Abstract

A precise statistical definition is established for the geometric “shape” of a randomly distorted wavefront. Relationships between the phase-structure function and the statistics governing the shape are derived. The most significant portion of wavefront distortion caused by atmospheric turbulence is a random tilting of the plane-wave front. A procedure is outlined for calculating the influence of wavefront distortion on optical systems. Estimates are formed of the effect of wave-front distortion on photographic resolution and optical heterodyne efficiency.

© 1965 Optical Society of America

Full Article  |  PDF Article

Corrections

D. L. Fried, "Errata: Statistics of a Geometric Representation of Wave-front Distortion.," J. Opt. Soc. Am. 56, 410-410 (1966)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-56-3-410

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961).
  2. See Ref. 1. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964). D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 54, 574A (1964).
    [CrossRef]
  3. It can be shown theoretically that phase fluctuation, because of the central limit theorem, has a Gaussian distribution. Since a Gaussian distribution is completely described (except for a mean value) by its second moment, and since D(r) is the second moment for differential-phase fluctuation (which difference has zero mean), we conclude that D(r) completely specifies the statistics of phase fluctuation.
  4. D. L. Fried, “The Effect of Wave-Front Distortion on the Performance of an Ideal Optical Heterodyne Receiver and an Ideal Camera,” presented at the Conference on Atmospheric Limitations to Optical Propagation at the U. S. National Bureau of Standards CRPL, 18–19 March 1965.
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
  6. A. Kolomogoroff in Turbulence, Classic Papers on Statistical Theory, edited by S. K. Friedlander and L. Topper (Interscience Publishers, Inc., New York, 1961), p. 151.
  7. W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Berlin/Vienna, 1961), Vol II, Eqs. (121.1) and (341.5a).
    [CrossRef]
  8. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1963), pp. 87, 106.

1964 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Fried, D. L.

D. L. Fried, “The Effect of Wave-Front Distortion on the Performance of an Ideal Optical Heterodyne Receiver and an Ideal Camera,” presented at the Conference on Atmospheric Limitations to Optical Propagation at the U. S. National Bureau of Standards CRPL, 18–19 March 1965.

Grobner, W.

W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Berlin/Vienna, 1961), Vol II, Eqs. (121.1) and (341.5a).
[CrossRef]

Hofreiter, N.

W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Berlin/Vienna, 1961), Vol II, Eqs. (121.1) and (341.5a).
[CrossRef]

Hufnagel, R. E.

Kolomogoroff, A.

A. Kolomogoroff in Turbulence, Classic Papers on Statistical Theory, edited by S. K. Friedlander and L. Topper (Interscience Publishers, Inc., New York, 1961), p. 151.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1963), pp. 87, 106.

Stanley, N. R.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

J. Opt. Soc. Am. (1)

Other (7)

It can be shown theoretically that phase fluctuation, because of the central limit theorem, has a Gaussian distribution. Since a Gaussian distribution is completely described (except for a mean value) by its second moment, and since D(r) is the second moment for differential-phase fluctuation (which difference has zero mean), we conclude that D(r) completely specifies the statistics of phase fluctuation.

D. L. Fried, “The Effect of Wave-Front Distortion on the Performance of an Ideal Optical Heterodyne Receiver and an Ideal Camera,” presented at the Conference on Atmospheric Limitations to Optical Propagation at the U. S. National Bureau of Standards CRPL, 18–19 March 1965.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

A. Kolomogoroff in Turbulence, Classic Papers on Statistical Theory, edited by S. K. Friedlander and L. Topper (Interscience Publishers, Inc., New York, 1961), p. 151.

W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Berlin/Vienna, 1961), Vol II, Eqs. (121.1) and (341.5a).
[CrossRef]

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1963), pp. 87, 106.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., 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.


