# Wideband Flat Radomes Using Inhomogeneous Planar Layers

Authors: Mohammad Khalaj-Amirhosseini
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

Inhomogeneous planar layers (IPLs) are optimally designed as flat radomes in a desired frequency range. First, the electric permittivity function of the IPL is expanded in a truncated Fourier series. Then, the optimum values of the coefficients of the series are obtained through an optimization approach. The performance of the proposed structure is verified using some examples.


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## Body

## 1. Introduction

Radomes are  sheltering structures to protect antennas
against severe weather such as high winds, rain, icing, and/or temperature
extremes [1]. In addition, radomes must not interfere with normal operation of
the antennas. Therefore, the input reflection of radomes must be negligible at
the usable frequency band of the protected antennas. Inhomogeneous planar layers
(IPLs) are widely used in microwave and antenna engineering [2–4]. In this
paper, we propose utilizing IPLs as flat radomes [5, 6] in a desired
frequency range. To optimally design IPLs, the electric permittivity function
of them is expanded in a truncated Fourier series, first. Then, the optimum
values of the coefficients of the series are obtained through an optimization
approach. The identical procedure has been used to optimally design IPLs as
impedance matchers between two different mediums previously [7]. Finally, the
usefulness of the proposed structure is verified using some examples.


## 2. Analysis of IPLs

In this section, the frequency domain equations of the IPLs
are reviewed. Figure1 shows a typical IPL with thickness d, whose left
and right mediums are the free space and whose electric permittivity function
is εr(z).
One way to fabricate the IPLs is to place several thin homogeneous dielectric
layers beside each other. It is assumed that the incidence plane wave
propagates obliquely toward positive x and z direction with an
angle of incidence 					θi
and electric filed
strength 		Ei. Also, two different polarizations are possible,
one is TM and the other is TE. Of course, we know that the wave radiated by an
antenna can be decomposed to many plane waves with different angle of
incidence.A typical IPL as a
flat radome (a) TE
polarization mode,  (b) TM polarization
mode.
(a)(b)The differential equations describing IPLs have nonconstant
coefficients and so, except for a few special cases, no analytical solution
exists for them. There are some methods to analyze the IPLs such as finite difference
[8], Taylor
’s series
expansion [9], Fourier series expansion [10], the equivalent sources method [11],
and the method of moments [12]. Of course, the most straightforward method to
analyze IPLs is subdividing them into K thin homogeneous layers with
thickness (1)Δz=dK≪λmin≅cfmaxmax(εr(z)) in which c is the  velocity of the light and fmax is the maximum analysis frequency. The ABCD parameters [13] of the IPL
are obtained from those of the layers as
follows: (2)[ABCD]=[A1B1C1D1]⋯[AkBkCkDk]⋯[AKBKCKDK],where the ABCD parameters of the kth layer are
as follows: (3)Ak=Dk=cos(Δθk),Bk=Zc2((k−0.5)Δz,θi)Ck=jZc((k−0.5)Δz,θi)sin(Δθk).   In (3), (4)Δθk=2πfcεr((k−0.5)Δz)−sin2(θi)Δzis  the electrical length of the
kth layer and 					Zc(z,θi)
is the characteristic impedance of the IPL, defined as the ratio of the
transverse electric field to the transverse magnetic field, given by
(5)Zc(z,θi)={μ0ε01εr(z)−sin2(θi),TE,μ0ε01εr(z)εr(z)−sin2(θi),TM.Finally, the input impedance and
reflection coefficient of the radome are determined as follows: (6)Zin(f,θi)=AZL(θi)+BCZL(θi)+D,Γin(f,θi)=Zin(f,θi)−ZS(θi)Zin(f,θi)+ZS(θi),
where (7)ZL(θi)=ZS(θi)={μ0ε01cos(θi),TE,μ0ε0cos(θi),TM
are the equivalent load and source impedances, respectively.


## 3. Synthesis of Radomes

In this section, a general method is proposed to optimally
design the IPLs as radomes. First, we consider the following truncated Fourier
series expansion  for the electric permittivity function(8)ln(εr(z)−1)=∑n=0NCncos(πnzd).
The reason to use logarithm function at the left of (8) is
to keep εr(z)≥1. An optimum designed radome has to have the input reflection
coefficient as small as possible in a desired frequency and incidence angle
range. Therefore, the optimum values of the coefficients
Cn in (8) can be obtained through minimizing the following error function
corresponding to M frequencies, J incidence angles, and two possible
polarizations TE and TM: (9)Error=12MJ∑POL=TETM∑m=1M∑j=1J|Γin(fm,θi,j)|2.
The defined error function should be restricted by some
constraints such as not having a significant reflection at all incidence angles
[0−θi,max], and easy
fabrication, respectively, as follows: (10)|Γin(fm,θi,j)|≤ρmax,∀m=1,2,…,M,∀j=1,2,…,J,POL.=TEandTM,(11)εr(z)≤(εr)max,(12)εr¯=1d∫0dεr(z)dz≥(εr)min,
where (εr)max is the
maximum value of εr(z), in the
fabrication step. It is noticeable that the constraint (12) is necessary to
avoid obtaining the wrong solution εr(z)=1 (the free apace) in the optimization process. Also, to enforce the designed
radomes to be symmetric, we have to use the following truncated Fourier series
instead of (8) for the electric permittivity function (13)ln(εr(z)−1)=∑n=0NCn′cos(2πnzd).
To solve the above constrained minimization problem, we can
use the fmincon. m file in the MATLAB program. fmincon uses a sequential
quadratic programming (SQP) method, in  which a quadratic programming (QP)
subproblem is solved at each of its iteration.


