PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
5.2 OPTICAL TURBULENCE IN THE ATMOSPHERE
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].
OPTICAL TURBULENCE IN THE ATMOSPHERE 83
stream or a culvert, when there is no velocity mixing, the flow is laminar. A protrusion into the stream creates velocity mixing that generates turbulent eddies. Turbulence occurs when the Reynold’s number,Re, exceeds
Re= Vl
ν (5.22)
whereVis the velocity,lis the dimension, andνis the viscosity in m2/s. The Navier–
Stokes partial differential equations for turbulence are generally too computationally demanding. So Kolmogorov developed a statistical approach [68] that is a mix of statistics and intuition, and is widely used today.
5.2.1 Kolmogorov’s Energy Cascade Theory
When the velocity exceeds the Reynold’s number, large size eddies appear of ap- proximate dimensionL0(Figure 5.1). Inertial forces break the eddies to decreasing sizes until a sizel0is reached at which the absorption rate is greater than the energy injection rate and the eddies disappear through dissipation. Hence, at any time, there are a discrete number of eddies ranging in size froml0toL0, the inertial range. Like whirlpools these eddies are somewhat circular and the air increases in velocity with radius and hence decreases in density with velocity. Lower density implies lower dielectric constant, so these eddies act like little convex lenses fluctuating around a mean orientation and of various sizes. Intuitively, such lenses cause a beam to wander and to spread more than that due to diffraction alone. From the cascade theory, struc- ture functions and corresponding spatial power spectra can be developed for velocity, temperature, and refractive index; the latter results in optical turbulence.
The structure function for velocity may be written as
DRR(R)= (V2−V1)2 =C2vR2/3 for l0 R L0 (5.23)
L0
l0
Energy injection
Dissipation FIGURE 5.1 Energy cascade theory of turbulence.
84 PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
whereCv2=2ξ2/3, the velocity structure constantCv, is a measure of the total energy andξis the average energy dissipation rate. For small-scale eddies, the smallest size l0≈η=ν3/ξ, so the minimum size is less for strong turbulence. In contrast, the largest size is larger for strong turbulence becauseL0∝ξ1/2.
From the equations in Section 5.1, the 3D spatial power spectrum for a locally homogeneous isotropic random field can be written as
RR(κ)=0.033C2vκ−11/3 for l/L0 κ 1/ l0 (5.24) Note that in equation (5.24), the 2/3 power law in the structure function for the locally homogeneous isotropic random field becomes a 5/3 power law for the 1D spectrum that becomes a characteristic (of Kolmogorov models) −11/3 power law in a 3D isotropic spectrum, as discussed in the text following equation (5.18).
Similarly, a spectrum may be written for temperature fluctuations as
T(κ)=0.033CT2κ−11/3 for l/L0 κ 1/ l0 (5.25) whereCTis the temperature structure constant.
Optical turbulence is associated with refractive index written by separating out the meann0= n(R, t):
n(R, t)=n0+n1(R, t) (5.26) Normally, the variations of refractive index due to turbulence are much slower than the time taken for a beam to travel through it at the speed of light, so we can neglect time to obtain
n(R)=n0+n1(R) (5.27)
In the optical (including IR) wave bands, an empirical equation for refractive index is [4]
n(R)=1+77.6+10−6(1+7.52×10−3λ−2)P(R)
T(R) (5.28)
whereP(R) is pressure in millibars,T(R) is temperature in kelvin, andλis optical wavelength in micrometers. As dependence on wavelength is weak, we can letλ= 0.5m to obtain
n(R)=1+79×P(R)
T(R) (5.29)
The refractive indexn(R) inherits the scales and range of velocity.
