1

Impact of Coated Ground Plane Edge Diffractions
on Amplitude Patterns of Circular Apertures
Nafati A. Aboserwal1 , Constantine A. Balanis1 , Life Fellow, IEEE, and Craig R. Birtcher1

Abstract—The impact of finite dielectric-covered ground plane
edge diffractions on the amplitude patterns of circular apertures
is investigated. The model is based on the Geometrical Optics
(GO) and the Uniform Theory of Diffraction (UTD) for an
impedance wedge. The circular aperture antenna is mounted on
square and circular finite ground planes that are coated with a
thin lossy dielectric layer. The predictions based on the GO/UTD
model are validated by comparisons with experimental results
and simulated data.
Index Terms—Coated aperture antenna, geometrical optics,
diffraction, impedance wedge, surface waves.

I. I NTRODUCTION
The GTD/UTD [1], [2] has been widely and successfully
employed to solve a wide variety of perfectly conducting
electromagnetic problems involving diffraction by edges on
perfectly conducting surfaces. Its extension to non-perfectly
conducting surfaces may provide a significant improvement to
the application of ray methods. The non PEC wedge uniform
GTD formulation was rigorously derived by asymptotically
evaluating the exact solution given by Maliuzhinets [3] using
the method of Sommerfeld. It provides a uniform highfrequency solution to the diffraction by a wedge with uniform
but different impedances on each face. The asymptotic solution
represents the incident, reflected, diffracted, and surfaces wave
fields. The surface wave exists over a finite angular range and
only for certain surface impedances [4]. The impedance wedge
UTD formulation is achieved by introducing suitable multiplying factors. Although these factors involve a special function
which is difficult to calculate for a general exterior (nπ) wedge
angle, numerically tractable expressions are available when n
is a rational number. In particular, this function can be easily
calculated for the four special cases n = 1/2, 1, 3/2, 2 [3].
Over many years considerable effort has been devoted
to examine aperture antennas mounted in PEC finite/infinite
ground planes [5]. However the impact of edge diffractions
from a coated finite ground plane on the radiation performance
of an aperture antenna has not received adequate attention.
This paper addresses this problem using the impedance wedge
solution to account for the edge diffractions. For this problem,
initially it is assumed that the coated aperture is mounted on
an infinite coated ground plane. Then the amplitude patterns of
the finite coated ground plane can be modified considerably
by the diffractions from the edges, especially in regions of
very low field intensity, such as the backward radiation.
1 School of Electrical, Computer and Energy Engineering, Arizona State
University, Tempe, AZ 85287-5706. naboserw@asu.edu; balanis@asu.edu;
craig.birtcher@asu.edu.

II. D IELECTRIC - COVERED A PERTURE A NTENNAS
For space applications, aperture antennas are covered by a
dielectric material to protect them from environmental conditions. The presence of the dielectric cover, however, impacts
the radiation characteristics [6], [7]. The spectral domain
method was applied to aperture antennas mounted on infinite
ground planes [8], and on infinite dielectric-covered ground
planes [9]. This method, previously developed to determine
the fields radiated from an aperture in a coated cylinder [10],
is extended to obtain the solution for the fields radiated by an
aperture in a coated ground plane. In particular, the radiation
structure considered is an infinite perfectly conducting plane
with a circular aperture excited by a specified tangential
electric field distribution. The entire plane is generally covered
by a thin dielectric layer of complex permittivity, complex
permeability, and thickness t, as depicted in Fig. 1.
The fields, produced by the field distribution over the
aperture, are found by solving a two-region boundary-value
problem. The first region is 0 ≤ z ≤ t and the second
region is t ≤ z ≤ ∞. The fields inside the waveguide
are uniquely defined by a discrete TE vector potential. The
fields inside the dielectric coating are represented by standing
waves and are described by integral transforms of TE and TM
potentials [9]. The boundary conditions at the aperture, and at
the outer surface of the dielectric coating, are used to solve
for the unknowns. For the same aperture excitation, the far
fields radiated by a coated aperture are related to those of an
uncoated by simple multiplicative functions which depend on
the parameters of the dielectric coating and the off-axis angle
(θ).
Eθ (r, θ, ϕ) = f (θ) Eθ0 (r, θ, ϕ)
(1)
Eϕ (r, θ, ϕ) = g(θ) Eϕ0 (r, θ, ϕ)

(2)

where Eθ and Eϕ are the electric field components of the covered aperture, and Eθ0 and Eϕ0 are the electric field components
of the uncovered aperture. For the definitions of f (θ) and g(θ)
functions, refer to [8] and [9].
When the permittivity of the dielectric layer equals the
permittivity of free space, or when the thickness of the
dielectric coating t is zero, (1) and (2) reduce to those for
the uncoated case.
The antenna under consideration consists of a circular
waveguide, opening onto a finite, perfectly electric conducting flat ground plane coated with a thin dielectric layer of
thickness t, and complex permittivity and permeability, as
shown in Fig. 1. The circular waveguide dominant TE11 mode
is assumed over the circular aperture, and the higher order
reflected modes are assumed to be negligible.

