Abstract

In short-exposure imaging through turbulence, there is some probability that the image will be nearly diffraction limited because the instantaneous wave-front distortion over the aperture was negligible. A number of years ago in a rather brief paper, Hufnagel (1966) argued heuristically that the probability of getting a good image would decrease exponentially with aperture area. This paper undertakes a rigorous quantitative analysis of the probability. We find that the probability of obtaining a good short-exposure image is Prob ≈ 5.6 exp[−0.1557 (D/r0)2] (for D/r0 ≥ 3.5), where D is the aperture diameter and r0 is the coherence length of the distorted wave front, as defined by Fried (1967). A good image is taken to be one for which the squared wave-front distortion over the aperture is 1 rad2 or less. The analysis is based on the decomposition of the distorted wave front over the aperture, in an orthonormal series with randomly independent coefficients. The orthonormal functions used are the eigenfunctions of a Karhunen-Loève integral equation. The integral equation is solved using a separation of variables into radial and azimuthal dependence. The azimuthal dependence was solved analytically and the radial, numerically. The first 569 radial eigenfunctions and eigenvalues were obtained. The probability of obtaining a good short-exposure image corresponds to a hyperspace integral in which the spatial dimensions are the independent random coefficients in the orthonormal series expansion. It is equal to the probability that a randomly chosen point in the hyperspace will lie within a hypersphere of unit radius, the points in the hyperspace being randomly chosen in accordance with the product of independent Gaussian probability distribution—one distribution for each dimension. The variance of these distrbutions is directly proportional to the eigenvalues of the Karhunen-Loève equation. This hyperspace integral (involving up to several hundred dimensions) has been evaluated using Monte Carlo techniques.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Hufnagel, in Restoration of Atmospherically Degraded Images (National Academy of Sciences, Washington, D. C., 1966), Vol. 3, Appendix 2, p. 11.
  2. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave-front,” Proc. IEEE 55, 57 (1967).
    [Crossref]
  3. D. L. Fried, “Statistics of a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427 (1965).
    [Crossref]
  4. W. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  5. D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front compensative systems,” J. Opt. Soc. Am. 66, 193 (1976).
    [Crossref]

1976 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave-front,” Proc. IEEE 55, 57 (1967).
[Crossref]

1965 (1)

Fried, D. L.

Greenwood, D. P.

Hufnagel, R. E.

R. E. Hufnagel, in Restoration of Atmospherically Degraded Images (National Academy of Sciences, Washington, D. C., 1966), Vol. 3, Appendix 2, p. 11.

Tatarski, W. I.

W. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave-front,” Proc. IEEE 55, 57 (1967).
[Crossref]

Other (2)

R. E. Hufnagel, in Restoration of Atmospherically Degraded Images (National Academy of Sciences, Washington, D. C., 1966), Vol. 3, Appendix 2, p. 11.

W. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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 (1)

FIG. 1
FIG. 1

Probability of obtaining a good short-exposure image. The probability is plotted on a logarithmic scale against the aperture diameter divided by r0 ratio squared. A straight line on this graph shows an exponential dependence of the probability on the aperture area. The data plotted corresponds to the values in Table II, with the spread due to the fact that Monte Carlo integral evaluation was used. The straight–line fit to the data matches Eq. (47).

Tables (2)

Tables Icon

TABLE I Eigenvalue listing, B n 2 = B p , q 2. Eigenvalues are ranked by magnitude, with ordinal number n assigned according to rank. Where the eigenvalue is degenerate (i.e., q ≠ 0), two values of n are assigned. This is indicated in the (p, q) numbering by listing both a positive and a negative value for q. The p values shown for each eigenvalue represent an ordinal ranking by magnitude for each value of the azimuthal number q. The eigenvalues are listed along with the cumulative sum of the eigenvalues. Whenever the eigenvalue is degenerate, the corresponding cumulative sum represents the addition of the eigenvalue twice.

Tables Icon

TABLE II Probability of obtaining a good short–exposure image.

Equations (56)