For statistically homogeneous fields,R=R2−R1, so covarianceBn(R1,R2)= Bn(R1,R1+R)= n1(R1)n2(R1+R) (Section 5.1). If the field is also isotropic,
OPTICAL TURBULENCE IN THE ATMOSPHERE 85
R= |R1−R2|, the structure function for refractive index can be written by analogy with equation (5.12) as
Dn(R)=2[Bn(0)−Bn(R)]=
C2nR−04/3R2 0≤R l0
C2nR2/3 l0 R L0 (5.30) where the strength of the turbulence is described by the refractive index structure parameterCn2in units of m−2/3. The inner scale isl0=7.4η=7.4(ν3/ξ)1/4.CT2in equation (5.25) may be obtained by point measurements of the mean square temper- ature difference with fine wire thermometers.Cn2can be computed fromC2Tas
Cn2=
79×10−6P(R) T(R2)
2
CT2 (5.31)
From Ref. [4], values ofCn, measured over 150 m at 1.5 m above the ground, range from 10−13for strong turbulence to 10−17for weak turbulence.
5.2.2 Power Spectrum Models for Refractive Index in Optical Turbulence
The following models enable generation of synthetic turbulence that may be used to estimate the effects of turbulence on optical propagation and allow test with adaptive optics.
• Power Spectrum for Kolmogorov Spectrum: From Section 5.2.1, the Kolmogorov spectrum for refractive indexnhas the same form as that for velocity; therefore, replacingCvin equation (5.24) withCngives
n(κ)=0.033C2nκ−11/3 for 1/L0 κ 1/ l0 (5.32) To compute phase screens to represent the effects of turbulence in space, we need to take an inverse Fourier transform of the spectrum. So we would like the model to have a range 0 κ ∞. Unfortunately, extending the lower (large- scale) limit in equation (5.32) to zero wave number or spatial frequency is not possible becausenblows up to∞asκ→0 because of the singularity atκ=0, and this makes the evaluation of the Fourier integral impossible. Truncating the spectrum at the large-scale limit where frequency is lowest is undesirable because equation (5.32) shows that the energy in the turbulence is largest at this limit, leading to large abrupt transitions that provide an unrealistic spectrum.
• Tatarski Spectrum and Modified Von Karman Spectra: The Kolmogorov spec- trum can be modified to extend above and below the limits by suitable mul- tipliers to equation (5.32). For extension into the high wave numbers (spatial frequencies) past the small-scale limitκ=1/ l0, caused by dissipation, we add
86 PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
a Gaussian exponential proposed by Tatarski [155]. For extending into the lower wave number (low spatial frequency) past the large-scale limitκ=1/L0, we di- vide by a factor (κ2+κ02), whereκ0=C0/L0withC0=2π, 4π, or 8πdepend- ing on the application [4]. The reason for the variety in selectingC0is that above the large-scale limitL0, the field is no longer statistically homogeneous—the eddies may be distorted by atmospheric striations—so approximation is mainly to make the equations more tractable. The resulting modified Von Karman spec- trum is
n(κ)= 0.033C2n (κ2+κ20)11/6exp
−κ2
κ2m for κm=5.92/ l0, κ0=2π/L0
(5.33)
5.2.3 Atmospheric Temporal Statistics
When wind blows across the path of a laser beam in the atmosphere, the turbulence will move at wind speedV⊥across the beam. Because the eddy pattern takes many seconds to change, the motion of noticeable wind L0/V⊥ is much faster. This is known as the frozen turbulence hypothesis and is similar to cloud patterns caused by air turbulence passing across the sky. Therefore, we can convert spatial turbulence due to eddy currents into temporal turbulence that incorporates the effects of the component of wind at right angles to the path.
u(R, t+τ)=u(R−V⊥τ, t) (5.34)
5.2.4 Long-Distance Turbulence Models
For an optical beam traversing many miles, the strength of turbulence may vary over the path. For example, if a beam moves over the land past cliffs over sea, the turbulence strength will vary because the land and sea have different temperatures and wind striking the cliff will generate strong turbulence. Such situations may be handled by partitioning the path into regions, each with its own constant turbulence.
Further in numerical computations, the effect of turbulence in each region may be equated to a phase screen. Such phase screens were implemented around a sequence of optical disks for a hybrid test facility for the airborne laser (Section 12.2.7.2).