IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 2, FEBRUARY 2003
273
Generating a Plane Wave With a Linear
Array of Line Sources
Randy Haupt, Fellow, IEEE
Abstract—Creating a plane wave across an antenna under test
is important for accurate antenna measurements. This paper optimizes the location and weightings of an array of line sources in
order to approximate a plane wave at a given location in space.
The amplitude and phase ripples across a desired test aperture are
optimized to be much less than that of a uniform array with the
same number of elements. Results are presented for a nine-element
array with optimized amplitude and phase weights, with optimized
weights and spacing in the -direction, and with optimized weights
and spacing in the and directions. The optimized approximate
plane wave is a significant improvement over a uniform array or a
single line source.
Index Terms—Antenna measurements, array, genetic algorithm,
near field, plane wave.
I. INTRODUCTION
A
NTENNA measurements in the far field require that the
phase and amplitude variations across the antenna under
test (AUT) be within specified tolerances. The IEEE standard
for nonlow sidelobe antennas requires the separation of the
transmit antenna and the AUT (diameter ) to be [1]
(1)
or
, where is the wavefor a maximum phase error of
length. The ratio of the maximum field amplitude to the minimum field amplitude across the AUT at a distance prescribed
by (1) is approximately
(2)
Thus, applications that require moving the plane wave region
inside an anechoic chamber or removing the reflector from the
chamber need another alternative to the cumbersome compact
range. One possibility of generating the plane wave in the near
field is to build an array that can project a plane wave at a prescribed distance. Focusing assumes the array concentrates the
field at a specific point in space. Antenna focusing has been applied to antenna measurements [2] and medical treatments [3].
Ricardi [4] and Hansen [5] show that the minimum spot size of
a uniform array antenna is about 0.35 .
This paper presents a method for designing an array that creates an approximate plane wave over a prescribed area at a set
distance. Sometimes, particularly for antenna measurements, a
constant amplitude and phase must be generated across an AUT.
Whereas focusing concentrates the field at a point, plane wave
generation attempts to minimize the amplitude and phase deviations across an area. Section II describes the array model used,
and the objective function developed from the model. Next, the
objective function is optimized for several scenarios at
from a nine-element generating array. The optimized array
designs do a nice job of approximating a plane wave over the
AUT. Increasing the degrees of freedom available to the array
designer improves the quality of the generated “plane wave.”
II. ARRAY FORMULATION
The model used for plane wave generation in this paper is a
linear array of line sources that are infinite in the -direction.
The relative radiation pattern of this array in the – plane at
is given by
(3)
, and can be igIn general, (2) is close to one, since
nored. The amplitude and phase is a constant over any cylinder
with a line source antenna at the center point. As the separation
between the transmit antenna and the AUT increases, the maximum amplitude and phase deviation across the AUT decreases.
Thus, getting far away from the antenna flattens the amplitude
and phase fluctuations across the AUT.
If it is impractical to get far enough away from the transmit
antenna to satisfy (1), then some other way is needed to approximate a plane wave in the near field. A common way of
generating a near-field plane wave in the near field is the compact range. The compact range reflector is large and immobile.
Manuscript received May 30, 2001; revised December 4, 2001.
The author is with the Department of Electrical and Computer Engineering,
Utah State University, Logan, UT 84322 USA.
Digital Object Identifier 10.1109/TAP.2003.809082
is the zeroth-order Hankel function of the second
where
;
is
;
is the
kind; is
is the location of observation point;
location of element ;
is the number of elements in the array; and
is the
amplitude and phase weighting of element .
The field at the AUT is a function of the number of elements,
element locations, and element weights.
A nine-element array has the field amplitude and phase as
shown in Figs. 1 and 2. Sidelobes begin to appear in the amplitude response as increases. The 180 degree shift between sidelobes appear as “kinks” in the phase plot in Fig. 2. In Fig. 2, the
phase is reasonably flat directly in front of the radiating aperture.
Hansen states that the power radiated by an antenna is contained
in a corrugated tube that has the same diameter as the antenna
from the array. For
[6]. This tube extends about
.
the nine-element array,
0018-926X/03$17.00 © 2003 IEEE
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 2, FEBRUARY 2003
Fig. 1. Amplitude of the field due to a uniform array (d = 0:5 ) of 9 line
sources.