2

and circular ground planes, where the side of the square is
equal to the diameter of the circular, is carried out.
The incident field at points Qd1 and Qd2 , as shown in Fig.
2, is found from (1)-(2) after substituting θ = π/2 and r = w,
where w is the half width of the square ground plane or the
radius of the circular ground plane, and ρc1 and ρc2 are the
distances between the caustic at the diffraction points (Qd1
and Qd2 ) and second caustic of the diffracted ray [4].
Fig. 1. Two-dimensional geometry of a circular aperture antenna mounted
on a coated perfectly conducting plane.

III. I MPEDANCE S URFACE B OUNDARY C ONDITIONS
The increasing application of lossy/lossless dielectric materials increases the need for suitable methods to characterize the
response of these materials, including the scattering effects of
edge/wedges and discontinuities. The approximate impedance
boundary conditions are an effective approach to model the
surface impedance. This model is good for conductor and
lossy materials where there is some penetration of the field.
The impedance surface boundary conditions, introduced by
Leontovich in 1940’s [11], are widely used to simulate the
material properties of a scatterer; they can be very helpful in
simplifying the analytical and numerical solutions of scattering
problems. The surface impedance concept is a popular and
efficient tool in computational electromagnetics. It has been
widely used to analyze electromagnetic problems for which
the material properties or surface characteristics are important.
The impedance boundary condition is a useful approximation
for many physical problems because it includes the material properties without explicitly involving the fields within
the material. It represents approximate relations between the
electromagnetic fields and the quantities at the surface of the
conductor, so that the conducting region does not need to
be included in the mesh and can be represented by surface
impedance boundary conditions (SIBCs) [12], [13].
The normalized equivalent surface impedance may be obtained from a simple transmission line model [14]. By considering the coating as a section of a transmission √
line of length
t and normalized characteristic impedance η = µ1 ε0 /µ0 ε1 ,
terminated with the zero impedance of the PEC, the input
impedance, corresponding to the normalized equivalent surface
impedance of a obliquely incident plane wave, is
√
µ0 ε0
ηeq = jη tan(k1 t sin ϕt ) sin ϕt = 1 −
cos2 ϕi (3)
µ1 ε1
√
where k1 = ω µ1 ε1 is the phase constant.
IV. D IFFRACTED FIELDS
The two-dimensional geometry of aperture antennas
mounted on square/circular coated ground planes is shown in
Fig. 1. Both the square and circular ground planes are treated
similarly. Far-zone E- and H-plane amplitude patterns are analytically calculated for ground planes covered with a dielectric
layer following the procedure described in [4]. To examine the
impact of the dielectric-covered ground plane geometry on the
E- and H-plane amplitude patterns, a comparison of the square

Fig. 2. Diffraction mechanism by edges of dielectric-covered ground planes.

For the far-field observations, the field diffracted from the
diffracting points Qd1 and Qd2 , for either the square or the
circular ground plane, after suppressing the spatial variation
factor (e−jkr /r), can be written as
√
(
π )
Eθd1 = Eθi (Qd1 )Dh L1 , ψ1 , 0, , 2
ρc1 e+jkw sin θ (4)
2
√
(
π )
Eθd2 = Eθi (Qd2 )Dh L2 , ψ2 , 0, , 2
ρc2 e−jkw sin θ (5)
2
for the far-zone E plane, and
√
(
π )
Eϕd1 = Eϕi (Qd1 )Ds L1 , ψ1 , 0, , 2
ρc1 e+jkw sin θ (6)
2
√
(
π )
Eϕd2 = Eϕi (Qd2 )Ds L2 , ψ2 , 0, , 2
ρc2 e−jkw sin θ (7)
2
for the far-zone H plane; for the definitions of L1 , ψ1 , ρc1 ,
L2 , ψ2 , and ρc2 , refer to [15].
Ds and Dh are, respectively, the soft and hard polarization
impedance wedge diffraction coefficients [16] which depend
implicitly on the surface impedances modelling of the wedge
faces [4]. Because these coefficients were derived assuming
that the antenna was mounted on the lower face of the wedge,
all angles in this paper must be referenced to the underside
face of the wedge. Because the incident field for the H-plane
vanishes at the grazing angle, the rate of change of the incident
field at the point of diffraction is used [4].
For the circular ground plane, the circular edge behaves as
a continuous ring radiator [4], which leads to the formation
of a caustic where the diffracted field are infinite. Therefore,
a caustic correction is needed for angles at and near the axis
of the antenna. Using the method of equivalent currents and
wedge diffraction coefficients, the caustic problem can be
corrected [4]. In this method, equivalent magnetic and electric
currents are created along the edge of the ground plane. Then
radiation integrals are used to obtain the fields due to these
currents, which correct the diffracted fields at and near the
symmetry axis of the antenna.

