Abstract

The variance of the log-amplitude of a laser beam is evaluated for a horizontal propagation path through the atmosphere. The treatment is based upon results obtained by Schmeltzer. It is found that the log-amplitude variance can be separated into two factors, one of which is simply the log-amplitude variance of a spherical wave, as derived by Tatarski. The second factor, which contains the dependence upon the size α0 of the transmitted beam, can be written as a function of 02/z, where k is the wave number and z is the path length. This second factor shows a significant oscillation around 02/z = 1 when the transmitted beam is collimated, and starts to roll off strongly for 02/z>1 when the transmitted beam is focused at a range z.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961).
  2. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).
  3. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  4. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [Crossref]
  5. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).
  6. Equation (2.2) is obtained from Schmeltzer’s Eq. (7.8) with τ set equal to zero to convert a covariance to a variance.
  7. A. Kolmogorov, in Turbulence, Classic Papers on Statistical Theory, Ed. by S. K. Friedlander and L. Topper (Interscience Publishers, Inc., New York, 1961), p. 151.
  8. D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966), Eq. (2.21).
    [Crossref]
  9. I. Goldstein and et al., Proc. IEEE 53, 1172 (1965).
    [Crossref]
  10. J. I. Davis, Appl. Opt. 5, 139 (1966).
    [Crossref] [PubMed]
  11. A. Erdelyi and et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, Eq. 4.3(1).
  12. A. Erdelyi and et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, p. 373.
  13. A. Erdelyi and et al., Higher Transcendental Functions (McGraw-Hill Book Co., New York, 1953), Vol. I, Chaps. 2 and 5.
  14. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (9.43).
  15. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (7.94).
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).
  17. D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).
    [Crossref]
  18. Although the Rytov approximation in the explicit form used by Tatarski is not apparent in Schmeltzer’s work, a careful reading of the latter work, especially Secs. 3 and 5, makes it apparent that the results Schmeltzer has obtained are equivalent to those he would have obtained with a formal invocation of the Rytov approximation.

1967 (2)

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
[Crossref]

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

1966 (3)

1965 (1)

I. Goldstein and et al., Proc. IEEE 53, 1172 (1965).
[Crossref]

1964 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).

Cloud, J. D.

Davis, J. I.

Erdelyi, A.

A. Erdelyi and et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, Eq. 4.3(1).

A. Erdelyi and et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, p. 373.

A. Erdelyi and et al., Higher Transcendental Functions (McGraw-Hill Book Co., New York, 1953), Vol. I, Chaps. 2 and 5.

Fried, D. L.

Goldstein, I.

I. Goldstein and et al., Proc. IEEE 53, 1172 (1965).
[Crossref]

Hufnagel, R. E.

Kolmogorov, A.

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

Schmeltzer, R. A.

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Stanley, N. R.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (9.43).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (7.94).

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

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

I. Goldstein and et al., Proc. IEEE 53, 1172 (1965).
[Crossref]

Quart. Appl. Math. (1)

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Other (11)

Equation (2.2) is obtained from Schmeltzer’s Eq. (7.8) with τ set equal to zero to convert a covariance to a variance.

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

A. Erdelyi and et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, Eq. 4.3(1).

A. Erdelyi and et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, p. 373.

A. Erdelyi and et al., Higher Transcendental Functions (McGraw-Hill Book Co., New York, 1953), Vol. I, Chaps. 2 and 5.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (9.43).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (7.94).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).

Although the Rytov approximation in the explicit form used by Tatarski is not apparent in Schmeltzer’s work, a careful reading of the latter work, especially Secs. 3 and 5, makes it apparent that the results Schmeltzer has obtained are equivalent to those he would have obtained with a formal invocation of the Rytov approximation.

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

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).

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

The dependence of the normalized log–amplitude variance Cl(0)/Cls(0) upon the normalized transmitter size, Ω = 02/z. k is the optical wave number of the transmitted radiation, z is the propagation path length, and α0 is the standard deviation of the amplitude variance across the transmitter aperture. α0 is, in a sense, the radius of the transmitter. The transmitter antenna is filled with a gaussian distribution such as is normally emergent from a laser. Cl(0) is the variance of the atmospheric turbulence-induced fluctuations of the log–amplitude measure at a range z, for the specified transmitter diameter, and Cls(0) is the corresponding variance if the transmitter diameter is vanishingly small (a spherical-wave transmitter). Results are shown for the two cases in which the transmitted beam is collimated, and in which it is focused on the measurement plane.

