Abstract

A new technique (speckle interferometry) has been developed by Gezari, Labeyrie, and Stachnik, which allows the measurement of stellar diameters from a series of photographs obtained from large-aperture ground-based telescopes. The series of photographs is processed to obtain the Weiner spectrum of the photographic image, i.e., the ensemble-averaged modulus-squared Fourier transform obtained from the series of images. Gezari, Labeyrie, and Stachnik have measured stellar diameters as small as 0.05, about 20 times better than is usually possible. In this paper, mathematical expressions are obtained for the Wiener spectrum of the image of a point source. As is well known, the Wiener spectrum of the image of an extended, incoherently radiating object, is expressible as a product of this point-source spectrum and the object spectrum. Calculations are performed using the Rytov approximation and assuming that the underlying atmospheric turbulence is describable by a Kolmogorov spectrum. Asymptotic closed-form expressions are obtained for angular frequencies much less than, and much greater than, the conventional seeing limit. In the latter case, the Wiener spectrum is found to be proportional to the optical transfer function.

© 1973 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).
  2. D. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  3. The seeing angle can be estimated from Fried’s definition (Ref. 2) of infinite-aperture long-exposure resolution and is typically of the order of 1 arc second for zenith viewing.
  4. It may be shown that <τ(0)> = 1, because the atmosphere does not absorb light, but only redistributes it. Hence, IT= <∫I(x)dx> (i.e., the averaged integrated irradiance of the image) is unaffected by the turbulence. If <|τ(0)|2> = 1, so that the normalized <|τ(f)|2> equals the unnormalized <|τ(f)|2>, this would imply that <IT2>is unaffected by the turbulence. This statement is, in general, not true, because <IT2>≠<IT>2—i.e., the atmosphere produces scintillation. Nevertheless, it is approximately true if geometric optics is valid and/or the aperture averaging effectively eliminates the scintillation. This latter condition will, in general, be satisfied for values of D much greater than r0.
  5. Cf. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).
  6. L. S. Taylor, J. Opt. Soc. Am. 58, 705 (1968).
    [Crossref]
  7. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, NTIS, Springfield, Va., 1971), p. 235.
  8. R. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964), p. 56.
    [Crossref]
  9. Reference 2, p. 1374.
  10. Reference 7, p. 239, Eqs. 36a and 37.
  11. Reference 2, p. 1377.
  12. Reference 2, p. 1366.
  13. Reference 7, pp. 74–102.
  14. R. Hufnagel, Restoration of Atmospherically Degraded ImagesWoods Hole Summer Study, Vol. 2, App. 3 (Defense Documentation Center, Alexandria, Va., 1966).
  15. Reference 5, Sec. 10.4.
  16. D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5(3), 187 (1972).
    [Crossref]
  17. D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1–L5 (1April1972).
    [Crossref]

1972 (2)

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5(3), 187 (1972).
[Crossref]

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1–L5 (1April1972).
[Crossref]

1971 (1)

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

1968 (1)

1966 (1)

1964 (1)

Born, M.

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

Dryden, G.

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5(3), 187 (1972).
[Crossref]

Fried, D.

Gezari, D.

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1–L5 (1April1972).
[Crossref]

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Hufnagel, R.

R. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964), p. 56.
[Crossref]

R. Hufnagel, Restoration of Atmospherically Degraded ImagesWoods Hole Summer Study, Vol. 2, App. 3 (Defense Documentation Center, Alexandria, Va., 1966).

Korff, D.

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5(3), 187 (1972).
[Crossref]

Labeyrie, A.

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1–L5 (1April1972).
[Crossref]

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Miller, M.

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5(3), 187 (1972).
[Crossref]

Stachnik, R.

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1–L5 (1April1972).
[Crossref]

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Stanley, N. R.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, NTIS, Springfield, Va., 1971), p. 235.

Taylor, L. S.

Wolf, E.