3

For the square ground plane, the edge diffractions do not
contribute significantly to the H-plane radiation pattern in the
backlobe region. However, the edge diffractions in the E-plane
are much more intense and contribute more significantly to the
overall pattern, above and below the ground plane. Therefore,
one needs to include the contributions from the E-plane edge
diffractions using the method of equivalent currents [4].
The surface wave components, and their corresponding surface wave transition fields, are necessary to provide continuity
of the fields, even in a smaller angular region where they exist;
their formulas are given in Chapter 14 of [4].
V. PATTERN VALIDATION
A model for the circular and square dielectric-covered
ground planes with the circular waveguide mounted at the
center has been constructed. The aperture antenna is assumed
to be exicted by the TE11 -mode circular waveguide. The
width of the coated square ground plane and the diameter of
the coated circular ground plane are 30.48 cm. The relative
permittivity (εr ) and the loss tangent (tan δ) of the coating
material are, respectively, 2.9 and 0.02 at 10 GHz. The
thickness of the dielectric layer, made from polycarbonate
(Lexan), is 0.25 cm and the normalized surface impedance
is 0.7480 + j0.4894 calculated using (3). The validity of the
radiation pattern formulation over the main beam and the near
and far sidelobes has been verified by calculating the farzone E- and H-plane amplitude patterns of the aperture. The
frequency at which the measurements were performed is 10
GHz. The diameter of the aperture is 2.38 cm. Measurements
and numerical data based on diffraction techniques were also
compared with the HFSS simulations.
Figs. 3-4 exhibit the far-zone E- and H-plane amplitude
patterns of a circular aperture mounted, respectively, on circular and square dielectric-covered ground planes. Because the
amplitude patterns of interest are in the principle planes, the
most significant diffractions come from two diffraction points
which are diametrically opposite to each other and along the
principle planes [4]. Good agreement between GO/UTD calculations, experiments, and HFSS simulations is indicated. The
amplitude patterns in the E-plane are more broad than those in
the H-plane as shown in Figs. 3-4. The ripples in the amplitude
patterns, especially in the backlobe region, are attributed to the
impedance edge diffractions. The ripples shown in the patterns
are due of the constructive and destructive interference of the
diffractions from the diametrically opposite diffraction points.
In the E-plane, these ripples are more significant because the
incident electric field at the point of diffraction is more intense
in this plane than in the H-plane.
In addition, the amplitude patterns of the same antennas
without coating in the E and H planes [15] are compared to
those obtained for the coated case. It can be seen that the
amplitude patterns of the coated ground planes have stronger
radiation especially in the back and far sidelobes because of
the existence of surface waves. The ripples in the forward
lobe of the coated-antenna patterns are more severe because
the edge excitations are very strong to have a significant effect
on the overall pattern.

Fig. 3. Far-zone E- and H-plane amplitude radiation patterns of a circular
aperture antenna mounted on a circular coated ground plane at 10 GHz.

Fig. 4. Far-zone E- and H-plane amplitude radiation patterns of a circular
aperture antenna mounted on a square coated ground plane at 10 GHz.

4

Although the side of the square is equal to the diameter
of the circular, it is clear from the patterns of Figs. 3,4 that
the E- and H-plane amplitude patterns of the circular ground
plane are greater than those of the square ground plane at and
near the antenna axis (θ = 180◦ ). These are due to the ring
radiator [4] which contributes about an additional 8-10 dB.

