PROCEEDINGS OF THE IEEE, VOL. 58, NO.

10, OCTOBER 1970 1523

A Survey of Clear-Air Propagation Effects

Relevant to Optical Communications
ROBERT S. LAWRENCE AND JOHN W. STROHBEHN, MEMBER, IEEE

Abstract-The theory and observations of the optical propagation n = refractive index

effects of the clear turbulent atmosphere are reviewed, with particu-
lar attention to those characteristics most important to the designer
no =ensemble average of n
of an optical communication system. Among the phenomena con- n, =n-no
sidered are the variance. probability distribution, spatial covariance, P =atmospheric pressure
aperture smoothing, and temporal power spectrum of intensity
fluctuations, and similar quantities for phase fluctuations and angle
R = autocorrelation function
of arrival. R , =radius of curvature of a wavefront
I. INTRODUCTION S = phase of the electric vector of the optical wave
S = integrated intensity over an aperature

T HIS SURVEY treats the effects of the clear atmo-

sphere upon light waves, with particular attention to
the effects of turbulence. The results of atmospheric
stratification and the normal variations with height of the
T = Kelvin temperature
uI = wind velocitycomponent normal to the optical path
u1I = wind velocitycomponent parallel to the optical path
W, = radius of an optical beam
atmosphere are also described, but we ignore any absorp- w =temporal power spectrum
tion by gaseous constituents and make no mention of 2 =zenith angle (= 4 2 - 6,)
scattering or absorption by aerosols or precipitation. Ef- a = angle of arrival
fects that aremost important to thedesign ofnarrow-beam a = angle between windand direction of propagation
optical communications systems operating betweentwo 6 = aperture-averaging factor
points on the ground orbetween the ground and space are 6, =apparent elevation angle above the horizon
emphasized. Much of our motivation in preparing this sur- (=x/2-2)
vey arises from the needs expressedclearly in the report [ 1] IC = 2x11, spatial wavenumber
of ajoint M.1.T.-NASA workshop. Rationalized MKS K, = 2Tt/10
units are used, and the following symbols are employed. K O =27C/Lo
A =amplitude of the electric vector of the optical wave i,= optical wavelength
B = autocovariance or covariance function v = ratio of specific heats, z 1.4 forair
b =normalized autocovariance or covariance function p=separation, parallel to the wavefront, of points of
b =separation of interferometer elements observation
C =curvature of a ray p= ratio of receiver aperture radius to Fresnel-zone
C; = refractive-index structure parameter radius
D = aperture diameter oTnI= variance of log-intensity
D, = phase structure function 0,' =variance of log-amplitude

E = electric vector of the optical wave z =astronomical refraction (total bending) of a ray
F = two-dimensional spatial power spectrum (0 = three-dimensional spatial power spectrum
h =height above ground x = log A , the log-amplitude of the optical wave
I = A', intensity (or irradiance) of the optical wave Y =log E
k = 2 x 1 5 wavenumber of the optical wave R = k Wg/2L.
L = path length, or distance to the turbulent layer
Lo = outer scale of turbulence In Section I1 we briefly mention the magnitudes of the
1 = 2 n / ~scale
, sizeof an irregularity various effects and in later sections consider them ingreater
1, =inner scale of turbulence detail. Section I11 describes the atmospheric model we are
N = (n- 1) lo6, refractivity in parts per million using to derive the various optical results, Sections IV, V, VI,
and VI1 discuss both theory and observations of optical.
Manuscript received May 17,1970. Thiswork was supported in part by intensity fluctuations, optical phase fluctuations, and angle-
the Atmospheric Sciences Section, National Science Foundation, under of-arrival effects, respectively.'
Grant GA 1465.
R. S. Lawrence is with the ESSA Research Laboratories, Boulder,
Colo. 80302. After this manuscript had been prepared, we learned of a similar
J. W. Strohbehn is with the Department of Radiophysics, Dartmouth effort, at the time unpublished, by Brookner [95]. Many readers will find
College, Hanover, N. H. 03755. his work also of interest.
1524 PROCEEDINGS OF THE IEEE, OCrOBER 1970

11. SUMMARY VALUE

OF NUMERICAL separated by distance p , parallel to the wavefront (the
“phase structure function”) D&), varies as p5I3 and in-
Optical IntensityFluctuations
creases linearly with path length and with q2, the strength
A smooth andfeatureless optical wave or beam traversing of the turbulence. See Table I11 for formulas.
more than a few meters of ordinarily turbulent atmosphere
has its energy redistributed and exhibits fluctuations in in- Angle of Arrival
tensity known as “scintillations.” The logarithm of the in- Systematic bending of a ray passing through the entire
tensity of the distorted wave has a Gaussian probability atmosphere between earth and space depends strongly on
density function, so the strength of the scintillations is de- the elevation angle 8, of the source as seen by the terrestrial
scribed in terms of “log-intensity variance.” Log-intensity observer. The sense is always such as to make the source
variance increases with the strength of the turbulence and appear higher above the horizon than it really is. For a sea-
as the 11/6 power of the path length until it reaches a value level observer with standard atmospheric pressure and
of approximately 2.4, then it “saturates” and no longer in- temperature, the total bending (or “astronomical refrac-
creases. In fact, it slightly decreases. For paths a few meters tion”) decreases from about 10 mrad when 8, =0 to zero at
above the ground, saturation may occur within a kilometer the zenith. For 8, > 170 mrad, the astronomical refraction
or two;for earth-to-space paths, saturationoccurs only for z x 0.28 cot 8, mrad. The value of z decreases nearly linearly
elevation angles less than a few degrees.Formulas are given with pressure as the observer rises above sea level.
in Table I. The downward bending of anoptical beam on a graund-
The instantaneous random pattern of scintillations, ‘bb- to-ground path is related to the vertical gradient of refractive
served in a plane parallel to the wavefront, displays spatial index. For standard sea-level conditions, the curvature of a
Fourier components with a variety of scales, but !he pre- beam is
dominant scale size is the Fresnel-zone size @, where i dT
is the optical wavelength and L is the path length or, for C = 32.6 + 0.93 -
dh
prad/km,
space-to-earth paths, the distance between the observer and
the turbulent atmospheric layer. See Table I1 for formulas. where dT/dh is the temperature lapse rate in K/km.
Predominant scalesizes thus vary from about 1 cmfor Short-term variation in the angle of arrival is best de-
paths 1 km long to 7 or 8 cm for slant paths through the scribed in terms of the phase structure function, as shown
entire atmosphere. Since the scintillation pattern is random, quantitatively in [53]. The variation on earth-to-space
if a receiving aperture receives a number N of statistically paths typically amounts to a few microradians, whileit
independent portions of the pattern corresponding to the may often beten or more times greater on horizontal paths.
predominant scalesize, the log-intensity variance of the
received signal will be reduced by a factor N . This effect is
known as “aperture smoothing.” 111. THEATMOSPHERIC MODEL
The temporal frequency of the variation of intensity ob- This section describes the clear-air atmospheric proper-
served at a fixed point depends upon the transverse com- ties most important when .considering optical communica-
ponent of wind that moves the turbulence across the beam. tions systems. Quite arbitrarily, we divide the discussion
The predominant frequency is obtained by dividing wind into “large-scale’’ and “small-scale” variations. We de-
velocity by Fresnel-zone size. For most paths it is usual for scribe as large scale those phenomena that affect the average
the important temporal frequencies to occupy the range position of a light beam measured over a period of, say, 10
from 1 to 100 Hz. seconds ; and we describe as small scale those irregularities
Until saturation occurs, the log-intensity variance of that, for example, distort an image, broaden a laser beam,
scintillations varies inverselyas the 7/6 powerof the optical or cause rapid twinkling and dancing of a star. The dividing
wavelength. Although available observations are contradic- line between these scales isa meter or less.
tory, the level of saturation probably varies little with wave- Large-Scale Variations
length. Details of the scintillation patternare relatively
broad-band, differing little with a wavelength change of a The largest scales in the atmosphere are the general de-
few tens of nanometers, but they become quite uncorrelated crease with height of atmospheric density and the curvature
when the wavelength changes by a factor of 2. of the atmospheric layer as it clings to the spherical earth.
An optical ray arriving at the earthfrom a star bends down-
ward as it passesthrough an atmosphere of steadily increas-
Optical Phase Fluctuations ing refractive index. This effect isparticularly significant for
While the intensity scintillations are caused by turbulence stars near the horizon or for near-horizontal ground-to-
with a scale near the Fresnel-zone size, phase fluctuations ground optical paths.
are produced by the largest scales in the turbulence and so The variation of atmospheric refractive index with height
depend on the “outer scale” of the turbulence and on the above sea level can be computed from The U.S . Standard
nature of the turbulence-generating mechanisms and the Atmosphere [2] adopted by the International Civil Aviation
height above ground. Organization (ICAO). The refractive index n of the atmo-
Until the size of the outer scale is reached, the mean- sphere is conveniently and customarily described in terms of
square difference of the phase observed at two points the refractivity M inparts per million. TheseN units are re-
LAWRENCE AND PROPAGATION
STROHBEHN: CLEAR-AIR EFFECTS 1525

lated to n by N =(n - 1)106.For optical wavelengths, and covariance functions of n(r), realizing all the time that
N z 79 PIT. (1) the useof spectra requires the assumption of statistical
homogeneity (or spatial stationarity) of the atmosphere.
Here P is the atmospheric pressure in millibars and T is the In fact, the atmosphere is notably nonstationary, i.e., the
temperature in Kelvin.N is roughly equal to290 at sea level, statistics of its fluctuations vary from time to time and from
varying with wavelength by about 10 percent over the visible place to place. A method of analysis involving structure
band and with humidity by about 1 percent. The details of functions and theassumption of only locally homogeneous
these variations have been reviewed by Owens [3 J. in Sec- statistics has been devised and is partially successfulin
tion VIwe discuss the application of these relationships meeting this problem. Tatarski [8], [9] deals with this a p
and of the model atmosphere to the calculation of the sys- proach at length, and a summary of the pertinent formulas
tematic bending of an optical beam. isgivenby Strohbehn [lo]. Assuming stationarityand
The actual atmosphere is usually less smooth than the isotropy, we then define the autocovariance function B,(r)
ideal model of the U. S . Standard Atmosphere. Particularly of the refractiveindex and its Fourier transform, the
characteristic are layers having great horizontal extent and spectrum @,(K), as
and a vertical temperature gradient that changes abruptly.
BAT) = (n1(rl)n1(r1 + T ) )
Their structure is usually measured with standard balloon-
borne radiosonde equipment having time constants that
limit the spatial resolution to about15 meters. Thus we have
only meager knowledge of howsharply defined such layers
may be. We do know, however, that “looming” can occa- and
sionally occur, a phenomenon in which a light ray is re-
fracted with a curvature equal to orgreater than thecurva-
ture of the Garth. We shall see in Section VI that a positive
vertical temperature gradient of 134 K/km is required to where IC=2x11 is a spatial wavenumber and 1 is the cor-
produce such ducting. Thayer [4], while interpreting UFO responding scale size.
sightings, displayed a number of radiosonde profiles which, Refractive-index fluctuations are caused almost exclu-
when corrected for sensor lag, illustrate such sharp layers sively by fluctuations in temperature, since pressure varia-
1 or 2 km above ground. He infers that even sharper gra- tions are relatively small and are rapidly dispersed. The
dients must sometimes exist but points out the difficulty of small-scalesizes andshort lifetime of the temperature
making the necessary measurements. Lane [5], using a fluctuations ensure that they are adiabatic, so that
high-speed thermometer mounted on an airplane, mea-
sured an elevated layer at 2.6 km with a total temperature 6P -
_ v -,6T
--
change of 4 K and a vertical temperature gradient that he P V-1 T
estimates to be about 1000 K/km. Richter [ 6 ] has observed
radar scattering layers less than 1 meter thick. where the ratio of speciiic heats v = c p / c v x 1.4 for air.
From such hints emerges the picture of thin sharp gradi- Differentiating (1) and eliminating 6P, wesee that the
ents of refractive index, often well above ground, extending refractive-index fluctuations are proportional to the tem-
horizontally for many kilometers and persisting for hours perature fluctuations :
at atime. Visual observation from aircraft passing through
-79 P
the boundary of a haze layer occasionally discloses a more 6 N = --6T.
or less regular undulation of the layer. Ludlam [7] sug- V - 1 T2
gested that the breaking of such waves produces much of
the turbulence that occurs a kilometer or more above The temperature fluctuations originate from large-scale
ground. phenomena, such as heating of the earth’s surface, but are
broken and mixed by the wind until they exist at all scales.
Small-Scale Variations Since wind velocity fluctuations control the temperature
variations, we shall fist briefly describe the statistics of the
Turbulence is almost always superposed on the large- random velocityfield. More detailed descriptions are
scale variations of the atmosphere and produces random available in many places [ll], [12] ;the two books by Tatar-
fluctuations in the density and therefore in the refractive ski [8], [9] cover aspects particularly relevant to propaga-
index. If weare willing to neglect the minor optical Doppler tion.
shifts produced by the irregular turbulent velocities, we The kinetic‘energyof turbulence is usually introduced by
need consider only the spatial structure of the turbulence such large-scale phenomena as wind shear or convection
and may write the random variation of the refractive index from solar heating of the ground.Therefore, it isreasonable
as to assume that the turbulent energy is introduced through
the action of scale sizes hrger than some minimum value
44 = no + n16-1; no = (n(r)>,
Lo, called the outer scale of turbulence, which corresponds
where the sharp brackets denote an ensemble average. to a wavenumber K~ =27c/L0.For wavenumberssmaller
For brevity and clarity, we define and work with spectra than K~ the form of the energy spectrum cannot be specified
1526 PROCEEDINGS OF THE IEEE,
OCTOBER 1970

