1. 1
Efficient Mode-Matching Analysis of
Two-Dimensional Scattering by Periodic Array of
Circular Cylinders
Yong Heui Cho and Do-Hoon Kwon
Abstract—Based on a common-area concept and insertion
of an infinitesimal PMC (Perfect Magnetic Conductor) wire,
a new mode-matching method for mixed coordinate systems
is proposed for the analysis of periodic magnetodielectric,
PEC (Perfect Electric Conductor), and PMC circular
cylinder arrays. Our scattering solutions were computed
and they showed favorable agreements with other known
results in terms of power scattering coefficients, resonant
frequency of nano-structures, and sum rule of extinction
width. The low-frequency solutions of periodic arrays were
also formulated and compared with the full-wave counterparts.
Index Terms—Electromagnetic diffraction, periodic struc-
tures, mode-matching methods, electromagnetic resonance,
nanowires, sum rule.
I. Introduction
APeriodic array of dielectric circular cylinders is a
canonical diffraction structure and has been exten-
sively studied in [1]-[5]. Since most of dielectric gratings
can be approximately regarded as circular cylinders, it
is of practical interest to obtain the simple yet analytic
scattering solutions of a periodic two-dimensional (2D)
dielectric cylinder array. In [1] and [2], diffraction by
periodic circular cylinders was analyzed with infinite
summation of circular cylindrical wave functions that
represent the electromagnetic fields of a single circular
cylinder. This fundamental approach has been widely
utilized in the analyses of periodic gratings [3]-[5]. A
variety of numerical methods for periodic dielectric and
metallic structures were discussed in [3] as well. Recently,
periodic gratings with very small radius have been used
for nano-structures to invoke the resonance characteristics.
This type of periodic gratings is called a nanorod [6]
or nanowire [7]. In optics, a nanorod structure is a
fundamental geometry to design various optical devices
such as biosensors, nanoantennas [8], polarization selec-
tive surface, terahertz transmission lines, and near-field
scanning optical microscopes.
In this work, a novel mode-matching technique for
mixed coordinate systems is proposed to present the
This work was supported in part by the US Army Research Office
Grant No. W911NF-12-1-0289.
Y. H. Cho is with the School of Information and Communication
Engineering, Mokwon University, Doanbuk-ro 88, Seo-gu, Daejeon,
302-729, Korea (e-mail: yongheuicho@gmail.com).
D.-H. Kwon is with the Department of Electrical and Computer
Engineering, University of Massachusetts Amherst, Amherst, MA,
01003, USA (e-mail: dhkwon@ecs.umass.edu).
simple yet exact scattering equations for a periodic 2D
circular cylinder array. A standard mode-matching tech-
nique in [3], [9] is still used to formulate the analytic
dispersion and scattering equations for rectangular [10]
and circular cylindrical [11] structures. Using staircase
approximation and generalized scattering matrix, open
periodic waveguides with arbitrary unit-cell can be ap-
proximately analyzed in the Cartesian coordinate system
[12]. A complex mode-matching method combined with
the perfectly matched layer (PML) [13] opened a new
way to get the reflection and transmission spectra of
optical waveguide interconnects. In addition, the fast scat-
tering analyses of electrically large substrate integrated
waveguide (SIW) devices were performed with a mode-
matching technique using a cylindrical mode expansion
[14]. Although a standard mode-matching technique is
well-known, it is difficult to apply this method to a
geometry with both rectangular and circular cylindrical
features [15]-[18]. To analyze such a geometry, we need
to introduce a common-area concept proposed in [15].
The common area concept has been widely used to
analyze waveguide problems [16]-[18]. In this approach, the
fields expanded in the rectangular and circular cylindrical
coordinate systems should have a so-called common area
where all boundary conditions are consistently satisfied.
Since our periodic problem requires the periodic boundary
condition on each side, our geometry is surely different
from those in [16]-[18] where the PEC (Perfect Electric
Conductor) region is essential to apply a mode-matching
technique based on the common area. In our periodic case,
there is no PEC region to match the boundary conditions
and get a set of simultaneous equations. Therefore, we will
use an infinitesimal PMC (Perfect Magnetic Conductor)
wire to overcome this difficulty and match the boundary
conditions of the TE-mode (Ez = 0, Hz = 0). It should
be noted that the infinitesimal PMC wire is transparent
and does not interact with the TE-mode incident field.
