GENERATING PLANE WAVES FROM A LINEAR
ARRAY OF LINE SOURCES
Randy Haupt
Electrical and Computer Engineering
Utah State University
4120 Old Main Hill
Logan, UT 84322-4120
435-797-2841
H (2)
0 (⋅) = zeroth order Hankel function of second kind
k = 2/ R = distance from line source to center of test aperture
Abstract
Usually, a plane wave is approximated by increasing the
distance between the transmit antenna and the antenna
under test. The phase error across the test aperture
increases from zero at the center to a maximum at the
edges. Sometimes it is difficult to separate the transmit
antenna from the test antenna enough to keep the phase
error within acceptable tolerances. In these cases, it
would be useful to be able to generate a plane wave
across the test aperture at the closer distance. This
paper presents an approach to generating a plane wave
across an antenna under test that is at an arbitrary
distance from a linear array of line sources. The
placement of the line sources, as well as the phase and
amplitude of each element is optimized to create an
approximate plane wave over a specified area.
If the line source radiates to a 2 test aperture at a distance
R, then the ratio of the amplitudes at the center to the edge
as a function of R is given by the solid line in Figure 1. The
difference between the phase (in radians) at the center and
the edge as a function of R is the dashed line in Figure 1.
The amplitude ratio falls off much faster than the phase
difference up to about R = 1.5. After that, the phase
difference has a steeper slope. Thus, very close to the
transmitting source, the amplitude across the test aperture
determines the quality of the plane wave, whereas farther
from the transmitting source, the phase is the more critical
factor.
2.5
1. Introduction
Sometimes testing an antenna in the far field is not
feasible due to size constraints in the anechoic chamber or
out door antenna range. The IEEE definition of the far field
is based upon the phase variations of the transmitted wave
across an antenna under test. If the transmit antenna is a
phased array, then control of the phase and amplitude of the
transmitted wave at the test aperture is possible. The goal of
this investigation is to design an array that can create an
approximate plane wave at a receive antenna in the “near
field.”
The relative field radiated by a single line source is
given by
E(R) = H (2)
0 (kR)
where
phase and amplitude ratios
array, antenna measurement, near field, plane wave
phase difference in radians
amplitude ratio
2
1.5
1
0.5
0
0
2
4
6
distance in λ
8
10
Figure 1. The ratio of the amplitude at the center of the
2 test aperture to the amplitude at the edge, and the
difference between the phase at the center and the edge
due to a single line source.
Adding several line sources creates an array whose
radiated field is given by
35
distance = 10 λ
distance = 20 λ
distance = 35 λ
relative phase in radians
30
N
E(R) = ¦ a n e jδ n H 0(2) (kR n )
25
n=1
20
where
N = number of elements
a n = amplitude of element n
15
10
δ n = phase of element n
R n = distance from element n to plane
Adjusting the phase of the radiating elements can focus the
field in the near field, i.e. the phase of the radiated field
adds to zero at a point. This near field focusing of arrays
has been used for antenna pattern measurements [2].
Medical applications that require heating tissue with
microwaves has also required phased array focusing [3].
Details on near field focusing of array antennas can be
found in [4], [5], and [6].
Focusing requires the phase of the transmitting
elements to be equal at a point in space. The purpose of this
paper is to present results of creating a local plane wave in
the near field of an array of transmitting line sources.
2. Near Field of an Array of Line Sources
A nine element linear array of line sources spaced 0.5
apart on the x-axis has a far field distance of 2D2/
40.
Figures 2 and 3 show the relative field amplitude and phase
across a 14 line parallel to the array and starting at the
center of the array. As expected, the field variations become
less as the distance from the array increases. The goal is to
find a nine element linear array that will produce the field
variations that occur at 35 at a closer distance, for instance
10.
relative field strength in dB
0
0
2
4
6
x in λ
8
10
12
14
Figure 3. Phase variations at 10, 20, and 35 from a nine
element linear array with 0.5 spacing. The phase is
measured at a line parallel to the plane of line sources
and starting at the center of the array.
The objective function for optimization returns the
maximum deviation of the electric field (amplitude and/or
phase) in a plane that is parallel to the line sources at a
distance zp from the array. The quality of the plane wave is
a function of the distance from the array, the width of the
desired plane wave, element positions, and element
weights.
One judging criterion for how well the field generated
by the array approximates a plane wave is a phase quality
factor given by
ε p = 1 − δ max − δ min / 2π
where δ max − δ min
is the maximum phase deviation over
the desired interval. Assuming the maximum phase
deviation is 2, then for the desired zero phase deviation,
p=1, and for the maximum deviation of 2, p=0. The
width of the plane wave is small enough to have all phase
variations less than 2. A quality factor for the amplitude is
0
-5
-10
ε a = amin / amax
-15
where amin / amax is the ratio of the minimum field
amplitude to the maximum field amplitude over the desired
interval. These quality factors appear in the plots in the next
section.
