Content uploaded by Mousa I Hussein
Author content
All content in this area was uploaded by Mousa I Hussein
Content may be subject to copyright.
UNCLASSIFIED
Defense Technical
Information
Center
Compilation
Part
Notice
ADP015062
TITLE:
Electromagnetic
Scattering by
a
System
of
Dielectric
Spheres
Coated
With
a
Dielectric
Shell
DISTRIBUTION:
Approved
for
public release,
distribution
unlimited
This
paper
is
part
of
the
following report:
TITLE:
Applied Computational Electromagnetics
Society
Journal.
Volume
18,
Number
4,
November
2003.
Special
Issue
on
ACES
2003
Conference.
Part
1
To
order
the
complete
compilation
report,
use:
ADA423296
The
component part
is
provided
here
to
allow
users access
to
individually authored
sections
f
proceedings,
annals,
symposia,
etc.
However,
the
component
should
be considered
within
[he
context
of
the overall
compilation
report
and
not
as
a
standalone technical
report.
The
following component part numbers
comprise the
compilation
report:
ADP015050
thru
ADP015064
UNCLASSIFIED
ACES
JOURNAL, VOL.
18,
NO. 4,
NOVEMBER
2003
91
Electromagnetic
Scattering
by
a
System
of
Dielectric
Spheres Coated
With
a
Dielectric
Shell
AK.
Hamid
M.
I.
Hussein
M.
Hamid
Department
of
Electrical and
Department
of
Electrical Engineering
Department
of
Electrical
Engineering
Computer
Engineering
United
Arab
Emirates
University University
of
South
Alabama
University
of
Sharjah
P.O.
Box
17555,
AlAin,
United
Arab
Mobile,
AL 36688, U.S.A
P.O.
Box
27272,
Sharj
ah,
U.A.E
Emirates
email: mhamid(@.usouthal.edu
email:
akhamidt@Shariah.ac.ae
email:
MIHussein(uaeu.ac.ae
Abstract
with
a
dielectric layer.
In
this
paper,
we
extend
the
solution
of
scattering by
two
dielectric
spheres
covered
with
a
Analytical
solution
is
derived
to the
problem
of
scattering
of
dielectric
shell
[9]
to the
case
of
scattering by
a
system
of
electromagnetic plane wave
by
an
array
of
dielectric
spheres dielectric spheres each
covered with
a
dielectric
shell.
The
each
coated
with
a
dielectric
shell.
The incident,
scattered
solution
to
this
problem
has
many
practical
applications
and
transmitted
electric
and
magnetic fields
are
expressed
in
since,
for
example,
it
may be
used
to
study
the scattering
by
terms
of
the
vector
spherical
wave
functions.
The
vector complex
objects simulated
by
a
collection
of
spheres
[12],
spherical
translation addition theorem
is
applied
to
impose
and
it
may
also
be
used
to
check
the
accuracy
of
numerical
the
boundary conditions
on
the
surface
of
various
layers.
solutions.
Numerical results
are
computed
and
presented
graphically
From
the
design
point
of
view, the
backscattering
cross
for the
radar cross
sections
of
several
configurations
of
section
of
an
array
of
N
dielectric
coated
spheres
can
be
spheres
system with multi dielectric
layers.
controlled
to
exploit
multiple
resonances
by
optimizing
the
multivariables
of
the
system.
These
include
the size
and
1.
Introduction
location
of
each
sphere,
number
of
dielectric
layers
coating
each
sphere
as
well
as
the
thickness
and
relative
dielectric
Many authors have
studied
the
scattering
of
electromagnetic constant
of
each
layer
as
already
done
for conducting
plane
wave
by
a
dielectric sphere
coated
with
a
dielectric
cylinders
[13].
shell.
Aden
and
Kerker
[1]
obtained
analytical
expressions
to
the
scattering
of
electromagnetic plane
wave by
a
2.