A common-area concept and the insertion of infinitesi-
mal PMC wires allows us to obtain the analytic scattering
relations of a periodic 2D circular cylindrical array in
both rectangular and circular cylindrical features. Using
the final set of simultaneous equations for a periodic
2D circular cylinder array, we can obtain the resonance
characteristics of nano-structures in the optical band and
the sum rule for the extinction width in the microwave
band.

2. 2
T
Region (I)
Region (II)
Region (IV)
x
y
z
2a
iθ
Incidence
11,µε
22,µε
Region (III)
Periodic
boundary
condition
Periodic
boundary
condition
Fig. 1. Unit-cell of a periodic two-dimensional circular cylinder
array in a free space
II. Mode-Matching in Mixed Coordinate Systems
Consider a periodic 2D circular cylinder array in a free
space shown in Fig. 1. In order to compute the scattering
characteristics of an infinite number of circular cylinders,
we introduce periodic boundary conditions at x = ±T/2.
Fig. 1 illustrates a unit-cell of the infinite number of
circular cylinders. In the following development, an e−iωt
time convention is assumed and omitted throughout. An
incident electric field in region (I) (ρ > T/2, |x| ≤ T/2, y >
0) is given by
Ei
z(x, y) = eik1(sin θix−cos θiy)
, (1)
where k1 = ω
√
µ1 1, θi is the angle of incidence of the
TE-mode (Ez = 0, Hz = 0) plane wave with respect to
the φ-axis, and 0 ≤ θi ≤ π/2. For convenience, we define
a transmitted field in region (IV) (ρ > T/2, |x| ≤ T/2, y <
0)
Et
z(x, y) = eik1(sin θix−cos θiy)
(2)
as the incident field evaluated behind the array. Utilizing
a standard mode-matching method and the Floquet anal-
ysis, we represent the electric fields for regions (I) to (IV)
as
EI
z (x, y) =
∞∑
m=−∞
Amei(Tmx+ξmy)
(3)
EII
z (ρ, φ) =
∞∑
m=−∞
[BmJm(k1ρ) + CmNm(k1ρ)]
× eimφ
(4)
EIII
z (ρ, φ) =
∞∑
m=−∞
DmJm(k2ρ)eimφ
(5)
EIV
z (x, y) =
∞∑
m=−∞
Emei(Tmx−ξmy)
, (6)
where Am to Em are unknown modal coefficients, Tm =
k1 sin θi + 2mπ/T, ξm =
√
k2
1 − T2
m, k2 = ω
√
µ2 2,
ρ =
√
x2 + y2, φ = tan−1
(y/x), and Jm(·), Nm(·) are the
mth-order Bessel functions of the first and second kinds,
respectively. It should be noted that the electric fields in
regions (I) to (IV) are formulated in the mixed coordinate
systems based on a common-area concept [15].