Fig. 3.
Optimized amplitude and phase weights of a nine-element array (d =
at a distance of 20 .
0:5 ) for an AUT of 10
(7)
(8)
Fig. 2. Phase of the field due to a uniform array (d = 0:5 ) of 9 line sources.
It is possible to solve (3) for the optimal weights using a direct
approach. Reformulating (3) as a matrix equation yields
..
.
..
.
..
.
..
.
..
.
(4)
) for
Solving this matrix equation (use least squares if
the optimum weights results in a “plane wave” with a very flat
amplitude and phase. Unfortunately, the “plane wave” occurs in
a null of the radiated field. This result is unacceptable, because
low power illuminates the AUT and high power illuminates objects surrounding the AUT. Also, the resulting optimal weights
are very sensitive to errors.
In order to create an approximate plane wave not in a null
of the radiated field, numerical optimization is used to find an
array configuration that produces minimal amplitude and phase
variations over the AUT at a specified distance. Here, the objective function for minimizing the phase and amplitude of the
field oscillations is given by
(5)
where
(6)
that sample the area of the desired plane
for points
wave. The values of , , and are found from (3). compares the maximum phase deviation across the AUT to the maximum phase variation across that same AUT if it were at the distance given by (1). compares the ratio of the maximum amplitude value to the minimum value across the AUT to the ratio of
the maximum amplitude to the minimum amplitude across that
same AUT due to a line source. There are physical limitations
on how well this objective function approximates a plane wave.
The distance between the source array and the AUT as well as
the size of the AUT will limit the amplitude and phase deviations. The number of elements and their locations have dramatic
effects on the quality of the plane wave. Changing the values of
the weighting constants , , and determine the shape of the
plane wave and trade off the flatness of the amplitude and the
flatness of the phase. Very flat amplitude and phase variations
are possible by optimizing with only the first two terms. Unfortunately, the resulting plane wave is in a deep null in the field
pattern, as occurred in the direct matrix solution. The third term,
, was added to keep the optimal solution out of a null. This
term compares the maximum field value of a uniform array to
the maximum value of the optimized array.
III. CREATING A PLANE WAVE IN THE NEAR FIELD
All the examples presented here are based upon a nine-element array (element spacings and weights are symmetric relative to the center element) of line sources in the – plane. The
with an amcenter element of the array is set at
plitude of 1 and a phase of 0. The amplitude and phase tapers
as well as the element locations in and are assumed to be
symmetric about the center element. Weights are restricted to
values of amplitude between 0 and 1 and phase between 0 and
. Spacing between the elements in the -direction is limited
to be greater than or equal to 0.5 and less than 1.5 . As long
HAUPT: GENERATING A PLANE WAVE WITH A LINEAR ARRAY OF LINE SOURCES
275
TABLE I
THE AMPLITUDE QUALITY FACTORS FOR A SINGLE LINE SOURCE AND
A NINE-ELEMENT UNIFORM ARRAY FOR VARIOUS DISTANCES AND
FREQUENCIES FOR AN AUT 10 WIDE
TABLE II
THE PHASE QUALITY FACTORS FOR A SINGLE LINE SOURCE AND
A NINE-ELEMENT UNIFORM ARRAY FOR VARIOUS DISTANCES
AND FREQUENCIES FOR AN AUT 10 WIDE
Fig. 4. Field phase of an optimized nine-element array (d = 0:5 ) across an
AUT of 10 at a distance of 20 .
as the number of elements is relatively small, local optimization
techniques work well.
Some quality factors are needed to judge how well the field
generated by the array approximates a plane wave. A quality
factor for the phase is given by
(9)
is the maximum phase value over the desired inwhere
is the minimum phase value over the desired interval and
terval. Assuming the maximum phase deviation is , then for
, and for the maximum
the desired zero phase deviation,
. The width of the plane wave area is
deviation of ,
kept small enough to limit phase variations to less than . A
quality factor for the amplitude is
(10)
is the minimum field amplitude over the desired inwhere
is the maximum field amplitude over the desired
terval and
, then
, and when there are
interval. When
. A value close to one
large amplitude fluctuations, then
is desired for both quality factors.
from the
The AUT is 10 wide and is placed at
transmit array. This distance is about 1/10 the far-field distance
of a nine-element uniform array with element spacing of 0.5
. The field is sampled every
across the AUT and the minimum and maximum values of the amplitude and phase are used
in (8) and (9) to calculate the quality factors. For comparison,
Tables I and II list the quality factors for a single line source
Fig. 5. Field amplitude of an optimized nine-element array (d = 0:5 ) across
an AUT of 10 at a distance of 20 .
and nine-element uniform linear array along the -axis with an
element spacing of 0.5 as a function of frequency and . In
general, the line source has flatter amplitude at the AUT than
the uniform array, while the uniform array has a flatter phase at
the AUT than the line source.