Equations (129)

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

D ( r ) = [ ϕ ( x ) - ϕ ( x ) ] 2 ,
r = x - x ,
a + b x + c y + d x 2 + e x y + f y 2 .
W ( x , D ) { 1 , if x < R 0 , if x > R .
F 1 ( x ) ( π R 2 ) - 1 2
F 2 ( x ) ( π R 4 / 4 ) - 1 2 ( x ) ,
F 3 ( x ) ( π R 4 / 4 ) - 1 2 ( y ) ,
F 4 ( x ) ( π R 6 / 12 ) - 1 2 ( x 2 + y 2 - R 2 / 2 )
F 5 ( x ) ( π R 6 / 6 ) - 1 2 ( x 2 - y 2 ) ,
F 6 ( x ) ( π R 6 / 24 ) - 1 2 ( x y ) ,
d x W ( x , D ) F μ ( x ) F ν ( x ) = δ μ ν ,
δ μ ν { 1 , if μ = ν 0 , if μ ν .
Φ ( x ) = μ = 1 a μ F μ ( x ) ,
Δ ( 1 / π R 2 ) d x W ( x , D ) [ ϕ ( x ) - Φ ( x ) ] 2 .
( / a μ ) Δ = 0.
a μ = d x W ( x , D ) ϕ ( x ) F μ ( x ) .
( a C ) 2 ( a 1 ) 2 ,
( a L ) 2 ( a 2 ) 2 + ( a 3 ) 2 ,
( a S ) 2 ( a 4 ) 2 ,
( a Q ) 2 ( a 4 ) 2 + ( a 5 ) 2 + ( a 6 ) 2 .
Φ j ( x ) = μ = 1 n j a μ F μ ( x ) ,
Δ j ( 1 / π R 2 ) d x W ( x , D ) [ ϕ ( x ) - Φ j ( x ) ] 2 .
Δ = 1 π R 2 d x W ( x , D ) ϕ 2 ( x ) - 2 π R 2 μ = 1 a μ d x W ( x , D ) ϕ ( x ) F μ ( x ) + μ , ν = 1 a μ a ν 1 π R 2 d x W ( x , D ) F μ ( x ) F ν ( x ) .
Δ = 1 π R 2 d x W ( x , D ) ϕ 2 ( x ) - 2 π R 2 μ = 1 a μ d x W ( x , D ) ϕ ( x ) F μ ( x ) + 1 π R 2 μ = 1 ( a μ ) 2 .
a μ = d x W ( x , D ) ϕ ( x ) F μ ( x ) ,
Δ = 1 π R 2 d x W ( x , D ) ϕ 2 ( x ) - 1 π R 2 μ = 1 ( a μ ) 2 .
Δ = 1 π R 2 d x W ( x , D ) ϕ 2 ( x ) - 1 π R 2 d x d x W ( x , D ) × W ( x , D ) μ = 1 F μ ( x ) F μ ( x ) ϕ ( x ) ϕ ( x ) .
Δ j = 1 π R 2 d x W ( x , D ) ϕ 2 ( x ) - 1 π R 2 μ = 1 n j ( a μ ) 2 ,
Δ j = 1 π R 2 d x W ( x , D ) ϕ 2 ( x ) - 1 π R 2 d x d x W ( x , D ) × W ( x , D ) i = 1 n j F μ ( x ) F μ ( x ) ϕ ( x ) ϕ ( x ) .
( a L ) 2 = Δ C - Δ L ,
( a S ) 2 = Δ L - Δ S ,
( a Q ) 2 = Δ L - Δ Q .
1 π R 2 d x W ( x , D ) ϕ 2 ( x ) = 1 π R 2 d x d x W ( x , D ) × W ( x , D ) μ = 1 n j F μ ( x ) F μ ( x ) ϕ 2 ( x ) .
1 π R 2 d x W ( x , D ) ϕ 2 ( x ) = 1 2 π R 2 d x d x W ( x , D ) × W ( x , D ) μ = 1 n j F μ ( x ) F μ ( x ) [ ϕ 2 ( x ) + ϕ 2 ( x ) ] .
r = 1 2 ( x + x ) ,
r = x - x ,
Δ j = 1 2 π R 2 d r d r W ( r + 1 2 r , D ) W ( r - 1 2 r , D ) × μ = 1 n j F μ ( r + 1 2 r ) F μ ( r - 1 2 r ) D ( r ) .
j ( r , D ) = d r W ( r + 1 2 r , D ) W ( r - 1 2 r , D ) × μ = 1 n j F μ ( r + 1 2 r ) F μ ( r - 1 2 r ) .