By enforcing the tangential field continuities at ρ = a
and ρ = T/2, we can constitute the simultaneous equa-
tions for the TE-mode. Multiplying the equations for
the Ez- and Hφ-field continuities at ρ = a by e−ilφ
(l = 0, ±1, ±2, · · · ) and integrating from φ = 0 to φ = 2π
gives
BmJm(k1a) + CmNm(k1a) = DmJm(k2a) , (7)
1
η1
[BmJm(k1a) + CmNm(k1a)] =
1
η2
DmJm(k2a) , (8)
where η1,2 =
√
µ1,2/ 1,2 and (·) denotes the differentia-
tion with respect to the argument k1,2ρ. Then,
Bm = B(0)
m Dm (9)
Cm = C(0)
m Dm , (10)
where
B(0)
m =
πk1a
2
[
Jm(k2a)Nm(k1a)
−
η1
η2
Jm(k2a)Nm(k1a)
]
(11)
C(0)
m =
πk1a
2
[
η1
η2
Jm(k2a)Jm(k1a)
− Jm(k2a)Jm(k1a)
]
. (12)
Similarly, multiplying the equations for the Hφ-field con-
tinuity at ρ = T/2 by e−ilφ
(l = 0, ±1, ±2, · · · ) and
integrating from φ = 0 to φ = 2π yields
Dl =
1
H
(0)
l
[ ∞∑
m=−∞
(AmKml + EmLml) + Sφ,l
]
, (13)
where
Kml = ik1e−ilφm
Jce
l
(
k1T
2
, −φm
)
(14)
Lml = ik1eilφm
Jce
l
(
k1T
2
, π + φm
)
(15)
Jce
m (x, φ) =
∂Je
m(x, φ)
i∂x
(16)
Je
m(x, φ) = im
[
πJm(x)
− 2
∞∑
n=−∞
ei(2n+1)φ (−1)n
2n + 1
Jm+2n+1(x)
]
(17)
Sφ,l = 2πk1e−ilθi
J−l(k1T/2) (18)
H
(0)
l = 2πk1
[
B
(0)
l Jl (k1T/2) + C
(0)
l Nl (k1T/2)
]
, (19)
and φm = tan−1
(ξm/Tm). In the next step, we multiply
the Ez-field continuity equation at ρ = T/2 by e−ilφ
(l =

3. 3
0, ±1, ±2, · · · ) and integrate on the two intervals, φ =
(0, π) and φ = (π, 2π). Combining (13) and the Ez-field
continuity equation, we obtain the final set of simultaneous
equations for Am and Em as
∞∑
m=−∞
[
Am
(
Iml −
2πE
(0)
l
H
(0)
l
Kml
)
+ Em
(
Jml −
2πE
(0)
l
H
(0)
l
Lml
) ]
=
2πE
(0)
l
H
(0)
l
Sφ,l − Sz,l , (20)
∞∑
m=−∞
[
Am
∞∑
k=−∞
(−1)k−l
KmkD
(0)
kl
+ Em
( ∞∑
k=−∞
(−1)k−l
LmkD
(0)
kl − Jml
) ]
= SIV
z,l −
∞∑
k=−∞
(−1)k−l
Sφ,kD
(0)
kl , (21)
where
Iml = e−ilφm
Je
l
(
k1T
2
, −φm
)
(22)
Jml = eilφm
Je
l
(
k1T
2
, π + φm
)
(23)
E
(0)
k = B
(0)
k Jk(k1T/2) + C
(0)
k Nk(k1T/2) (24)
D
(0)
kl =
E
(0)
k
H
(0)
k
Ge(k − l) (25)
Ge(m) =
(−1)m
− 1
im
(26)
SIV
z,l = il
e−ilθi
Je
l
(
k1T
2
,
3π
2
− θi
)
(27)
Sz,l = 2πe−ilθi
J−l(k1T/2) . (28)
After obtaining Am and Em, we can get Dm with (13).
Similarly, Bm and Cm are obtained with Dm based on
(9)–(10).
The Ez-field continuity at ρ = T/2 should be enforced
on two different intervals, φ = (0, π) and φ = (π, 2π) to
obtain the additional boundary condition. Since we have
five sets of unknown modal coefficients, Am to Em in (3)
to (6), we should get five tangential electric and magnetic
boundary conditions at ρ = a and T/2 to solve the final
simultaneous equations. Applying the Ez-field matching
condition on two intervals means that the PMC wires exist
at (x, y) = (±T/2, 0) in Fig. 1. Although the PMC wires
should be at (x, y) = (±T/2, 0) to match the boundary
conditions, no magnetic current can flow through the PMC
wires due to the TE-mode excitation, thus indicating that
the lines can be ignored.
Since the geometry in Fig. 1 is composed of magnetodi-
electric materials, we can easily obtain the simultaneous
scattering equations of the TM-mode (Ez = 0, Hz = 0)
based on the duality of the Maxwell’s equations. In view
of the duality theorem, we need to replace → µ, µ → ,
¯Ee → ¯Hm, and ¯He → − ¯Em. Then, (20)–(21) can be used
for the TM-mode with substituting → µ, µ → . The
simultaneous equations for the PEC and PMC materials
are given in Appendix A based on (20)–(21).
When 1 = 2 and µ1 = µ2, we get the result of Am =
Em = 0 which means that there is no scattered field in
regions (I) and (IV). This behavior partly confirms the
validity of our simultaneous equations, (20)–(21).