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

Astrophys. J. (1)

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1–L5 (1April1972).
[Crossref]

Bull. Am. Astron. Soc. (1)

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5(3), 187 (1972).
[Crossref]

Other (11)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, NTIS, Springfield, Va., 1971), p. 235.

The seeing angle can be estimated from Fried’s definition (Ref. 2) of infinite-aperture long-exposure resolution and is typically of the order of 1 arc second for zenith viewing.

It may be shown that <τ(0)> = 1, because the atmosphere does not absorb light, but only redistributes it. Hence, IT= <∫I(x)dx> (i.e., the averaged integrated irradiance of the image) is unaffected by the turbulence. If <|τ(0)|2> = 1, so that the normalized <|τ(f)|2> equals the unnormalized <|τ(f)|2>, this would imply that <IT2>is unaffected by the turbulence. This statement is, in general, not true, because <IT2>≠<IT>2—i.e., the atmosphere produces scintillation. Nevertheless, it is approximately true if geometric optics is valid and/or the aperture averaging effectively eliminates the scintillation. This latter condition will, in general, be satisfied for values of D much greater than r0.

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

Reference 2, p. 1374.

Reference 7, p. 239, Eqs. 36a and 37.

Reference 2, p. 1377.

Reference 2, p. 1366.

Reference 7, pp. 74–102.

R. Hufnagel, Restoration of Atmospherically Degraded ImagesWoods Hole Summer Study, Vol. 2, App. 3 (Defense Documentation Center, Alexandria, Va., 1966).

Reference 5, Sec. 10.4.

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 (11)

Fig. 1
Fig. 1

Schematic arrangement of the GLS experiment. Laser A irradiates frame from film roll B and transmitted light is focused by lens C onto film D. Irradiance of the image recorded on film D is |τ(f)|2 (for an unresolved object). Film roll B is turned from frame to frame without changing film D. If each frame of roll B represents an ensemble member of the atmosphere, then the signal at D is N〈|τ(f)|2〉, where N is the number of frames. The variance of 〈|τ(f)|2〉 is the noise, and the signal-to-noise ratio varies as √N.

Fig. 2
Fig. 2

Illustration of the manner in which a collection of random speckle patterns, when averaged, tends to a uniform spot.

Fig. 3
Fig. 3

Schematic representation of the four-circle overlap. Only the shaded area contributes to the integral of Eq. (20).

Fig. 4
Fig. 4

Turbulence parameter r 0 as a function of observer altitude and elevation angle. The Hufnagel night model has been used for the turbulent atmosphere.

Fig. 5
Fig. 5

〈|τ(f)|2〉 vs λf/D for D / r 0 = 1.17, 11.7, and 38.4, corresponding to D ~ 15 cm, 1.5 m, and 5 m for 0.9-arc-second seeing angle.

Fig. 6
Fig. 6

〈|τ(f)|2〉 vs λf/D for λ f / r 0 = 1.5 (upper curve) and for λ f / r 0 = 6 (lower curve).

Fig. 7
Fig. 7

〈|τ(f)|2〉 vs λf/D for λ f / r 0 = 15, 30, and 60.

Fig. 8
Fig. 8

q2τ0(q) vs q, illustrating the efficiency of the system for determining information at fixed value of λ f / r 0. Optimum lens diameter is one for which λf/D = 0.58.

Fig. 9
Fig. 9

Minimum-useful diameter for observing angular detail Δθ.

Fig. 10
Fig. 10

Maximum-obtainable 〈|τ(f)|2〉 vs λ f / r 0.

Fig. 11
Fig. 11

Illustration of the formation of the speckle pattern. Two points at distance Δu apart on the lens give rise to fringes of angular separation λ/Δu at the film. The angular frequency f is the inverse of this angle. Hence, Δu ~ λf.