C ( r , D ) = ( 1 / π ) { 2 cos - 1 ( r / D ) - 2 ( r / D ) [ 1 - ( r / D ) 2 ] 1 2 } W ( r , 2 D ) ,
L ( r , D ) = ( 1 / π ) { 6 cos - 1 ( r / D ) - [ 14 ( r / D ) - 8 ( r / D ) 3 ] × [ 1 - ( r / D ) 2 ] 1 2 } W ( r , 2 D ) ,
S ( r , D ) = ( 1 / π ) { 8 cos - 1 ( r / D ) - [ 24 ( r / D ) - ( 80 / 3 ) ( r / D ) 3 + ( 32 / 3 ) ( r / D ) 5 ] × [ 1 - ( r / D ) 2 ] 1 2 } W ( r , 2 D ) ,
Q ( r , D ) = ( 1 / π ) { 12 cos - 1 ( r / D ) - [ 44 ( r / D ) - 64 ( r / D ) 3 + 32 ( r / D ) 5 ] × [ 1 - ( r / D ) 2 ] 1 2 } W ( r , 2 D ) .
Δ j = 1 R 2 0 D r d r j ( r , D ) D ( r ) ,
( a L ) 2 = 1 R 2 0 D r d r [ C ( r , D ) - L ( r , D ) ] D ( r ) ,
( a S ) 2 = 1 R 2 0 D r d r [ L ( r , D ) - S ( r , D ) ] D ( r ) ,
( a Q ) 2 = 1 R 2 0 D r d r [ L ( r , D ) - Q ( r , D ) ] D ( r ) .
D ( r ) = A r 5 / 3 ,
r 0 ( 6.88 / A ) 3 / 5 .
D ( r ) = 6.88 ( r / r 0 ) 5 / 3 .
u = r / D ,
j ( r , D ) = j ( u , 1 ) ,
Δ j = 27.5 ( D / r 0 ) 5 / 3 0 1 u d u j ( u , 1 ) u 5 / 3 .
0 1 z 2 α + 1 ( 1 - z 2 ) β d z = 1 2 B ( α + 1 , β + 1 ) ,
0 1 z α cos - 1 ( z ) d z = 1 2 ( α + 1 ) B ( α + 2 2 , 1 2 ) ,
I j = 0 1 u 8 / 3 j ( u , 1 ) d u ,
I C 3.68 × 10 - 2 ,
I L 4.73 × 10 - 3 ,
I S 3.96 × 10 - 3 ,
I Q 2.29 × 10 - 3 ,
Δ C 1.013 ( D / r 0 ) 5 / 3 ,
Δ L 0.1301 ( D / r 0 ) 5 / 3 ,
Δ S 0.1090 ( D / r 0 ) 5 / 3 ,
Δ Q 0.0630 ( D / r 0 ) 5 / 3 .
( a L ) 2 0.883 ( D / r 0 ) 5 / 3 ,
( a S ) 2 0.0211 ( D / r 0 ) 5 / 3 ,
( a Q ) 2 0.0671 ( D / r 0 ) 5 / 3 .
Δ * = 4 π ( D j * ) 2 d x W ( x , D j * ) [ ϕ ( x ) - Φ j ( x ) ] 2 .
A j * ( π / 4 ) ( D j * ) 2 .
D C * 0.992 r 0 ( Δ * ) 3 / 5 ,
D L * 3.40 r 0 ( Δ * ) 3 / 5 ,
D S * 3.79 r 0 ( Δ * ) 3 / 5 ,
D Q * 5.26 r 0 ( Δ * ) 3 / 5 .
A C * / n C 0.77 r 0 2 ( Δ * ) 6 / 5 ,
A L * / n L 3.03 r 0 2 ( Δ * ) 6 / 5 ,
A S * / n S 2.82 r 0 2 ( Δ * ) 6 / 5 ,
A Q * / n Q 3.63 r 0 2 ( Δ * ) 6 / 5 .
1 R 2 0 D r d r j ( r , D ) r 6 / 3 = { R 2 / 2 if j = C 0 if j C
K 0 ( r , D ) = d r W ( r + 1 2 r , D ) W ( r - 1 2 r , D ) ,
K 0 ( r , D ) = d r W ( r + 1 2 r , D ) W ( r - 1 2 r , D ) r 2 ,
K 2 ( r , D ) = d r W ( r + 1 2 r , D ) W ( r - 1 2 r , D ) r 4 ,
K 3 ( r , D ) = d r W ( r + 1 2 r , D ) × W ( r - 1 2 r , D ) ( r · r ) 2 ,
