Abstract

The irradiance fluctuations of beams which propagate through a turbulent medium can be used to infer the spatial spectrum of the turbulent scatterers. The propagation of Goubau-type beam waves in a random medium is used to show that amplitude fluctuations depend upon the transmitter aperture size, observation distance, and wavelength. Most of the results are presented in terms of a parameter, αL, which is inversely proportional to the Fresnel number, πW02/Lλ. A Monte Carlo numerical procedure is used to obtain the normalized mean-square log-amplitude fluctuation as a function of the inverse Fresnel number for various magnitudes of the turbulence inner scale. The applicability of these procedures to measured turbulence spectra is also discussed.

© 1969 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).
  2. L. A. Chernov, Wave Propagation in a Random Medium (McGraw–Hill Book Co., New York, 1960).
  3. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [Crossref]
  4. F. P. Carlson and A. Ishimaru, J. Opt. Soc. Am. 59, 319 (1969).
    [Crossref]
  5. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  6. F. P. Carlson, Ph.D. thesis, University of Washington (1967).
  7. Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Sci. 3, 287, (1968).
  8. Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
    [Crossref]
  9. D. L. Fried and J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
    [Crossref]
  10. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [Crossref]
  11. D. L. Fried, J. Opt. Soc. Am. 57, 980 (1967).
    [Crossref]
  12. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).
  13. G. Goubau and F. Schwering, Proc. IEEE Trans.,  AP-9, 248 (1961).
  14. G. Goubau, Electromagnetic Waves, R. E. Langer, Ed. (University of Wisconsin Press, Madison, 1962).
  15. G. Goubau, Electromagnetic Theory and Antennas (Macmillan Co., London, 1963).
  16. A. Ishimaru, Radio Sci. 4. No. 4, 295 (1969).
    [Crossref]
  17. See Ref. 1, p. 139.
  18. The merits of the Rytov procedure are discussed by J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
    [Crossref]
  19. See Ref. 1, p. 140.
  20. Large αL corresponds to the spherical-wave case and αL → 0 corresponds to the plane-wave case.
  21. αL = N−1, where N is the Fresnel number, πW02/Lλ.
  22. P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 69, 527 (1969).
    [Crossref]
  23. P. Beckmann, Radio Science J. Res. Natl. Bur. Std/USNC–URSI 69D, 629 (1965).
    [Crossref]
  24. J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, Inc., New York, 1964).
  25. L. S. Taylor, J. Opt. Soc. Am. 58, 1418 (1968).
    [Crossref]
  26. L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  11, 1674 (1960).
  27. L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  8, 1252 (1960).
  28. M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

1969 (3)

F. P. Carlson and A. Ishimaru, J. Opt. Soc. Am. 59, 319 (1969).
[Crossref]

A. Ishimaru, Radio Sci. 4. No. 4, 295 (1969).
[Crossref]

P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 69, 527 (1969).
[Crossref]

1968 (4)

L. S. Taylor, J. Opt. Soc. Am. 58, 1418 (1968).
[Crossref]

The merits of the Rytov procedure are discussed by J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Sci. 3, 287, (1968).

Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
[Crossref]

1967 (5)

1965 (2)

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

P. Beckmann, Radio Science J. Res. Natl. Bur. Std/USNC–URSI 69D, 629 (1965).
[Crossref]

1964 (1)

1961 (1)

G. Goubau and F. Schwering, Proc. IEEE Trans.,  AP-9, 248 (1961).

1960 (2)

L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  11, 1674 (1960).

L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  8, 1252 (1960).

Asakura, T.

Beckmann, P.

P. Beckmann, Radio Science J. Res. Natl. Bur. Std/USNC–URSI 69D, 629 (1965).
[Crossref]

Carlson, F. P.

F. P. Carlson and A. Ishimaru, J. Opt. Soc. Am. 59, 319 (1969).
[Crossref]

F. P. Carlson, Ph.D. thesis, University of Washington (1967).

Chernov, L. A.

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

Deitz, P. H.

P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 69, 527 (1969).
[Crossref]

Fried, D. L.

Goubau, G.