Equations (60)

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

| I ( f ) | 2 = | O ( f ) | 2 | τ ( f ) | 2 ,
τ ( f ) = B d x U * ( x ) U ( x ) exp [ 2 π i f · x ] ,
| τ ( f ) | 2 = B 2 d v d v W ( v ) W ( v λ f ) W ( v ) W ( v λ f ) × U ( v ) U * ( v λ f ) U * ( v ) U ( v λ f ) ,
W ( v ) = 1 if υ | v | < D / 2 ( D denotes lens diameter ) = 0 if υ > D / 2 ,
l 0 7 / 3 L 0 5 / 3 / λ 2 ( L / 21 ) 2 ,
l 0 = ( 10 9 H ) 1 3 , L 0 = ( 4 H ) 1 2 ,
| τ ( f ) | 2 = B 2 d v d v W ( v ) W ( v λ f ) W ( v ) W ( v λ f ) × exp { [ l ( v ) + l ( v ) + l ( v λ f ) + l ( v λ f ) ] + [ i ϕ ( v ) ϕ ( v λ f ) ϕ ( v ) + ϕ ( v λ f ) ] } ,
W ( v ) = 1 if | v | υ < D / 2 = 0 if υ > D / 2 .
| τ ( f ) | 2 = B 2 d v d v W ( v ) W ( v λ f ) W ( v ) W ( v λ f ) × exp { l ( v ) + l ( v ) + l ( v λ f ) + l ( v λ f ) } × exp i { ϕ ( v ) ϕ ( v λ f ) ϕ ( v ) + ϕ ( v λ f ) } .
A l exp { l ( v ) + l ( v λ f ) + l ( v ) + l ( v λ f ) } = exp [ 4 l ¯ + 1 2 { l ( v ) l ¯ ) + ( l ( v ) l ¯ ) + ( l ( v λ f ) l ¯ ) + ( l ( v λ f ) l ¯ ) } 2 ] ,
[ l ( a ) l ¯ ] [ l ( b ) l ¯ ] C l ( a , b ) = C l ( | a b | ) ,
l ¯ = C l ( 0 ) .
A l = exp [ 2 C l ( 0 ) + 2 C l ( Δ υ ) + 2 C l ( λ R f ) + C l ( | Δ v + λ f | ) + C l ( | Δ v λ f | ) ] ,
D l ( | a b | ) = 2 ( C l ( 0 ) C l ( | a b | ) ) = [ l ( a ) l ( b ) ] 2 ,
A l = exp 4 C l ( 0 ) [ exp D l ( λ f ) D l ( Δ υ ) + 1 2 ( D l ( | Δ v + λ f | ) + D l ( | Δ v λ f | ) ) ] .
A ϕ = exp i [ ϕ ( v ) ϕ ( v λ f ) ϕ ( v ) + ϕ ( v λ f ) ] .
A ϕ = exp [ D ϕ ( λ f ) + D ϕ ( Δ υ ) 1 2 D ϕ ( | Δ v + λ f | ) 1 2 D ϕ ( | Δ v λ f | ) ] ,
[ d v d v W ( v ) W ( v ) W ( v ) W ( v ) Q ( Δ v , 0 ) ] 1 .
| τ ( f ) | 2 = d v d v W ( v λ f ) W ( v ) W ( v ) W ( v λ f ) Q ( Δ v , f ) / d v d v W ( v ) W ( v ) W ( v ) W ( v ) Q ( Δ v , 0 ) ,
Q ( Δ v , f ) exp [ D ( λ f ) + D ( Δ υ ) + 1 2 D ( | Δ v λ f | ) + 1 2 D ( | Δ v λ f | ) ] × exp [ D ϕ ( | Δ v + λ f | ) + D ϕ ( | Δ v λ f | ) ]
