The fluctuations of the angle of arrival of a collimated beam that has propagated through atmospheric turbulence to a reflector and back have been the subject of ...
1264
J. Opt. Soc. Am. A/Vol. 4, No. 7/July 1987
J. H. Churnside and R. J. Lataitis
Angle-of-arrival fluctuations of a reflected beam in
atmospheric turbulence James H. Churnside and Richard J. Lataitis Wave Propagation Laboratory, National Oceanographic and Atmospheric Administration/Environmental Research Laboratories, Boulder, Colorado 80303 Received March 3, 1986; accepted February 19, 1987 The statistics of the angle-of-arrival fluctuations
are studied for the case of a laser beam reflected from a curved
surface in a uniformly turbulent atmosphere. The variance for the direct beam and for the reflected beam and the covariance for the two beams are calculated for arbitrary divergence of the illumination and radius of curvature of
the reflector. Experimental results support the conclusions that the reflected-beam angle-of-arrival fluctuations are very sensitive to small deviations from perfect collimation and that the correlation of the fluctuations in the direct and reflected beams is high.
1.
INTRODUCTION
path is assumed, the extension to path-position-dependent
The fluctuations of the angle of arrival of a collimated beam that has propagated through atmospheric turbulence to a reflector and back have been the subject of some study, and Mironov and Nosovl derived an expression for the variance of the image centroid displacements
in the focal plane of a
telescope. The two problems are nearly the same. The correlation between the angle of arrival and the ratio of the image centroid to the focal length of the telescope is 0.992, and the variance of the former quantity is about 1.07 times that of the latter.2 Mironov and Nosov concluded that the angular fluctuations of the reflected beam have a variance four times that of a beam undergoing a single pass along the same path. Lukin et al.3 extended the analysis to show that the factor of 4 in the variances reduced to a factor of 2 for small
telescope apertures. These results were experimentally verified over a short (2-m) path.
More recent work 4 has ex-
tended the theory to include the effects of the diameters of the transmitted beam and the reflector. A related investi-
turbulence levels is straightforward tion 4. 2.
and is discussed in Sec-
THEORY
The analysis is based on the geometry shown in Fig. 1. A
beam of diameter D and focal range ft is assumed to propagate from the transmitter toward the reflector at a distance L. The reflector is assumed to have some arbitrary radius of curvature so that the reflected beam may have a different focal range fr. In the case of a plane reflector, it is clear that fr is given by ft - L. This geometry will be used to find the variance of the angular fluctuations that would be observed at the reflector and then that of those fluctuations that would be observed back at the transmitter after reflection. To proceed, consider a linear refraction-index graqient at a position z along the path and of thickness dz. Treating this thin section of refractive gradient as a thin prism, we see that it will produce a tilt at the reflector given by
gation 5 has considered intensity fluctuations for similar ge-
ometries. Aksenov et al. 6 extended the analysis to conditions of strong turbulence, where saturation of the intensity fluctuations occurs. They found no difference between the angleof-arrival theories for the weak-turbulence regime and the strong-turbulence regime. As long as the reflector and the telescope are large enough to intercept the entire beam, the
da = An(z) dz, w(L)
where da is the induced tilt, An is the difference between the refractive indices at the opposite edges of the beam, and w(L) is the beam diameter at the reflector. Summing over contributions from refractive gradients all along the path yields an arrival angle of
variance is increased by a factor of 4 on reflection back to the source.
With the exception of Ref. 3, these investigations have been purely theoretical. An attempt to verify these predictions experimentally ran into difficulties that were traced to the effects of slight deviations from perfect collimation. In this paper, an analysis is presented for the ratio of the variance of the angular fluctuations of the reflected beam to that of the direct beam, including the effects of arbitrary curvature on the transmitted beam and on the reflected beam. A geometrical-optics formulation is used; therefore the results hold for beams that are large enough that diffraction effects
can be neglected. Although uniform turbulence along the 0740-3232/87/071264-09$02.00
(1)
a = w(L) JL An(z)dz.