Tables (2)

Tables Icon

Table I Collimated-beam propagation.

Tables Icon

Table II Focused-beam propagation.

Equations (36)

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

u 0 ( r ) ] z = 0 = exp ( - ρ 2 / 2 α 0 2 + i k ρ 2 / 2 R ) ,
C l ( 0 ) = - ( k 2 / 8 π ) Re ( 0 z d s 0 d σ Φ ( σ 1 2 ; s ) × exp [ ( σ / 4 ) γ ( z , s ) ] { exp [ ( σ / 4 ) γ ( z , s ) ] - exp [ ( σ / 4 ) γ * ( z , s ) ] } ) ,
Φ ( σ ; s ) = d r exp ( i σ · r ) C n ( r ; s ) .
Φ ( σ ; s ) = 8.16 C N 2 σ - 11 / 3 .
γ ( z , s ) = [ 2 ( z - s ) / i k ] [ ( s - i k α 2 ) / ( z - i k α 2 ) ] ,
1 / α 2 = 1 / α 0 2 - i k / R .
C l ( 0 ) = ( 8.16 / 8 π ) k 2 C N 2 Re [ 0 z d s 0 d σ σ - 11 / 6 × ( exp { 1 2 σ Re [ γ ( z , s ) ] } - exp [ 1 2 σ γ ( z , s ) ] ) ] .
0 x λ ( e - μ x - e - ν x ) d x = Γ ( λ + 1 ) ( μ - λ - 1 - ν - λ - 1 ) ,             for             Re ( λ ) > - 2 ,             Re ( μ ) > 0 ,             Re ( ν ) > 0 ,
- Re [ γ ( z , s ) ] = [ ( z - s ) 2 / α 0 2 ] / [ ( z / α 0 2 ) 2 + k 2 ( 1 + z / R ) 2 ] ,
C l ( 0 ) = ( 8.16 / 8 π ) k 2 C N 2 Γ ( - 5 6 ) × Re [ 0 z d s ( { - 1 2 Re [ γ ( z , s ) ] } 5 / 6 - [ - 1 2 γ ( z , s ) ] 5 / 6 ) ] .
γ ( z , s ) = 2 ( z - s ) i k s - i k α 0 2 z - i k α 0 2 .
Ω = k α 0 2 / z ,
x = ( s - i k α 0 2 ) / ( z - i k α 0 2 ) .
γ ( z , s ) = [ 2 z ( 1 - i Ω ) / i k ] x ( 1 - x ) ,
Re [ γ ( z , s ) ] = - [ 2 z Ω ( 1 - i Ω ) 2 / k ( 1 + Ω 2 ) ] ( 1 - x ) 2 ,
d s = z ( 1 - i Ω ) d x .
C l ( 0 ) = ( 8.16 / 8 π ) k 7 / 6 z 11 / 6 Γ ( - 5 6 ) C N 2 × Re [ ( 1 - i Ω ) 8 / 3 ( 1 + Ω 2 ) 5 / 6 Ω 5 / 6 - i Ω / ( 1 - i Ω ) 1 ( 1 - x ) 5 / 3 d x - i 5 / 6 ( 1 - i Ω ) 11 / 6 - i Ω / ( 1 - i Ω ) 1 x 5 / 6 ( 1 - x ) 5 / 6 d x ] .
- i Ω / ( 1 - i Ω ) 1 ( 1 - x ) 5 / 3 d x = 3 8 ( 1 - i Ω ) - 8 / 3 .
U 1 x p - 1 ( 1 - x ) q - 1 d x = Γ ( p ) Γ ( q ) / Γ ( p + q ) - ( U p / p ) F 2 1 ( p , 1 - q ; p + 1 ; U ) ,
C l ( 0 ) = 8.16 8 π k 7 / 6 z 11 / 6 Γ ( - 5 6 ) C N 2 Re { 3 8 ( Ω 1 + Ω 2 ) 5 / 6 - i 5 / 6 ( 1 - i Ω ) 11 / 6 [ Γ ( 11 / 6 ) Γ ( 11 / 6 ) Γ ( 11 / 3 ) - 6 11 ( - i Ω 1 - i Ω ) 11 / 6 F 2 1 ( 11 6 , - 5 6 ; 17 6 ; - i Ω 1 - i Ω ) ] } .