D ( a ) = D l ( a ) + D ϕ ( a ) .
D ϕ ( a ) = D ( a ) if D 2 λ L ( near field ) , D ϕ ( a ) = 1 2 D ( a ) if D 2 λ L ( far field ) .
D ( a ) = 6.88 ( a r 0 ) 5 / 3 ,
r 0 = [ 0.42 k 2 0 L C n 2 ( z ) d S ] 3 / 5 ,
Q ( Δ v , f ) = exp D ( λ f ) exp ( 6.88 { ( Δ υ r 0 ) 5 / 3 1 2 [ ( | Δ v + λ f | r 0 ) 5 / 3 + ( | Δ v + λ f | r 0 ) 5 / 3 ] } ) .
Δ v = v v , v + = v + v ,
| τ ( f ) | 2 = d Δ v Q ( f , Δ v ) S ( f , Δ v ) / [ d υ W 2 ( υ ) ] 2 ,
S ( f , Δ v ) = d v + W ( Δ v + v + 2 λ f 2 ) W ( Δ v + v + 2 ) × W ( v + Δ v 2 λ f 2 ) W ( v + Δ v 2 ) ,
W ( Δ v + v + 2 λ f 2 ) = 1 when | Δ v + v + 2 λ f | < D ,
v + 1 = 2 λ f Δ v , v + 2 = Δ v , v + 3 = 2 λ f + Δ v , v + 4 = Δ v .
λ f D q , Δ v D y , a [ q 2 + y 2 + 2 q y cos θ ] 1 2 , α = D r 0 , θ c sin 1 q , y c | ( 1 q 2 ) 1 2 sin θ q cos θ | , τ ( q ) τ ( f ) , S ( q , y ) S ( f , Δ v ) , Q ( q , y ) Q ( f , Δ v ) .
S = 1 2 [ cos 1 a a ( 1 a 2 ) 1 2 ] .
S = 1 2 [ sin 1 l B A + sin 1 l B D l B A × ( 1 l 2 B A ) 1 2 l B D ( 1 l 2 B D ) ] + 2 [ s ( s l B A ) ( s l B D ) ( s l D A ) ] 1 2 ,
l B A = { [ 1 2 cos ( θ cos 1 y ) q 2 y cos θ ] 2 + [ 1 2 sin ( θ cos 1 y ) + ( 1 q 2 ) 1 2 2 y sin θ ] 2 } 1 2 , l D B = { [ 1 2 cos ( θ cos 1 y ) q 2 ] 2 + [ 1 2 sin ( θ cos 1 y ) ( 1 q 2 ) 1 2 2 ] 2 } 1 2 , l D A = { y 2 + ( 1 q 2 ) 2 y ( sin θ ) ( 1 q 2 ) 1 2 } 1 2 .
Q ( q , y ) = exp [ 6.88 ( q ) 5 / 3 ( α ) 5 / 3 ] × exp ( 6.88 { ( y ) 5 / 3 ( α ) 5 / 3 1 2 [ ( | y + q | ) 5 / 3 ( α ) 5 / 3 + ( | y q | ) 5 / 3 ( α ) 5 / 3 ] } ) .
τ ( f ) LE 2 = exp ( 6.88 q 5 / 3 α 5 / 3 ) τ 0 2 ( q )
τ ( f ) SE 2 = exp ( 6.88 q 5 / 3 α 5 / 3 ) ( 1 q 5 / 3 ) τ 0 2 ( q ) ,
τ 0 2 ( q ) = 4 π 2 ( cos 1 q q ( 1 q 2 ) 1 2 ) 2
exp { 3.44 ( λ r 0 ( Δ θ ) A ) 5 / 3 } 0.1 ,
Δ θ A λ 0.91 r 0 .
λ f r 0 0.91 Δ θ A Δ θ ,
| τ ( f ) | 2 : α q 1 L i m i t
