Abstract

Tatarski has found the frequency spectra for the amplitude, phase, and phase-difference fluctuations of an infinite plane wave propagating through turbulence. Many practical optical beams, used in atmospheric studies, closely resemble point sources, for which the spherical-wave theory is more applicable. The same spectra, calculated for spherical waves, reveal contributions at higher frequencies for amplitude scintillations, nearly identical phase results, and a phase-difference spectrum with no nulls, in contrast with the plane-wave results. Comparison with recent data is shown.

© 1971 Optical Society of America

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References

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  1. R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [CrossRef]
  2. Formulas derived here in the context of optical propagation, where n is function of the random temperature field, extend directly to microwave or acoustic propagation with n interpreted as a function of the humidity fields or temperature and wind fields, respectively (see Refs. 3–5).
  3. R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
    [CrossRef]
  4. S. F. Clifford and J. W. Strohbehn, Proc. IEEE AP-18, 264 (1970).
  5. S. F. Clifford and E. H. Brown, J. Acoust. Soc. Am. 48, 1123 (1970).
    [CrossRef]
  6. V. I. Tatarski, Wave Propagation in a Turbulent Atmosphere (Nauka, Moscow, 1967) (in Russian).
  7. The phase fluctuations, because they are most sensitive to the largest turbulent eddy scales, have an undefined spatial covariance function for a Kolmogorov spectrum of turbulence. Therefore, the phase power spectrum should be defined in terms of structure functions (Ref. 1) that are less sensitive to large scale variations. The two definitions give the same spectrum when the covariance function exists.
  8. Tables of Integral Transforms, edited by A. Erdelyi(McGraw–Hill, New York, 1959), Vol. 1, p. 43, No. 1.
  9. Reference 8, Vol. 1, p. 311, No. 30.
  10. Reference 8, Vol. 2., p. 234, No. 12.
  11. The integral in Ref. 9 converges for λ−ρ≥2 if Re(a)=0, a condition satisfied by Eq. (A1).
  12. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. Govt. Printing Office, Washington, 1964; Dover, New York, 1965).
  13. Reference 8, Vol. 2, p. 200, No. 95.
  14. Reference 12, p. 363, No. 9.1.75.
  15. Reference 8, Vol. 1, p. 43, No. 2.
  16. Reference 12, p. 361, No. 9.1.44.

1970 (3)

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

S. F. Clifford and J. W. Strohbehn, Proc. IEEE AP-18, 264 (1970).

S. F. Clifford and E. H. Brown, J. Acoust. Soc. Am. 48, 1123 (1970).
[CrossRef]

1969 (1)

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

Brown, E. H.

S. F. Clifford and E. H. Brown, J. Acoust. Soc. Am. 48, 1123 (1970).
[CrossRef]

Clifford, S. F.

S. F. Clifford and J. W. Strohbehn, Proc. IEEE AP-18, 264 (1970).

S. F. Clifford and E. H. Brown, J. Acoust. Soc. Am. 48, 1123 (1970).
[CrossRef]

Harp, J. C.

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Lee, R. W.

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

S. F. Clifford and J. W. Strohbehn, Proc. IEEE AP-18, 264 (1970).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Atmosphere (Nauka, Moscow, 1967) (in Russian).

J. Acoust. Soc. Am. (1)

S. F. Clifford and E. H. Brown, J. Acoust. Soc. Am. 48, 1123 (1970).
[CrossRef]

Proc. IEEE (3)

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

S. F. Clifford and J. W. Strohbehn, Proc. IEEE AP-18, 264 (1970).

Other (12)

Formulas derived here in the context of optical propagation, where n is function of the random temperature field, extend directly to microwave or acoustic propagation with n interpreted as a function of the humidity fields or temperature and wind fields, respectively (see Refs. 3–5).

V. I. Tatarski, Wave Propagation in a Turbulent Atmosphere (Nauka, Moscow, 1967) (in Russian).

The phase fluctuations, because they are most sensitive to the largest turbulent eddy scales, have an undefined spatial covariance function for a Kolmogorov spectrum of turbulence. Therefore, the phase power spectrum should be defined in terms of structure functions (Ref. 1) that are less sensitive to large scale variations. The two definitions give the same spectrum when the covariance function exists.

Tables of Integral Transforms, edited by A. Erdelyi(McGraw–Hill, New York, 1959), Vol. 1, p. 43, No. 1.

Reference 8, Vol. 1, p. 311, No. 30.

Reference 8, Vol. 2., p. 234, No. 12.