(2)
If there is no average gradient of refractive index, the variance of a is equal to its second moment and can be written as 2=
2
(L)
J J
dzldZ 2 (An(zl)
An(Z 2 )),
(3)
where the angle brackets denote an ensemble average and the order of the averaging and integration operations has been reversed. © 1987 Optical Society of America
J. Ii. Churnside and R. J. La'taitis
Vol. 4, No. 7/July 1987/J. Opt. Soc. Am. A
Transmitter
Reflector
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is intercepted is considered, and D represents the projection of the reflector aperture back onto the transmitter. We can use Eq. (7) for both cases, provided that we let
D =Dt
forD,>Dt i-
D -
forDr,
L
(13)
where X is the optical wavelength, for the geometric-optics approximation used here to be valid. In a similar manner, we find that an index-of-refraction gradient at position z along the path will affect both the direct beam and the reflected beam, to produce a tilt back at the transmitter of da= Ant(z)
+ Afnr(Z)Wr(O) w
where Dt' is the diameter of the measurement aperture at the transmitter. Using these and the same approximations and numerical integration used in the direct-beam case gives the variance of angle-of-arrival fluctuations of the reflected beam: a
=
1 2.92 C 2LD-
X
-
1(iX
X f'du 0
X 1 2
(0)
J010 rL L
I (
D )1/3
8 AD
Ofr
L
fir (D )2
2 D3I 22/3 {1 _ LA
fry
- UL
I- L
I - (I - U) L
ft 11- U L
ft I _ -L
fr
ft
ft
I
-
(I - U) L fr
We see that, for the reflected
beam, 2 = Orr
ft DIL~~~~~J
Jr
Calculation of the variance of the angle-of-arrival fluctuations proceeds as before.
8
-
I (15)
2
(I-
sults in a total tilt angle of a = w(O) J|L[nt(z) + An,(z)Idz.
[3(D)
( ftr
(14)
where Ant is the refractive-index difference across the transmitted beam, Anr is the difference across the reflected beam, and Wr(O)is the diameter of the reflected beam at the transmitter. Summing contributions along the entire path re-
3
dzldz2[(Ant(zl)
UL L1- + D
n-(1-U) 5/3
I
An,(Z 2 )) ll
+ (Anr(ZI) Anr(Z2)) + 2(Ant(zl) Anr(Z2 ))],
(16)
LatU
D: I[1 (I
_
) Lt]|
5/3-
(21)
J. H. Churnside and R. J. Lataitis
Vol. 4, No. 7/July 1987/J. Opt. Soc. Am. A
Evaluation of the final integral is straightforward, ues of L/Jt and L/Jr are known.
1267
once val-
The ratio of the angle-of-arrival variance of the reflected beam from Eq. (24) to that of the transmitted beam from Eq.
Perhaps the most generally useful case of Eq. (21) is one in which neither the transmitted beam nor the reflected beam comes to a focus between the transmitter and the reflector.
(11) is plotted in Fig. 3 as a function of the path length divided by the focal range fr. The ratio goes to zero for a widely diverging return beam (L/fr - -A), as one would
In this case, we have
expect from the result for the angular fluctuations of a diverging transmitted beam. For the combination of a collimated beam and a plane reflector (L/Jr = 0 in the figure), the variance ratio is 4, as predicted by other researchers. The
1U L > 0
for 0
1-(1-U)->0
u _1,
for0u
fr
1,
(22)
and most of the absolute value signs in Eq. (21) can be
dropped. To simplify further, consider the case in which the reflector and both measurement apertures are all larger than the beams incident upon them. In this case, all beam geometries are determined by the transmitter aperture Dt as
curve in this region is very steep, however, suggesting that measurement of a ratio of 4 is very sensitive to beam geome-
try. For the return beam focused at the transmitter, the geometric-optics approximation used here predicts an infinite ratio, which, as before, demonstrates the limitation of this approximation. In this case, the geometric-optics approximation is valid for
follows:
The second case of interest is that of a plane reflector.