coated ground planes. It is apparent that the back lobe radiation
at θ = 180◦ is stronger for thinner coatings. As shown in Fig.
5(c), by increasing the relative permittivity of the coating, the
amplitude level of the diffracted field at θ = 180◦ decreases
for both geometries. Based on the simulated data, increasing
either the thickness or relative permittivity of the dielectric
coating leads to less radiation in the back lobe region.
VI. C ONCLUSIONS
The impact of the finite dielectric-covered ground plane
edge diffractions on the amplitude patterns of a circular aperture is investigated. The diffraction of electromagnetic waves
for impedance wedges (half plane with two face impedances
for our case) is investigated. The UTD diffracted fields and the
GO fields for a dielectric-covered PEC plane are vectorially
combined to determine the far-zone amplitude patterns in the
E- and H-planes. The study indicates that the finite coated
ground plane edge diffractions do impact the main forward
lobe pattern, especially for the E-plane. Their most significant
contribution appears in far side and back lobes. All these
results are indicators that the finite edge contributors must
be considered to obtain more accurate amplitude radiation
patterns. Moreover, the impact of the ground plane size,
the coating thickness, and the relative permittivity has been
investigated in this work.
R EFERENCES

Fig. 5. Amplitude pattern level at θ = 180◦ due to: (a) size of ground plane,
(b) coating thickness, and (c) relative permittivity.

The radiation amplitude in the back region of the aperture
antenna mounted on coated ground planes depends significantly on the size and geometry of the ground plane. Fig.
5(a) indicates, as expected, that the amplitude pattern level at
θ = 180◦ decreases monotonically with increasing the ground
plane size for both geometries, circular and square. Because
of the ring radiator of the circular ground plane, its amplitude
in the back lobe is more than that for the square ground plane.
As the size increases, the edge diffractions do not introduce
any noticeable ripples in the forward region, and the diffraction
level in the back region becomes very comparable to the noise
level in the detection system. The periodicity of the ripples in
the patterns depends upon the size of the ground plane, while
the amplitude of the ripples depends upon the strength of the
edge excitation.
The amplitude pattern level at θ = 180◦ for different coating
thickness is also displayed in Fig. 5(b) for circular and square

[1] J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Amer.,
Vol. 52, pp. 116-130, February 1962.
[2] R. G. Kouyoumjian and P. H. Pathak, “A Uniform Geometrical Theory of
Diffraction for an Edge in a Perfectly Conducting Surface,” Proceedings
of the IEEE, Vol. 62, No. 11, pp. 1448–1461, November 1974.
[3] G. D. Maliuzhinets, “Excitation, Reflection and Emission of Surface
Waves from a Wedge with Given Faces Impedances,” Sov. Phys. Doke.,
Vol. 3, pp. 752-755, March 1958.
[4] C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. : John
Wiley and Sons, NJ, 2012.
[5] C. A. Balanis, “Pattern Distortion Due to Edge Diffraction,” IEEE Trans.
Antennas Propagat., Vol. 18, No. 4, pp. 561-563, July 1970.
[6] A. K. Bhattacharyya and D. L. Sengupta, Radar Cross Section Analysis
and Control, 3rd ed. : Artech House, MA, 1991.
[7] M. D. Deshpande and P. D. Prabhakar, “Analysis of Dielectric Covered Infinite Array of Rectangular Microstrip Antennas,” IEEE Trans.
Antennas Propagat., Vol. 35, No. 6, pp. 732-736, June 1987.
[8] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. : John
Wiley and Sons, NJ, 2005.
[9] C. M. Knop and G. I. Cohn, “Radiation from an Aperture in a Coated
Plane,” Radio Sci. J. Res., Vol. 68D, No. 4, pp. 363-378, April 1964.
[10] J. Wait and W. Mientka, “Calculated Patterns of Slotted Elliptic-Cylinder
Antennas,” Appl Sci. Res., Vol. 7, No. 1, pp. 449-462, December 1959.
[11] M. A. Leontovich, “Approximate Boundary Conditions for the Electromagnetic Field on the Surface of a Good Conductor, Investigation of
Propagation of Radio Waves, Part III,” Academy of Sciences, pp. 5-12,
Moscow, 1948.
[12] T. B. A. Senior and J. L. Volakis, “Approximate Boundary Conditions
in Electromagnetics,” IEEE Press, U.K., 1995.
[13] T. Griesser, “High-frequency Electromagnetic Scattering from Imperfectly Conducting Structures,” Ph.D. dissertation, Arizona State University, Tempe, August 1988.
[14] R. Collin, Foundations for Microwave Engineering, 2nd ed. : McGrawHill, NY, 1992.
[15] N. Aboserwal, C. Balanis, and C. Birtcher, “Impact of Finite Ground
Plane Edge Diffractions on Radiation Patterns of Aperture Antennas,”
Progress In Electromagnetics Research B, Vol. 55, pp. 1-21, 2013.
[16] R. Tiberio, G. Pelosi, and G. Manara, “A Uniform GTD Formulation
for the Diffraction by a Wedge with Impedance Faces,” IEEE Trans.
Antennas Propagat., Vol. 33, No. 8, pp. 867-873, August 1985.