Using the scattered fields, (3) and (6), we can define the
reflectance, transmittance, and absorbance, respectively,
as
Rtot =
Pr
Pi
=
M−
∑
m=−M+
ρm (29)
Ttot =
Pt
Pi
=
M−
∑
m=−M+
τm (30)
Atot = 1 − Rtot − Ttot , (31)
where Pi, Pr, Pt are incident, reflected, and transmitted
powers per unit-cell in Fig. 1, respectively, M± = [k1T(1±
sin θi)/(2π)], [x] is the maximum integer less than x,
ρm = |Am|
2
[
ξ∗
m
k1 cos θi
]
(32)
τm = |Em + δm0|
2
[
ξ∗
m
k1 cos θi
]
, (33)
δml is the Kronecker delta, (·)∗
is the complex conjugate
of (·), and [·] is the real part of (·).
III. Low-Frequency Solutions
Even though the exact simultaneous equations, (20)–
(21), can be used to analyze the low-frequency behaviors
when frequency approaches zero, it is convenient and nu-
merically efficient to deduce new low-frequency solutions
based on (20)–(21). Taking the low-frequency limit of
(51)–(52) in Appnedix A yields the following simplified

4. 4
low-frequency equations of the PEC cylinders:
∞∑
m=−∞
[
(Am + δm0)
( ∞∑
k=−∞
KmkD
(E)
kl − Iml
)
+ (Em + δm0)
∞∑
k=−∞
Km,−kD
(E)
kl
]
= k1
{
Ti1+l
[
e−ilθi
Jce
l (0, π/2 − θi)
]
− 2
∞∑
k=−∞
UkD
(E)
kl
}
, (34)
∞∑
m=−∞
[
(Am + δm0)
∞∑
k=−∞
(−1)k−l
KmkD
(E)
kl
+ (Em + δm0)
( ∞∑
k=−∞
(−1)k−l
Km,−kD
(E)
kl − Im,−l
) ]
= −2k1
∞∑
k=−∞
(−1)k−l
UkD
(E)
kl . (35)
When k1 → 0, the low-frequency forms of the parameters
in (34)–(35) are given by
Kml ∼
4π|m|
T
×
{
(−i) · si(1 − l, |m|π) for m > 0
i · si∗
(1 + l, |m|π) for m < 0
(36)
K0l ∼ k1i1−l
eilθi
Jce
l (0, θi − π/2) (37)
Iml ∼ −2 ×
{
si(−l, |m|π) for m ≥ 0
si∗
(l, |m|π) for m < 0
(38)
Um = πe−imθi
J−m(0)
− i ·
[
eim(θi−π/2)
Jce
m (0, θi − π/2)
]
(39)
D
(E)
ml ∼
T
4π
Ge(m − l)
×
{
1
|m|
1−(2a/T )2|m|
1+(2a/T )2|m| for m = 0
− log
(2a
T
)
for m = 0
(40)
Jce
m (0, φ) =
e−imφ
i(m2 − 1)
[1 + (−1)m
]
× (m cos φ + i sin φ) (41)
si(m, a) = −
1
2
∫ π
0
ei[a exp(iφ)+mφ])
dφ . (42)
Eqn. (42) is a generalized sine integral such that si(0, a) =
si(a) and its recurrence relation is given by
a · si(m + 1, a) − im · si(m, a)
=
(−1)m
e−ia
− eia
2
. (43)
The sine integral si(a) is defined in [19, Eqn. (5.2.5)].
The simultaneous equations, (34)–(35), can be repre-
sented in a matrix form as
M(E)
[
Am + δm0
Em + δm0
]
= k1S(E)
, (44)
TABLE I
TM-mode power reflection coefficient of the zeroth-order Floquet
mode ρ0 with the same parameters in Fig. 2
T/λ0 M = 3 M = 7 M = 11 M = 21 Lattice sum [4]
0.4 0.02825 0.02703 0.02703 0.02703 0.0271
0.5 0.02416 0.02273 0.02273 0.02273 0.0222
0.6 0.01356 0.01259 0.01258 0.01258 0.0133
0.7 0.002776 0.002588 0.002586 0.002586 0.002
0.8 0.001326 0.001279 0.001282 0.001282 0
0.9 0.3029 0.4820 0.4824 0.4824 0.534
1.0 0.008885 0.004154 0.004193 0.004193 0.0043
where M(E)
and S(E)
are corresponding scattering and
source matrices of (34)–(35), respectively. Since M(E)
and
S(E)
are independent of k1, Am + δm0 and Em + δm0 are
linearly proportional to k1.