J ( m , n ; r , D ) = 2 0 R - 1 2 r d p - [ R 2 - ( R + 1 2 r ) 2 ] 1 2 + [ R 2 - ( R + 1 2 r ) 2 ] 1 2 d q p m q n ,
L ( m , n ; r , D ) = r / D 1 d v v m ( 1 - v 2 ) ( n + 1 ) / 2 .
i = 1 n C F i ( x ) F i ( x ) = 1 π R 2 ,
i = 1 n L F i ( x ) F i ( x ) = 1 π R 4 ( 4 r 2 - r 2 + R 2 ) ,
i = 1 n S F i ( x ) F i ( x ) = 1 π R 6 [ 12 r 4 + ( 6 r 2 - 8 R 2 ) r 2 + ( 3 4 r 4 - 4 R 2 r 2 + 4 R 4 ) - 12 ( r · r ) 2 ] ,
i = 1 n Q F i ( x ) F i ( x ) = 1 π R 6 { 18 r 4 + ( - 3 r 2 - 8 R 2 ) r 2 + [ ( 9 / 8 ) r 4 - 4 R 2 r 2 + 4 R 4 ] - 6 ( r · r ) 2 } .
C ( r , D ) = ( 1 / π R 2 ) K 0 ( r , D ) ,
L ( r , D ) = ( 1 / π R 4 ) [ 4 K 1 ( r , D ) + ( R 2 - r 2 ) K 0 ( r , D ) ] ,
S ( r , D ) = ( 1 / π R 6 ) [ 12 K 2 ( r , D ) + ( 6 r 2 - 8 R 2 ) K 1 ( r , D ) + ( 3 4 r 4 - 4 R 2 r 2 + 4 R 4 ) K 0 ( r , D ) - 12 K 3 ( r , D ) ] ,
Q ( r , D ) = ( 1 / π R 6 ) { 18 K 2 ( r , D ) + ( - 3 r 2 - 8 R 2 ) K 1 ( r , D ) + [ ( 9 / 8 ) r 4 - 4 R 2 r 2 + 4 R 4 ] K 0 ( r , D ) - 6 K 3 ( r , D ) } .
K 0 ( r , D ) = J ( 0 , 0 ; r , D ) W ( r , 2 D ) ,
K 1 ( r , D ) = [ J ( 2 , 0 ; r , D ) + J ( 0 , 2 ; r , D ) ] W ( r , 2 D ) ,
K 2 ( r , D ) = [ J ( 4 , 0 ; r , D ) + 2 J ( 2 , 2 ; r , D ) + J ( 0 , 4 ; r , D ) ] W ( r , 2 D ) ,
K 3 ( r , D ) = r 2 J ( 2 , 0 ; r , D ) W ( r , 2 D ) ,
r 2 = p 2 + q 2 ,
r 4 = p 4 + 2 p 2 q 2 + q 4 ,
( r · r ) 2 = r 2 p 2 .
v = ( 2 p + r ) / D ,
J ( m , n ; r , D ) = 4 n + 1 R m + n + 2 r / D 1 d v ( v - r D ) m ( 1 - v 2 ) ( n + 1 ) / 2 .
J ( 0 , 0 ; r , D ) = 4 R 2 [ L ( 0 , 0 ; r , D ) ] ,
J ( 2 , 0 ; r , D ) = 4 R 4 [ L ( 2 , 0 ; r , D ) - 2 ( r / D ) L ( 1 , 0 ; r , D ) + ( r / D ) 2 L ( 0 , 0 ; r , D ) ] ,
J ( 0 , 2 ; r , D ) = 4 3 R 4 [ L ( 0 , 2 ; r , D ) ] ,
J ( 4 , 0 ; r , D ) = 4 R 6 [ L ( 4 , 0 ; r , D ) - 4 ( r / D ) L ( 3 , 0 ; r , D ) + 6 ( r / D ) 2 L ( 2 , 0 ; r , D ) - 4 ( r / D ) 3 L ( 1 , 0 ; r , D ) + ( r / D ) 4 L ( 0 , 0 ; r , D ) ] ,
J ( 2 , 2 ; r , D ) = 4 3 R 6 [ ( 2 , 2 ; r , D ) - 2 ( r / D ) L ( 1 , 2 ; r , D ) + ( r / D ) 2 L ( 0 , 2 ; r , D ) ] ,
J ( 0 , 4 ; r , D ) = 4 5 R 6 [ L ( 0 , 4 ; r , D ) ] .