G. Goubau and F. Schwering, Proc. IEEE Trans.,  AP-9, 248 (1961).

G. Goubau, Electromagnetic Waves, R. E. Langer, Ed. (University of Wisconsin Press, Madison, 1962).

G. Goubau, Electromagnetic Theory and Antennas (Macmillan Co., London, 1963).

Gracheva, M. E.

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

Hufnagel, R. E.

Ishimaru, A.

Keister, M. P.

Kinoshita, Y.

Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
[Crossref]

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Sci. 3, 287, (1968).

Lumley, J. L.

J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, Inc., New York, 1964).

Matsumoto, T.

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Sci. 3, 287, (1968).

Mevers, G. E.

Panofsky, H. A.

J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, Inc., New York, 1964).

Schmeltzer, R. A.

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

Schwering, F.

G. Goubau and F. Schwering, Proc. IEEE Trans.,  AP-9, 248 (1961).

Seidman, J. B.

Stanley, N. R.

Strohbehn, J. W.

The merits of the Rytov procedure are discussed by J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Suzuki, M.

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Sci. 3, 287, (1968).

Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).

Taylor, L. S.

Tsvang, L. R.

L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  11, 1674 (1960).

L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  8, 1252 (1960).

Wright, N. J.

P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 69, 527 (1969).
[Crossref]

Izvestia ANSSSR, Geophys. Ser. 1960 (2)

L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  11, 1674 (1960).

L. R. Tsvang, Izvestia ANSSSR, Geophys. Ser. 1960,  8, 1252 (1960).

J. Opt. Soc. Am. (9)

Proc. IEEE (1)

The merits of the Rytov procedure are discussed by J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Proc. IEEE Trans. (1)

G. Goubau and F. Schwering, Proc. IEEE Trans.,  AP-9, 248 (1961).

Quart. Appl. Math. (1)

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

Radio Sci. (2)

A. Ishimaru, Radio Sci. 4. No. 4, 295 (1969).
[Crossref]

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Sci. 3, 287, (1968).

Radio Science J. Res. Natl. Bur. Std/USNC–URSI (1)

P. Beckmann, Radio Science J. Res. Natl. Bur. Std/USNC–URSI 69D, 629 (1965).
[Crossref]

Radiofizika (1)

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

Other (10)

J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, Inc., New York, 1964).

See Ref. 1, p. 139.

F. P. Carlson, Ph.D. thesis, University of Washington (1967).

G. Goubau, Electromagnetic Waves, R. E. Langer, Ed. (University of Wisconsin Press, Madison, 1962).

G. Goubau, Electromagnetic Theory and Antennas (Macmillan Co., London, 1963).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).

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

See Ref. 1, p. 140.

Large αL corresponds to the spherical-wave case and αL → 0 corresponds to the plane-wave case.

αL = N−1, where N is the Fresnel number, πW02/Lλ.

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Figures (5)

F. 1
F. 1

Beam-wave filter functions vs a normalized spatia wave number, q = K ( L / k ) 1 2, for various Fresnel numbers. Dashed line is the Kolmogorov inertial subrange, 6,040 (√q)−8/3 = Φn(√q)√q×103.

F. 2
F. 2

Filter-function maxima as a function of Fresnel number.

F. 3
F. 3

Filter-function maxima as function of normalized wave number.

F. 4
F. 4

Normalized mean-square amplitude fluctuations vs Fresnel number for various inner scales l0 with C l 0 = 4 π 2 ( 0.033 C n 2 ) L 11 / 6 k 7 / 6 .Case considered corresponds to L = 8 km and λ = 6328 Å.

F. 5
F. 5

Normalized mean-square amplitude fluctuations versus inner scale l0 for various Fresnel numbers. Case considered utilizes L = 8 km and λ = 6328 Å.