Formulation
of the
Problem
dielectric sphere coated
with
a
concentric
spherical
shell
of
different dielectric materials,
while
Scharfman
[2]
presented
Consider
a
linear
array
of
N
dielectric
spheres
each
coated
numerical
results
for
the
special
case
of
a
small
electrical
with
a
dielectric
shell
and
having different radii
and
unequal
radius
(ka<l)
dielectric
coated conducting
sphere.
It
was
spacing
with
centers lying
along
the
z
axis,
as
shown
in
Fig.
found
in
those
early studies
that
the
presence
of
dielectric
1.
Electromagnetic
plane
wave
of
unit
electric
field
coatings
leads
to
substantial
increase
in
the
backscattering
intensity,
whose
propagation
vector k
lies
in
the
xz
plane
cross
section
for
an
appropriate
choice
of
the
dielectric and makes
an
angle
a
with
the
zaxis,
is
assumed
to be
constant
and
thickness
of
the
coating
relative
to
that
of
incident
on
the
spheres.
Its
incident
electric and magnetic
uncoated
sphere.
Further,
Wait
has
extended
the
solution
to
fields are
the
case
of
scattering
by
a
radially
inhomogeneous
sphere
k 
[3],
while
a
numerical
solution
using
the
method
of
Ei =ek
y
(1)
moments
obtained
by
MedgyesiMitschang
and
Putnam
for
the
case
of
dielectriccoated
concentric
sphere
[4].
More
H
=
eJkr(cos
a
isinoti)
(2)
recently,
an
exact solution
of
electromagnetic plane
wave
?7
scattering
by
an
eccentric
multilayered
sphere
was
with
k
being
the
wave
number,
i,5,
and
2
are the
unit
developed
by
Lim and
Lee
[5].
Numerous
papers
on
the
vectors along
the
x, y
and
z
axes,
respectively,
and q/
is
the
scattering
from
systems
of
spheres
of
various natures
in
surrounding medium
intrinsic
impedance.
The
incident
close
proximity
have been
treated
by
numerous
researchers
electric and
magnetic
fields
may be
expanded
in
terms
of
[611].
elect
or ae fios
a
b
e
en ter
of
Up
to
now,
there
has
been
no
analytical
or
numerical
spherical
vector
wave
functions around
the
center
ofthe
pth
solution
to
the
problem
of
scattering
of
electromagnetic
sphere
as
plane
wave
by
an
array
of
conducting
spheres
each coated
10544887
©
2003
ACES
92
ACES
JOURNAL,
VOL.
18,
NO.
4,
NOVEMBER
2003
m
= n
m =(
(r,, I
[P,(m,n)N,(r,,8,p,)
(3)
77H
(r,p,,p,)j
E
[AE,,(m,n)MWm(rP,p,bp)
(10)
fl
n=1
m=n
n=I
m=
+Q,(~n)H,',)(r,0P,0)] n n
+Amp
(m,
n)N
(,)
(rp,gOp,
op)
]
where
AEp
(m,n),
AMp(m,n)
are
the unknown scattered
field
77Hi(r,,,O,,p)=
ji [P(mn
'n)M.,)(rp'Op'q5p) (4)
coefficients. To
express
the
scattered
fields
from
the
qth
"
='n
+
Q,
(m,n)NmI
(rp,O,,
p)
sphere
in
the
coordinate
system
of
the
pth
sphere,
we
apply
the
spherical
vector translation addition
theorem
for
where
M
and
are
the
spherical vector
wave
translation
along
the
zaxis
[15], i.e.,
functions
of
the
first
kind
representing
incoming
waves
W(3)(r,,0 q>=
[A'
(d )M(
1
)(r
0
(1
associated
with the
spherical
Bessel
function, while
V=r
mn
) m
p
' kp
Pp(m,n)
and
Qp(m,n)
are the
incident
field
expansion
+
my(d
coefficients defined
in
[78,14].
The
field
in
the
region
II
r 
can be
also
expressed
in
terms
of
the
vector
spherical
wave
N(q',q,(kq)=
[A,,(dpq)N(v)(r'Opqp)
(12)
functions
of
the first and
third kinds.
