Abstract

The relationship between the statistics of log-amplitude fluctuations and irradiance fluctuations due to atmospheric turbulence is derived. This is used to evaluate the effect of use of a large aperture diameter in reducing the variance of a fluctuating signal. Curves for the reduction factor are presented. From these, an irradiance-fluctuation correlation distance is evaluated. This distance, unlike the correlation distance for log-amplitude fluctuations, is found to be a function of the log-amplitude variance. A particular example of the application of these results to a space-to-ground communications systems performance is worked out.

© 1967 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 Company, Inc., New York, 1961), Chap. 13.
  2. As a measure of aperture averaging Tatarski computes a quantity G, defined by his Eq. (13.28), with results presented in his Fig. 36. The equation defining G can be cast in the formG=ln{16π∫01exp{σ2[bA(Dx)-1]}[cos-1x-x(1-x2)12]xdx},where D is the aperture diameter, σ2 is the variance of the log-intensity, and bA(Dx) is what we have called the normalized log-amplitude covariance function [which we have denoted by Cl(Dx)/Cl(0)]. (The factor 1/σ2 which Tatarski has in front of the logarithm in Eq. (13.28) is “absorbed” into the integrand in our expression by the minus one in the exponent. The minus one is absent in Tatarski’s expression.) We need only note that σ is non-negative and that [bA(Dx)−1] is non-positive for all values of D and x. The exact physical significance of bA and σ need not concern us. The derivative of G with respect to σ is∂G∂σ=∫012σ[bA(Dx)-1] exp{σ2[bA(Dx)-1]}×[cos-1x-x(1-x2)12]xdx∫01 exp{σ2[bA(Dx)-1]}[cos-1x-x(1-x2)12]xdxThe integrand in the numerator is everywhere negative, while the integrand in the denominator is everywhere non-negative. Obviously, ∂G/∂σ is negative. For any particular value of D, the value of G should be decreased when σ is increased. This condition is clearly contradicted by Tatarski’s Fig. 36.
  3. The shot noise σN is intended to refer to that (quantum) noise which would be present if there were no atmospherically induced irradiance fluctuations. Even a modestly sophisticated system can avoid confusing the atmospherically induced irradiance fluctuations with the signal variations which are meant to carry information. Simple voice-controlled intensity modulation at the transmitter, of course, results in the atmospherically induced signal variations appearing as noise, as has been considered to be the case by J. I. Davis, Appl. Opt. 5, 139 (1966), but binary coding, or AM–FM modulation permits voice information to be transmitted without the atmospheric effects coming through as a spurious signal, i.e., as noise. Only the quantum fluctuations cannot be eliminated. σN refers only to this noise.
    [Crossref] [PubMed]
  4. D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).
    [Crossref]
  5. The log-amplitude l(x), normally computed, equals the logarithm of the ratio of the amplitude measured to the amplitude which would be measured if there were no refractive-index variation in the propagation path. Because energy must be conserved, and refractive-index variations do not dissipate optical energy, in the propagation of an infinite plane wave or a spherical wave, the measured average irradiance, I0, corresponds to the irradiance which would be observed in the absence of atmospheric turbulence. For infinite-plane-wave propagation, the amplitude computed from the square root of I0 can be related to the amplitude which would exist in the absence of turbulence. For propagation of a collimated beam, it cannot. Thus, if we wished to apply Eq. (1.1) to propagation of a laser beam, I0 could not be interpreted as the measured average irradiance, but rather as the irradiance that would be measured if there were no turbulence.
  6. Ref. 1, p. 209 and p. 210.
  7. G. E. Mevers and et al., J. Opt. Soc. Am. 55, 1575A (1965).
  8. The quantity (4z/k)12 is a unit of length which arises in the analysis of the log-amplitude statistics for horizontal propagation of an infinite plane wave. It is useful in normalizing correlation distance so that results can be expressed in nondimensional form, as is developed in Ref. 4. Tatarski uses (λz)12 for this same purpose. For propagation from space to the ground, the normalization distance is (4h0 secθ/k)12 where θ is the zenith angle of the apparent source, k is the wave number, and h0=3200 m is a quantity which characterizes the vertical distribution of atmospheric turbulence.
  9. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [Crossref]

1966 (2)

1965 (2)

G. E. Mevers and et al., J. Opt. Soc. Am. 55, 1575A (1965).

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Other (5)

The quantity (4z/k)12 is a unit of length which arises in the analysis of the log-amplitude statistics for horizontal propagation of an infinite plane wave. It is useful in normalizing correlation distance so that results can be expressed in nondimensional form, as is developed in Ref. 4. Tatarski uses (λz)12 for this same purpose. For propagation from space to the ground, the normalization distance is (4h0 secθ/k)12 where θ is the zenith angle of the apparent source, k is the wave number, and h0=3200 m is a quantity which characterizes the vertical distribution of atmospheric turbulence.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961), Chap. 13.