in general, since it depends on surface geography and local models for the atmosphere under many conditions. They do
meteorological conditions. Verylikely, the turbulence is not represent the actual atmosphere at alltimes or all
neither homogeneous nor isotropic in this range. Near the places but it seems impossible to comment usefully on their
ground, the value of the outer scale is thought to be com- range of validity. Bolgiano [18] predicted that the buoy-
parable with the height above ground. In the upper atmo- ancy effects that sometimes become important atlow wave-
sphere Lo is often 100 meters or more, but in the presence of numbers should produce a steeper spectrum. Lawrence,
stratified layers, perhaps as thin as a few meters, Lo can Ochs, and Clifford [19] observed flatter spectra near the
not greatly exceed the layer thickness. ground, presumably caused by the direct introduction of
As the spatial wavenumber increases above IC, and the turbulent energy at scale sizes smaller than Lo.
scale size becomessmaller than the distance to the ground In contrast to theforegoing, a Gaussian spectrum and its
or to the nearest inversion layer, the turbulence tends to corresponding Gaussian autocovariance function are fre-
become isotropic and homogeneous. Under these condi- quently used for mathematical convenience. A serious
tions, both theory and experiment indicate a K - 11/3 de- problem is associated with the use of the Gaussian model
pendence for the spectrum. The theory of this dependence, for optical propagation ; we believe its use should be dis-
originally proposed by Kolmogorov [13] in 1941, is based couraged. The Gaussian model contains only a single scale
on a number of assumptions: the energy is input at small size, although at least two independent scales are important
wavenumbers and dissipated at much larger wavenumbers, for optical effects. We shall see that the small-scale fluctua-
the Reynolds number must be much greater than unity, and tions are primarily responsible for intensity scintillation
the scale sizesare small enough so that buoyancy effects are effects, andthe large-scale fluctuations produce optical
negligible. In this wavenumber range, the inertial subrange, phase effects. A Gaussian model having too few adjustable
the Reynolds number is so high for the large eddies that parameters ties thesetogether in an arbitrary andunrealistic
they are unstable and break apart. Thusthey transfer energy way and engenders misleading and erroneous. predictions.
to smaller scales (higherwavenumbers) down to the “inner So far we have discussed the form of the refractive-index
scale of turbulence” I,, a scale of only a few millimeters, spectrum of the turbulent atmosphere but have made only
where the Reynolds number approaches unity andthe the most general references to the actual values of the pa-
energy is dissipated into heat. The spectrum in the viscous rameters. The adjustable parameters of the model are as
dissipation region for whichIC > K, = 5.92/1, is considerably follows: 1) C;, the refractive-index structure parameter
steeper than in the inertial subrange, but its form is un- (sometimes optimistically called the structure “constant”)
certain. which is a measure of the intensity of the refractive-index
Temperature fluctuations which, as we haveseen, are fluctuations ; 2) I,, the inner scale of turbulence, below
directly pertinent to optical propagation were considered which viscousdissipation steepens the spectrum ;3) Lo, the
by Obukhov [14], Corrsin [15], and Pond et al. [16]. They outer scale of turbulence, above which the energy is pre-
found thatthe temperature fluctuations obey the same sumed to be introduced into the turbulence. Between 1,
spectral law as the velocity fluctuations. It should be and Lo lies the “inertial subrange” where the Kolmogorov
remembered that while the forms of the spectra are identi- model predicts a spectral slope of - 11/3. The slope may
cal, their magnitudes are not necessarily related. Strong be considered to be a fourth parameter if we wish to consider
mechanical turbulence tends to produce a well-mixed cases near the ground where the energy is introduced in a
atmosphere with an adiabatic lapse rate so that the tem- wide variety of scale sizes.Alternatively, and perhaps more
perature fluctuations become very weak[17]. Conversely, a properly, such a case may be described as one where the
stable atmosphere with little windmaypossess strong outer scale Lo has decreased to nearly equal the inner scale
thermal gradients. Henceforth, weuse the word “turbu- I,, thereby destroying the inertial subrange. Let us consider
lence” to refer only to the temperature fluctuations, not to these parameters in turn.
the velocity as in the usual fluiddynamic sense. The refractive-index structure parameter C,’ is measured
Assuming that the form of the temperature spectrum, in units of Its value varies from io-” or less
and hence the refractive-index spectrum, is the same as that when the turbulence isextremely weak, toor more
of the velocity spectrum, Tatarski [9] used the following when the turbulence, generated near the ground in direct
form for the refractive-index spectrum : sunlight, is strong. InsufEcient measurements exist to per-
mit us to predict with confidence the mean value and vari-
@“(IC)= 0.033 C,’IC-’~’~ exp ( - K ’ / K ; ) , (3)
ance of C,2 under specified meteorological and geographical
with the following three conditions. 1) ~,1,=5.92. 2) The conditions. Except for occasional individual measurements
equation is a poor approximation for K 6 K,, i.e., for large near the ground, estimates are based largely upon optical
scales. 3) The equation is a reasonable approximation for observations and, as we shall see, substantial uncertainty
IC 2 IC,,,,i.e., for small scales. The refractive-index structure exists in the deduction of C,’ from such observations. A
parameter Cl is a measure of the -@tensityof the refractive- commonly used representation (Fig. 1) is the modelof
index fluctuations. Tatarski’s spectrum has a singularity at. C.”suggested by Hufnagel [20]. T h ~ model s is derived from
IC =0, so it does not possess an autocovariance function, but stellar scintillation measurements and from the few avail-
this difficulty can be avoided [lo]. able direct temperature-fluctuation measurements, usually
The preceding spectra are reasonable approximations or within 30 meters of the ground.Although Hufnagel’s model
LAWRENCE
EFFECTS
AND PROPAGATION
STROHBEHN: CLEAR-AIR 1527

are unsatisfactory because of the uncertainty of the effect

of the supporting structure. Gray and Waterman [21] and
Strohbehn [22] have proposed-measuring the inner scale
by measuring light intensity fluctuations over a short path.
Limited measurements by Gray and Waterman have given
values of I , from 5 to 12 mm.
The outerscale Lo is also uncertain. If we define Lo to be
lrn 1 0 0 rn IO krn
that size above which the turbulence becomes nonisotropic
ALTITUOE A W E UloVNo and depends on the details of the generating mechanism,
Fig. 1 . Model of refractive-indexstructureparameterproposed by then certainly t o must not exceed some reasonable fraction
Hufnagel [20]. Solid linesrepresentnormalatmospheric conditions of the distance from the ground. The value of Lo may also
for (in decreasing order)a sunny day, a clear night, anda dawn-dusk
minimum. The dashed line indicates typical (not average) effects to depend on the strength of the turbufence. Values frequently
be expected from such meteorological disturbancesas frontal activity. used for Lo are about 100 meters or 1/5 the height above
the ground, whichever is less. Ochs [23] has measured the
4 structure function of temperature fluctuations
metersat 1.6
above flat ground and found the function to increase
>-
tm u
steadily up to spacings of about-60 cm, where it saturates.
20 This observation suggests 1/3 the height above ground as
P5
ZE
2
a reasonable value for Lo.
w3
9
0 I v . GENERAL COMMENTS ON THE THEORY
4 . We
toattempt
do not present the derivations of the
theoretical results in any comprehensive way. Instead, the
>
final theoretical results are presented with a discussion of
F5 2 the assumptions and limitations involved. These results
ZE
A3 are then compared with observations, and we try to point
s out where we believe theory and experiment agree well,
0 where theory and experiment disagree, or where there is
2.0 insufficient information on either the observational or the
N
theoretical side.
-Q In this section
review
a brief of the theoretical area is pre-
: IO sented. These comments are restricted to the fluctuations
#-E small-scale
caused by variations in the refractive index.
N C
0
From a theoretical viewpoint there are two features that
0
w 06 I2 la w cause difficultiesin the problem
line-of-sight
of propagation
LOCAL TIME, SEPTEMBER 8. 1968 of optical waves through the atmosphere. The first d8i-
Fig. 2. Log-intensityvarianceversustime of dayanditsrelationship to culty is that the atmosDheric refractive index is a random

tion O J K ) which has beendiscussedin Section 111. The

second difficulty isthat for a number of situations the linear
is commonly used, other workers haveused the simpler theory based on applying a perturbation method to the
exponential model with scale heights of from 2 to 8 km. wave equation is inadequate. At the present time there has
In Fig. 2 the bottomcurve shows the typical diurnal varia- been little progress in extending the linear theory.
tion of C: measured with temperature sensors 1.6 meters There are several different theoretical approaches to the
above the ground. Particularly notable are the increase in general problem of the propagation of waves in a random
C: and its variability during the sunny hours of the day, medium; excellent discussions of the mathematical aspects
and the very low values near sunrise and sunset when the are given by Keller [24], [25] and Hoffman [26]. The ap-
temperature lapse rate near the ground was adiabatic. proach that appears to be the most fruitful has been in di-
The inner scale of turbulence Zo can be calculated from rectly solving the wave equation. This method is discussed
the kinematic viscosity of air and from the rate of viscous in detail by Chernov [27] and Tatarski [8], [9]. Though
dissipation of turbulent energy. From such calculations, 1, there are a great many useful results in the monograph by
is usually considered to be approximately 1 or 2 mm near Chernov and in the first monograph by Tatarski, their
the ground, increasing to about 1 cm at the tropopause. claims for the range of validity of some of the derived ex-
Direct measurements of Io with small temperature probes prssims are tooliberal. The more recent work ofTatarski
1528 PROCEEDINGS OF THE IEEE,
OCTOBER 1970