)-
D'= Dt (1-
(23)
ft
Evaluation of the integral in Eq. (21) can then proceed with the followingresult: 3
,J 2 = 2 pœ2C, 2r,n -1/3 -- n -- tL2
- r
- .vvnt
L)(I
f (
(
(27)
fr
D = Dt,
The ratio of the angle-of-arrival variance of the reflected beam from Eq. (26) to that of the transmitted beam from Eq. (10) is plotted in Fig. 4 as a function of the path length L divided by the focal range ft. In this case, a widely diverging
transmitted beam produces a variance ratio of = 22/3+ _ 2.42,
-
L2 f r)
~fit
Ut2
(28)
6
and the fluctuations of the reflected beam remain higher than those of the transmitted beam. With a collimated transmitter (L/ft = 0), the ratio equals 4. The curve in Fig. 4 is even steeper than that in Fig. 3 at this point, however, suggesting that the ratio can be very sensitive to miscollimation of the transmitted beam.
L 8/3]
+ fr ( -L )5/3[I- (I- L)8/]
10
4 (1- L8/3 - 2-2/3 1 + (I
-
L) (1 - L)]8/3 8
( ft )(
2[
fr) (24)
ft 1 -frl}
6
except for two special cases represented by
fr = ±(f -L).
(25)
In these cases, the u dependence of one term or the other within the integrand vanishes, and the integral of that term must be evaluated differently. The case, represented by the plus in Eq. (25) corresponds to the physically important case of a plane reflector. For this case, we have Ur2=
2LDt 113 2.92C,,
3 )
[ -
-2
4
2
] 0
16 (
)5/3
_
L
I5/3.
-4
(26)
To illustrate these results, it is useful to consider two specific cases. First, consider a collimated transmitter.
-3
-2
-1
0
1
L/fr Fig. 3. Ratio of reflected- to transmitted-beam angle-of-arrival variances Ur2 /at 2 for a collimated transmitter as a function of the
path length L divided by the reflected-beam focal range f,.
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+3 fr) J. H. Churnside and R. J. Lataitis
J. Opt. Soc. Am. A/Vol. 4, No. 7/July 1987
10 0r2= 2.92Cn2LDt-1/3f 1 8
(2_-L)
2L)/ 3 + (_ L)"/3]
~~~~~~~~~~~~~.(31) L (1) 5/3
+
Or 2 at 2
-
-
8
6
(
8 1,L (1 _LfL5/3
22/3
The ratio of this variance to the angle-of-arrival variance of the transmitted
4 -
beam is plotted as a dashed line in Fig. 3.
Here, the ratio for a collimated transmitter and a widely diverging reflected beam is given by 8/3.
This is the com-
monly accepted value for the ratio of spherical-wave angleof-arrival variance to plane-wave angle-of-arrival variance when both are measured over the same aperture diameter. With a plane reflector, Eq. (26) is valid for 2L < ft < a. For negative values of ft, the variance is given by
2 -
0-
-4
-3
-2
-1
0 0.5
3
2
/3+
~Dt
2(
a 2 = 3 2.9 2Cn2L
-13
(
)
L/ft Fig. 4. Ratio of reflected to transmitted-beam angle-of-arrival variances ar2 /at2 for a plane reflector as a function of the path length L divided by the transmitted-beam
+
If the measurement apertures and the reflector are not all large, things are more complicated. In our experiment, for example, the transmitter, the reflector, and the measurement apertures were all the same size. Under these conditions D =D D =DDt
L (1 I
for ft > L, for ft < 0,
L 8/3
(1-L 8/3-
focal range ft.
L 2- 3ft
X
L 1- ft
1-2-
2/3)
2(
L2
8/3
L 1 - 2-
L
(29)
L)5/3
L)1/3 (1
L)8/3 f/)
-
2
-
L)8/3} ftt
ft
1- L
1- L
and 22/3L
for ft L,
Dt = Dt
(1
-
L)2 (3
- 2)
-
forft>L, frL, fJr>L, D Dt
_-L
forft