Hence
the
electric
and
+B
I
(12)
magnetic fields
may
be
written
as
+ v
(dpq"
Man
p(rOp
1 p0
where
Am,
(dpq)and
Bmv
(dpq)
are the
translation
... .
[AE(mn),
(r
0
0) (5)
coefficients
of
the
spherical
vector
translation
addition
+A
Ep(mnnl)N(rp
,9P,/)
(
theorem. To
determine the unknown scattered
field
+A'mp(mn,n)jW.((r
0pO,)
coefficients,
we
apply
the
boundary
conditions
on
the
+
A"Mp
(m,n)M(m)(rp,Op,
p)] various
interfaces,
i.e.,
[A
~
~ ~ ~
Z
Ep ?)m
FpO)F
E(bp,
Op,
0p)
+ A'
ES(bp,Op,0p)
=
p.×
El(bb,'9p,0p
) (3
 +
A
Ep(m'?1)MW')
(rp'P
P))
(6) I Ns
+A'Mp(m,n)lM4(r
n( ,,p,,
()
FpX[Hj(bp,Op,)
p)+
p
H
o(bp,P)
7
II(bp,pqpD)
(14)
ZL P=P
+p
x
Eii(ap,Op,
Op)
=
7p
x
Ei(ap,Op,
p)
(15)
where
ApE
(r,
n),
A
pM
(m,n),
A
pE
(m,n),
and
p
xHi(ap,
Op,
Op)=
xH,(apIOpIp)
(16)
A
pM
(n,n)
are
the field
expansion coefficients,
while
Substituting
the
appropriate
field
expansion expressions in
(3)
j(3)
equations
(13)
to
(16),
and
applying
the
orthogonality
n(3)
and
)Vm
are
the
vector
spherical
wave
functions
of
m
n
n properties
of
spherical
vector
wave
functions
and
the
third kind
representing outgoing
waves
associated
with
eliminating
the
transmission
coefficients
we
obtain
the
spherical Hankel
function. The subscripts
E
and
M
N
denote
transverse
magnetic (TM)
and
transverse
electric
AEp(m,n)=v,(pp)Pp(in,n)+_.
[A.
(dpq)A,,,(m,v)
(17)
waves
(TE),
respectively.
The
field
in
region I
of
the
pth q p )Av( )
sphere
may be
written
in
terms
of
the
vector
wave
functions
+B..(dpq)Ap(m,v)]
of
the
first kind,
i.e.,
N
of
te
fist
knd,
~e.,A~p(m,n)=u,(pp)Qp(m,n)+Yj
[A.,(dpq)A,,P(m
,' v)
(18)
q= =1
ErP0)ZZ
[Ap(nz,n)Nm.1
( p, , qkp
,) (7) q*P
,
....
0)+
B
";
(dpq)AEp
(m,
v)]
Amn(,pp
,where
vn ( p
p)
and
u,,
(pp)are
the
electric
and magnetic
H, I
[A,,(nln)A)(rP,0P,,
) (8, ) scattered
field
coefficients
for
a
single
dielectric
sphere coated
n=
m=l
[M.
(8)
with
a
dielectric
layer
[1,9].
Equations
(17)
and
(18)
may
be
+AMp
(m,n)N.,n)(r,,O,
,)]
written
in
matrix
form
for the
purpose
of
computing
the
scattered
field
coefficients,
i.e.
where
AEP
and AMP
are
the
unknown
transmitted
A
=L
+TA
(19)
coefficients.
Finally,
the
scattered
electric
and
magnetic
where

and
E
are
column matrices
for
the
unknown
fields from
the
pth
sphere
are
expanded
as
scattered
and
incident
field
coefficients, respectively,
and

m=n
ES(rp'O9p'bp)=Z
I
[AE(mn)N'(rP,9P,,)
T
is
a
square
matrix
which
contains
the
translation
addition
= .... [ ( M" ,
(9)
coefficients.
+AMp(in,n)M
(rp,0p,p,,)]
Once
the
scattered
field
is
computed
from
equation
(19),
the
normalized bistatic
cross
section
can
be
obtained
as
in
[16].