and a paper by Strohbehn [ 2 8 ] discuss the limitations in called the log-amplitude and the imaginary part is the phase.
the theory of the earlier works. A different but extremely With this substitution, the wave equation becomes
interesting approach has been introduced by Lee and Harp
[29]based on phase screens. Their method leads to the same V2Y(r) + VY(r). V Y ( r ) + k2n2(r)= 0. (6)
results but is more physical in its formulation. The problem is to solve (6) for the fluctuations in log-
We assume in the derivations that we are concerned with amplitude and phase.Since aperturbationapproach is
an atmosphere having no free charges, that the magnetic used, the results are valid onlyif the perturbations are
small.
permeability is constant, and that the time dependence of The primary quantities that have been studied are the
the electricalfield is given by (-expiot).With these assump variances and the spatial covariance functions in a plane
tions the vector wave equation becomes perpendicular to the direction of propagation. The major
emphasis has been on phase and log-amplitude fluctuations,
+
V Z E k2n2E + 2V(E V log n) = 0,
1 (4) with some attention to angle-of-arrival, intensity, and co-
herence functions. For mathematical simplicity most of the
where k=2a/;1 and iis the wavelength of the electromag- work has dealt with plane-wavepropagation ; only recently
netic wave. The last term, which contains the interaction have the spherical-wave and beam-wave cases beenstudied
between orthogonal components of the field, produces in much detail.
depolarization effects. Strohbehn and Clifford [30] calcu- When a plane wave propagates through a statistically
lated the average power in the depolarized component and homogeneous random medium, there are certain symmetry
showed that it is typically160 dB weaker than the power in properties that aremathematically convenient. In any plane
the incident wave [ l o ] .Saleh [31] was unsuccessful in his perpendicular to the direction of propagation the statistics
attempts to measure the depolarized signal witha sensitivity at any one point in the plane are the same as at any other.
of 45 dB. Therefore, the spatial covariance function depends only on
A practical possibility associated with the small depohri- the separation of the two points and noton their location in
zation of an optical signal when propagating through the the plane. In this case, for an isotropic medium, the spatial
“turbulent” atmosphere might be to use polarization mod- covariance function may be written as a two-dimensional
ulation in optical communications systems. The amount Fourier expansion, where
of “polarization” noise introduced by the atmosphere is
almost negligible, whilethere is very significant“amplitude
modulation” and “phase modulation” noise. However, we
are in no position to comment on other problem areas, such
as the amount of depolarization caused by particle scatter-
ing or the technical problems in polarization modulation.
In any event, the lack of depolarization is a great theoreti-
cal advantage in that the vector wave equation (4)may be
reduced to ascalar wave equation
Here F A X ) is the two-dimensional spectral density of the
V Z E+ k2nZE= 0. (5)
fluctuations in f, where f might represent the log-amplitude
For simplicity innotation, we assume that the wave isplane or phase. For spherical-waveor beam-wave propagation, the
polarized in the z direction, i.e., E = E,, and is propagating in expansion (7) is no longer valid sincethe symmetry has been
the x direction. We define E = A exp (is),where A is the destroyed. However, under the approximations thatare
amplitude of the wave and S is the phase. Note that since usually made, expressionssimilar to (7) are often used. In a
the exp ( - iot) term has been eliminated, the amplitude and number of works the quantity derived is the two-dimen-
phase refer to quantities that would not change with time in sional spectral density function of the desired quantity, for
a frozen medium. Further simplification would lead to a example, log-amplitude fluctuations. Equation (7) is then
geometrical-opticsapproach thatignores diffraction effects. used to find the covariance function or the variance
While thatapproach gives reasonable results for phase 0; = B/(O).
fluctuations, its results for amplitude fluctuations are valid The derivation of the spectral density functions of ampli-
only for L << l;/)., where Io is the inner scale of turbulence tude and phase from (6) is extremely involved and includes a
and L is the length of the propagation path. For typical number of approximations. The clearest derivations appear
parameters, this leads to restrictions of from 10 to 100 in the works of Tatarski [ 8 ] , [ 9 ]and are not repeated here.
meters for the path length. For this reason we consider only In all the derivations a number of approximations are made.
the full-wave solution, whichincludes diffraction effects. First, itisusually assumed that i<<lo and i3L/l:<< 1 ,
The geometrical-optics results may be found as a limiting where Io is the inner scale of turbulence. These are not im-
case. portant restrictions at optical wavelengths.
Most of the literature since 1960 has followed the Russian The only assumption that is definitely violated in some
approach of using the so-called Rytov method, which sub- cases isthe basic one that permits a perturbationexpansion
stitutes E = exp (‘Y) into the scalar wave equation (5). If we to be applied to the wave equation. As has been shown [32],
write Y = x + is,where x =log A, then the real part of Y is [9], [33], there are cases of practical interest where the
LAWRENCE AND STROHBEHN: CLEAR-AIR PROPAGATION EFFECTS 1529

expansion convergesveryslowly and the perturbation

method is no longer useful.
A major problem has been the difficulty in comparing Normally, : a is calculated but tJk1 is measured.
theory with experiment. Until 1966 almost all of the theory When we present the theoretical results, there are a num-
centered on plane-wave propagation, for which the only ber of different cases of interest, separated by the type of
applicable experimental arrangement involves stellar or wave originally transmitted at the source. The three cases
satellite sources. For space-to-ground paths, the detailed that we consider are an infinite plane wave ;a spherical wave,
distribution of refractive-index fluctuations is unknown i.e., a point source; and a beam wave. The beam wave that
and is certainly not uniform. The more recently proposed has been considered by almost all workers is the so-called
and lesswell-developed theories of spherical-wave and Gaussian beam. At the transmitter the electricfieldis
beam-wave propagation can, of course, be compared with assumed to obey the formula
ground-to-ground experiments over accessible paths. Even
here, direct measurements of temperature fluctuations
have seldombeen supplied in adequate detail to permit
critical tests of the theories.
E,(O, y , z) = A exp
{-(A+- i - (y2
2 3
+2 )
} (8)

where W, is the radius of the beam and R, is the radius of

V. OPTICAL INTENSITY FLUCTUATIONS curvature of the wavefront. If R,= 00, then the beam is
Probability Distribution and Variance of Lag-Intensity collimated; if R , equals the path length, then the beam has
its focal point at the receiver.
Theory: Most of the theoretical investigations have con- For each of the three cases we usually distinguish be-
centrated on the fluctuations in the log-amplitude x or the tween two types of propagation conditions. First we assume
log-intensity (more properly, the log-irradiance). In apply- that the atmosphere is statistically homogeneous and
ing Rytov's method, the scalar wave equation in the electric second that the medium is smoothly varying,i.e., at the
field E was transformed to an equation in Y = log E. The spatial wavenumbers of interest the spectral shape remains
solution of (6)for Yl = Y - (Y)can be written as an integral constant along the path butthe intensity of the fluctuations
equation. Application of the central limit theorem to this varies slowly along the path so
integral equation leads to the prediction of a normal proba-
bility distribution for Y,, i.e., for x=ln (AIA,,) and for SI, On(X,K) = c.z(xplp'(lc).
the deviation of the phase from its mean value. If instea'd of
transforming the scalar wave equation a perturbation ex- A further distinction is made : the first equations given are
pansion had been applied to E, the same procedure would the general cases where the shape of the spectral density
have led to a normal probability distribution for E,. This function of the refractive-index fluctuations is not specified.
distribution leads to a Riee-Nakagami distribution for the Where possible the results are also given for some specific
amplitude A, in contrastto the log-normal distribution form of the spectral density function, which is closely re-
predicted by Rytov's method. For very small variances in x lated to the forms discussed in Section 111. In all cases we
the difference betweenthe log-normal and Rice-Nakagami assume that the refractive-index fluctuations are isotropic.
distributions issmall.However, as the variance in x be- Though most of the equations could be generalized, there
comes moderate or large, the difference between these dis- is verylittle known about the nonisotropic case.
tributions becomes easily recognizable. The specific equations are given inTable I, assuming that
The preceding comments are based on a theoretical ap- the variance of the intensity fluctuations is measured by a
proach valid when the variance of x, denoted t~;, is small, point receiver. In practice a receiving aperture much smaller
i.e.,less than 0.64. When t~,' exceeds 0.64, the theoretical than ,/% should adequately represent a point.
results are rather scarce and quite controversial. DeWolf The most important restriction on these results is that
[34] originally predicted a Rice-Nakagami distribution must remain small for the equations to be valid. Based
in this region, but more recently [35] he seems to favor the on plane-wave theory, Tatarski [32], [9] derives 0,' << 1, or
log-normal. A physical discussion pointing out the differ- tJk1<<4, and claims experimental verification has shown
ences is gven by Strohbehn [lo]. This argument tends to good agreement if c,' < 0.64, or a;nr<2.5. It is apparent that
favor the log-normal distribution. Note that if the proba- this type of restriction isplaced on all the results, inde-
bility distribution of the amplitude is log-normal, so is the pendent of the type of source; however, the numerical co-
probability distribution of intensity. A Rice-Nakagami efficient may differ depending on the source distribution.
distribution describes an electric field E that contains a The only theoretical work has considered plane waves,
constant component plus a component with a normal while the experimental work has been primarily with spheri-
probability distribution. In the limitwhere the constant cal waves.Unfortunately, the restriction is not a minor one.
component is zero, the amplitude obeys a Rayleigh distri- For horizontal propagation paths close to the ground,
bution, and the intensity follows an exponential probability where high values of C l are often observed, the theoretical
distribution. expressions for c;nI may exceed 2.5 for paths as short as
Since intensity (or irradiance) is I = A2,the variance of the 500 meters. For vertical or slant paths, where the quantity
log-intensity is equal to four times the variance of the log- C,Z varies along the path, there is less chance of exceeding
amplitude : this limit.
1530 PROCEEDINGS OF THE IEEE,
OCTOBER 1970

TABLE I
VARIANCE
OF LOG-INTENSITY

Case Random
General Medium Locally
Random
Isotropic Medium*

Kc)
,

Plane wave, homogeneous ufnr= 8x2k2L

medium j:( 1 - -SUI
K‘L ’
- @ , ( K ) K dK (Tl) ufn1= 1.23C:k7/6L”16, << L

Plane wave, smoothly

varying medium
u&, = 16xZk2 dqC:(q)
Jo Jo
1’
dKdioYK) lrn Jo

x sin’ K2(L - 7) (T3) u;, = 38.41;7’3joLC:(q)(L - q ) , dq, L << &I. (T73

Spherical wave, smoothly = 16x2k2jOL

dqCi(q)j: dKdio’(K)
varying medium

uLI = 46.71; 7/3j0LCf(q)(!y(L - q), dq,

rr. rm r

where
where
1 - a,L + {(K: + a:)L - a,jq a,@ - 7)
Y1 =
(1 - Or2L)’ i- (KlL)’
Pt =
(1 - a,L)’ + (a,L), ,F,(a, c ; z ) = @(a, c; z) is the Kummer function
a1 = A/(xW:) at = l/Ro
[1 + (a1L)2x](l -x)
a = a, + ia, p = (y2 + z2)’I2 = distance from beam center
g(a,L) = jol Re[(l - x)’ -i
KlL 1 5’6

dx

#,(a, b ; c ; z ) is a hypergeometric function

Beam wave, smoothly

varying medium
u;, = 8n‘jOLdqC:(q)j: dxu[J0(i2y,rp) u; I = 4n2(0.033)[- r(- 5/6)]k716L’ 1/6 jol dqc:(q)GA(q)