-20
-25
distance = 10 λ
distance = 20 λ
distance = 35 λ
-30
-35
0
5
2
4
6
x in λ
8
10
12
14
Figure 2. Amplitude variations at 10, 20, and 35 from a
nine element linear array with 0.5 spacing. The
amplitude is measured at a line parallel to the plane of
line sources and starting at the center of the array.
3. Results
The first example attempts to generate a plane wave
across a 4 aperture at a distance of 10 from the five
element transmitting array. Figure 4 show the result of the
genetic algorithm optimization of the element positions.
The optimized amplitude and phase weights appear in
2
phase in radians
Figure 5. Figures 6 and 7 show the field phase and
amplitude at 10 from the center of the transmitting array.
Vertical dashed lines indicate the width of the desired plane
wave. The optimized array generates a superior plane wave
compared to a uniform array with elements spaced 0.5
apart. The single line source has a flatter amplitude
compared to the optimized array, but the optimized array
has a flatter phase.
1
0
-1
-2
2
optimized array, εp=0.89
line source, εp=0.73
uniform array, εp=0.75
-3
1.5
-6
1
z in λ
0.5
-4
-2
0
x in λ
2
4
6
Figure 6. Optimized phase radiated by a five element
array at a distance of 10.
0
-0.5
-1
-1.5
-2
-1
0
x in λ
1
2
Figure 4. Optimized element positions of a five element
array projecting an approximate plane wave over a 4
aperture at a distance of 10.
amplitude in dB
10
-2
5
0
optimized array, εa=0.83
line source, ε =0.97
a
uniform array, εa=0.55
-5
-6
amplitude or phase
2
amplitude
phase in radians
1.5
0.5
1
2
3
element
4
-2
0
x in λ
2
4
6
Figure 7. Optimized amplitude radiated by a five
element array at a distance of 10.
1
0
0
-4
5
6
Figure 5. Optimized element weights of a five element
array projecting an approximate plane wave over a 4
aperture at a distance of 10.
The second example attempts to generate a plane wave
across a 4 aperture at a distance of 10 from the nine
element transmitting array. Figure 8 show the result of the
genetic algorithm optimization of the element positions.
The optimized amplitude and phase weights appear in
Figure 9. Figures 10 and 11 show the field phase and
amplitude at 10 from the center of the transmitting array.
Again, the optimized array generates a superior plane wave
compared to a uniform array with elements spaced 0.5
apart. The single line source has a flatter amplitude
compared to the optimized array, but the optimized array
has a flatter phase. Increasing the number of elements
resulted in a better approximation to a plane wave.
3
15
amplitude in dB
2
10
z in λ
1
0
-1
-2
optimized array, εa=0.95
line source, εa=0.97
uniform array, ε =0.32
5
a
0
-3
-2
0
x in λ
-5
-6
2
Figure 8. Optimized element positions of a nine element
array projecting an approximate plane wave over a 4
aperture at a distance of 10.
-4
-2
0
x in λ
2
4
6
Figure 11. Optimized amplitude radiated by a five
element array at a distance of 10.
4. Conclusions
The best way to create a plane wave across a test
aperture is to put distance between the transmit and receive
antennas. This paper demonstrates that controlling the
element positions and weights of the transmitting array can
produce a reasonable approximation to a plane wave in the
near field. More elements in the transmit array allows
greater control over the transmitted field and a better
approximation to a plane wave.
amplitude
phase in radians
amplitude or phase
4
3
2
1
References
0
0
2
4
6
8
10
element
Figure 9. Optimized element weights of a nine element
array projecting an approximate plane wave over a 4
aperture at a distance of 10.
3
phase in radians
2
1
0
-1
optimized array, εp=0.93
line source, εp=0.73
uniform array, εp=0.97
-2
-3
-6
-4
-2
0
x in λ
2
4
6
Figure 10. Optimized phase radiated by a nine element
array at a distance of 10.
[1] IEEE Standard Definitions of Terms for Antennas, NY:
IEEE Press, 1993.
[2] W.K. Bartley, "Near-field antenna focussing," Goddard
Space Flight Center Report X-811-75-183, Greenbelt, MD,
Aug 75.
[3] A. J. Fenn, C. J. Diederich, and P. R. Stauffer, "An
adaptive-focusing algorithm for a microwave planar
phased-array hyperthermia system," The Lincoln
Laboratory Journal, Vol. 6, No. 2, 1993, pp. 269-288.
[4] L. J. Ricardi, "Near field characteristics of a linear
array," Electromagnetic Theory and Antennas, E.C. Jordan,
ed., New York: Pergamon Press, 1963.
[5] R. C. Hansen, "Minimum spot size of focused
apertures," URSI Symposium on EM Theory, Delft,
Pergamon, Press, 1965, pp. 661-667.
[6] R. C. Hansen, "Focal region characteristics of focused
array antennas," IEEE AP-S Trans., Vol. 33, No. 12, Dec
85, pp. 1328-1337.