The first example optimizes the weights of a nine-element
linear array along the -axis with an element spacing of
to produce a “plane wave” across an AUT aperture width
of 10 at a distance of 20 . The resulting field across the AUT
and
. In this case,
is better than
has
is better than the single line source.
the uniform array, and
Fig. 3 shows the optimized amplitude and phase weights for the
array, and Figs. 4 and 5 show the amplitude and phase values
of the projected field. The optimized field amplitude is an improvement over the uniform array, but the optimized field phase
is about the same as that of the uniform array. The quality factors for amplitude and phase are graphed over a 20% bandwidth
about the center frequency in Fig. 6. This plot shows that the
optimized weights are reasonably robust over a wide bandwidth.
The quality factors for amplitude and phase are graphed over
a 20% bandwidth about the center frequency in Fig. 11. Again,
is
the optimized field is robust over a 20% bandwidth but
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 2, FEBRUARY 2003
Fig. 6. The amplitude and phase quality factors of the field radiated by the
optimized nine-element array (d = 0:5 ) for an AUT of 10 at a distance of
20.
Fig. 7. Optimized amplitude and phase weights of a nine-element array with
nonuniform spacing in the x-direction for an AUT of 10 at a distance of 20 .
more sensitive to frequency than
for the uniformly spaced
optimized array.
Adding another degree of freedom, element spacing in the
-direction, produces the optimized element weights and spacings shown in Figs. 7 and 8. The projected plane wave across the
and
. Both
AUT at a distance of 20 has
and is better than the uniform array. Figs. 9 and 10 show
the amplitude and phase values of the projected field at the AUT.
Optimizing the element spacing produces a significant improvement to the quality of the “plane wave.” The quality factors for
amplitude and phase are graphed over a 20% bandwidth about
the center frequency in Fig. 11. Again, the optimized field is robust over a 20% bandwidth but is more sensitive to frequency
than for the uniformly spaced optimized array.
Optimizing the weights and element spacings in the -and
-directions produces the element weights and spacings shown
, the projected plane wave has
in Figs. 12 and 13. At
and
. These efficiencies are better than
those of a uniform array. Figs. 14 and 15 show the amplitude
and phase values of the projected field. The optimized field amplitude is an improvement over the uniform array, but the opti-
Fig. 8. Optimized element spacings in the x-direction of a nine-element array
for an AUT of 10 at a distance of 20 .
Fig. 9. Field phase of an optimized (weights and element spacing in
x-direction) nine-element array (d = 0:5 ) across an AUT of 10 at a
distance of 20 .
Fig. 10. Field amplitude of an optimized (weights and element spacing in
x-direction) nine-element array (d = 0:5 ) across an AUT of 10 at a distance
of 20 .
mized field phase is about the same as that of the uniform array.
The quality factors for amplitude and phase are graphed over
a 20% bandwidth about the center frequency in Fig. 16. This
and
are more sensitive to frequency
plot shows that the
HAUPT: GENERATING A PLANE WAVE WITH A LINEAR ARRAY OF LINE SOURCES
Fig. 11. The amplitude and phase quality factors of the field radiated by the
(weights and element spacing in x-direction) optimized nine-element array
(d = 0:5 ) for an AUT of 10 at a distance of 20 .
Fig. 12. Optimized amplitude and phase weights of a nine-element array with
nonuniform spacing in the x and z -directions for an AUT of 10 at a distance
of 20 .
277
Fig. 14. Field phase of an optimized (weights and element spacing in x and
z -directions) nine-element array (d = 0:5 ) across an AUT of 10 at a
distance of 20 .
Fig. 15. Field amplitude of an optimized (weights and element spacing in x
and z -directions) nine-element array (d = 0:5 ) across an AUT of 10 at a
distance of 20 .