- cos (v K2)]lH~2@~o)(K) (T16) where

GA(x) = Re [(y2 + iy,)(l - x) + S:]5/6 - [yz(l - x) + Si]’*

S, = (k/L)1‘210/5.92 (TW11
~~ ~

For the locally isotropic random medium @,(~)=0.033Cf(x)~-

113 exp ( - K ’ / K ~ ) .For the homogeneous medium C:(x)is a constantindependent of x.
t C(x)=j; cos ( d / 2 ) dt, S(x)= jg sin (ntz/2)dt.
3 Results according to Ishimaru PSI, [40].Receiver a distance p from beam center, beam collimated (R, = a).
11 Receiver at beam center, any value of R,.
LAWRENCE AND STROHBEHN: CLEAR-AIR PROPAGATION EFFECTS 1531

The results for plane- and spherical waveshavebeen

derived by a number of different workers and are well ac-
cepted. For locally isotropic turbulence it is apparent that
the functional form of the equations for the two cases is the
same. However, the numerical coefficients differby a factor
of about 2.5, i.e., a spherical wave has smaller variance. The
beam-wave problem has been attacked more recently, and
the expressions are considerably more complex. For this
reason the different results have not been carefully com-
Fig. 3. Dependence of normalized log-intensity variance ofJukx9
pared. The first comprehensive paper on beam-wave (where the subscript s denotes the variance calculated for a spherical
propagation was by Schmeltzer [36], who gave some rea- wave) on n o r m a k d transmitter size R Solid curvesrepresenta
sonably general expressions for the mean value of the homogeneous pathand, for them, Q = k W ; / ( Z ) . Thedashedcurve
representsslantpropagationthroughtheentireatmosphere. In the
electromagnetic field and for the covariance functions of slant case, Q=kWt/(ul, csc e,), where h, is the “scale height” of the
log-amplitude and phase. These results were left as double atmosphereand 8, is theanglefromthe horizon. FromFriedand
integrals that were not evaluated. His derivation uses the Seidman [37], [38].
approximation that the observation point is in the far field,
i.e., the path length however, in the case where the reduction would be most
useful,i.e.,highvalues of itis doubtful that itwould
occur. In.this case, in a very short distance the phase front
of the wave is so perturbed that the concept of a focused
where Wo is the width of the Gaussian beam at the transmit- beam becomes meaningless.
ter and R , is the curvature of the wavefront. With this One way to look at this problem is to compare the phase
approximation, the terms in hisexpressions are only weakly fluctuations introduced by the atmosphere with the differ-
dependent on the position of the observation point relative ence inphase that distinguishes between a collimated beam
to the center of the beam; therefore, the dependence is and a focused beam. For a beam focused at a point at some
dropped, i.e., all calculations are done on the beam axis. distance L, the difference in phase at the transmitter com-
For a collimated beam,where R , = CL),(9) becomes pared with a collimated beam is 6 S r $ k p 2 / L , where p is
L >> k W ; / 2 . For a 1-cm beam, L must exceed 1 km ; for a the transverse distance from the beam axis. Clearly, phase
5-cm beam, L>> 12 km at optical wavelengths. Therefore, fluctuation introduced by the atmosphere must be much less
for a number of cases of interest this approximation is-not than 6s for the concept of a focused beam to hold. The
met. Fried and Seidman [37] and Fried [38] have taken phase fluctuations may be estimated from the phase struc-
Schmeltzer’s expressions and evaluated specific cases. Since ture function DJp) (see Section VI) ; therefore, D J p ) < 6s’.
in the beam-wave case there are additional parameters, the Using (T35) (‘T” means the equation appears in a table)
beam width and its wavefront curvature, it is difficult to for a plane wave, we have
portray the results in any general manner. They assumed a 2.92C,2k2LpSi3 <ip4k2/L2.
K- l u 3 formfor @“(K) and investigated the two specific
cases wherethe beam is collimated (R, = 00) and where the Therefore,
beam is focused at the receiver (R, =L). They introduced a c,2 < A p 7 ”L3.
parameter R = k Wg/(2L),which is the ratio of the distance
to the far fielddivided by the path length. Note thatSchmelt- For a 5-cm beam and a 1-km path, C; must be less than
zer made the approximation R << 1 in deriving his formulas 10- l 3 to consider focusing a beam. Notice that
which would appear tolimit Fried and Seidman’s results to the permissible valueof C; decreases as L3.
R<< 1 also. However, though it is not specifically stated, it Fried [38] also considered the reduction in scintillation
appears that theabove restriction can be removed if we are for a collimated Gaussian beam propagating from the
interested in effects only on the beam axis. For a collimated ground to a receiver in space and its dependence on zenith
beam and a given source diameter, the variance g;nI starts angle and aperturesize. His results are plotted as the dashed
out being equal to that of a plane wave forL << k W$2, i.e., line in Fig. 3. In this case, as the optics size increases, the
for short paths where the beam appears as a plane wave (see predicted scintillations decrease substantially. This is a
Fig. 3). WhenL >>kWg/2, i.e., the observation point is well distinct contrast to a uniform path where the scintillations
into the far field,the variance becomes very close to thatof a increase,a difference that seems somewhat hard to reconcile.
spherical wave, which isreasonable. For thefocused case in Ishimaru’s approach [39], [40]is somewhat different than
the near field the calculations give a sharp decrease in Schmeltzer’s inthat he extends the idea of spectral represen-
As the path length increases and goes into the far field tations. His final equations also include the effectof the
of the transmitter, the variance of the log-intensity again observation point being off the center of the beam. However,
approaches that of a spherical wave. One point of caution a quantitative evaluation of the magnitude of this effect
should begiven,however.These results wouldseem to is not given. Ishimaru also predicts a strong reduction in the
indicate that one method of reducing the scintillations scintillations when the beam is focused. Fig. 4 is a graph
(G;“ I ) would be to use a focused beam and a large aperture ; from Ishimaru showing the dependence of o;nI as a func-
1532 PROCEEDINGS OF THE IEEE, OCTOBER 1970

Y 4~ I I I I

ov I 1
0 2 3
' (aILilm PFiEDlCTED UXi-INTENSITYMRIANCE

Fig. 4. Variance of log-intensity as a function of (alL)"", where Fig. 5. Theory and observation of saturation of scintillations compared
a1= A/(x@,) .and W, is the beam radius [39]. The vertical scale was withvaluespredictedfromtemperature fluctuations using (T4) or
added after privatecorrespondence. (T10).The lower dashed curve is the theoretical result from Tatarski[9]
[ut, from (Il)]; the upper dashed curve is the theoretical result from
deWolf [35] [o$ from (lo)]. Solid curvesareaveraged observations
tion of alL)11/6.For a given aperture size, i.e., a constant replotted fromGracheva,Gurvich,and Kallistratova [W]. Curve 1,
value of a, =A/(nW;), the curve shows how the variance spherical wave; curve 2, plane wave (5O-cmdiam collimator);curve 3,
should increase with path length. Again, we urge a note of white light. The shaded area includes all theobservations by Ochs [94]
over a 1Wmeter path. If Ochs' measurements are restricted to times
caution in using these curves. When C,Z is reasonably high, of moderate wind(2 to 6 m/s), the mean of his observations agrees well
it seems that the phase fluctuations would be large enough with curve 1.
to seriously affect the beam parameters.
As indicated previously, the equationsfor the variance of
geometrical-optics approachthat also led tosaturation
the log-intensity fluctuations are valid only when o& I <<4,
despite the .neglect of diffraction effects. In this case his
Attempts at verifying experimentally (T4) or (T10) have
final equation was
shown that for small values of ofn,there is good agreement
between theory and experiment. However,when o;n, is
greater than about 2.5, as predicted by the preceding equa-
tions, the experimental value of o& I appears to saturate and
o.fn,= 4[ 1 - (1 + 3 0 9 - 7

remain almost constant. Therefore, the theoreticians know 4

not only that the above expressions are of limited validity 0; = - n2L3 O p , ( K ) K 5dlc.
3
but also what type of relation they should expect to find.
Despite this insight, very little progress has been made
This curve is also plotted in Fig. 5.
theoretically as of this writing. A number of workers have
In summary, at this time there is no well-accepted general
tried using the renormalization techniques of field theory
theory for amplitude fluctuations in the saturation region.
[41]-[45], [34], [35], which approach is exceedingly com-
Though there are some reSults for the variance of the log-
plex mathematically, involving the selective summing and
intensity, no method has yet predicted other quantities of
neglecting of very complicated terms in a series representa-
interest, such as the amplitude covariance function or phase
tion. There is still no general agreement among workers
fluctuations.
over which terms must be kept. The only worker that ap-
ExperimentalObservationsCompared withTheory: We
pears to derive a saturation-like curve for the variance of the
have grouped the experimental observations into two cate-
log-intensity fluctuations using this method is deWolf, and
gories. First, there are those measurements that we believe
his results are controversial. In his second paper [34] he
may be meaningfully compared with theory. In those experi-
predicted a saturation effect but also concluded that the
ments the optical configuration isclose enough to the
probability distribution of the amplitude fluctuations
should follow a Rice-Nakagami curve. The experimental theoretical models, and the meteorological parameters are
known well enough so that such a comparison should
evidence does not seem to bear this out, andin a more recent
either prove or disprove the theory. Second, the bulk of the
paper [35] he favors a log-normal distribution while keep-
measurements are those that have been taken under such
ing his saturation curve. His work leadsto the following ex-
conditions that a comparison is meaningless, but they do
pression for the variance of the log-intensity:
provide numerical values showing the magnitude of the
" 1 effects.
of,= 1
n = O (n + m)'
The only experimental configuration for which the me-
teorological parameters can be obtained is a short hori-
m = 1/[1 - exp (-G:)] zontal path close to the gound. A number of experiments of
2 - 1.23C;k7/6L11/6
OR - (10) this type have been conducted, originally in the U.S.S.R.
[47)-[50], [28], and more recently in the West [51]-[54].
where o i is the variance calculated by the first-order per- These experiments attempt to verify (T4) or (T10). Fig. 5
turbation theory. This curve, which leads to a saturation shows some experimental data [52], [50] indicating a
value of 1.65, is plotted in Fig. 5. decrease in the measured variance after the onset of satura-
Tatarski has tried two other approaches toward pre- tion. On thehorizontal axis isthe value Cfnr calculated from
dicting saturation. In [46] hetried a rather crude approach (T4) or (TlO),where C; was measured withhigh-speed
that gave a saturation curve, but the final value was an temperature sensors. On the vertical axisis the experi-
undetermined constant. In another attempt [9] he used a mental value of of, 1. As is apparent, there is good agreement
LAWRENCE AND PROPAGATION
STROHBEHN: CLEAR-AIR EFFECTS 1533

