PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
5.1 STATISTICS INVOLVED
Owing to random fluctuations of refractive index in space, whenever a pulse of light from a laser propagates through the atmosphere, it takes a slightly different physical path. The collection, called ensemble, of different paths in space composes a continu- ous random field [4]. Each path is considered to be a realization of the random field in space. A randomfieldfor which realizations are in space contrasts a randomprocess for which realizations are in time. Because of randomness, the path for a realization is unpredictable, so we use averages or expected values that we label witha(in some texts E{}is used). For tractability, we initially focus on first- and second-order statistics: the mean and autocovariance. The mean for a random field m(R) is the ensemble average at a point in space identified by its vector locationR(x, y, z) over all realizations or paths:
m(R)= u(R) (5.1)
The spatial autocovariance function (second-order statistic) provides the expected relationship between the fields at two different locationsR1andR2:
Bu(R1,R2)= [u(R1)−m(R1)][u∗(R2)−m∗(R2)] (5.2) When the mean is zero, m(R)=0, Bu reduces to the autocorrelation function, a special case of the autocovariance function.
STATISTICS INVOLVED 79
IfBudoes not depend on the absolute value ofR(x, y, z) but only on the difference R=R2−R1orR2=R1+R, we have a statistically homogeneous field in space.
(This is the equivalent of a wide sense stationary process in random processes.) Bu(R1,R2)=Bu(R1−R2)= u(R1)u∗(R1+R) − |m|2 (5.3) Often fields are invariant under rotation or statistically isotropic: independent of di- rection ofR,R = |R2−R1|. This allows us to reduce the 3D vector location to a 1D scalar, the distanceRfrom the axis of a beam. Then,
Bu(R1,R2)=Bu(R) (5.4)
5.1.1 Ergodicity
In practice, when we shine a laser beam through the atmosphere, we have only a single realization. Statistics for this one beam are easier to compute than those for an ensemble of beams. For example, a 3D space average is
Bu(R)= lim
x→∞
1 L
L
0
u(R)dR (5.5)
whereL is the path length. This is the 3D space equivalent of a time average in a random process.
Similarly, for the space average for autocovariance, Bu(R1,R2)= lim
x→∞
1 L
L
0
[u(R1)−m(R1)][u∗(R2)−m∗(R2)]dR (5.6) Often we assume that the random field is ergodic, which means that the ensemble average at a point in space for a specific realization or path can be replaced by the space average for a single path. Ergodicity is difficult to prove except for simple cases, but for optical turbulence most of the time, ergodicity turns out to be a reasonable assumption. For an ergodic random field, the space averageu(R) is the same as the ensemble averageu(R):
u(R)= u(R) (5.7)
and the space average of autocovarianceBu(R1,R2) is the same as the ensemble average for the autocovarianceBu(R1,R2):
Bu(R1,R2)= Bu(R1,R2) (5.8) Note that the space average for one realization is easier to measure than the ensemble average.
80 PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
5.1.2 Locally Homogeneous Random Field Structure Function
In practice, for atmospheric turbulence, the mean is not constant over all space: ve- locity, temperature, and refractive index vary because the wind blows harder in some places. However, the mean is constant in a local region. We write
u(R)=m(R)+u1(R) (5.9)
wherem(R) is the local mean that depends onRandu1(R) represents the random fluctuations about the mean. Locally homogeneous random fields are characterized by structure functions [4]. Such behavior frequently arises in random processes where it is labeled a random process with stationary increments [135], for example, a Poisson random process. Stationary may be qualified by wide sense when only autocovariance and mean are involved.
The structure function that characterizes locally homogeneous atmospheric turbu- lence is written as [4]
Du(R1,R2)=Du(R)= [u1(R1)−u1(R1+R)]2 (5.10) where we setu(R1)≈u1(R1). Expanding equation (5.10) for a locally homogeneous random field gives
Du(R)= u1(R1)u∗1(R1) + u1(R1+R)u∗1(R1+R)
−2u1(R1+R)u1(R1) (5.11)
If the random field is also isotropic,R= |R|,Du(R) is related toBu(R)
Du(R)=2[Bu(0)−Bu(R)] (5.12) 5.1.3 Spatial Power Spectrum of Structure Function
For high-frequency microwaves and optics, we often measure power because sen- sors may not be fast enough to record phase information. Furthermore, for random processes or fields, the phase may be of no consequence. Also, extracting signals from noise is often accomplished by spectral filtering to remove noise frequencies.