Hamid,
et
al.:
Scattering
by
a
System
of
Dielectric
Spheres
Coated with
a
Dielectric
Shell
93
for the
outer layer
is
3
and
the
electrical
separation
between
3.
Numerical
Results
the
centers
of
the
spheres
is
3.0
(touching).
In
order
to
check
the
validity
of
our computer program,
4.
CONCLUSIONS
several
numerical tests
were
conducted
and the
results
compared
favorably
with
previously published
results
[7
We
have
obtained
an
analytic
solution
of
the
problem
of
8,11].
These
tests
included
the
limiting
cases
of
(i)
an
array
scattering
by
an
array
of
dielectric
spheres each coated
with
of
dielectric spheres
obtained
by
setting
kb
=ka,
E
=I
or
a
dielectric
shell.
The
boundary
conditions
are
satisfied
at
various interfaces
with
the
help
of
the
vector
translation
61ir
=
Eir
(ii)
an
array
of
conducting
spheres
each
coated
addition theorem. The system
of
equations
was
written
in
with
a
single
dielectric
layer
obtained
by
setting
EIr
=
oo
and matrix form
while
the
scattered
field
coefficients
were
obtained
by
matrix inversion.
Numerical
results
were
(iii)
an
array
of
conducting
spheres
obtained by setting
presented
for
different numbers
of
spheres,
angles
of
E£r
=
0o
and
kb
=ka
or
Ellr
=1.
incidence, electrical
separation,
and
relative
dielectric
In
this
paper,
we
presented
numerical results
for
different
constant.
For
the general
case
of
spheres orientation,
the
sphere arrays
to
show
the
dependence
of
the
radar
cross
reader
may
find
more details
in
[8].
section
on
various
parameters characterizing
the
geometry,
material
properties,
and incidence
angles.
Fig.
2
shows the
ACKNOWLEDGMENT
normalized bistatic
cross
section
versus the
scattering angle
0
for
a
system
of
three identical
spheres
in
the
E and
H
The
first
author
wishes
to
acknowledge
the
support provided
planes. The electrical
radii
of
the
outer
and
inner
spheres
are
by
University
of
Sharjah. The second
author
wishes
to
ka=2.0
and
kb=2.5,
respectively,
while
the
electrical
acknowledge
the
support
of
the
United
Arab Emirates
separation between successive
spheres
is
kd=7.0,
and the
University,
while the
third
author
wishes
to
acknowledge
relative dielectric
permittivity
of
the
inner dielectric layer
is
the financial
support
of
the
University
of
South
Alabama
3.0
and
the
outer
is
air.
The
purpose
of
this
comparison
is
to and the
National
Science
Foundation.
check
the
accuracy
of
the
computer
code
for
the
dielectric
sphere
case
[8]
as
a
special
case
of
the dielectric
spheres
except
the
relative
dielectric
permittivity
of
the
dielectric
layer
is
set
equal to
unity.
The parameters
of
Fig.
3 are
similar
to
Fig.
2
except
that
the
dielectric
layer
has
a
value
Z
of
2.
We
can see
that
the
number
of
resonances
in E
plane
is
increased. Figs.
4
and
5
have the same
parameters
as
in
Fig.
3
except
that
the
number
of
spheres
is
increased
to five
and eight,
respectively.
We can
see
that the
number
of
P
resonances
also
increases
with
the
number
of
spheres.
xP
Fig.
6
shows
the
normalized backscattering
cross
section
versus
the
electrical
distance
(kd), which ranges from
8
(touching)
to
15.5
for
end
fire
incidence
and
the
number
of
dpq
spheres
is
five.
The
electrical
radii
of
outer
and
inner
spheres
are
ka=4.0
and
kb=3.0,
repectively,
while
the
xq

relative dielectric
permittivity
of
the
inner dielectric layer
is
3.0
and for the
outer
layer
is
2.
Fig.
7
is
similar
to
Fig.
6
except
the
number
of
spheres
is
increased
to
8.