The integral in Ref. 9 converges for λ−ρ≥2 if Re(a)=0, a condition satisfied by Eq. (A1).

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. Govt. Printing Office, Washington, 1964; Dover, New York, 1965).

Reference 8, Vol. 2, p. 200, No. 95.

Reference 12, p. 363, No. 9.1.75.

Reference 8, Vol. 1, p. 43, No. 2.

Reference 12, p. 361, No. 9.1.44.

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Figures (4)

Fig. 1
Fig. 1

Temporal power spectrum of the log-amplitude fluctuations for both plane- and spherical-wave propagation. Each curve is normalized to the applicable log-amplitude variance, 〈χ2〉, and plotted vs frequency ratio Ω=f/f0, with f 0 = v ( 2 π λ L ) - 1 2.

Fig. 2
Fig. 2

Temporal power spectrum of the phase fluctuations times frequency ratio, Ω=f/f0, for both plane- and spherical-wave propagation. Each curve is normalized to the applicable log-amplitude variance, 〈χ2〉. f 0 = v ( 2 π λ L ) - 1 2.

Fig. 3
Fig. 3

Normalized frequency spectrum for intensity or log-amplitude fluctuations vs log(f/f0), f 0 = v ( 2 π λ L ) - 1 2. The plane- and spherical-wave cases are normalized to their appropriate log-amplitude variance 〈χ2〉=∫Wx(f)df.

Fig. 4
Fig. 4

Theoretical spherical-wave frequency spectrum of the phase-difference fluctuations (solid line) vs log of the frequency ratio f/f1, where f1=v/ρ, v is the transverse wind, and ρ the spacing of two sensors. The curve is normalized such that it also represents the normalized angle-of-arrival spectrum. Also shown are recent data for four different spacings. □—ρ=0.3 cm, ●—ρ=3 cm, △—ρ=15 cm, and ○—ρ=30 cm.

Tables (1)

Tables Icon

Table I Comparison between the plane- and spherical-wave asymptotic forms of the log-amplitude, phase, and phase-difference temporal power spectra.a

Equations (56)