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

Prob 5.6 exp [ - 0.1557 ( D / r 0 ) 2 ] .
W ( r , D ) = { 1 , if r 1 2 D , 0 , if r > 1 2 D ,
ϕ ¯ = ( 1 4 π D 2 ) - 1 d r W ( r , D ) ϕ ( r ) ,
α = ( 1 64 π D 4 ) - 1 d r W ( r , D ) r ϕ ( r ) .
φ ( r ; D ) = ϕ ( r ) - ϕ ¯ - α · r .
D ( r - r ) = ϕ ( r ) - ϕ ( r ) 2 .
D ( r - r ) = 6.88 ( r - r / r 0 ) 5 / 3 .
r 0 = ( 0.423 k 2 path d s C N 2 ( s / L ) 5 / 3 ) - 3 / 5 ,
C φ ( r - r ; D ) = φ * ( r ; D ) φ ( r ; D ) .
d r W ( r , D ) f n ( r ; D ) f n ( r ; D ) = { 1 , if n = n , 0 , if n n .
φ ( r ; D ) = n β n f n ( r ; D ) .
β n = d r W ( r ; D ) f n * ( r , D ) φ ( r ; D ) .
β n * β n = { B n 2 ( D ) , if n = n , 0 , if n n ,
E = d r W ( r , D ) φ * ( r , D ) φ ( r ; D ) f n ( r ; D ) .
E = d r W ( r , D ) C φ ( r - r ; D ) f n ( r ; D ) .
E = d r W ( r ; D ) φ ( r ; D ) × n β n * f n * ( r ; D ) f n ( r ; D ) .
E = φ ( r ; D ) n β n d r W ( r , D ) f n * ( r ; D ) f n ( r ; D ) = φ ( r ; D ) β n * .
E = n β n f n ( r ; D ) β n * = n f n ( r ; D ) β n * β n * = B n 2 ( D ) f n ( r ; D ) .
d r W ( r , D ) C φ ( r - r ; D ) f n ( r ; D ) = B n 2 ( D ) f n ( r ; D ) .
C φ ( r - r ; D ) = [ ϕ ( r ) - ϕ ¯ - α · r ] * [ ϕ ( r ) - ϕ ¯ - α · r ] = ϕ * ( r ) ϕ ( r ) - ϕ * ( r ) ϕ ¯ - ϕ ( r ) ϕ ¯ * + ϕ ¯ * ϕ ¯ - φ * ( r ) α · r - ϕ ( r ) α ¯ * · r + ϕ ¯ * α · r + ϕ ¯ α · r + α ¯ * · r α · r .
x = r / D ,
x = r / D ,
C φ ( r - r ; D ) = ( D / r 0 ) 5 / 3 C ( x - x ) .
f n ( r ; D ) F ( x ) ,
B n 2 ( D ) = D 2 ( D / r 0 ) 5 / 3 B n 2 ,
W ( r ; D ) = M ( x ) = { 1 , if x 1 2 , 0 , if x > 1 2 ,
d x M ( x ) C ( x - x ) F n ( x ) = B n 2 F n ( x ) .
C φ ( r - r ; D ) = ( D / r 0 ) 5 / 3 { - G 0 ( ( r / D ) - ( r / D ) ) + G 1 ( r / D ) + G 1 ( r / D ) - G 2 + ( r / D ) cos θ G 3 ( r / D ) + ( r / D ) cos θ G 3 ( r / D ) - ( r / D ) ( r / D ) cos θ G 4 } ,
G 0 ( x ) = 3.44 x 5 / 3 ,
G 1 ( x ) = 3.44 ( 1 4 π ) - 1 0 1 / 2 d x x × 0 2 π d θ ( x 2 + x 2 - 2 x x cos θ ) 5 / 6 ,
G 2 ( x ) = 8 0 1 / 2 d x x G 1 ( x ) ,
G 3 ( x ) = 3.44 ( 1 64 π ) - 1 0 1 / 2 d x x 2 0 2 π d θ cos θ × ( x 2 + x 2 - 2 x x cos θ ) 5 / 6 ,
G 4 ( x ) = 64 0 1 / 2 d x x 2 G 3 ( x ) .
C ( x - x ) = G 0 ( x - x ) + G 1 ( x ) + G 1 ( x ) - G 2 + x cos θ G 3 ( x ) + x cos θ G 3 ( x ) - x x cos θ G 4 .
x ( x , θ ) ,
F n ( x ) R p q ( x ) exp ( i q θ ) .
n ( p , q ) ,
K q ( x , x ) = x 0 2 π d θ C ( x , x , θ ) exp ( i q θ ) ,
B n 2 B p , q 2 .
d x B ( x ) C ( x - x ) F n ( x ) = 0 1 / 2 d x x 0 2 π d θ C ( x , x , θ - θ ) R p q ( x ) exp ( i q θ ) = exp ( i q θ ) 0 1 / 2 d x K q ( x , x ) R p q ( x ) = B p , q 2 R p q ( x ) exp ( i q θ ) .
0 1 / 2 d x K q ( x , x ) R p q ( x ) = B p , q 2 R p q ( x ) .
K 0 ( x , x ) = - x 0 2 π d θ G 0 ( [ x 2 + x 2 - 2 x x cos θ ] 1 / 2 ) + 2 π x [ G 1 ( x ) + G 1 ( x ) - G 2 ] ,
K ± 1 ( x , x ) = x 0 2 π d θ G 0 ( [ x 2 + x 2 - 2 x x cos θ ] 1 / 2 exp ( i θ ) + π x [ x G 3 ( x ) + G 3 ( x ) - x x G 4 ] ,
K q ( x , x ) = x 0 2 π G 0 ( [ x 2 + x 2 - 2 x x cos θ ] 1 / 2 ) exp ( i q θ ) ,             q = ± 2 , ± 3 , ± 4 , .
n = 1 B n = 0.1056.
n = 1 569 B n = 0.1047 = 0.9915 n = 1 B n ,
Δ 2 = ( 1 4 π D 2 ) - 1 d r W ( r , D ) φ ( r ; D ) 2 .
Δ 2 = ( 1 4 π D 2 ) - 1 n , n β n * β n d r W ( r , D ) f n * ( r ) f n ( r ) .
Δ 2 = ( 1 4 π D 2 ) - 1 n β n * β n = n S n 2 ,
S n = [ ( 1 4 π D 2 ) - 1 β n * β n ] 1 / 2 .
σ n 2 = S n 2 .
σ n 2 = ( 1 4 π D 2 ) - 1 β n * β n = ( 1 4 π D 2 ) - 1 B n 2 ( D ) .
σ n 2 = ( 4 / π ) ( D / r 0 ) 5 / 3 B n 2 .
Prob ( good short - exposure image ) = Prob ( n S n 2 1 rad 2 )
Prob ( good short - exposure image ) = n = 1 limits d S n ( 2 π σ n 2 ) - 1 / 2 × exp ( - 1 2 S n 2 / σ n 2 ) .
Prob 5.6 exp [ - 0.1557 ( D / r 0 ) 2 ] ( if D / r 0 3.5 )