Similar to the PEC case, the low-frequency solutions for
the PMC and magnetodielectric materials are given by
∞∑
m=−∞
[
Am
( ∞∑
k=−∞
KmkD
(p)
kl − Iml
)
+ Em
∞∑
k=−∞
Km,−kD
(p)
kl
]
= Sz,l − SIV
z,l −
∞∑
k=−∞
Sφ,kD
(p)
kl , (45)
∞∑
m=−∞
[
Am
∞∑
k=−∞
(−1)k−l
KmkD
(p)
kl
+ Em
( ∞∑
k=−∞
(−1)k−l
Km,−kD
(p)
kl − Im,−l
) ]
= SIV
z,l −
∞∑
k=−∞
(−1)k−l
Sφ,kD
(p)
kl , (46)
where p = M for PMC, p = 0 for magnetodielectric
material,
D
(M)
ml ∼
T
4π
Ge(m − l)
×
{
1
|m|
1+(2a/T )2|m|
1−(2a/T )2|m| for m = 0
8
(k1T )2[(2a/T )2−1] for m = 0
(47)
D
(0)
ml ∼
T
4π
Ge(m − l)
×



1
|m|
1+
µ1
µ2
+
“
1−
µ1
µ2
”
(2a/T )2|m|
1+
µ1
µ2
−
“
1−
µ1
µ2
”
(2a/T )2|m|
for m = 0
8
(k1T )2
h“
1− 2
1
”
(2a/T )2−1
i for m = 0
. (48)
IV. Discussions
To verify our mode-matching formulations in mixed
coordinate systems, the scattering equations, (20)–(21),
were computed to get the resonance characteristics of a
periodic 2D dielectric cylinder array.
Fig. 2 and Table I show the power reflection coefficient
of the zeroth-order Floquet mode ρ0 in (32) with respect

5. 5
0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
Normalized period, T/λ0
Powerreflectioncoefficient,ρ0
M = 3
M = 7
M = 11
M = 21
Lattice sum [4]
Fig. 2. TE-mode power reflection coefficient of the zeroth-order
Floquet mode ρ0 versus normalized period T/λ0 with θi = 0, a =
0.3T, 1 = 0, 2 = 2 0, and µ1 = µ2 = µ0
400 600 800 1000 1200 1400 1600 1800 2000
0
0.2
0.4
0.6
0.8
1
Wavelength [nm]
TE−modereflectance,Rtot
M = 11
M = 3
M = 1
Fig. 3. TE-mode reflectance of TiO2 nanorods versus wavelength
with θi = 0, a = 40 [nm], T = 783.6 [nm], 1 = 0, µ1 = µ2 = µ0,
and 2 obtained from [20]
to the normalized period. In addition, the convergence
characteristics of modal coefficients are shown in Fig. 2
and Table I, where M denotes the truncated number
of modes in regions (I) or (IV). As M increases, the
power reflection coefficient approximately converges to the
results computed by the lattice sum technique [4]. Table
I indicates that the cases of M = 3 [m = 0, ±1 in (20)–
(21)] and M = 7 [m = 0, ±1, ±2, ±3] converge to that of
M = 21 with errors less than 7.2% and 0.2%, respectively,
when T/λ0 ≤ 0.8. For the TM-mode case in Table I, we
used the duality form of (20)–(21) with which 2 and µ2
are replaced with µ2 and 2, respectively.
Figs. 3 and 4 illustrate the characteristics of scattered
powers for an array of nanorods composed of TiO2 (di-
electric) and silver (metal). We computed the reflectance
(29) and transmittance (30) based on (20)–(21) using the
300 350 400 450 500
0
0.2
0.4
0.6
0.8
1
Wavelength [nm]
NormalizedTM−modepower
Reflectance, Rtot
Transmittance,T
tot
Absorbance, Atot
Fig. 4. Normalized TM-mode powers of silver nanowires versus
wavelength with M = 11, θi = 0, a = 70 [nm], T = 375 [nm],
1 = 0, µ1 = µ2 = µ0, and 2 taken from [21]
refractive index (n = n + iκ, 2 = n 2
) data [20], [21].