As a measure of aperture averaging Tatarski computes a quantity G, defined by his Eq. (13.28), with results presented in his Fig. 36. The equation defining G can be cast in the formG=ln{16π∫01exp{σ2[bA(Dx)-1]}[cos-1x-x(1-x2)12]xdx},where D is the aperture diameter, σ2 is the variance of the log-intensity, and bA(Dx) is what we have called the normalized log-amplitude covariance function [which we have denoted by Cl(Dx)/Cl(0)]. (The factor 1/σ2 which Tatarski has in front of the logarithm in Eq. (13.28) is “absorbed” into the integrand in our expression by the minus one in the exponent. The minus one is absent in Tatarski’s expression.) We need only note that σ is non-negative and that [bA(Dx)−1] is non-positive for all values of D and x. The exact physical significance of bA and σ need not concern us. The derivative of G with respect to σ is∂G∂σ=∫012σ[bA(Dx)-1] exp{σ2[bA(Dx)-1]}×[cos-1x-x(1-x2)12]xdx∫01 exp{σ2[bA(Dx)-1]}[cos-1x-x(1-x2)12]xdxThe integrand in the numerator is everywhere negative, while the integrand in the denominator is everywhere non-negative. Obviously, ∂G/∂σ is negative. For any particular value of D, the value of G should be decreased when σ is increased. This condition is clearly contradicted by Tatarski’s Fig. 36.

The log-amplitude l(x), normally computed, equals the logarithm of the ratio of the amplitude measured to the amplitude which would be measured if there were no refractive-index variation in the propagation path. Because energy must be conserved, and refractive-index variations do not dissipate optical energy, in the propagation of an infinite plane wave or a spherical wave, the measured average irradiance, I0, corresponds to the irradiance which would be observed in the absence of atmospheric turbulence. For infinite-plane-wave propagation, the amplitude computed from the square root of I0 can be related to the amplitude which would exist in the absence of turbulence. For propagation of a collimated beam, it cannot. Thus, if we wished to apply Eq. (1.1) to propagation of a laser beam, I0 could not be interpreted as the measured average irradiance, but rather as the irradiance that would be measured if there were no turbulence.

Ref. 1, p. 209 and p. 210.

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Figures (3)

Fig. 1
Fig. 1

Dependence of the normalized variance of irradiance, CI(0)/I02, upon the log-amplitude variance Cl(0).

Fig. 2
Fig. 2

Dependence of the aperture-averaging factor, Θ, upon the normalized aperture diameter, D / ( 4 z / k ) 1 2. The calculated curves, shown for various values of Cl(0), the log-amplitude variance, are based upon the statistics of propagation of an infinite plane wave of wave number k, traveling a horizontal path of length z.

Fig. 3
Fig. 3

Dependence of the aperture-averaging factor, Θ, upon the normalized aperture diameter, D / ( 4 h 0 sec θ / k ) 1 2. The calculated curves, shown for various values of Cl(0), the log-amplitude variance, are based upon the statistics of propagation of an infinite plane wave of wave number k, traveling from space to the ground and arriving with a zenith angle θ. (h0=3200 m).

Tables (4)

Tables Icon

Table I Aperture-averaging factor Θ, for various values of the normalized collector diameter D / ( 4 z / k ) 1 2, and values of the log-amplitude variance Cl(0). Calculated values are based on the statistics of propagation of an infinite plane wave of wave number k, over a horizontal path of length z. The negative number in parenthesis following each entry is the power of ten by which the entry (i.e., the three digit number) is to be multiplied. Thus, 3.35(−4) denotes 3.35×10−4.

Tables Icon

Table II Aperture-averaging factor Θ, for various values of the normalized collector diameter D / ( 4 h 0 sec θ / k ) 1 2, and values of the log-amplitude variance Cl(0). Calculated values are based on the statistics of propagation of an infinite plane wave of wave number k, traveling from space to the ground and arriving with a zenith angle θ (h0=3200 m). The negative number in parenthesis following each entry is the power of ten by which the entry (i.e., the three digit number) is to be multiplied. Thus, 3.35(−4) denotes 3.35×10−4.

Tables Icon

Table III Normalized irradiance-fluctuation coherence distance d 0 / ( 4 z / k ) 1 2 for various values of the log-amplitude covariance Cl(0) based on the statistics of propagation of an infinite plane wave of wave number k traveling a distance z over a horizontal path.

Tables Icon

Table IV Normalized irradiance-fluctuation coherence distance, d 0 / ( 4 h 0 sec θ / k ) 1 2, for various values of the log-amplitude covariance, Cl(0) based on the statistics of propagation of an infinite plane wave of wave number k, traveling from space to the ground and arriving with a zenith angle θ. (h0=3200 m).

