r 1 ) I ( r - r 1 ; r 1 ) d r I ( r - r 1 ; r 1 ) d r ,
S A B = δ r A · δ r B ( δ r A ) 2 1 / 2 ( δ r B ) 2 1 / 2
S A B = d r d r r · r I ( r ; 0 ) I ( r ; r 1 ) [ d r d r r · r I ( r ; 0 ) I ( r ; 0 ) ] 1 / 2 × [ d r d r r · r I ( r ; r 1 ) I ( r ; r 1 ) ] - 1 / 2 .
I ( r ; 0 ) = τ ( f ; 0 ) exp ( 2 π i f · r / R ) d f , I ( r ; r 1 ) = τ ( f ; r 1 ) exp ( 2 π i f · r / R ) d f ,
S A B = f · f τ ( f ; 0 ) τ * ( f ; r 1 ) f = f = 0 [ f · f τ ( f ; 0 ) τ * ( f ; 0 ) ] f = f = 0 1 / 2 × [ f · f τ ( f ; r 1 ) τ * ( f ; r 1 ) ] f = f = 0 - 1 / 2 .
S A B = f · f τ ( f ; 0 ) τ * ( f ; r 1 ) f = f = 0 f · f τ ( f ; 0 ) τ * ( f ; 0 ) f = f = 0 .
P = [ 1 T 0 T τ ( f ; t ) d t ] [ 1 T 0 T τ * ( f ; t ) d t ] ,
P = 1 R θ 0 0 R θ 0 ( 1 - r 1 R θ 0 ) τ ( f ; 0 ) τ * ( f ; r 1 ) d r 1 ,
P = P τ ( f ; 0 ) 2 = 1 R θ 0 0 R θ 0 ( 1 - r 1 / R θ 0 ) τ ( f ; 0 ) τ * ( f ; r 1 ) d r 1 τ ( f ; 0 ) 2 .
ψ 1 ( r 1 ; v 1 ) ψ 1 * ( r 2 ; v 2 ) = ( 2 π k ) 2 0 κ d κ 0 z d z 1 Φ n ( κ ; z 1 ) J 0 ( κ | ( 1 - z 1 z ) v + z 1 z r | ) ,
ψ 1 ( r 1 , v 1 ) ψ 1 ( r 2 , v 2 ) = - ( 2 π k ) 2 0 κ d κ 0 z d z 1 Φ n ( κ ; z 1 ) × J 0 ( κ | ( 1 - z 1 z ) v + z 1 z r | ) exp [ - i κ 2 z 1 ( z - z 1 ) 2 k z ] .
U 2 = 4 ( 2 π k ) 2 0 z d z 1 0 κ d κ Φ n ( κ ; z 1 ) { 1 - J 0 ( κ λ f ( 1 - z 1 z ) ) - J 0 ( κ | ( 1 - z 1 z ) v - + z 1 z r | ) + 1 2 J 0 ( κ | ( 1 - z 1 z ) ( v - + λ f ) z 1 z r | ) exp [ - i κ 2 z 1 ( z - z 1 ) 2 k z ] + 1 2 J 0 ( κ | ( 1 - z 1 z ) ( v - - λ f ) + z 1 z r | ) exp [ i κ 2 z 1 ( z - z 1 ) 2 k z ] } ,
Φ n ( κ ; z 1 ) = 0.033 C n 2 ( z 1 ) κ - 11 / 3 .
U 2 = 24 5 ( 2 π k ) 2 0 d z 1 [ 0.033 C n 2 ( z 1 ) ] [ 2 - 8 / 3 Γ ( 1 / 6 ) Γ ( 11 / 6 ) { l 5 / 3 + a 5 / 3 } - d 5 / 6 4 Γ ( 1 / 6 ) × { e 5 i π / 12 F 1 1 ( - 5 / 6 ; 1 ; i b 2 / 4 d ) + e - 5 i π / 12 F 1 1 ( - 5 / 6 ; 1 ; - i c 2 / 4 d ) } ] ,
l = λ f ( 1 - z 1 z ) , a = | ( 1 - z 1 z ) v - + z 1 z r | , d = z 1 ( z - z 1 ) 2 k z , b = | ( 1 - z 1 z ) ( v - + λ f ) + z 1 z r | , c = | ( 1 - z 1 z ) ( v - - λ f ) + z 1 z r | ,
F 1 1 ( a ; b ; z ) = n = 0 Γ ( a + n ) Γ ( b ) Γ ( a ) Γ ( b + n ) z n n ! .
U 2 = 24 5 ( 2 π k ) 2 0 d z 1 [ 0.033 C n 2 ( z 1 ) ] · 2 - 8 / 3 Γ ( 1 / 6 ) Γ ( 11 / 6 ) [ l 5 / 3 + a 5 / 3 - 1 2 b 5 / 3 - 1 2 c 5 / 3 ] = 2 ( 2.915 ) k 2 0 d z 1 C n 2 ( z 1 ) [ ( λ f ) 5 / 3 + v - + z 1 δ ψ 5 / 3 - 1 2 v - + λ f + z 1 δ ψ 5 / 3 - 1 2 v - - λ f + z 1 δ ψ 5 / 3 ] .
2 δ ψ i 2 exp { - 1 2 U 2 } δ ψ = 0 = exp { - 1 2 U 2 } [ ( U 2 δ ψ i ) 2 - 2 U 2 δ ψ i 2 ] .
2 δ ψ i 2 τ ( q ; 0 ) τ * ( q , z δ ψ ) δ ψ = 0 = 4 π 2 d y S ( q , y , θ ) exp { - 1 2 U 2 ( 0 ) } · M i ,
M i = A i 2 - ( B i + Δ i ) ,
A i = 5 3 ( 6.88 ) α 5 / 3 C n 2 d z 1 [ d z 1 ( z 1 D ) C n 2 ( y 2 / 3 ê i - 1 2 ( y ê i + q ĝ i ) ( y 2 + q 2 + 2 q y cos θ ) - 1 / 6 - 1 2 ( y ê i - q ĝ i ) ( y 2 + q 2 - 2 q y cos θ ) - 1 / 6 ) ] , B i = 5 3 ( 6.88 ) α 5 / 3 C n 2 d z 1 [ d z 1 ( z 1 D ) 2 C n 2 { y - 1 / 3 ( 1 - 1 3 ĥ i ) - 1 2 ( y 2 + q 2 + 2 q y cos θ ) - 1 / 6 [ 1 - 1 3 ( y 2 + q 2 + 2 q y cos θ ) - 1 ( y ê i + q ĝ i ) 2 ] - 1 2 ( y 2 + q 2 - 2 q y cos θ ) - 1 / 6 [ 1 - 1 3 ( y 2 + q 2 - 2 q y cos θ ) - 1 ( y ê i - q ĝ i ) 2 ] } ] , Δ i = 6.88 α 5 / 3 [ q 5 / 3 + y 5 / 3 - 1 2 ( y 2 + q 2 + 2 q y cos θ ) 5 / 6 - 1 2 ( y 2 + q 2 - 2 q y cos θ ) 5 / 6 ] ,
ê = ( cos θ sin θ ) ,             g = ( 1 0 ) ,             h = ( cos 2 θ sin 2 θ ) .