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Modal Method based on subsectional Gegenbauer polynomial Expansion for lamellar gratings: Weighting function, convergence and stability

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The Modal Method by Gegenbauer polynomials Expansion (MMGE) has been recently introduced for lamellar gratings by Edee [8]. This method shows a promising potential of outstanding convergence but still suffers from instabilities when the number of polynomials is increased. In this work, we identify the origin of these instabilities and propose a way to remove them.
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Progress In Electromagnetics Research, Vol. 133, 17–35, 2012
MODAL METHOD BASED ON SUBSECTIONAL GEGEN-
BAUER POLYNOMIAL EXPANSION FOR LAMELLAR
GRATINGS: WEIGHTING FUNCTION, CONVERGENCE
AND STABILITY
K. Edee
1, *
, I. Fenniche
1
, G. Granet
1
, and B. Guizal
2
1
Universit´e Blaise Pascal, LASMEA, UMR-6602-CNRS, BP 10448, F-
63000 Clermont-Ferrand, France
2
Laboratoire Charles Coulomb, UMR 5221 of the CNRS, University of
Montpellier 2, France
Abstract—The Modal Method by Gegenbauer polynomials Expan-
sion (MMGE) has been recently introduced for lamellar gratings by
Edee [8]. This method shows a promising potential of outstanding
convergence but still suffers from instabilities when the number of
polynomials is increased. In this work, we identify the origin of these
instabilities and propose a way to remove them.
1. INTRODUCTION
Among the numerical methods developed for the analysis of lamellar
diffraction gratings, modal metho ds play an important role because of
their great versatility and relative effectiveness. In the classical modal
method [1, 2], the eigenvalues are obtained by solving a transcendental
equation. In other modal methods, the eigenmodes and propagation
constants are generally obtained by searching the eigenvalues and
eigenvectors [3–8] of a matrix which is derived from the Maxwell’s
equations by using the method of moments [9]. Mathematically
speaking, for one-dimensional gratings with piecewise homogeneous
media, and plane wave excitation, the eigenmodes are solutions
of the Helmholtz equation subject to boundary conditions at the
interfaces between two media and to the pseudo-periodicity condition.
Numerically, the rate of convergence of the method depends on how
the matrix from which eigenvalues are sought takes into account the
continuity relations. Indeed, one of the main differences b etween
Received 13 June 2012, Accepted 20 July 2012, Scheduled 16 October 2012
* Corresponding author: M. Kofi Edee (kof 8@hotmail.fr).
18 Edee et al.
the variants of modal methods is the choice of the expansion and
test functions. In a previous paper [10], we have developed a modal
method based on Gegenbaueur polynomials expansions (MMGE) and
emphasized that the main advantage of such an approach is that
continuity relations can be written in an exact manner. We have
shown through various examples that this method outperforms other
modal methods. However, we found that under its original form, the
MMGE suffers from some instabilities for large values of polynomials
degree. Even if it is not, for a certain class of grating problems,
necessary to use a large number of polynomials in order to have very
reliable results (because the MMGE converges very rapidly), it is of
fundamental importance to identify the origin of instabilities and find
a way to remove them. In [10], it has already been highlighted that
the instabilities were linked with the way we calculated the inner
product required by the Galerkin method. In the present work, we
track precisely the origin of the numerical problems and study the
influence of the weighting function appearing in the inner products.
It is shown that introducing this latter makes the calculation of inner
products analytical in one hand and ensures unconditional stability
on the other hand; on the contrary to our first implementation. The
Gegenbauer polynomials of degree m denoted by C
Λ
m
differ from each
other through a parameter Λ. Special cases where Λ is equal to 0.5
corresponds to the Legendre polynomial and as Λ approaches 0, these
polynomials are Chebychev polynomials. In addition, we investigate
the influence of Λ on the rate of convergence for the two fundamental
cases of polarization.
2. STATEMENT OF THE PROBLEM AND THE
FRAMEWORK OF MMGE
The MMGE consists in defining a partition
i
corresponding to the
different homogeneous subintervals of (which is nothing but the
elementary period of the structure), that can be, possibly, divided
into layers
ij
, 1 j N
i
. For example, in Figure 1, is subdivided
into two homogeneous subintervals
1
and
2
and each subinterval
i
, (i = 1, 2) contains N
i
layers.