We can see
"c
1
that the location
of
the
maximum peaks
did
not
change
by
increasing
the
number
of
spheres
for
both
cases.
x
Furthermore,
the
magnitude
of
the
normalized
backscattering
cross
section
at the
maximum
peaks K
increased
with
increasing
number
of
spheres.
In
Figs.
8
and
9
we
have
plotted
the
normalized
backscattering
cross
as
a
function
of
the angle
of
incidence
a,
which
ranges from
0
to 90
degrees for
a
system
of
three
Fig.
1:
Geometry
of
the
scattering
problem.
and
eight
spheres.
The
electrical radii
of
the
outer
and
inner
spheres
are
ka=l.5
and
kb=l.0,
repectively,
while the
relative
permittivity
of
the
inner
dielectric layer
is
4
while
94
ACES
JOURNAL,
VOL.
18,
NO.
4,
NOVEMBER
2003
10' 10'
102
HP
102HPlane
1'E Plane 10'
10' 10'
.1
10
10
10*
1
10
10. 10 2
3
10.
10" 0 20 40 60 00 100 120 140 100 10
0 20 40 60 80 100 120 140 160 180 Scattering angle (0)
Scattering angle (0)
Fig.
2:
Normalized bistatic
cross
section
patterns
for
Fig.
4:
Normalized
bistatic
cross
section patterns
for
five
three
identical dielectric
spheres
each
covered
with
identical
dielectric
spheres
each
covered
with
a
dielectric
dielectric layer
with
ka=2.0,
kb=2.5, kd=7.0,
layer
with
ka=2.0,
kb=2.5,
kd=7.0,
a
=0,gr
=3.0,
and
a
=0,
gir
=3.0, and
Elir
=1.0.
In
the
Eplane
(
=r/2)
Er=2.
and
Hplane
(=
0).
10'
103
EP
lane
F_
Pla
2
. HPlane
10'
. HPae10'
10, 10
10 1 "
10' 10
, 10
10 0 10100 20 40 60 80 100 120 140 160 180
0 20 40 60 80 100 120 140 160 10
Scattering angle (0) Scattering angle (0)
Fig.
3:
Normalized
bistatic
cross
section
patterns
for
Fig.
5:
Normalized
bistatic
crosssection patterns
for
eight
three
identical
dielectric
spheres
each
covered
with
identical
dielectric
spheres
each
covered
with
a
dielectric
dielectric
layer
with
ka=2.0,
kb=2.5,
kd=7.0,
layer
with
ka=2.0,
kb=2.5,
kd=7.0,
a
=0,c1r
=3.0,
and
a
'
=0,
Ir
=3.0,
and
E8ir
=2.0.
Ei1
=2.0.
Hamid,
et
al.:
Scattering
by
a
System
of
Dielectric
Spheres Coated
with
a
Dielectric
Shell
95
1.8
1.69
1.4
1.2 7
1 6
0 .8  6
0.6
4
0.4 3
0.2
2
9 10 11 12 13
14
15
kd
Fig.
6:
Normalized
backscattering
cross section
versus
0 10 20 30 40 50 60 70 80 90
electrical separation (kd)
for
endfire incidence
and
a
linear
array
of
five
identical dielectric
spheres each
Fig.
8:
Normalized
backscattering
cross section versus
covered
with
a
dielectric layer
with:
ka=4.0,
kb=3.0,
aspect
angle
a
for
a
linear
array
of
three identical
a
=
0.0,
Eir
=3.0,
and
Elir
=2.0.
dielectric
spheres
each
covered
with
a
dielectric
layer
with
ka=l.5,
kb=1.0,
kd=3.0,
Elr
=4.0,
and
E11r
=3.0.
2.5
180
160
2
140
1.5 120
100
1 0
60
0.5 40
OL ~20
5 10 11 12 13 14 15 0
kd 0 10
20 30 40
50
60
70 80 90
Fig.
7:
Normalized
backscattering
cross section
versus
electrical
separation
(kd)
for
endfire incidence
and
a
Fig.