The geometrical parameters of the nanorod array for Fig.
3 were taken from [6]. The refractive index of TiO2 for
an ordinary ray was obtained as a form of the Sellmeier
equation from [20]. The resonant wavelength from our
mode-matching analysis is around 814.6 [nm], which is
very close to 800 [nm] predicted in [6]. The convergence
results show that a dominant-mode solution with M = 3
(m = 0, ±1) yields a very good approximation for λ >
700 [nm]. This means that three modes (M = 3) are
enough to predict resonance behaviors of TiO2 properly.
The dominant-mode M = 1 (m = 0) only gives favorable
results when λ > 1400 [nm]. In Fig. 4, the TM-mode
reflectance, transmittance, and absorbance of a silver
nanowire array are shown. Since the measured refractive
index of silver is complex (κ = 0) [21], the resonance peak
in Fig. 4 is not as sharp as that in Fig. 3, where the loss
of TiO2 is assumed to be zero (κ = 0) [20]. The resonance
peak of reflectance in Fig. 4 is at 374.1 [nm] which is very
close to the approximate value of 377 [nm] in [7]. This
confirms that our approach is valid for nano-structures.
As a second example, TM-mode scattering by an array
of PEC cylinders in the microwave frequency band is
considered. For an array of closely spaced cylinders with
T = 50 [mm], a = 20 [mm], Fig. 5 plots the zeroth-order
Floquet mode reflectance ρ0 and transmittance τ0 with
respect to frequency and compares them with the nu-
merical solutions obtained using the commercial analysis
package HFSS from Ansys. The mode-matching solution
was obtained from the dual configuration of the TE-
mode. For both incident angles θi = 0, 30◦
considered in
Fig. 5, mode-matching and HFSS results show an excellent
agreement, validating the proposed solution methodology.
It is noted that grating lobes begin to appear at 6 [GHz]
and 4 [GHz] for the θi = 0 and θi = 30◦
cases, respectively,
and the mode-matching solution recovers transmission and

6. 6
0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Reflectanceandtransmittancespectra ρ0
: θi
=0
τ0
: θi
=0
ρ0
: θi
=30°
τ0
: θi
=30
°
HFSS
Fig. 5. The zeroth-order Floquet mode reflectance and transmit-
tance spectra for a PEC cylinder array with M = 11, T = 50 [mm],
and a = 20 [mm] for two different incident angles θi = 0 and 30◦.
Circled data points (◦) represent simulation results obtained using
Ansys HFSS.
reflection coefficients accurately and efficiently.
A sum rule [22], [23] relates the dynamic scattering
characteristics integrated over all frequency to the static
and low-frequency scattering responses. The sum rule is a
useful tool to estimate the fundamental limit of practical
antennas [23]. Let S0 denote the complex scattering
coefficient of the zeroth-order Floquet mode. From the
field definitions, (2) and (6), S0 = E0 for the PMC array
in the TE-mode case (and the PEC array in the TM-
mode via duality). Appropriately for 2D configurations,
let the extinction width σext be defined as the sum of the
scattering width σs and the absorption width σa (which is
equal to zero for lossless scatterers). The optical theorem
for doubly-periodic scatterers [22] can be extended to
obtain the optical theorem for singly-periodic 2D arrays
as
σext(k1) = σs + σa = −2T · [S0] cos θi . (49)
If we define a function ρext(k1) = S0/k2
1, the sum rule for
σext for doubly-periodic scatterers [22] is modified to read
∫ ∞
0
σext(k1)
k2
1
dk1 = −T cos θi
∫ ∞
−∞
ρext(k1) dk1
= πT · [Res(ρext, k1 = 0)] cos θi . (50)
where ρext(k1) is analytic in the complex upper half-
plane ( [k1] > 0), [·] is the imaginary part of (·),
and Res[f(z), z = 0] is the residue of f(z) at z = 0.