We will denote by N
the number of subintervals. The
eigenfunctions X
p
(x) of the L-operator (L
T E
= k
2
2
x
+ ν
2
(x) and
L
T M
= k
2
ν
2
(x)
x
ν
2
(x)
x
+ ν
2
(x) with k = 2π) are described in
each homogeneous layer
ij
as follows:
¯
¯
X
i,j
p
®
=
N
X
n=1
a
i,j
n,p
¯
¯
b
i,j
n
®
, (1)
Progress In Electromagnetics Research, Vol. 133, 2012 19
Figure 1. The grating configuration: one period is depicted.
where |X
i,j
p
i is the restriction of the eigenfunction |X
p
i to the
homogeneous layer
ij
, characterized by its refractive index ν
i
and
N is the number of basis functions on each homogeneous layer. It is
of fundamental importance to note that |X
i,j
p
i satisfies:
(i) the Helmholtz equation identical for both T E and T M
polarizations:
L
i,j
¯
¯
X
i,j
p
®
= β
2
p
¯
¯
X
i,j
p
®
, (2)
with
L
i,j
=
1
k
2
d
2
dx
2
+ ν
2
i
. (3)
(ii) the following boundary conditions:
for each layer
ij
of the same subinterval
i
, (i, j)
[1 : N
1] × [1 : N
i
], the boundary equations obtained
by writing the continuity of the tangential components of
the electromagnetic field at the interfaces (x
i,j
)
j[1 : N
i
1]
separating two adjacent layers,
ij
and
i,j+1
:
X
i,j
p
(x
i,j
) = X
i,j+1
p
(x
i,j
), (4a)
"
dX
i,j
p
dx
#
x=x
i,j
=
"
dX
i,j+1
p
dx
#
x=x
i,j
, (4b)
at the interfaces separating two adjacent subintervals
i
and
i+1
, i.e., for j = N
i
, i [1 : N
1],
X
i,j
p
(x
i,j
) = X
i+1,1
p
(x
i,j
), (5a)
1
η
i
"
dX
i,j
p
dx
#
x=x
i,j
=
1
η
i+1
"
dX
i+1,1
p
dx
#
x=x
i,j
, (5b)
20 Edee et al.
where η
i
= 1 for T E polarization and η
i
= ν
2
i
for T M
polarization.
for the last subinterval
i
, i.e., when i = N
, Eq. (4) are
written for the first layers (Ω
ij
)
ij
, (i, j) {N
}×[1 : N
i
1]
and the pseudo-periodic condition for j = N
i
:
X
i,N
i
p
(d) = e
ik sin θd
X
1,1
p
(0), (6a)
1
η
i
"
dX
i,N
i
p
dx
#
x=d
=
e
ik sin θd
η
1
"
dX
1,1
p
dx
#
x=0
, (i = N
). (6b)
From Eq. (1), Eq. (2) may be written as follows:
N
X
n=1
a
i,j
n,p
L
i,j
¯
¯
b
i,j
n
®
= β
2
p
N
X
n=1
a
i,j
n,p
¯
¯
b
i,j
n
®
. (7)
Finally, by projecting Eq. (7) on the basis functions
³
|b
i,j
m
i
´
m[1 : N2]
,
we obtain:
L
i,j
[N2]×[N]
a
i,j
[1 : N],p
= β
2
p
G
i,j
[N2]×[N]
a
i,j
[1 : N],p
, (8)
where
L
i,j
[N2]×[N]
=
1
k
2
D
i,j
[N2]×[N1]
h
G
i,j
[N1]×[N1]
i
1
D
i,j
[N1]×[N]
+ ν
2
i
G
i,j
[N2]×[N]
, (9)
a
i,j
[1 : N],p
is a column vector formed by the coefficients a
i,j
n,p
, n [1 : N]:
a
i,j
[1 : N],p
=
h
a
i,j
1,p
, . . . , a
i,j
N,p
i
t
, (10)
and
G
i,j
[M]×[Q]
=
£
hb
i,j
m
, b
i,j
q
i
¤
, (11a)
D
i,j
[M]×[Q]
=
"
hb
i,j
m
,
db
i,j
q
dx
i
#
. (11b)
The subscripts of the matrices denote their size; for example in
Eq. (11), (m, q ) [1 : M] × [1 : Q]. We are, thus, led to the
computation of the eigenvalues β
2
p
and their associated eigenvectors
a
i,j
[1 : N2],p
of a matrix with dimension N
max
× N
max
, with
N
max
= (N 2)
N
X
i=1
N
i
. (12)
Progress In Electromagnetics Research, Vol. 133, 2012 21
We chose the basis functions formed by the Gegenbauer
polynomials [11] C
Λ
m
(ξ) defined over the interval [1, 1] as follows:
C
Λ
m
(ξ) =
1
Γ(Λ)
[m/2]
X
q=0
(1)
q
Γ(Λ + m q)
(q + 1)!(1 + m 2q)!