Fig. 13. Optimized element spacings in the x and z -directions of a
nine-element array for an AUT of 10 at a distance of 20 .
Fig. 16. The amplitude and phase quality factors of the field radiated by the
(weights and element spacing in x and z -directions) optimized nine-element
array (d = 0:5 ) for an AUT of 10 at a distance of 20 .
than the previous two optimized arrays. Figs. 17 and 18 show
the fields of the optimized array at the AUT for three values of
. The approximate plane wave shows a small degradation as
increases. Moving closer to the array, however, quickly degrades the quality of the plane wave across the AUT.
Arrays can be optimized for various widths of the AUT and
different values of . The quality factors of the optimized arrays
increases and get worse as
decreases. If the
improve as
array has variable amplitude and phase settings, then the “plane
wave” can be adjusted for changes in and frequency.
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 2, FEBRUARY 2003
provement to the projected plane wave close to the source array,
but offers no improvement over a uniform array far from the
source array. Optimizing the element locations as well as the
amplitude and phase tapers provides significant improvement
to the projected plane wave at close and far distances. Tradeoffs exist in the ripples in the phase and the ripples in the amplitude of the projected plane wave. Adjusting the -location of
the elements improves the quality of the projected “plane wave,”
but makes the array configuration more sensitive. Since out of
plane elements would produce some blockage of the elements
in the rear, adjusting the -spacing of the elements is the recommended procedure and the most practical to implement.
REFERENCES
Fig. 17. Field phase at three distances from the array of an optimized (weights
and element spacing in x and z -directions) nine-element array (d = 0:5 )
across an AUT of 10 at a distance of 20 .
Fig. 18. Field amplitude at three distances from the array of an optimized
(weights and element spacing in x and z -directions) nine-element array (d =
0:5 ) across an AUT of 10 at a distance of 20 .
IV. CONCLUSION
Creating an approximate plane wave across a test AUT is best
done by sufficiently separating the source antenna and the antenna under test. If this distance is not reasonable for the testing
situation, then some alternatives exist to approximate a plane
wave with an array. Optimizing the amplitude and phase tapers
of a linear array with equal element spacing offers some im-
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1993.
[2] W. K. Bartley, “Near-Field Antenna Focusing Rep.,” Goddard Space
Flight Center, Greenbelt, MD, X-811-75-183.
[3] A. J. Fenn, C. J. Diederich, and P. R. Stauffer, “An adaptive-focusing
algorithm for a microwave planar phased-array hyperthermia system,”
Lincoln Lab. J., vol. 6, no. 2, pp. 269–288, 1993.
[4] L. J. Ricardi and R. C. Hansen, “Comparison of line and square source
near-fields,” IEEE Trans. Antennas Propagat., vol. AP-11, pp. 711–712,
Nov. 1963.
[5] R. C. Hansen, “Minimum spot size of focused apertures,” in URSI Symp.
EM Theory, Delft, The Netherlands, 1965, pp. 661–667.
, “Focal region characteristics of focused array antennas,” IEEE
[6]
Trans. Antennas Propagat. , vol. 33, pp. 1328–1337, Dec. 1985.
Randy Haupt (F’00) received the B.S. degree in electrical engineering from the
USAF Academy, the M.S. degree in electrical engineering from Northeastern
University, Boston, MA, the M.S. degree in engineering management from
Western New England College, and the Ph.D. degree in electrical engineering
from the University of Michigan.
He is a Professor and Department Head of Electrical and Computer
Engineering at Utah State University, Logan. He was a Professor of Electrical
Engineering at the USAF Academy and Professor and Chair of Electrical
Engineering at the University of Nevada, Reno. He was a project engineer for
the OTH-B radar and a research antenna engineer for Rome Air Development
Center. His research interests include genetic algorithms, antennas, radar,
numerical methods, signal processing, fractals, and chaos. He has numerous
journal articles, conference publications, and book chapters on antennas, radar
cross-section and numerical methods, and is coauthor of the book Practical
Genetic Algorithms (Wiley: New York, 1998). He has eight patents in antenna
technology.
Dr. Haupt is a member of Tau Beta Pi, Eta Kappa Nu, URSI Commission B,
and the Electromagnetics Academy. He was the Federal Engineer of the Year in
1993.