Equations (32)

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ψ 1 ( r ) = k 2 2 π E 0 ( r ) υ η 1 ( r ) E 0 ( r ) e i k ( r r ) ( r r ) d υ ,
E ( r ) / E 0 ( r ) = e ψ 1 ( r )
[ 2 + k 2 ( 1 + η 1 ) 2 ] E = 0 .
E 0 ( r ) = A 1 + i α x exp { i k x y 2 + z 2 W 0 2 [ 1 + i α x ] } ,
α = λ / π W 0 2 ,
χ ( L , y , z ) = 1 2 0 L d x [ exp ( i K 2 γ y + i K 3 γ z ) × H ( L , x , K 2 , K 3 ) + exp ( i K 2 γ * y + i K 3 γ * z ) × H * ( L , x , K 2 , K 3 ) ] d ν ( K 2 , K 3 , x ) ,
η 1 ( x , y , z ) = exp ( i k 2 y + i K 3 z ) d ν ( K 2 , K 3 , x ) .
γ = ( 1 + i α x ) / ( 1 + i α L )
H ( L , x , K 2 , K 3 ) = i k exp { [ i γ ( L x ) / 2 k ] ( k 2 2 + k 3 2 ) } .
B A ( L ; y 1 , z 1 ; y 2 , z 2 ) = χ ( L , y 1 , z 1 ) χ ( L , y 2 , z 2 ) .
y 1 = y 2 and z 1 = z 2 .
B A = π 2 0 L d η 0 K d K { [ J 0 ( K P ) + J 0 ( K P * ) ] | H | 2 + J 0 ( K Q ) H 2 + J 0 ( K Q * ) H * 2 } Φ n ( K ) ,
P = [ ( γ y 1 γ * y 2 ) 2 + ( γ z 1 γ * z 2 ) 2 ] 1 2
Q = γ [ ( y 1 y 2 ) 2 + ( z 1 z 2 ) 2 ] 1 2
χ 2 = 2 π 2 0 L d η 0 K d K [ J 0 ( i 2 γ 2 K ρ ) | H | 2 + Re ( H 2 ) ] Φ n ( K ) ,
γ 2 = α ( L η ) / [ 1 + ( α L ) 2 ] and ρ = [ y 2 + z 2 ] 1 2 .
χ 2 = 2 π 0 F A ( K ) K d K ;
F A ( K ) = π k 2 L Φ n ( K ) f A ( K ) ,
f A ( K ) = 1 L 0 L d η { exp [ α ( L η ) 2 K 2 k ( 1 + ( α L ) 2 ) ] × [ 1 cos ( ( 1 + α 2 η L ) ( L η ) K 2 k ( 1 + ( α L ) 2 ) ) ] }
f A ( K ) = 1 2 K ( π k L ) 1 2 { 1 b erf [ K ( L b k ) 1 2 ] e ( K 2 L / 4 k b ) b 1 ( erf [ i k 2 ( L k b 1 ) 1 2 ] + erf [ i k 2 ( L k b 1 ) 1 2 + K ( L b 1 k ) 1 2 ] ) } ,
b = α L / ( 1 + ( α L ) 2 )
b 1 = L / ( 1 i α L ) .
f A ( K ) = 1 2 k ( π k L ) 1 2 [ 1 + ( α L ) 2 ( α L ) 2 ] 1 4 ( [ 1 + ( α L ) 2 ] 1 4 2 exp ( K 2 4 α k ) cos ( K 2 L 4 k θ 2 ) ) k K 2 L exp ( K 2 L b k ) [ 1 2 b + sin ( K 2 b k α 2 θ ) ] ,
θ = tan 1 α L .
Φ n ( K ) = { 0.033 K 11 / 3 C n 2 , 2 π L 0 < K < 2 π l 0 0 , otherwise ,
χ 2 = 4 π 2 ( 0.033 C n 2 ) L 11 / 6 k 7 / 6 ( 2 π / L 0 ) ( L / k ) 1 2 ( 2 π / l 0 ) ( L / k ) 1 2 d p p 8 / 3 × 0 1 d x exp ( b p 2 x ) sin 2 [ p 2 2 ( x x 2 α L b ) ] ,
p 2 = K 2 L / k
x = η / L .
C l 0 = 4 π 2 ( 0.033 C n 2 ) L 11 / 6 k 7 / 6 ,
χ 2 C l 0 = 0.1785 / L 0 0.1785 / l 0 d p p 8 / 3 0 1 d x exp { b p 2 x } sin 2 [ p 2 2 ( x x 2 α L b ) ] ,
C l 0 = 2.72 .
C l 0 = 4 π 2 ( 0.033 C n 2 ) L 11 / 6 k 7 / 6 .