| τ ( f ) | 2 = ( 4 π ) 2 0 2 π d θ G ( q , θ ) y d y exp [ 6.88 α 5 / 3 ( q 5 / 3 + y 5 / 3 1 2 | q + y | 5 / 3 1 2 | q + y | 5 / 3 ) ] S ( y , θ , q ) ,
G ( q , θ ) q cos θ + ( 1 q 2 sin 2 θ ) 1 2 .
| τ ( f ) | 2 64 π 2 0 π / 2 d θ 0 G y d y S ( q , y , θ ) ( exp 6.88 α 5 / 3 q 5 / 3 × [ 1 ( q y ) 1 3 ( 5 6 5 18 cos 2 θ ) ] ) { τ 0 2 ( f ) + 64 ( 6.88 ) π 2 ( α q ) 5 / 3 q 1 3 × 5 6 0 π / 2 d θ 0 G d y y 2 3 S ( q , y , θ ) ( 1 cos 2 θ 3 ) } × exp 6.88 ( α q ) 5 / 3 .
5 6 0 π / 2 d θ 0 G d y y 2 3 S ( q , y , θ ) ( 1 cos 2 θ 3 ) = 5 6 0 π / 2 d θ 0 G y d y S ( q , y , θ ) [ y 1 3 ( 1 cos 2 θ 3 ) ] .
| τ ( f ) | 2 { τ 0 2 ( f ) + 64 ( 6.88 ) π 2 ( α q ) 5 / 3 q 5 / 3 ( 0.94 ) × 0 π / 2 d θ 0 G y d y S ( q , y , θ ) } exp 6.88 ( α q ) 5 / 3 ,
| τ ( f ) | 2 = α q 1 L i m i t
| τ ( f ) | 2 = 64 π 2 0 π / 2 d θ 0 G y d y exp { 6.88 ( α y ) 5 / 3 × ( 1 ( y q ) 1 3 [ 5 6 5 18 cos 2 θ ] ) } S ( q , y , θ ) = 64 π 2 { 0 π / 2 d θ 0 G y d y [ exp 6.88 ( α y ) 5 / 3 ] S ( q , y , θ ) + 0 π / 2 d θ 0 G y d y [ exp 6.88 ( α y ) 5 / 3 ] S ( q , y , θ ) ( α y ) 5 / 3 × ( y q ) 1 3 [ 5 6 5 18 cos 2 θ ] } .
S ( q , y , θ ) S ( q , 0 , 0 ) .
| τ ( f ) | 2 16 π τ 0 ( q ) { π 2 0 G y d y exp [ 6.88 ( α y ) 5 / 3 ] + ( 5 6 · π 2 5 18 · π 4 ) 0 G d y y 3 exp [ 6.88 ( α y ) 5 / 3 ] × ( 6.88 ) ( α 5 / 3 q 1 3 ) } 8 τ 0 ( q ) { ( 6.88 ) 6 / 5 α 2 0 α G u d u exp [ u 5 / 3 ] + 25 36 ( 6.88 ) 7 / 5 1 α 7 / 3 q 1 3 0 α G d u u 3 exp [ u 5 / 3 ] } .
| τ ( f ) | 2 q 2 τ 0 ( q ) ( α q ) 2 [ 0.435 + 0.278 ( α q ) 1 3 ] .
D min 1.44 λ ( Δ θ ) 1 .
D 2 ( 6 cm ) 2
D 2 ( 15 cm ) 2
| I ( f ) 2 | = | τ ( f ) 2 | O ( f ) | 2 .
N = ( λ / r 0 ) 2 4 ( λ / D ) 2 = D 2 4 r 0 2 .
I ( x ) = | | I ( f ) | 2 e i 2 π f · x d f | 2 = | d f d x 1 d x 2 I ( x 1 ) I ( x 2 ) e i 2 π f · ( x 1 x 2 x ) | 2 = | d x 1 I ( x 1 ) I ( x 1 x ) | 2 .
I ( x ) | A 2 ( C ( x ) ) | 2 ,
| τ ( f ) | 2 τ ( f ) SE 2 , λ f r 0 ,