So we need to compute the power spectrumu(κ) from the structure functionDu(R) for locally homogeneous isotropic fields. Relation between a space function and its spectrum is usually defined using Fourier transforms (Section 3.1.2). This requires that the space function be absolutely integrable [133]:
∞
−∞u(R)d(R)<∞ (5.13)
STATISTICS INVOLVED 81
Unfortunately, equation (5.13) is not valid for a random field or process. The Fourier–
Stieltjes (or Riemann–Stieltjes) transform solves this problem [4, 135] and produces the well-known Wiener–Khintchine theorem.
For wide sense (mean and covariance) stationary random processes (in time), the autocovarianceBx(τ) and the power spectral density (or power spectrum)Sx(ω) are a Fourier transform pair:
Sx(ω)= 1 2π
∞
−∞Bx(τ) exp{−iωτ}dτ= 1 π
∞
0
Bx(τ) cos(ωτ)dτ Bx(τ)=
∞
−∞Sx(ω) exp{iωτ}dω=2 ∞
0
Sx(ω) cos(ωτ)dω (5.14) where the cosine equations arise in the second equalities because power and autoco- variance are real and even.
In space, for statistically homogeneous isotropic random fields, from equa- tions (5.14), the autocovariance Bu(R) and 1D spatial power spectrum Vu(κ) are similarly related by
Vu(κ)= 1 2π
∞
−∞Bu(R) exp{−iκR}dR= 1 π
∞
0
Bu(R) cos(κR)dR Bu(R)=
∞
−∞Vu(κ) exp{iκR}dκ=2 ∞
0
Vu(κ) cos(κR)dκ (5.15) whereκis the spatial angular frequency,κ=2πfs(space equivalent toω=2πf).
We now find the relationship between the 1D spatial power spectrumVuin equa- tion (5.15) and the spatial power spectrum for a 3D statistically homogeneous isotropic medium (for which radiusR= |R2−R1|). Note that these are not the same. For a statistically homogeneous medium (not isotropic), the 3D autocovarianceBu(R) and the 3D spatial power spectral density are related by the Wiener–Khintchine theorem:
u(κ)= 1
2π
3 ∞
−∞exp{−iκ·R}Bu(R)d3R Bu(R)=
∞
−∞exp{iκ·R}u(κ)d3κ (5.16) If the medium is also isotropic, we setR= |R2−R1|andκ= |κ|in equations (5.16) to get [4]
u(κ)= 1 2π2κ
∞
0
Bu(R) sin(κR)RdR Bu(R)= 4π
R ∞
0
u(κ) sin(κR)κdκ (5.17)
82 PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
Therefore, the 3D spatial power spectrum for a locally statistically homogeneous isotropic medium,u(κ) in equation (5.17), is related to the 1D spatial power spectrum for a locally statistically homogeneous medium,Vu(κ) in equation (5.15), by
u(κ)= − 1 2πκ
dVu(κ)
dκ (5.18)
From equation (5.18), we see the origin of the −11/3 power rule that arises in models of turbulence: If 1D spectrum Vu∝κ−5/3, then the 3D spectrum u∝
−(5/3)V−8/3/κ= −(5/3)V−11/3. Note thatu(κ) in equation (5.17) has a singu- larity atκ=0, so the convergence ofBu(R) in equation (5.17) require restrictions in the rate of increase ofu(κ) asκ→0 [4].
In practice, we allow a beam to propagate from one plane to another at separation zso that we need the 2D spatial power spectrumFu(κx, κy,0, z) between two planes zapart [4]:
Fu(κx, κy,0, z)= ∞
−∞u(κx, κy, κz) cos(κzz)dκz
u(κx, κy, κz)= 1 2π
∞
−∞Fu(κx, κy,0, z) cos(κzz)dκz (5.19) The structure function for a 3D locally homogeneous random field,Du(R), equa- tion (5.11), is related to its 3D spatial power spectrum u(κ), equation (5.16), by [4, 60, 155]
Du(R)=2 ∞
−∞u(κ)[1−cos(κ·R)]d3κ (5.20) If the field is also isotropic, we have the Wiener–Khintchine equation pair
u(κ)= 1 4π2κ2
∞
0
sin(κR) κR
d dR
R2 d
dRDu(R)
dR Du(R)=8π
∞
0
κ2u(κ)
1−sin(κR) κR
dκ (5.21)
We note that some approaches to optical turbulence use the spatial power spectrum and some the structure function. We compare the two approaches in Refs. [88, 90].