for C T<; 2.5,

~ ~ but the variance saturates near 2.5. Different function for irradiance fluctuations B,(p). Fortunately, if
saturation values are quotedby different workers, but these we accept the log-normal probability distribution for the
values appear the most reliable. The two theoretical curves amplitude fluctuations, discussedinsome detail previ-
of Tatarski anddeWolf are shown for comparison. Gurvich ously, the relationship between B,(p) and B,(p) is reasonably
etal. [49] havemeasured the ratio of the log-intensity straightforward. Let
variance of a plane wave and a spherical wave, obtaining a
value of2.3 compared to the theoretical value of 2.4.
x ln(A/A,) = j +
x l , j = (x)
Numerous measurements [47E[53] of the probability dis- I = A’ = A i exp (21 + 2x1)
tribution of the intensity have shown good agreement with a B,(Pl, P 2 ) = (Xl(Pl)XI(P2)), (12)
log-normal distribution over short paths and fairly good
agreement even in the saturation region. Fitzmaurice et al. where we have assumedthat the mean value of x is not zero
[55] compared scintillations at 0.632 and 10.6pm and ( A , could be chosen such that j = O ) . Then if x follows a
found good agreement with the wavelength effectspredicted normal probability distribution,
in Table I.
Experimental attempts to verify Fried and Seidman’s M P ) ) = 4 exp [ m ) ] exp [2(x:(P))]
predictions [37] about the difference between a focused and (I(Pl)I(P2)) = A 3 P l ) A i ( P , ) exp [2;c(P1) + 2X(P2)]
a collimated beamledKhmelevtsov and Tsvyk [56] to
conclude that there was no difference. Their measurements * exp P [ ( x : ( P l ) ) + < x : ( P 2 ) ) + 2B,(Pl, P,)])’. (13)
were inthe weak-scattering region and, althoughtheir paper Defining the covariance function as
lacks the details required to judge the quality of the work,
their result seems to agree with our contention that the BI(P1, P 2 ) = ( I ( P A I ( P 2 ) ) - <I(Pl))<I(P2)), (14)
concept of a focused beam isnot very meaningful for optical
propagation. we have
General Observations: Fig. 2 shows a one-day sample of BI(P1, P 2 ) = A3P1)-4i(P2)exp [2X(P1) + 2X(P2)]
the log-intensity variance at 490 and lo00meters on a path
close to the ground, compared with the measured value of . exp [2<x:(P1)) + 2(X:(P2))]{exP [4B,(P,, P2)] - 1). (15)
C i . The typical diurnal variation of scintillation consists of Defining a normalized covariance function by
a broadmaximum throughout daylight hours, sharp minima
near sunrise and sunset, and highly variable effects at night.
The daytime maximum reaches a broad smoothpeak soon
after noon,but cloud cover, either intermittentor con- we have
tinuous, produces a sharpdrop in scintillation activity.
Activity increases with wind velocity forlight winds but de-
creases again as the wind becomes strong, mixing the air to
produce an adiabatic lapse rate. Sharp minima at sunrise
and sunset occur when the lapse rate, changing because of If the transmitted wave is plane, or if the statistics at p1and
the rapidly varying solarradiation, passesthrough the p2 are the same, then the formula reduces to
adiabatic condition. The middle curve inFig. 2 is typical of
the diurnal behavior of scintillation. Along a 1000-meter
path, duringthe middle of the day, the variance has become
saturated and is actually smaller at lo00 meters than at490 If (x:) e< 1, then (16) and (17) become
meters! For paths longer than 10oO meters it is difficult to
calculate the expected variance since the meteorological
parametersare rarely statistically homogeneous.Where
most of the path is not close to the ground, the variance and
tends to be muchlower, This may beattributed to two facts.
First, the turbulence normally decreases sharply with
distance above the ground. Second, for spherical waves it is
only near the middle of the path that turbulence is effective
Since the equationsderived theoretically are for B,(p) or
in producing scintillations [see (T13)]. Measurements by
B,(p,, p2), and since these equations are valid whether or
Ochs et al. [57] over a 45-km path at night gave values of
not the amplitude distribution is log-normal, these are the
c:,,% 1.2.
equations given in Table 11. In comparing Table I1 with
experimental observations, which are usually of the co-
SpatialCovarianceFunctions f o r Irradiance and L o g - variance of intensity (irradiance) fluctuations, we may use
Amplitude the formulas, recognizing the assumptionsmadewhen
Theory: Theoretically the spatial covariance function they were derived.
normally calculated is for log-amplitude fluctuations B,(p), The approximationsinvolved in deriving the equationsof
but experimentally there is more interest in the covariance Table I1 are essentially the same as those used in deriving
1534 PROCEEDINGS OF THE IEEE, OCTOBER 1970

TABLE I1
COVARIANCE
OF LOG-AMPLITUDE

Case General Random Medium Locally Isotropic Random Medium

1 - 2.36rG)516 + 1.71 1
b,@) = + ... lo << p << ( Z ) 1 ' 2

Spherical wave, B,@) = 4n2k2f: d ~ d ~ ( ~ ) ~ ~ ~ d ~ l J ~ ( y )

homogeneous medium
K211(L -
x sin2 [ 2kL ] (T26)

x sin2 [ utL '1

K ~ ~-( L
(T27)

Beam wave, smoothly Let @,(IC)= C:(q)@i0'(~)

in (Tu)). (T31) No cases worked out.
varying medium

@,(K) / ~ ( - K ~ / K : ) . For homogeneous medium, Cf(x)= constant.

=0.033 C : ( X ) K - ' ~exp
t @=(~)=0.033
C~(X)K-~~'~.
2 p1=(yl,zl), p2=(y2, zz) are the observation points referred to the center of the beam. Other symbols defined in Table I.
LAWRENCE AND PROPAGATION
STROHBEHN: CLEAR-AIR EFFECTS 1535

[57], is plotted in Fig. 6. This covariance curve maybe

duplicated by (T29) only through the use of a nonunifom
distribution of turbulence, specifically by assuming all the
turbulence to be concentrated near the ends of the path, a
reasonable situation for the mountaintop-to-mountaintop
path that was used.
Deitz [59] has published photographs of the scintillation
Fig. 6. Observed covariance function of a mountaintop-to-mountaintop
pattern produced by a ground-to-ground laser beam, which
path (dashed line) [57] and for saturation conditions over a uniform demonstrate the striking variation and complexity of the
path (circles) [94] as compared with the solid curve calculated from patterns. Sometimes thepatterns consist of amorphous
M8). areas of random sizes and shapes, but frequently they have
a reticulated appearance with sharp lines bounding large
the expressions for the variance of the log-intensity. Again, polygonal areas.
the most critical restriction is that 0:" I must be lessthan 2.5, Observations of spatial covariance are also available for
or 0,' less than 0.64. Note thatfor valuesof CT?,, 22 there may starlight. Both Gaviola [60] and Mikesell et al. [61] have
be considerable error in using (19) in place of (17). The ex- published photographs of "shadow bands," the intensity
pressions for the log-amplitude covariance function for variations of light incident on the earth's surface from a
plane and spherical waves are reasonably well accepted. In single star. The shadow bands tend to be elongated rather
the beam-wave case, to our knowledge, there are no exam- than isotropic random variations. The width of the bands
ples worked out for the case of locally isotropic turbulence. is sometimes as small as 3 or 4cm but is more often 7 or 8
The major differencesbetween the plane-wave and the cm. Thus, if one's eyes are slightly converged by focusing
spherical-wave log-amplitude covariance functions are in on a nearby object, the two images of a star areoften seen
the point of the zero crossing and in the depth of the mini- to scintillate independently. Interpreting the width of the
mum. In the beam-wave case it seems reasonable to expect shadow band as the width of the covariance function, we
that if Wg>>IL and the wave is collimated or diverging, see from (T22) that the distance from the ground-based
then the covariance function for the beam-wave case will observer to the effective midpoint of the turbulent path is
be similar to thecases presented. about 2 km for 3-cm shadow bands and 12 km for 8 c m
For spherical or plane waves it is often easier to evaluate bands. Even if the stellar-scintillation observations were
the integrals numerically. This can be done evenin the made at angles of 50" or 60" from the zenith, the scintillation
smoothly varying medium if the model is mathematically must have arisen from elevated layers-high in the atmo-
reasonable. Specific casesfor light propagating through the sphere.
atmosphere have been worked out by Tatarski [8], [9] and
Fried and Cloud [58]. In the beam-wavecase there are Aperture Smoothing
double integrals to evaluate, unless some reasonable ap- Theory: The preceding results are appropriateonly when
proximations are found. No theoretical results now exist the intensity is measured by a point receiver. If the receiving
for thelog-amplitude covariance function in the saturation
aperture is lessthan some critical size, such that theintensity
region.
fluctuations are correlated over its area, then the aperture
Experimental Observations Compared with Theory: The will behave as a point detector. As the aperture size in-
most striking prediction of the equations for the covariance
creases so that the intensity fluctuations are uncorrelated
function of log-intensity is that the covariance will fall to
over the aperture,then the intensity is a spatial average over
zero in a distance given by the Fresnel-zone size (A!,)'''.
some finite region of space, and the fluctuations tend to
The validity of these equations when the variance is not
decrease. This is a well-known experimental effect in as-
saturated has been shown by Tatarski [8], [9] over 500- to tronomical observations with a telescope. Tatarski con-
2000-meter paths and by Ochs et al. [57] on 5.5-km and sidered the problem in his first monograph [8] but he re-
15-km paths. Fitzmaurice et al. [55] in a two-wavelength vised his work in his later manuscript [9] and his results
experiment also indicated good agreement with the wave- agree with those of Fried [62].
length dependence. Most of these measurements use a The pertinent parameter in these calculations is defined
spherical wave, and no experimental work has beenre-
in terms of the quantity
ported whichspecifically relates to beam-waveeffects.
There are notheoretical predictions and very few measure-
ments of the covariance function in the saturation region. s = S j d Y dz WY,2) K Y , 21, (20)
Fig. 6 shows data collected by Ochs [52] over a l-km path
when saturation existed. Similar experiments by Gracheva where I is the intensity of the light falling on a point (y,z),
[48] show an increasing broadening of the covariance and W(y, z ) is the weighting function over the aperture.
,
curves as the theoretical value for c?, increases. Both Tatarski and Fried have used the simple assumption
General Observations: Over nonuniform paths the char- that W k , z ) is equal to one over a circle of diameter D and
acter of the covariance function may change markedly; a zero elsewhere. Using S , we maydefine the aperture-
typical example for a 45-km path, taken from Ochs et al. averaging factor 8 by
1536 PROCEEDWGS OF THE IEEE,OCTOBER 1970

which isthe ratioof the actualvariance of S to the variance

if the intensity were perfectly correlated over the aperture.
“I I
They find 0 05 I 15
D/2
(XLY2