9:
Normalized
backscattering
cross section
versus
linear
array
of
eight
identical
dielectric
spheres
each
aspect
angle
afor
a
linear
array
of
eight
identical
covered
with
a
dielectric layer
with:
ka=4.0, kb=3.0, dielectric
spheres
each
covered
with
a
dielectric
layer
with
a=
0.0,
eI,
=3.0,
and
EjIr
=2.0.
ka=l.5,
kb=1.0, kd=3.0,
Ei,
=4.0,
and
Elir
=3.0.
REFERENCES
4.
MedgyesiMitschang,
L.N, and
Putnam,
J.M.,"
Electromagnetic scattering
from
axially
1.
Aden,
A.L.,
and
Kerker,
M.,
"Scattering
of
inhomogeneous bodies
of
revolution,"
IEEE
electromagnetic
waves from
two
concentric
Trans. Antennes Propag.,
vol. AP32,
no.
8,
pp.
spheres",
J.
Appl.
Phys.,
22,
pp.
12421246,
797806,
1984.
1951.
5.
Kyutae,
L.,
and
Lee, S.L.,"
Analysis
of
2.
Scharfman,
H,
"Scattering
from
dielectric coated
Electromagnetic scattering
from
an
eccentric
spheres
in
the
region
of
the
first resonance",
J. multilayered
sphere," IEEE
Trans.
Antennes
Appl.
Phys.,
25,
pp.
13521356,
1954.
Propag.,
vol.
AP43,
no.
11,
pp.
13251328,
3.
Wait,
J.R.,
"Electromagnetic scattering from
a 1995.
radially
inhomogeneous
spheres," Appl.
Sci.
6.
Bruning,
J. H.,
and
Lo.,
Y.,
1971,
"Multiple
Res.,
vol.
10,
p.
441,
1963.
scattering
of
EM
waves
by
spheres: Parts
I
96
ACES
JOURNAL,
VOL.
18,
NO.
4,
NOVEMBER
2003
and
II",
IEEE
Trans.,
AP19,
pp.
378400,
2000
he
was
with the
faculty
of
electrical engineering
at
1971.
King
Fahd
University
of
Petroleum
and
Minerals,
7.
Hamid,
AK.,
Ciric
I.R.,
and
Hamid, Dhahran,
Saudi
Arabia.
Since
Sept.
2000
he
has
been
an
M.,"Multiple scattering
by
a
linear array
of
associate
Prof.
in
the
electrical\electronics
and
computer
conducting
spheres",
Can.
J.
Phys.,
vol.
68,
pp.
engineering
department
at
the
University
of
Sharjah,
11571165,
1990.
Sharjah,
United
Arab
Emirates.
His
research
interest
8.
Hamid,
AK.,
Ciric
I.R.,
and
Hamid,
M.,
includes
EM wave
scattering
from
two and
three
"Iterative Solution
of
The Scattering
by
an
dimensional
bodies,
propagation
along waveguides
with
Arbitrary
Configuration
of
Conducting
or
discontinuities,
FDTD
simulation
of
cellular phones,
and
Dielectric
Spheres",
IEE
Proc.,
Part
H,
vol.
148,
inverse
scattering
using
pp.
565572,
1991.
Neural
Networks.
9.
Hamid,
AK.,
Ciric
I.R.,
and
Hamid,
M.,
"Analytic
Solutions
of
The
Scattering
by
Two
Mousa
I.
Hussein
received
the
B.Sc.
degree
in
electrical
Multilayered Dielectric Spheres",
Can.
J.
Phys.,
engineering
from
West
Virginia
Tech,
USA,
1985,
M.Sc.
vol. 70,
pp.
696705,
1992.
and
Ph.D.
degrees from
University
of
Manitoba,
10.
Comberg,
U.,
and
Wriedt,
T."
Comparison
of
Winnipeg,
MB,
Canada,
in
1992
and
1995,
respectively,
scattering
calculations for aggregated
particles both
in
electrical
engineering.