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W g ( f ) = 4 0 d τ cos ( 2 π f τ ) R g ( τ ) ,
R g ( τ ) = g ( r , t ) g ( r , t + τ ) .
B g ( ϱ ) = g ( r , t ) g ( r + ϱ , t ) .
g ( r , t + τ ) = g ( r - v τ , t ) .
R g ( τ ) = g ( r , t ) g ( r - v τ , t ) = B g ( v τ ) .
W g ( f ) = 4 0 d τ cos ( 2 π f τ ) B g ( v τ ) .
B x , S ( ρ ) = 2 π 2 k 2 0 L d z 0 d K K Φ n ( K ) J 0 ( K ρ z L ) × { 1 cos [ K 2 z ( L - z ) k L ] } ,
Φ n ( K ) = 1 ( 2 π ) 3 d 3 r B n ( r ) e i K · r
B n ( r ) = n 1 ( r 1 ) n 1 ( r 1 + r ) .
g ( r , t + τ ) = g [ r - v ( L / z ) τ , t ] .
R g ( τ , z ) d z = g ( r , t ) g [ r - v ( L / z ) τ , t ] = d B g [ v ( L / z ) τ ] .
R g ( τ ) = 0 L d z R g ( τ , z ) = B g ( ) B g ( v τ ) d B g ( v L τ z ) .
R g ( τ ) = B g ( v τ ) .
R x , S ( τ ) = B x , S ( v τ ) = 2 π 2 k 2 0 L d z 0 d K × K Φ n ( K ) J 0 ( K v τ ) { 1 cos [ K 2 z ( L - z ) k L ] }
W x , S ( f ) = 8 π 2 k 2 0 L d z 2 π f / v d K × K Φ n ( K ) [ ( K v ) 2 - ( 2 π f ) 2 ] - 1 2 × { 1 cos [ K 2 z ( L - z ) k L ] }
Φ n ( K ) = 0.033 C n 2 K - 11 / 3 ,
W x , S ( f ) = 0.132 π 2 k 2 3 L 7 / 3 C n 2 v - 1 Ω - 8 / 3 0 1 d u × 0 d σ σ - 1 2 ( σ + 1 ) 11 / 6 { 1 cos [ - 1 4 Ω 2 ( σ + 1 ) ( 1 - u 2 ) ] } ,
W x , S ( f ) = 2.19 k 2 3 L 7 / 3 C n 2 v - 1 Ω - 8 / 3 R e × [ 1 F 2 2 ( 1 , - 5 6 ; 3 2 , - 1 3 ; i 1 4 Ω 2 ) ( 4 / 11 ) Γ ( - 4 3 ) ( - i 1 4 Ω 2 ) 4 3 F 1 1 ( 1 2 , 17 / 6 , i 1 4 Ω 2 ) ] ,
F 2 2 ( a 1 , a 2 ; b 1 , b 2 ; z ) = n = 0 Γ ( a 1 + n ) Γ ( a 2 + n ) Γ ( b 1 ) Γ ( b 2 ) Γ ( b 1 + n ) Γ ( b 2 + n ) Γ ( a 1 ) Γ ( a 2 ) z n n !
F 2 2 ( a 1 , a ; b 1 , a ; z ) = F 1 1 ( a 1 , b 1 , z ) ,
δ ρ S ( t ) = S ( r , t ) - S ( r + ϱ , t ) ,
R δ S ( τ ) = δ ρ S ( t ) δ ρ S ( t + τ ) .
R δ S ( τ , z ) d z = [ S ( r - v L z τ , t ) - S ( r + ϱ - v L z τ , t ) ] × [ S ( r , t ) - S ( r + ϱ , t ) ] .
R δ S ( τ , z ) d z = 2 d B S ( v L z τ ) - d B S ( ϱ - v L z τ ) - d B S ( ϱ + v L z τ ) ,
R δ S ( τ ) = 0 L R δ S ( τ , z ) d z = [ 2 B S ( v τ ) - B S ( ϱ - v τ ) - B S ( ϱ + v τ ) + 2 B S ( ) - B S ( - ) - B S ( ) ] .
R δ S ( τ ) = 2 π 2 k 2 0 L d z × 0 d K K Φ n ( K ) { 1 + cos [ K 2 z ( L - z ) k L ] } × [ 2 J 0 ( K v τ ) - J 0 ( K ρ z L - K v τ ) - J 0 ( K ρ z L + K v τ ) ] .
W δ S ( f ) = 32 π 2 k 2 0 L d z sin 2 ( π ρ f z v L ) 2 π f / v d K × ( K Φ n ( K ) [ ( K v ) 2 - ( 2 π f ) 2 ] - 1 2 × { 1 + cos [ K 2 z ( L - z ) k L ] } ) .
W δ S ( f ) = 0.264 π 2 k 2 3 L 7 / 3 C n 2 v - 1 Ω - 8 / 3 × 0 1 d u [ 1 - cos ( π ρ f u v ) cos ( π ρ f v ) ] × ( 0 d σ σ - 1 2 ( σ + 1 ) 11 / 6 { 1 + cos [ - 1 4 Ω 2 ( σ + 1 ) ( 1 - u 2 ) ] } ) ,
W δ S ( f ) = 4.38 k 2 3 L 7 / 3 C n 2 v - 1 Ω - 8 / 3 × 0 1 d u [ 1 - cos ( π ρ f v ) cos ( π ρ f u v ) ] × [ 1 + G ( 1 4 Ω 2 u 2 ) ] ,