Fig. 7. Relativedecreaseinthevariance of intensityfluctuations of a

planewavewhenreceived by an aperture of diameter D, assuming
I, << (X)*< Lo. From Tatarski [9, p. 3741.
If we assume the intensity follows a log-normal distribution,
we may use (17)to get
exp[4B,(p)]
16 -1 covariance function that falls sharply in the first 10 cm but
e = 3Jo dp exp [4B,(O)] - 1
then fails to decrease further for spacings as large as 1
meter. They attribute this effect to some unexplained fea-
ture of beam-wave propagation, as opposed to spherical-
wave propagation forwhich the theory was developed. We
suspect it is simply the result of the concentration of the
Fig. 7 is agraph from Tatarski whoassumed that turbulence near the ends of the path, the only places where
( x 2 ) << 1 and so used (19) forb,(p). An equation similar to the beam approached within 15 meters of the ground. This
(T22) was used forB,(p). Fried presents a family of curves would then be the same effect as seen by Ochset al. [57] on
by working with (23); but since the expressions he used for their 45-km path (see Fig. 6).
B,(p) are notvalid for( x 2 ) >> 1, some of his curvesare mis-
Hohn [66] used 4.5-km and 14.5-km paths and observed
leading. He evaluated the space-to-ground case, but the little or nosmoothing for aperturesup to8 cm. Here again,
only curves thatare meaningful are for ( x 2 ) e< 1. (In the explanation may be that the path was not uniform or
Fried’s paper, C,(O) corresponds to (x2).) Note that (22) that saturationexisted. The resulting large-scale covariance
and (23) are reasonably general for a circular aperture as function would prevent aperture smoothing for modest
long as the log-normal distribution for the intensity is apertures.
correct, and therefore maybeusedin any casewhere There have been numerous papers concerned with the
B,(p) is known. dependence of stellar scintillations on zenith angle. As
Lutomirski and Yura [63] have attempted to extend the shown by Tatarski [8], [9], this dependence is strongly re-
range of the results in Fried’s paper by a curve-fitting pro- lated to the aperture size used. For large apertures, i.e.,
cedure. SinceFried’s calculations are validonlyin the D>> (ih)’/2 (h is the scale height of the atmosphere), the
region of weak fluctuations, theyextendhis results by variance of scintillations should vary as (sec Z ) 3 . For
assuming B,(p, L) is given by D e< (Ah)li2,the prediction for the dependence is (secZ)’ ‘I6.
Probably the theories would adequately describe most
observed scintillations of starlight, given detailed informa-
where B!(p, L ) is given by expression equivalent to (T22). tion aboutthe distribution of turbulence and winds
Although (24) was chosen so that it fit the experimental throughoutthe atmosphere. However,since the atmo-
results for p = 0, there is no evidence that it is a reasonable spheric parameters are not known in detail and since the
model for the covariance function, i.e., for p # 0. signal-to-noise ratio for starlight observations is poor,
Gracheva and Gurvich [64]used a heuristic approach comparisons between observation and theory can be only
based on Gracheva’s measurements [48] of B A p ) in the qualitative and statistical.
saturation region. Using the measured values, they inte- Poor seeing has always plagued astronomers and a huge
grated (22) numerically to estimate the aperture-averaging volume of literature has accumulated to describe and inter-
effects. Their results for a given aperture size indicate that pret the effects. Meyer-Arendt and Emmanuel [67]have
as the theoretical value of t$nI increases, the amount of surveyed this literature and present more than 300 refer-
smoothing decreases. ences with a brief discussion of each topic. Astronomical
Experimental Observations Compared with Theory: We “seeing” is an ill-defined function of scintillation, image
know of no definitive experiments over homogeneous paths quality, and image motion, and its effects depend on the
concerned with aperture averaging. aperture of the optical telescope. As a result, only a small
GeneralObservations: Fried etal. [65]used a slightly portion of the literature on astronomical seeing is usefulto
diverging laser beam over an 8-km path and found the log- our discussion of scintillation. Some of the most pertinent
amplitude variance to decrease nicely as the aperture in- are mentioned in the following.
creased from 1 mm to 10 cm, but further increase of the In accord with theory, Megaw[68] reported that rms
aperture to 89 cmproduced no further decrease in scintilla- fluctuations vary as (sec Z ) 3 / 2for zenith angles less than
tion and no change in the log-normal distribution of re- about 60”. He presented some datato show a weaker
ceived signal. As Fried et al. point out, this result implies a variation nearer the horizon. Siedentopf [69]found a varia-
LAWRENCE AND STROHBEHN: CLEAR-AIRPROPAGATION EFFECTS

tion as (sec Z)”’ at zenith angles greater thanabout

60”, but very near the horizon he observed deviations from
this law and referred to the effect as “saturation,” particu- = B,(UlT).
larly evident at the high-frequency end of the scintillation The definition of the frequency spectrum,
spectrum.
Observationsof scintillations forpropagation in the
w , ( f ) = 4Jomcos(2nft)R,(t) dt
opposite direction, from earth to space, are only just be-
coming available. Minott [70], observing with the satellite
GEOS-I1 a ground-based laser at a wavelength of 488 nm,
found the irradiance to be log-normally distributed and to
have a variance comparable with that of stellar scintilla-
=4
:1 COS (Znft)B,(v,t) dt,

gives the general relation between the spatial covariance

tions. function B,(p) and the frequency spectrum w , ( f ) under the
Many attempts have been made to determine the height assumption of “frozen” turbulence.
of the turbulent region responsible for scintillations by Tatarski [8], [9] has evaluated the frequency spectrum for
observing balloon-borne lights and comparing their scin- a plane wave propagating through amedium with a refrac-
tillations with those of nearby stars. For example, Mikesell tive-index spectrum given by (3). He finds
[71] observed suchlights 3 mm in diameterand found them
to scintillate only 1/3 to 1/10 as much as adjacent stars, even 0- 813
though the lights were carried to heights of 8 km. Proper w , ( f ) = 271 * 0.033C,2k7’6L11’6
-
fo
interpretation of this observation demandsattention to
(T13) which shows that even though the balloon may have 1.69 - Im
risen above all the turbulent atmosphere, the scintillations R2
could not be expected to become fullydeveloped. Thus the
atmospheric structure is only weakly determined by such
an experiment.

Temporal Power Spectrum of Scintillations

Theory: The frequency spectra of both intensity fluctua-
11
. ,F1(17/6, 10/3, -io2) li/,i. << L
2 (29)

where fo = 0J(27cnlL)’/~,R = f /fo, and ,F, is a hyper-

tions and log-amplitude fluctuations are of interest, but
geometric function (the Kummer function). As can be seen,
since the theoretical work has been concerned with log-am-
the frequency spectrum depends on the characteristic fre-
plitude fluctuations, we shall concentrate on them. When-
quency fo, which is the ratio of the perpendicular compo-
ever the conditions are such that (19) holds, then the fre-
nent of the wind velocity to the width of the first Fresnel
quency spectrum for intensity and log-amplitude fluctua-
zone. The scale sizes of the inhomogeneities primarily
tions will bethe same. When(19) does not hold, but reason-
responsible for the amplitude fluctuations are about the
able expressions for B,@) are known, the expression for
width of the first Fresnel zone. Defining To= l/fo, we find
the frequency spectrum ofthe intensity fluctuations should
that the characteristic time To is equal to the time required
be obtainable using (17) and numerical techniques.
for an inhomogeneity of size(27~iL)”~ to cross the path.For
The frequency spectrum ofthe intensity or log-amplitude
fluctuations is caused by the motion of the atmosphere.
R << 1 or R >> 1, (29) is approximately equal to
For our purposes this motion may be separated into three
1
groups: 1) mean motion of the wind, 2) changes in wind w , ( f ) = 0.15(x2) - [l 0.48R4l3+ + ...I,
R<<1 (30)
direction, and 3) internal “mixing” motions of the atmo- fo
sphere due to theevolution of turbulence. Tatarski [8], [9] or
appears to be the only author to have studied these effects
theoretically in any detail. The most straightforward ap-
proximation involves the useof Taylor’s hypothesis of
“frozen” turbulence, whichassumes that the temporal
variations of somequantity measured at a point are caused The dimensionless quantity F(R)= few,( f )/(x2) is plotted
by the uniform motion of the atmosphere past the point, in Fig. 8 where it is labeled F(R, 0).
and that internal motions of the atmosphere may be ne- Let us consider the conditions Tatarski requires for the
glected. For propagation problems, it is assumed that the above expressions to hold. Denote u, as the perpendicular
only motion of interest is perpendicular to the path. and v l l as the parallel component of the averagewind
Under the assumption of Taylor’s hypothesis, velocity. Since the fluctuations in the direction of the wind
areabout0.1,ifv,-O.lvli,thenthefluctuatingportionofthe
perpendicular component is of the same order asthe mean,
and Taylor’s hypothesis does not hold. Let u equal the
where u, is the average velocity component perpendicular magnitude of the average windspeed, 0, and 0 the standard
to the path. The time autocorrelation function is deviations of the perpendicular component and of the
1538 PROCEEDINGS OF THE IEEE,OCTOBER 1970

These calculations have not considered the influence of

the random motions of the medium caused by the evolution
of the turbulence. Such random motion may be ignored if
VI >> [ & ( E L ) ” 2 ] ” 3 , (34)
log
Fig. 8. Frequencyspectrum of intensityfluctuations as a function of
where E is the energy dissipation rate of the turbulence. Or
receiving
aperture
diameter p=nD2/(WL).From Tatarski [9, p. 3781. if u~ u~ is by OI w(Eb)1/3, leading to
Lo >> (LL)? (35)
magnitude of the wind speed, respectively, and a the angle
between the wind direction and the direction of propaga- Normally this last condition is always satisfied.
a =&T~. Tartarski [9] also calculates the frequency spectrum for a
tion. From the turbulence theory of Tatarsk : [ 9 ] ,:
Then receiver witha finite aperture. He determines the frequency
spectrum of the quantityS defined in (20).Let
oI
- = -,
0.8 cr
uI usina
-
Usually a/u-0.1; therefore, for a = 4 2 , oi/t‘, 10 percent.
For a 6 50”, crJul 21 and the frozen hypothesis must be where the subscript denotes the fluctuating component.
considered questionable. Defining
In the limit when a =0, however, Tatarski [ 9 ] has esti-
mated the frequency spectrum by considering the fluctua-
tions in the diffraction pattern of the wave, where the per-
ws( f ) = 4 :j cos (2xf t ) R S ( t ) d t ,

pendicular component of the velocity is considered to have he then defines a relative spectrum
a normal distribution with variance 0 2 . He calculates an
average spectrum, again assuming the same form for
@“(K), as

and shows that, in general, if f >> uJ(xD), i.e., for high fre-
quencies, the effect of the aperture is to average a number of
[[ $
1- sin $]e-.. & , ( X ~ ) X ~ /dx,
~ inhomogeneities and hence to reduce substantially the
spectrum compared with that of a point aperture. For
l;/A <<L, (32) f << uJ(xD), if the important spatial wavenumber K << 210,
the spectrum is approximately equal to that for a point
where aperture, but if the important spatial wavenumber K >>210,
the spectrum will also decrease substantially.
For weak fluctuations, where (19) holds, and assuming
- Tatarski finds for the plane-wave
@ n ( ~ ) = 0 . 0 3 3 C , 2 ~1/3,
0 1 = .f(L/k)’/2/a = f l f l , case
K,,(x) = a modifiedBessel function.
If R>> 1, then

(w,(f )) = 0.32 -
(x2> l;/A <<L. (33)
fl

For both cases considered (constant motion and completely where

random motion) for f >> f,, fo, the spectrum decreases as
f -‘I3 ;however, the coefficient in the second case issmaller.
Since
f x=
ku:l(4x2L)
n2=(fl f O l 2
p2 =nD2/21L
j;(w.(f)) d f = Jj,(f)d f = <x2>, z =LK2/k
D = aperture diameter.
this implies that the spectrum in the low-frequency region Note thatp is the ratioof the apertureradius to the radius of
must be higher for the completely random motion than for the first Fresnel zone. Fig.8 is a plot of
the constant motion (frozen turbulence).
In summary, Tatarski concludes that for cr <<uI, (29),(30), F(Q P ) = fows/<s>(f)/(X2)
and (33) hold, and the spectral peak is at f = fo. For cr- vi as p is varied.
the maximum of the spectrum begins to move toward zero, Experiments Compared with Theory: Experimental mea-
and for a >> vi this maximum beginsto grow. surements verifying the theory have been discussedby
STROHBEHN:
LAWRENCE AND CLEAR-AIR PROPAGATION EFFECTS 1539

