different from the assumed approximation, the 
analysis herein presented can be similarly 
applied to any other spectrum. 
We shall examine the two cases: 
Case I K,<«K, or n; <1, and 
Case II: K> K; or 1< Dj. 
Writing 
f(njn') 1 cos ri, 
¢(n,2") ™ ain? [ a 
then from equation (10) the spectra of the 
phase and amplitude fluctuations may be 
written 
+00 
Fo(m) _ ogy 9, (n*+ n) £.(250") gar. (21) 
f,(n,n') 
F,(n) rod AY? 
For each value of n the integral in (21) deter- 
mines the contribution to the power density 
spectra P,(n) and F,(n) of the power density 
from the “various wave numbers of y(n). It is 
observed that as n increases the function 
% (n'4 n) is shifted to the left on the n' axis. 
ais, as n increases the period of the functions 
f,(n,n') and f,(n,n*) increase with respect to n'‘, 
The first minimum of f_(n,n') and the first 
maximum of f (n,n') for a given n occur when 
n'= 1/(2n), 4while the spectral function 0, (n'+n) 
will terminate at n'= n,- n. If both of these 
points are plotted against n for case I and 
case II the results would appear as shown in 
Fig. 2. 
From the figure we see that for case I 
the first minimum of f_(n,n") and the first 
meximum of f,(n,n') on the n' axis will occur 
‘at higher values of n'* than at which the 
spectral function @ (n'+ n) terminates. 
Fig.3 illustrates the relationships that 
exist between #(n'+ n) and the functions 
f(n,n') and f (n,n") for case I, For these 
relationships we note that 
f.(n,n') = = 
and 
le 
8 e 
For the integrand involving f,(n,n') the 
maximum will fall near the origin or at low 
values of n'. For small n the contributions 
will come from the large size components of In 
For large n the contributions will come from 
the smaller components of $,, however, their 
effects will be considerably less since 9, (n" +n) 
is less in value and £.(n,n") falls off as 
n e 
n nt 
f,(n,n') ~ 
Since 9, (n'+ n) is nearly constant at 
the origin and increasingly decays until the 
rate is proportional to the -11/3 power of 
n* over the larger values of n'<¢n.- n, the 
integrand in (21) involving f,(n, f&") will 
have a maximum on the n' axis“between the 
35 
small values of n and at small 
middle to large values of n'<¢n,.—n for 
values of 
n' for large values of n. Thus, the chief 
contributions will occur from the middle 
to small size components of ¢. ° 
For case II we see from Fig.2 that the 
first minimum of f.(n,n') and the first max- 
imum of f (n,n') will generally occur at 
values of n' much smaller than the value 
n,- n at which the spectral function % (n'+n) 
terminates as shown in Fig.4. For both inte- 
grands the maximum for most values of n will 
occur over the large size components of the 
spectrum » eB8pecially for the integrand 
in the formula for the phase spectrum which 
has its maximum at n'= 0. For the integrand 
involving the function f,(n,n') the maximum 
will tend to lie over the range of large to 
middle size components of $.. The effects 
of the small size components as n increases 
is reduced by the fact that the average 1 
values of f_(n,n') and f,(n,n") fall of as n 
It can be Shown by examPles that the max- 
imum contributions to the amplitude fluctu- 
ation spectrum will come from components of 
4 whose dimensions are of orderVAL . 
We shall now examine the effects of the 
spectrum 9 (n) on the mean square fluctuation 
of the wave front normal direction. Defining 
the term in the curled brackets of the inte- 
grand of equation (19) as f,(n) we may write 
equation (19) as 
+00 
o = aut [to n? g.(n) nan. (22) 
Fig.5 shows the relationships between the 
spectral function § (n) and the function f(a) 
for case I. We observe that in this case 
f,(n)~ 8/3. Since 9, (n) decays as n to the 
-fi/3 power over the higher wave numbers, the 
integrand of (22) will be a maxinum over the 
largest size components of the spectrum Ine 
However, comparing this with case I for 
the phase fluctuations we observe that the 
contributions from the smaller size components 
of § are more significant for the wave front 
norm fluctuations because of the presence of 
the additional term n@, 
Fig. 6 shows the relationships between 
f,(n) and 9, (n) for case II. Except in the 
vicinity of very low wave numbers, the function 
f.(n) <= 4/3. Thus, the maximum contribution to 
the mean square fluctuation of the wave front 
normal direction will come from the large size 
components of N° 
V. CONCLUSIONS. 
1. The small components of the thermal variations 
principally affect the spectrum of the amplitude 
fluctuations when the smallest component of the 
thermal variations is greater than VAL. The 
Manner in which the small components affect the 