G = Γ ( 11 / 6 ) π 1 2 Γ ( 4 3 ) Re { exp [ - i 1 4 Ω 2 ( u 2 - 1 ) ] × 0 d σ σ - 1 2 ( σ + 1 ) 11 / 6 exp [ - i 1 4 Ω 2 ( u 2 - 1 ) ] } .
G cos [ 1 4 Ω 2 ( u 2 - 1 ) ] .
I 3 0 1 cos [ 1 4 Ω 2 ( u 2 - 1 ) ] d u
I 4 cos ( π ρ f v ) 0 1 cos ( π ρ f u v ) cos [ 1 4 Ω 2 ( u 2 - 1 ) ] d u .
[ I 1 + I 2 + I 3 + I 4 ] 0 1 d u [ 1 - cos ( π ρ f v ) cos ( π ρ f u v ) ] = [ 1 - sin ( 2 π ρ f / v ) ( 2 π ρ f / v ) ] ,             Ω π ρ f / v .
[ I 1 + I 2 + I 3 + I 4 ] 2 0 1 d u [ 1 - cos ( π ρ f v ) cos ( π ρ f u v ) ] = 2 [ 1 - sin ( 2 π ρ f / v ) ( 2 π ρ f / v ) ] , Ω π ρ f / v .
W δ S ( f ) = 0.033 C n 2 k 2 L v 5 / 3 × [ 1 - sin ( 2 π ρ f / v ) ( 2 π ρ f / v ) ] f - 8 / 3 ,             ρ ( λ L ) 1 2 ,
W δ S ( f ) = 0.066 C n 2 k 2 L v 5 / 3 × [ 1 - sin ( 2 π ρ f / v ) ( 2 π ρ f / v ) ] f - 8 / 3 ,             ρ ( λ L ) 1 2 .
W α ( f ) = 0.033 C n 2 k 2 L v 5 / 3 [ 1 - sin ( 2 π b f / v ) ( 2 π b f / v ) ] f - 8 / 3 ,             b ( λ L ) 1 2 ,
W α ( f ) = 0.066 C n 2 k 2 L v 5 / 3 [ 1 - sin ( 2 π b f / v ) ( 2 π b f / v ) ] f - 8 / 3 ,             b ( λ L ) 1 2 ,
D S ( ρ ) = 0.54 k 2 L C n 2 ρ 5 / 3 ,             l 0 ρ ( λ L ) 1 2 ,
D S ( ρ ) = 1.09 k 2 L C n 2 ρ 5 / 3 ,             ρ ( λ L ) 1 2 .
f W δ S ( f ) W δ S ( f ) d f = 16.5 [ 1 - sin ( 2 π f / f 1 ) ( 2 π f / f 1 ) ] ( f f 1 ) - 5 / 3 ,
I = Re ( 0 d σ σ - 1 2 ( σ + 1 ) 11 / 6 × { 1 exp [ - i 1 4 Ω 2 ( σ + 1 ) ( u 2 - 1 ) ] } ) ,
0 d σ σ - 1 2 ( σ + 1 ) 11 / 6 = ( π ) 1 2 Γ ( 4 3 ) Γ ( 11 / 6 ) .
( π ) 1 2 [ i 1 4 Ω 2 ( u 2 - 1 ) ] 1 / 6 exp [ - i 1 8 Ω 2 ( u 2 - 1 ) ] × W 2 / 3 , - 2 / 3 [ i 1 4 Ω 2 ( u 2 - 1 ) ] ,
F 1 1 ( a , b , x ) = n = 0 Γ ( a + n ) Γ ( b ) Γ ( b + n ) Γ ( a ) x n n ! .
( π ) 1 2 Γ ( 4 3 ) Γ ( 11 / 6 ) exp ( - x ) [ F 1 1 ( 1 2 , - 1 3 , x ) + Γ ( - 4 3 ) Γ ( 11 / 6 ) ( π ) 1 2 Γ ( 4 3 ) x 4 3 F 1 1 ( 11 / 6 , 7 / 3 , x ) ] ,
e - x F 1 1 ( a , b , x ) = F 1 1 ( b - a , b , - x )
I = ( π ) 1 2 Γ ( 4 3 ) Γ ( 11 / 6 ) Re [ 1 F 1 1 ( - 5 6 , - 1 3 , - x ) Γ ( - 4 3 ) Γ ( 11 / 6 ) ( π ) 1 2 Γ ( 4 3 ) x 4 3 F 1 1 ( 1 2 , 7 / 3 , - x ) ] .
W x , S ( f ) = 0.132 π 2 k 2 3 L 7 / 3 C n 2 v - 1 Ω - 8 / 3 ( π ) 1 2 Γ ( 4 3 ) Γ ( 11 / 6 ) R e { 0 1 d u I } .
0 1 d u I = 1 2 ( t ) 1 2 0 t d z ( t - z ) - 1 2 [ 1 F 1 1 ( - 5 6 , - 1 3 , i z ) Γ ( - 4 3 ) Γ ( 11 / 6 ) ( π ) 1 2 Γ ( 4 3 ) ( - i z ) 4 3 F 1 1 ( 1 2 , 7 / 3 , i z ) ] .
0 d τ cos γ τ [ J 0 ( β + α τ ) + J 0 ( β - α τ ) ] .
J 0 ( β + α τ ) + J 0 ( β - α τ ) = k = - J - k ( β ) J k ( α τ ) + k = - J k ( β ) J k ( α τ ) ,
J 0 ( β + α τ ) + J 0 ( β - α τ ) = 2 J 0 ( β ) J 0 ( α τ ) + 4 k = 1 J 2 k ( β ) J 2 k ( α τ ) .
and             2 J 0 ( β ) ( α 2 - γ 2 ) - 1 2 + 4 k = 1 ( - 1 ) k × J 2 k ( β ) T 2 k ( γ / α ) ( α 2 - γ 2 ) - 1 2 , 0 < γ < α , 0 , α < γ < ,
and             2 ( α 2 - γ 2 ) - 1 2 cos ( β γ / α ) , 0 < γ < α , 0 , α < γ < .