Equations (37)

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I ( x ) = I 0 exp [ ( 2 l ( x ) ] .
C l ( ρ ) = [ l ( x ) - l ] [ l ( x ) - l ] ,
ρ = x - x .
exp ( a g ) = exp [ a g + 1 2 a 2 ( g - g ) 2 ] ,
l = - C l ( 0 ) .
C I ( ρ ) = [ I ( x ) - I 0 ] [ I ( x ) - I 0 ] ,
C I ( ρ ) = I 0 2 { exp [ 2 l ( x ) ] - 1 } { exp [ 2 l ( x ) ] - 1 } .
C I ( ρ ) = I 0 2 { exp 2 [ l ( x ) + l ( x ) ] - exp [ 2 l ( x ) ] - exp [ 2 l ( x ) ] + 1 } .
exp { 2 [ l ( x ) + l ( x ) ] } = exp { 4 l + 2 [ l ( x ) + l ( x ) - 2 l ] 2 } .
[ l ( x ) + l ( x ) - 2 l ] 2 = 2 C l ( 0 ) + 2 C l ( ρ ) ,
C I ( ρ ) = I 0 2 { exp [ 4 C l ( ρ ) ] - 1 } .
C I ( 0 ) = I 0 2 { exp [ 4 C l ( 0 ) ] - 1 } .
W ( x , D ) = { 1 , if x D / 2 0 , if x > D / 2 } .
S = d x W ( x , D ) I ( x ) .
S ¯ = d x W ( x , D ) I ( x ) = d x W ( x , D ) I ( x ) .
S ¯ = ( π / 4 ) D 2 I 0 ,
σ S 2 = ( S - S ¯ ) 2 .
σ S 2 = { d x W ( x , D ) [ I ( x ) - I 0 ] } × { d x W ( x , D ) [ I ( x ) - I 0 ] } .
σ S 2 = d x d x W ( x , D ) W ( x , D ) C I ( ρ ) .
ϱ = x - x
ϱ = 1 2 ( x + x )
σ S 2 = d ϱ [ d ϱ W ( ϱ + 1 2 ϱ , D ) W ( ϱ - 1 2 ϱ , D ) ] C I ( ρ ) .
K 0 ( ρ , D ) = d ϱ W ( ϱ + 1 2 ϱ , D ) W ( ϱ - 1 2 ϱ , D ) = { D 2 2 { cos - 1 ( ρ D ) - ( ρ D ) [ 1 - ( ρ D ) 2 ] 1 2 } , if ρ D 0 , if ρ > D .
σ S 2 = 2 π 0 D ρ d ρ K 0 ( ρ , D ) C I ( ρ ) .
Θ = σ S 2 / ( π 4 D 2 ) 2 C I ( 0 ) .
Θ = 16 π D 2 0 D ρ d ρ exp [ 4 C l ( ρ ) ] - 1 exp [ 4 C l ( 0 ) ] - 1 × { cos - 1 ( ρ D ) - ( ρ D ) [ 1 - ( ρ D ) 2 ] 1 2 } ,
Θ = π 4 d 0 2 / π 4 D 2 .
d 0 = D Θ 1 2 .
( 4 h 0 sec θ / k ) 1 2 = 3.2 × 10 - 2 sec 1 2 θ = 3.2 × 10 - 2 m             ( θ = 0 ° ) , = 4.5 × 10 - 2 m             ( θ = 45 ° ) , = 6.4 × 10 - 2 m             ( θ = 60 ° ) ,
C l ( 0 ) = 0.73 sec 11 / 6 θ = 0.73             ( θ = 0 ° ) , = 1.38             ( θ = 45 ° ) , = 2.60             ( θ = 60 ° ) .
D / ( 4 h 0 sec θ / k ) 1 2 = 47 ( θ = 0 ° ) , = 33 ( θ = 45 ° ) , = 23 ( θ = 60 ° ) .
Θ = 2.4 × 10 - 4 ( θ = 0 ° ) , = 2.2 × 10 - 4 ( θ = 45 ° ) , = 1.7 × 10 - 4 ( θ = 60 ° ) .
σ S 2 / S ¯ 2 = Θ C I ( 0 ) / I 0 2 .
σ S 2 / S ¯ 2 = Θ { exp [ 4 C l ( 0 ) ] - 1 } .
σ S 2 / S ¯ 2 = ( 0.065 ) 2 ( θ = 0 ° ) , = ( 0.81 ) 2 ( θ = 45 ° ) , = ( 2.37 ) 2 ( θ = 60 ° ) .
G=ln{16π01exp{σ2[bA(Dx)-1]}[cos-1x-x(1-x2)12]xdx},
Gσ=012σ[bA(Dx)-1]exp{σ2[bA(Dx)-1]}×[cos-1x-x(1-x2)12]xdx01exp{σ2[bA(Dx)-1]}[cos-1x-x(1-x2)12]xdx