L ( 4 , 0 ; r , D ) = 1 16 cos - 1 ( r / D ) + [ 1 - ( r / D ) 2 ] 1 2 [ - 1 6 ( r / D ) 5 + ( 1 / 24 ) ( r / D ) 3 + 1 16 ( r / D ) ] ,
L ( 3 , 0 ; r , D ) = [ 1 - ( r / D ) 2 ] 1 2 [ - 1 5 ( r / D ) 4 + ( 1 / 15 ) ( r / D ) 2 + ( 2 / 15 ) ] ,
L ( 2 , 0 ; r , D ) = 1 8 cos - 1 ( r / D ) + [ ( 1 - ( r / D ) 2 ] 1 / 2 × [ - 1 4 ( r / D ) 3 + 1 8 ( r / D ) ] ,
L ( 1 , 0 ; r , D ) = [ 1 - ( r / D ) 2 ] 1 2 [ - 1 3 ( r / D ) 2 + 1 3 ] ,
L ( 0 , 0 ; r , D ) = 1 2 cos - 1 ( r / D ) + [ 1 - ( r / D ) 2 ] 1 2 [ - 1 2 ( r / D ) ] ,
L ( 2 , 2 ; r , D ) = 1 16 cos - 1 ( r / D ) + [ 1 - ( r / D ) 2 ] 1 2 [ 1 6 ( r / D ) 5 - ( 7 / 24 ) ( r / D ) 3 + 1 16 ( r / D ) ] ,
L ( 1 , 2 ; r , D ) = [ 1 - ( r / D ) 2 ] 1 2 × [ 1 5 ( r / D ) 4 - 2 5 ( r / D ) 2 + 1 5 ] ,
L ( 0 , 2 ; r , D ) = 3 8 cos - 1 ( r / D ) + [ 1 - ( r / D ) 2 ] 1 2 × [ 1 4 ( r / D ) 3 - 5 8 ( r / D ) ] ,
L ( 0 , 4 ; r , D ) = 5 16 cos - 1 ( r / D ) + [ 1 - ( r / D ) 2 ] 1 2 [ - 1 6 ( r / D ) 5 + ( 13 / 24 ) ( r / D ) 3 - 11 16 ( r / D ) ] .
1 R 2 0 D r d r j ( r , D ) r 2 = { R 2 / 2 if j = C 0 if j C .
r 2 = ( x - x ) · ( x - x ) = x 2 + y 2 + x 2 + y 2 - 2 x x - 2 y y .
r 2 = [ π 2 R 8 / 12 ] 1 2 [ F 1 ( x ) F 4 ( x ) + F 1 ( x ) F 4 ( x ) ] + π R 4 F 1 ( x ) F 1 ( x ) - ( π R 4 / 2 ) × [ F 2 ( x ) F 2 ( x ) + F 3 ( x ) F 3 ( x ) ] .
1 2 π R 2 d x d x W ( x , D ) W ( x , D ) r 2 μ = 1 n j F μ ( x ) F μ ( x ) = { R 2 / 2 if j = C 0 if j C ,
1 2 π R 2 d x d x W ( x , D ) W ( x , D ) r 2 i = 1 n j F i ( x ) F i ( x ) = 1 2 π R 2 d r r 2 j ( r , D ) = 1 R 2 0 D r d r r 2 j ( r , D ) .
1 R 2 0 D r d r j ( r , D ) r 2 = { R 2 / 2 if j = C 0 if j C ,
1 R 2 0 D r d r j ( r , D ) = 1 2 ,             ( for all j )
A = 2.91 ( 2 π λ ) 2 path of propagation d Λ C N 2
C N 2 = A L 0 - 2 3 .
A = 6.7 × 10 - 14 exp ( - h / 3200 ) ,
L 0 = 2 h 1 2 ,
D ( r ) = [ ϕ ( x ) - i l ( x ) ] - [ ϕ ( x ) - i l ( x ) ] 2 = [ ϕ ( x ) - ϕ ( x ) ] 2 + [ l ( x ) - l ( x ) ] 2 ,
Δ = 1 π R 2 d x W ( x , D ) [ ϕ 2 ( x ) + l 2 ( x ) ] - 1 π R 2 μ = 1 d x W ( x , D ) × { [ ϕ ( x ) - i l ( x ) ] a ¯ μ + [ ϕ ( x ) + i l ( x ) ] a μ } F μ ( x ) + 1 π R 2 μ , ν = 1 a ¯ μ a ν d x W ( x , D ) F μ ( x ) F ν ( x ) ,
a μ = d x W ( x , D ) [ ϕ ( x ) - i l ( x ) ] F μ ( x ) ,