8-km to 14-km range. Young [77] prepared a computer

program to calculate power spectra of stellar and planetary
scintillations for various apertures and source sizes and
used arbitrary verticalprofiles of turbulence and wind
velocity. He obtained good agreement withMikesell’s
observed spectra [71] without invoking thin turbulent
layers high in the atmosphere.
Minott [70] observed a light source carried aboard the
GEOS-I1 satellite and found spectra similar to those of
stellar scintillations. There was no evidence of a “pseudo-
f / fo wind” caused by the motion of the line of sight across the
Fig. 9. Normalized frequencyspectrum u(f)=fm&w(jJ df for in- turbulent structure of the upper atmosphere because the
tensity fluctuations. Curve 1, plane wave; curve 2, spherical wave; satellite moved so slowly that the actual winds aloft were
curve 3, theoretical curve for plane wave. The spectra are averages of comparable with the pseudowind.
about ten ditferent samples Lfo=uL/(2nAL)f].From Gracheva [48].
Variation of Scintillations with Wavelength
It is common knowledge thatstars near the horizon
Tatarski [8], [9] and Gurvich et al. [49]. Fig. 9 shows the scintillate with differentcolors at different times, whilestars
normalized frequency spectrum for both plane and spheri- higher in the sky show no such effect. Therefore, we might
cal waves as obtained by Gurvich. These measurements conclude that weak scintillations are correlated over a wave-
show that the predicted normalization frequency length ratio of at least 2 : 1, while stronger scintillations
fo =V J ( ~ R I L ) ’is/ ~very close to the actual peak, and they show less correlation. Neglecting saturation effects, Fried
appear to agree well with theory for short homogeneous [78] calculated that the correlation of log-amplitude should
paths. No theoretical predictions exist for the spectrum in drop to 0.5 for wavelengths differing by an octave. Fitz-
the saturation region, but Gracheva’s measurements [48] maurice [79] has recently measured the correlation of in-
show much more energy in the low-frequency region when tensity for various wavelength separations of from 10 to 50
compared with Fig. 9. percent and found the correlation to be about 0.6.
GeneralObservations: We have already mentioned the
discrepancies between the spatial covariance functions ob- VI. PHASE FLUCTUATIONS
served on long paths and the functions expected for ideal
Introduction
conditions. Similarly,we must expect that measurements of
temporal power spectra will frequently differ significantly In general, the same considerations arise when studying
from the ideal forms @veri above. Thus, it may not be sur- phase fluctuations as when studying intensity fluctuations,
prising that Ryznar [72] reports that the peak frequency of though there are a few important differences.We start
the spectrum is independent of path length, or that Hohn with (6), where the imaginary part of t,b is equal to S. For
[73] finds the spectra unaffected by variations in receiving intensity fluctuations the important scale sizes of the
aperture diameter from 5 to 80 mm and states that the ex- refractive-index inhomogeneities are about the width of a
pected variation with (AL)”’ does not hold. On the other Fresnel zone; for phase fluctuations the longer scales are
hand, Gilmartin and Homing[74] observed the peak of the more important. As a result, a geometrical-optics approach
fw(f) curve to be f,= Kv,(AL)- ‘Izwith K assuming values leads to results that are very similar to those of a diffrac-
of 2.5 to 2.8. Chu [75] measured the power spectrum at three tion theory. The importance of the large scales poses a prob-
wavelengths (0.63,3.5,and 10.6 pm) and found an essentially lem since this portion of the refractive-index spectrum does
exponential spectrum at each wavelength, the width of the not have a universal shape. Therefore, attempts to predict
spectrum decreasing with wavelengthbut more slowly than the variance of the phase fluctuations for the general case
A- 1 P . are meaningless. For this reason, much attention has been
Temporal power spectra of stellar scintillations have focused on the “phase structure function” DJp) which is
been measured by many observers [71], [76]. Scintillations less sensitiveto large scales, definedas follows :
observed through telescopes with aperture less than about
10 cm display spectra that are nearly constant from 0.1 to W P ) = D&l) = ([SlW - + dlZ). (39)
100 Hz and decrease steadily into the noise level at about Because of less theoretical and less experimental attention,
500 Hz. As the telescope aperture increases, the intensity of the range of validity of the expressions for phase fluctua-,
the scintillations is reduced at all frequencies in agreement tions is not as well understood as for intensity fluctuations,
withFig. 8. Above about 25 Hz the reduction proceeds though there is someindication that theexpressions should
faster than thereciprocal of the aperturediameter. Low fre- be valid for longer paths [lo].
quencies are enhanced as the star approaches the horizon. The probability distribution for phase fluctuations, based
Mikesell found that the higher frequencies of scintillation on the same arguments as for log-intensity, should be
are enhanced when the wind speed at heights of 8 to 14 km normal, at least where linear perturbation theory is valid.
increases. There is no evidence in his observations that the Note, however, that the fluctuations in phase about the
wind exactly at the tropopause is more closely related to mean value will be much greater than 27c. Therefore, while
scintillation frequency than wind at other heights in the the total phase deviation from the mean has a normal dis-
1540 PROCEEDINGS OF THE IEEE, OCTOBER 1970

tribution, the phase referred to a zero-to-2~band has a the frequency spectrum d e h e d in terms of covariance
distribution that is practically uniform. functions, provided the covariance function exists. The
spectral representation is
Spatial Structure Function of Phase
Table I11 shows equations for the phase structure function
for the same cases as before, i.e., plane wave, spherical wave,
Hs(7) = 2 j;
[l - cos(2nf7)]wS(f)df.(42)
and beam wave.Again we give no derivations, only ref-
erences.Exceptfor the numerical coefficients, the phase Using this approach, Tatarski[9] found for the general case
structure functions for plane waves and spherical waves are
similar. It seems reasonable to expect that the beam-wave
case will lie somewhere betweenthe two but, to our knowl-
edge, no such case has been worked out. Specific cases fora where
spherical wave propagating vertically through the atmo-
sphere have been published for the wave structure function
[80],[81]. The wave structure function is related to the
phase and log-amplitude structure functions by the relation
, ~ (K-IC'/IC;),
Assuming ~ ' , ( K ) = O . O ~ ~ Cexp - ~ ~ ~ ~hefinds
D(P) = DdP) + D,(P) (40)
0- 813
where wS(f ) = 211 . 0.033C,Zk7/6L1'16
-
fo
D,(P) = 2[B,(O) - B,(p)l.
Since the phase structure function is usually larger than the
I1.69 + Im RZ
log-amplitude structure function, the wave structure func-
tion is dominated by the phase term. Therefore, the results
for specific case for the wave structure function give some
insight into the behavior of the phase structure function.
The difficulties involvedin optical interferometry through
the open atmosphere have tended to discourage direct mea-
surements of the phase structure function. Bertolotti et al. ~ . one can see by
where R = f/fo and fo = v , / ( 2 ~ 1 L ) ' /As
[82] and Moldon [83] observed phase differences over4-km comparing (44)with (29X the expression for the frequency
urban paths 25 meters or more above the ground and con- spectrum for the phase fluctuations is the same as for log-
cluded that the phase structure function could not be con- amplitude fluctuations except for a minus sign. For R<< 1
strued to increase as p5l3 for any p >3 cm. Measurements of and R >> 1, (44)becomes
the modulation transfer function (a function of the wave 3.28. 10-2C,Zk2L~:13f-8/3. a<<1, (45)
\\,s(,f') =
structure function)by Djurle and Back [84]and by Coulman
[85] also failed to verify the p5I3 dependence,but the effect of ws(f) = 1 . 6 4 - 10-2C,2k2L~:/3f-8/3, R>> 1. (46)
scintillations confuses the interpretation of these measure- Note thatfor R>> 1, the spectra ws(f) and w,(f) agree. For
ments in terms of phase structure function. the two regionsR e< 1 and R >> 1, the values of w d f )differ
By contrast, Bouricius and Clifford [86] observed phase only by a factor of 2. However, inthe low-frequency region,
structure function over a folded50-meter path only 1.6 the form for @,(IC) is not known in general; therefore, (45)
meters above ground and verified an increasing power-law must be applied cautiously at frequencies less than u,/Lo,
dependence on p, though the exponent may be lessthan 5/3. where Lo is the outer scale of turbulence.
Temporal Power Spectrum of Phase For many applications the quantity of interest is not the
The procedure for finding the frequency spectrum of the frequency spectrum of the phase fluctuations but the spec-
phase fluctuations is similar to that forfinding the log- trum of the fluctuations of the phase difference measured at
intensity frequency spectrum, except that the formulas two points separated by a distance p. If we define the phase
involve the structure function instead of the covariance difference as
function. Again, the simplest approach is to use Taylor's s,s(t) = S1(r,t ) - S,(r + P, 0 , (47)
hypothesis, so that
Tatarski [9] shows that the covariance function of the phase
S,(r, t + 7) = S,(r - u17, t). (41)
difference is
Defining a time structure function Hs(7), we get
RdS(7) = (s,s(t)s,s(t + 7))
Hd7) ([S,(r, t + z) - &(r, t)]') = &(u17). = ${D& - ~ 1 7+
) D& + 117) - 2Ds(~l7)},(48)
The structure function may be written as a spectral r e p using once more the frozen-turbulence hypothesis. If p and
resentation, where the frequency spectrum ws(f) is equal to uI are parallel, then Tatarski calculates
LAWRENCE AND STROHBEHN: CLEAR-AIRPROPAGATION EFFECTS

TABLE 111
STRUCTURE
FUNCTIONOF PHASE

Case RandomGeneral Medium Locally Isotropic Random Medium*

Plane wave, homogeneous D,(p) = 4n2k2L

medium

Plane wave, smoothly )[: - J , ( K ~ ) ]

D,(p) = 8 ~ ~ k ~ [ ~ ~ d q C ~ ( qdKK[1
varying medium

X @ko’(K)COS2 ‘’1
[“‘(tk-
~ (T34)

Spherical wave,
homogeneous medium
D,(p) = 8n2k2/: dKd.(K)joLdq[ 1 - J0(7)] D,@) = 1.089k2Cfp5’3L- 2[BAO) - EJp)] V42N

Spherical wave, smoothly D,(p) = 8n2k2

varying medium
K ~ ~-( L
x cos2 [ 2kL ‘’1 (T41)

Beam wave, smoothly in (T44).

Let @ , ( K ) = Ci(qpko’(~) (T45) No caw worked out.
varying medium

* @,(~)=0.033 C~(X)K-”’~
exp ( - K ’ / K ~ ) except as noted. For homogeneous medium, Cf(x)= constant.
@,(~)=0.033C:K- ‘‘I3. Also, B,(p) given by (T26) and (T28).
1Symbols defined in Tables I and 11. Same comments as (Tl5) and(T30).
1542 PROCEEDINGS OF THE IEEE,OCTOBER 1970

w 6 ~ f )= 4 sin2 (c)ws(f). (49)

Variance of Angle-of-Arrival Fluctuations
Most of the theoretical work has been done by Tatarski
[8], [9]; the second work is more complete. For an inter-
Using(45) and (46),we h d that (49)becomes ferometer with separation b, the variance of the angle of
arrival a is