## 4. Examples and Results

We would like to design an IPL with thicknessd=2.5 cm (a practical and feasible chosen) as a radome in a frequency range DC to 8.0 GHz, considering θi,max=60°,ρmax=0.1=−20 dB, and (εr)max=10. Using the proposed optimization approach, considering N=10
spatial  harmonics, M=80 frequencies, J=2 incidence angles (0° and 60°), and (εr)min=1.05
or 1.10, two radomes were synthesized. The unknown coefficients of the
truncated Fourier series related to the synthesized radomes are written in
Table 1. Figures 2 and
3 illustrate the obtained electric permittivity function εr(z),
respectively, considering (εr)min=1.05 or 1.10.
Figures 4, 5,
6, and 7 illustrate the
magnitude of the input reflection coefficient |Γin(f)|
for TE and TM polarizations.
It is observed that the designed radomes have a good performance in both
desired frequency band and the incidence angle range. Meanwhile, the reflection
coefficient degrades with increasing the angle of incidence. To show the effect
of the thickness of IPLs, we increase d from 2.5 cm to 5, 7.5, and 10 cm. The unknown coefficients of the truncated Fourier series, the electric
permittivity function εr(z), and the
magnitude of the input reflection coefficient |Γin(f)| corresponding to θi=60° and considering (εr)min=1.05 are shown in  Table 2
and Figures 8, 9, and
10,  respectively.
It is observed that as the thickness of the IPL is chosen larger, the obtained electric permittivity function tends to a continuous function, whose property is matching between the air mediums and an intermediate medium. Also, it is seen that  the efficiency for
TM polarization is better than that for the TE polarization.Table 1
The unknown coefficients of the truncated Fourier
series ford=2.5 cm.
C0C1C2C3C4C5C6C7C8C9C10(εr)min=1.05−29.04624.689−5.8701−20.617−20.55129.9998.613711.6949−6.12177.65550.2282(εr)min=1.10−30.000−20.412−11.794−10.023−0.05213.02478.947312.820510.425215.449223.7096Table 2
The unknown coefficients of the truncated Fourier
series for(εr)min=1.05.
C0C1C2C3C4C5C6C7C8C9C10d=2.5 cm−29.04624.689−5.8701−20.617−20.55129.9998.613711.6949−6.12177.65550.2282d=5.0 cm−30.000−3.4358−19.617−1.3242−6.88440.920114.47351.935317.26402.184825.0426d=7.5 cm−8.5288−1.3020−1.3469−0.4938−0.92100.14051.29220.58391.26051.23457.2452d=10 cm−3.0284−0.0009−0.31820.0019−0.00760.00260.15340.00230.09760.00210.0659Figure 2
The electric permittivity functionεr(z),
considering (εr)min=1.05.Figure 3
The electric permittivity functionεr(z),
considering (εr)min=1.10.Figure 4
The magnitude of|Γin(f)|
for TE polarization, considering (εr)min=1.05.Figure 5
The magnitude of|Γin(f)| for TM
polarization, considering (εr)min=1.05.Figure 6
The magnitude of|Γin(f)|
for TE polarization, considering (εr)min=1.10.Figure 7
The magnitude of|Γin(f)|
for TM polarization, considering (εr)min=1.10.The electric
permittivity functionεr(z), considering (εr)min=1.05
(a) d=2.5 cm,  (b) d=5.0 cm,  (c) d=7.5 cm,  (d) d=10 cm.
(a)(b)(c)(d)Figure 9
The magnitude of|Γin(f)|
for TE polarization with θi=60°,
considering (εr)min=1.05.Figure 10
The magnitude of|Γin(f)|
for TM polarization with θi=60°,
considering (εr)min=1.05.


## 5. Conclusion

Inhomogeneous planar layers (IPLs) were optimally designed
as flat radomes in a desired frequency. First, the electric permittivity
function of the IPL is expanded in a truncated Fourier series. Then, the
optimum values of the coefficients of the series are obtained through an
optimization approach. The performance of the proposed structure is verified
using some examples. It was observed that the designed radomes have a good
performance in both desired frequency band and the incidence angle range, where
the efficiency for  TM polarization is better than that for the TE polarization.
Also, as the thickness of the IPL is chosen
larger, the obtained electric permittivity function tends to a
continuous function, whose property is matching between the air mediums and an intermediate medium. The proposed method can be extended for
IPLs, whose magnetic permeability is inhomogeneous solely or along with their
electric permittivity. Also, we can consider IPLs for spherical wavefronts
instead of planar ones in the future.



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*Source: 636047-2008-10-26.xml*