From
1995
to 1997,
he
was
based
on
different models,"
J.
of
Quantitative
with
research and
development group
at
Integrated
Spectroscopy
and
Radiative
Transfer,
vol.
63,
pp.
Engineering
Software
Inc.,
Winnipeg,
Canada,
working
149162,
1999. on
developing
EM
specialized
software
based
on the
11.
AK. Hamid,
M.I.
Hussein,
and
M.
Hamid,
Boundary Element
method.
In 1997
he
joined
the
faculty
"Radar Cross
Section
of
a
System
of
Conducting
of
engineering
at
Amman
University,
Amman,
Jordan,
as
Spheres Each
Coated
with
a
Dielectric
Layer,"
J.
an
Assistant
Professor.
Currently
Dr.
Hussein
is
an
of
Electromagnetic
Waves
and
Applications,
vol.
Associate
Professor
with the
Electrical Engineering Dept.
17,
pp.
431445,
2003.
at
the
United
Arab
Emirates
University. Dr.
Hussein
12.
Hamid,
AK.,"
Modeling
the
Scattering
from
a
research
interests include
computational
electromagnetics,
Dielectric
Spheroid
by
System
of
Dielectric
electromagnetic scattering,
antenna
analysis
and
design,
Spheres",
J.
of
Electromagnetic
Waves
and
EMI
and signal
integrity. microstrip
antennas, phased
Applications,
vol.
10,
no.5,
pp.
723729,
1996.
arrays,
slot
and
open
ended
waveguide
antennas.
13.
Hamid,
M.,
and
Rao,
T.C.K,"
Scattering
by
a
multilayered
dielectriccoated conducting
Michael Hamid
Graduated
from McGill
University in
cylinder",
International
J.
of
Electronics,
vol.
38,
Montreal
with
a
B.Eng.
degree
in
1960,
a
M.Eng.
degree
pp.
667673,
1975.
in
1962
and
from
the
University
of
Toronto
with
a
Ph.D.
14.
Stratton,
J.A.,
"Electromagnetic
Theory",
degree
in
1966,
all in
Electrical Engineering.
He
joined
(McGraw
Hill,
New
York),
1941.
the
University
of
Manitoba
in 1965
where he
became a
15.
Cruzan,
OR.,
"Translational
addition
theorems
Professor
of
Electrical
Engineering
and
head
of
the
for
spherical
wave
functions",
Quart.
Appl.
Antenna
Laboratory.
He was
a
visiting
Professor
at
the
Math.
,
20, pp.
3340.
1962.
Naval
Postgraduate
School
as
well
as
the
universities
of
16.
Hamid,
AK.,
Hussein,
M.I.,and
Hamid,
M.,
California
Davis
and Central
Florida
and
is
presently a
"Bistatic Cross Section
of
an
Array
of
Dielectric
Professor
of
Electrical
Engineering
at
the
University
of
Spheres
Each
Covered
with
a
Dielectric
Shell,"
South
Alabama.
He
is
a
past president
of
the
International
the
19
th
Annual
Review
of
Progress
in
Applied Microwave
Power
Institute,
a
Fellow
of
IEE
and IEEE
Computational
Electromagnetics,
Naval and
published
307
referred
articles and
25
patents.
Postgraduate
School,
Monterey, California,
U.S.A.,
pp.
7781,
March 2003.
A.K.
Hamid
was
born
in
Tulkarm, West
Bank,
on
Sept.
9,
1963.
He
received the
B.Sc.
degree
in
Electrical
Engineering
from
West
Virginia
Tech,
West Virginia,
U.S.A.
in
1985.
He
received
the
M.Sc.
and
Ph.D.
degrees
from
the
university
of
Manitoba, Winnipeg, Manitoba,
Canada
in
1988
and
1991,
respectively, both
in
Electrical
Engineering.
From 19911993,
he
was
with Quantic
Laboratories
Inc.,
Winnipeg, Manitoba,
Canada,
developing
two
and
three
dimensional electromagnetic
field
solvers
using
boundary
integral
method. From
1994