C l s ( 0 ) = - 8.16 8 π k 7 / 6 z 11 / 6 Γ ( - 5 6 ) C N 2 × Γ ( 11 / 6 ) Γ ( 11 / 6 ) Γ ( 11 / 3 ) Re ( i 5 / 6 ) ,
C l ( 0 ) / C l s ( 0 ) = [ ( 1 + Ω 2 ) 11 / 6 / cos ( 5 π / 12 ) ] × sin [ ( 11 / 6 ) tan - 1 ( 1 / Ω ) ] - { Γ ( 11 / 3 ) / [ Γ ( 11 / 6 ) Γ ( 11 / 6 ) cos ( 5 π / 12 ) ] } × { ( 3 / 8 ) [ Ω / ( 1 + Ω 2 ) ] 5 / 6 + ( 6 / 11 ) Ω 11 / 6 × Im [ F 2 1 [ 11 / 6 , - 5 / 6 ; 17 / 6 ; - i Ω / ( 1 - i Ω ) ] } .
lim α 0 γ ( z , s ) = 2 ( z - s ) / i k ,
lim Ω C l ( 0 ) / C l s ( 0 ) = 6 11 Γ ( 11 / 3 ) Γ ( 11 / 6 ) Γ ( 11 / 6 ) = 2.47332.
γ ( z , s ) = [ 2 ( z - s ) / i k z ] { s - i k α 0 2 [ ( z - s ) / z ] } .
x = ( z - s ) / z .
γ ( z , s ) = ( 2 z / i k ) x [ 1 - x ( 1 + i Ω ) ] ,
Re [ γ ( z , s ) ] = - ( 2 z Ω / k ) x 2 ,
d s = - z d x .
C l ( 0 ) = 8.16 8 π k 7 / 6 z 11 / 6 Γ ( - 5 6 ) C N 2 Re { Ω 5 / 6 0 1 x 5 / 3 d x - i 5 / 6 0 1 x 5 / 6 [ 1 - x ( 1 + i Ω ) ] 5 / 6 d x } .
0 1 x 5 / 6 [ 1 - x ( 1 + i Ω ) ] 5 / 6 d x = ( 1 + i Ω ) - 11 / 6 0 1 + i Ω y 5 / 6 ( 1 - y ) 5 / 6 d y .
0 U x p - 1 ( 1 - x ) q - 1 d x = U p p F 2 1 ( p , 1 - q ; p + 1 ; U ) ,
0 1 x 5 / 6 [ 1 - x ( 1 + i Ω ) ] 5 / 6 d x = ( 6 / 11 ) F 2 1 ( 11 / 6 , - 5 / 6 ; 17 / 6 ; 1 + i Ω ) .
F 2 1 ( a , b ; c ; z ) = [ Γ ( c ) Γ ( c - a - b ) / Γ ( c - a ) Γ ( c - b ) ] × z - a F 2 1 ( a , a - c + 1 ; a + b - c + 1 ; 1 - 1 / z ) + [ Γ ( c ) Γ ( a + b - c ) / Γ ( a ) Γ ( b ) ] ( 1 - z ) c - a - b × z a - c F 2 1 ( c - a , 1 - a ; c - a - b + 1 ; 1 - 1 / z )
C l ( 0 ) = ( 8.16 / 8 π ) k 7 / 6 z 11 / 6 Γ ( - 5 6 ) C N 2 ( ( 3 / 8 ) Ω 5 / 6 - Re { i 5 / 6 [ Γ ( 11 / 6 Γ ( 11 / 6 ) / Γ ( 11 / 3 ) ] ( 1 + i Ω ) - 11 / 6 - i [ Γ ( - 11 / 6 ) / Γ ( - 5 6 ) ] ( 1 + i Ω ) - 1 Ω 11 / 6 × F 2 1 [ 1 , - 5 / 6 ; 17 / 6 ; i Ω / ( 1 + i Ω ) ] } ) .
C l ( 0 ) C l s ( 0 ) = ( 1 + Ω 2 ) - 11 / 12 sin [ ( 11 / 6 ) tan - 1 ( 1 / Ω ) ] cos ( 5 π / 12 ) - 3 8 Γ ( 11 / 3 ) Ω 5 / 6 Γ ( 11 / 6 ) Γ ( 11 / 6 ) cos ( 5 π / 12 ) - 6 11 Γ ( 11 / 3 ) Γ ( 11 / 6 ) Γ ( 11 / 6 ) cos ( 5 π / 12 ) Ω 11 / 6 1 + Ω 2 × { Im [ F 2 1 ( 1 , - 5 6 ; 17 6 ; i Ω 1 + i Ω ) ] - Ω Re [ F 2 1 ( 1 , - 5 6 ; 17 6 ; i Ω 1 + i Ω ) ] } .