/~ - f
~ & f ) = 0 . 0 6 6 C , 2 k ~ L u :sin2 -8/3,p<<(IL)'I2. (51) where it has been assumed that a is small so that sin a w a.
Thus, the variance of the angle of arrival may be obtained
from the appropriate expressions for the phase structure
- In the low-frequencyregionfor f <<u,jp, w6S(f) function given in Table 111.
f 'wS(f ) , and i f p does not exceed Lo,then waS(f)- f -2/3. Tatarski [9] derives the angle-of-arrival variance for a
Although there is a singularity at f =0, Tatarski shows that telescope of diameter b, and the result differs from (53)'by
it does not contribute much energy to the spectrum of the only 3 percent. For spherical-wave propagation, Gurvich
fluctuations. Another point tonote is that (49) predicts et al. [49] give
zeros or nulls in the phasedifference spectrum. These nulls
result from assuming a constant wind speed, and fluctua- 0,' = 1.05C,Zb-'/3L, (AL)'/'<<B. (54)
tions in wind speed smear them out. If f >gu2crDp), then In (53) and (54), a corresponds to one component of the
the sin2 factor can be replaced by i,so that angle-of-arrivalfluctuations ;(54) is valid onlyas long as the
1 ..2 expressions used forDdp) are valid.
Gurvich and Kallistratova [87] measured the variance of
angle-of-arrival fluctuations under conditions where the
Note that all the specific cases for the above spectra are intensity fluctuations were saturatedandnot saturated.
calculated assuming plane-wave propagation. Though we Using a telescope witha constant aperture size, they verified
do not know of any calculations forspherical-wave or that (54) was correct over paths 125 and 500 meters long.
beam-wave cases, the approach is straightforward, but the The maximum values of cawere 15and 25 p a d , respectively.
mathematics may become cumbersome. Over a 1750-meterpath the predictions were still quite good,
The only measurements of the spectrum of fluctuations but the measured values were somewhat lower than pre-
of phase are those made incidentally to the structure func- dicted. Similar measurements by Gurvich et al. [49] gave
oa - 5 to 20 prad for a 650-meter path and 15 to 45 prad
tion measurements of Bouricius and Clifford [86] who, for
'
wind velocities of about 1 m s- and L= 50 meters, ob-
for a 6500-meter path during daytime. These measurements
indicate that the theoretical expressions for the phase
served nocontribution from frequencies above 15 H z .
structure function appear to be valid even when the ex-
Phasedifference spectra have not yetbeen measured di-
pressions for intensity are not. More comparisons of this
rectly butare intimately related to the angle-of-arrival
type are needed. Gurvich etal. also measured of,J~,',s,
spectra described in Section VII.
(the ratio of the plane-wave to spherical-wave variance) and
found a ratio of 2.42 compared with a theoretical value of
VII. ANGLE-OF-ARRIVAL FLUCTUATIONS
2.75.
Introduction Kerr [88] has tabulated theoretical valuesfor various
In considering angle-of-arrival fluctuations, we must dis- cases.
tinguish carefully among the different definitions of angle
of arrival, remembering that the angle of arrival is closely The Covariance Function and Spectrum of Angle-of-Arriual
tied to the equipment involved. Attempting to remove Fluctuations
equipment considerations, Strohbehnand Clifford[30] Strohbehn and Clifford[30] calculated the angle-of-
used the direction of the wave normal at the receiving point. arrival covariance function assuming thatthe angle of
With this definition, the angle of arrival is proportional to arrival corresponds to the normal to the wavefront. Their
the spatial derivative of the phase fluctuations. However, if results are of questionable practical value since most mea-
one is using an interferometer, then the angle of arrival is surement techniques do considerable averaging over the
the phase difference measured at two points separated by phase front.
some distance, call it b. Obviously, the parameter b affects The temporal power spectrum of angle-of-arrivalfluctua-
the measurements. If a telescope isused, the angle of arrival tions may be estimated by using some of the previous results.
is closely related to the average tilt of the wave across the We have mentioned that the variance of angle-of-arrival
aperture of the telescope. In this case spatial variations in fluctuations of an interferometer is almost identical to that
phase tend to average out if they are smaller than the of a telescope. Assuming that the same similarity holds for
diameter D of the apertur?. the spectrum, we must only findthe frequency spectrum for
STROHBEHN:
LAWRENCE AND CLEAR-AIR PROPAGATION EFFECTS 1543

index gradient observed within the first 100 meters above

ground. Their interest is the refraction of radio waves, but
their methods, tables, and graphs, with the humidity taken
as zero, all apply directly to the optical case. This arises from
the fact that the refractive index forradio waves isthe same
as that for light waves, exceptthat for the radio case a term
must be added to account for a strong water-vapor effect.
They havecompared in detail the results of the exponential
model that was corrected forsurface observations with
actual ray tracings through observed radiosonde profiles of
the atmosphere. For elevation angles 8, greater than about
10 mrad the simplified method can be trusted to yield
accuracies in total bending z of a small fraction of a milli-
radian.
f (Hz)
Downward bending of an optical ray also occurs on
ground-to-ground paths. The curvature C of the ray is given
Fig. 10. Normalizedfrequencyspectrum for angle-of-arrival fluctu-
ations over a 650-meterpath; vi = 2 mjs. The heavy curveis theoretical by the transverse gradient of refractive index or, for nearly
From Gurvich e? al. [49, p. 13681. horizontal rays, by the vertical refractive-index gradient.
Adopting a sign convention that downward curvature is
an interferometer. When considering phase fluctuations, positive and using N units so that curvature is measured in
however, we described the spectrum of the difference of the prad/km, we can differentiate (1) to obtain
phase fluctuations at two receivers separated by distance p .
Setting p =b makes the spectrum was( f)found in Section VI c = - -dN
=---
79 dP +--.79P dT
dh T dh T 2 dh
also the spectrum of the angle-of-arrival fluctuations of an
interferometer, provided the expressions for was(f) are For sea-level conditions, where the pressure is 1013.25
divided by kZ ;i.e., in (49) through (52), let mbar, the vertical pressure gradient is - 121 mbar . km-’,
and the temperature is 20°C; the ray curvature is related to
the temperature lapse rate (“C km- ’) by
dT
Gurvich et al. [49] measured the frequency spectrum of C = 32.6 + 0.93 -p a d . km-
dh
’.
angle-of-arrival fluctuations over a 650-meter path when
zl was 2 m * s-
i ’.
Fig. 10 compares their results with theory. Under these conditions the temperature gradient needed to
The value of &I for these measurements varied between 1 prevent bending (C=O) is - 35°C. km- ’. Although this
and 2.2 ; therefore, they were not in saturation. Excellent temperature lapse rate is muchgreater than thatfor a normal
agreement is shown. atmosphere ( - 6°C . km- I) or an adiabatic atmosphere
(- 10°C. km-I), even greater lapse rates can occur for
Astronomical Refraction and Beam Wander a short distance above a hot surface. Such conditions
A ray of starlight traveling to the earth’s surface under- produce upward curvature and cause a mirage. On the
goes a total bending t that depends on the refractive-index other hand, if the vertical temperature gradient becomes
profile of the atmosphere and on the elevation angle of the strongly positive,exceeding about 134°C km-’ as may
star above the observer’s horizon. This bending is known happen above a cold surface, the normal downward curva-
as “astronomical refraction.” For a grazing ray where the ’,
ture of the ray may exceed 157 p a d . km- the curvature
elevation angle of arrival Bo = 0, at sea level, a typical value of the earth’s surface. This produces another kind of
for z is 10 mrad (34 minute of arc, or slightly more than the mirage known as looming. The corresponding phenomenon
apparent angular diameter of the sun). For a surface refrac- in radio-wavepropagation is known as “ducting.” A review
tivity N, and for Bo> 170 mrad, the approximate formula of the theory and observation of mirages, withan extensive
t x N, cot 8, prad is valid to within 0.1 mrad. bibliography, was recently prepared by Viezee [91].
Garfinkel [89] prepared a computer program for calcu- Since the curvature of a light ray depends so strongly on
lating the astronomical refraction throughanarbitrary the vertical temperature gradient of the atmosphere, the
atmospheric profile, including the U. S. Standard Atmo- apparent position of a distant object or, what is equivalent,
sphere, applicable to all cases for -n/2 10, I4 2 . More the spot produced at a distance by a fixed laser, will vary
approximate and more convenient methods for estimating from time to time. Ochs and Lawrence [92] measured the
astronomical refraction are discussed and tabulated by Bean variations of beam curvature over paths from 5 to 45 km
and Dutton [W]. They fitted an exponential function of long. A typical24-hour measurement showed a diurnal
height tothe model atmosphere and made appropriate ’
variation of 30 prad . km- on a 15-km path. A laser beam
corrections to permit the inclusion of the initial refractive- wanders sufficiently over long paths toexceed the minimum
1544 PROCEEDINGS OF THE IEEE, OCTOBER 1970

beamwidth permitted by the turbulent atmosphere. Thus of velocity and temperature fluctuations in the atmospheric bound-
o p t i d systems using minimum beamwidth must employ a ary layer over the sea,” J . Atmos. Sci., vol. 23, pp. 376386, 1966.
[17] B. R. Bean, “Meteorological factors affecting the fine-scale structure
servo pointing system to remove variable refraction. Ochs of theradio and optical refractive index,”-in Tropospheric Wave
and Lawrence [92]trackedtheir beams at amaximum Propagation, Conf. Publ. 48. London: IEE, 1968.
angular rate of 7.5 pad - s-’ and were usually, but not al- R. Bolgiano, Jr., “A theory of wavelength dependence in ultrahigh
frequency transhorizon propagation based on meteorological con-
ways, able to maintain lock. When occasionally the nearly siderations,” J. Res. Nut. Bur. Stand., vol. 64D, pp. 231-237, 1960.
horizontal beam passes throughan elevated inversion layer R. S. Lawrence, G. R. Ochs, and S. F. Clifford, “Measurements of
of the kind describedin Section 111, the vertical deflections atmospheric turbulence relevant to optical propagation,” J. Opt.
Soc. Amer., vol. 6 0 , no. 6, pp. 82-30, 1970.
are greatly enhanced.In these cases nearly periodic vertical R. E. Hufnagel, “An improved model turbulentatmosphere,” in
oscillations are colll~~lon.
The maximum required tracking Restoration of Atmospherically Degraded Images. Washington,
rate is unknown; the spot position at the receiving plane D. C.: National Academy of Sciences, July 1966, vol. 2, pp. 1&18.
D. A. Gray and A. T. Waterman, Jr., “Measurement of fine-scale
may, in fact, sometimes be discontinuous. More observa- atmosphericstructure using an opticalpropagation technique,”
tions are required to permit predicting the reliability of a J . Geophys. Res., vol. 75, no. 6, pp. 1077-1083, 1970.
laser-beam tracking system. J. W. Strohbehn, “The feasibility of laser experiments for measuring
the permittivity spectrum of the turbulent atmosphere,” J . Geophys.
Motion of stellar images can be measured either with Res., vol. 75, no. 6, pp. 1067-1076, 1970.
smallaperturesand long exposures, in which case the G. R. Ochs, ESSA Res. Lab., Boulder, Colo., private communica-
motion results m a blurred image indistinguishable from tion, 1970.
J. B. Keller, “Wave propagation in random media,” in Proc. Symp.
&focusing, or with large apertures and short exposures, in in Applied Mathematics, G . Birkhoff, R. Bellman, and C. C.Lin, Eds.
which case the aperture-averaging effects produce multiple Providence, R. I . : American Mathematical Society, 1962, vol. 13,
overlapping images. Measurements of the latter type were pp. 227-246.
-, “Stochastic equations and wave propagation in random me-
made by Lese [93] with a fixed 0.9-metertelescope and ex- dia,” in Proc.Symp. in AppliedMathematics, R. Bellman, Ed.
posure times of a few milliseconds. Multipleexposures Providence, R. I.: American Mathematical Society, 1964, vol. 16,
produced images lying more or less on a straight line, and pp. 145-170.
W. C. Hoffman, “Wave propagation in a general random continuous
deviations kom the straight line were caused by random medium,” in Proc. Symp. in Applied Mathematics, R.Bellman, Ed.
atmospheric fluctuations. The mean valueof such deviations Providence, R. I.: American Mathematical Society, 1964, vol. 16,
was 3 p a d for zenith angles 8, I1 rad. Maximum excur- pp. 117-144.
L. A. Chernov, Wave Propagation in aRandomMedium. New
sions of about 5 5 pad occurredforthelargestzenith York: McGraw-Hill, p. 168, 1960.
angles. J. W. Strohbehn,“Comments on Rytov’s method,” J . Opt.Soc.
Amer., vol. 58, pp. 139-140, 1968.
R. W. Lee and J. C. Harp,“Weak scattering in random media, with
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