For θi = 0, Fig. 6(a) plots S0 at low frequencies using
different number of terms M = 3 and M = 11. At
both values of M, the full-wave solutions (51)–(52) and
the low-frequency solutions (45)–(46) show an excellent
agreement. In addition, Fig. 6(a) also shows that only
a small number of terms with M = 3 are enough to
obtain accurate low-frequency solutions. It is observed
that S0 = O(k1) and purely imaginary as k1 → 0. Hence,
the residue in (50) is related to the slope of [S0] in
Fig. 6(a). Fig. 6(b) shows the extinction width σext for
0 20 40 60 80 100 120 140 160 180 200
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Frequency [MHz]
Complexscatteringcoefficient,S0
imag
real
(51) & (52) with M=3
(51) & (52) with M=11
(45) & (46) with M=3
(45) & (46) with M=11
(a)
0 2 4 6 8 10 12
0
20
40
60
80
100
Frequency [GHz]
Extinctionwidth,σext
[mm]
θ
i
=0
θ
i
=30°
(b)
Fig. 6. Low-frequency scattering behavior of a PEC cylinder
array considered in Fig. 5 and the extinction scattering width σext.
(a) The zeroth-order Floquet mode scattering coefficient S0 at low
frequencies for the θi = 0 case. (b) The extinction width σext with
respect to frequency.
the two incident angles as a function of frequency from
(49). The sum rule (50) can be tested, where the integral in
(50) is obtained numerically from Fig. 6(b) and the residue
is obtained from the low-frequency solutions [Fig. 6(a) for
θi = 0]. With the numerical integration performed from
0 to 20 [GHz], (50) was found to be accurate with errors
less than 4.2% for the two cases considered. The accuracy
will improve as numerical integrations are performed over
a wider frequency range. This example provides another
validation of the full-wave and efficient low-frequency
solutions of the proposed analysis method.
V. Conclusions
A novel mode-matching method in mixed coordinate
systems was proposed to analyze plane-wave scattering
characteristics of periodic circular cylinders composed of
magnetodielectric, PEC, and PMC materials. Numerical
computations were performed to check the accuracy of our
analytic formulations. Numerical experiments show that
a three-term approximation (M = 3) is enough to obtain
accurate scattering results in most practical cases. For
instance, the result of M = 3 for a = 0.3T, 2 = 2 0,

7. 7
and T/λ0 ≤ 0.8 is within the maximum error of 7.2%
compared to that of M = 21. A three-term approximation
is also good for efficient computation of the low-frequency
field quantities. The proposed method has been tested
and validated for the resonance characteristics of nano-
structures in the optical regime and for the sum rule for
the extinction width in the microwave regime.
Appendix A: Equations for PEC and PMC
For the cases of PEC ( 2 → ∞ and µ2 = µ0) and
PMC ( 2 = 0 and µ2 → ∞) cylinders, the simultaneous
equations for magnetodielectric, (20) and (21), are refor-
mulated as
∞∑
m=−∞
[
Am
(
Iml −
2πE
(p)
l
H
(p)
l
Kml
)
+ Em
(
Jml −
2πE
(p)
l
H
(p)
l
Lml
) ]
=
2πE
(p)
l
H
(p)
l
Sφ,l − Sz,l , (51)
∞∑
m=−∞
[
Am
∞∑
k=−∞
(−1)k−l
KmkD
(p)
kl
+ Em
( ∞∑
k=−∞
(−1)k−l
LmkD
(p)
kl − Jml
) ]
= SIV
z,l −
∞∑
k=−∞
(−1)k−l
Sφ,kD
(p)
kl , (52)
where p = E for PEC, p = M for PMC,
H
(E)
k = 2πk1
[
Jk(k1T/2) −
Jk(k1a)
Nk(k1a)
Nk(k1T/2)
]
(53)
E
(E)
k = Jk(k1T/2) −
Jk(k1a)
Nk(k1a)
Nk(k1T/2) (54)
H
(M)
k = 2πk1
[
Jk(k1T/2) −
Jk(k1a)
Nk(k1a)
Nk(k1T/2)
]
(55)
E
(M)
k = Jk(k1T/2) −
Jk(k1a)
Nk(k1a)
Nk(k1T/2) (56)
D
(p)
kl =
E
(p)
k
H
(p)
k
Ge(k − l)
p=E,M
. (57)
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