(2ξ)
m2q
, (13)
where Λ > 1/2 and m denoted the degree of the polynomials. The
Gegenbauer polynomials C
Λ
m
are m degree orthogonal polynomials on
the interval [1, 1] satisfying:
C
Λ
m
, C
Λ
n
®
=
Z
1
1
¡
1 ξ
2
¢
Λ
1
2
C
Λ
m
(ξ)C
Λ
n
(ξ) = δ
nm
h
Λ
n
, (14)
where δ
nm
denotes the Kronecker’s symbol and
h
Λ
n
= π
1
2
C
Λ
n
(1)
Γ(Λ +
1
2
)
Γ(Λ)(n + Λ)
, (15)
with
C
Λ
n
(1) =
Γ(n + 2Λ)
Γ(2Λ)Γ(n + 1)
,
C
Λ
n
(1) = (1)
n
C
Λ
n
(1).
(16)
These relationships will be essential for the boundary conditions
associated with the transition points in the x direction. It is important
to note that the parameter N is referred to the number of basis
functions over each layer. Consequently, the highest degree of these
polynomials is N 1.
3. INNER PRODUCT CALCULATION
3.1. Computation without the Weighting Function:
Numerical Instabilities and Gamma Function
In a first attempt and for sake of lightening the computations, the
inner product described by Eq. (14) can be simplified by removing
the weighting function (1 ξ
2
)
Λ1/2
. This is the approach adopted
in [10] and which led to very good convergence rates. However, and as
we mentioned in the introduction, numerical instabilities arise when
the number of polynomials is increased. Obviously, the first idea
that comes to mind is that the origin of the instabilities might have
something to do with the removed weighting function. In general,
the instabilities of an algorithm based on a modal method may come
either from: (i) the fact that the modal decomposition is not able
to represent the actual fields (especially their discontinuities), (ii) the
22 Edee et al.
numerical computation of the modes and especially at the level of the
inner products computations, and (iii) the resolution of the algebraic
system stemming from the boundary conditions. In the present case,
we suspect the accuracy of evaluation of the inner products given in
Eq. (11) that use the inner product of Eq. (14) (under its simplified
form, i.e., by removing the weighting function). In order to clarify
the situation, we return to the construction of the matrix G which
elements are G
mn
=< C
Λ
m
|C
Λ
n
>. Let’s consider as an example the
case of Λ = 0.5, where the elements G
mn
computed by convolving
and integrating the polynomials must be perfectly equal to the inner
product defined by Eq. (14); i.e., if the polynomials are normalized by
p
h
Λ
n
, G
mn
must be equal to δ
mn
. In Eq. (17) we give the numerical
computation of G elements for (m, n) [15 : 17] × [15 : 17]:
G
[15 : 17],[15 : 17]
=
"
1.0000 0 0.0000
0 1.0000 0
0.0000 0 1.0000
#
. (17)
It can be seen that the results are satisfactory for this range of
integers. Nevertheless, when m or n increases, the results become
highly unstable as is shown in matrix (18):
G
[29 : 31][29 : 31]
= 10
4
"
0.0193 0 1.4470
0 0.1740 0
0.8932 0 6.4372
#
. (18)
This behavior suggests that the instabilities come from the
manipulation of the coefficients of Gegenbauer polynomials. We
verified and confirmed this fact through the numerical calculation of
C
Λ
n
(1) by use of the expression (13). The numerical evaluation of the
sum in Eq. (13) can be diverging. Indeed, this expression contains
Gamma functions which numerical expression leads to very large
values. The numerical calculation of the ratio between the numerator
and denominator appearing in the expression of the coefficients of
monomials (13) rapidly tends to infinity as a function of q and m,
while in reality the fraction tends to a finite value. The same behavior
is observed for C
Λ
m
(1), (dC
Λ
m
/dξ)
ξ=1,1
, i.e., all values of C
Λ
m
(ξ)
and (dC
Λ
m
/dξ)
ξ
, which are essential for boundary conditions. One
alternative to solve this problem, which was proposed in [10], consists
in increasing the number of homogeneous layers
ij
for each subinterval
i
. Indeed, by doing so the number of Gegenbauer polynomials needed
for the field description on each layer decreases. Another alternative,
which is presented in Subsection 3.2, consists in analytically computing
all the terms needed for the matrix of diffraction. This will be done by
taking into account the weighting function in the inner products and
by examining these ones case by case.
Progress In Electromagnetics Research, Vol. 133, 2012 23
3.2. Analytical Computation of the Inner Products with the
Weighting Function
The construction of the matrix of diffraction entailed the calculation
of the terms:
hC
Λ
m
, C
Λ
n
i =
R
1
1
(1 ξ
2
)
Λ
1
2
C
Λ
m
(ξ)C
Λ
n
(ξ),
hC
Λ
m
,
dC
Λ
n
i =
R
1
1
(1 ξ
2
)
Λ
1
2
C
Λ
m
(ξ)
µ
dC
Λ
n
(ξ).
The computation of hC
Λ
m
, C
Λ
n
i is simple and directly deduced, in closed
form, from Eq. (14). For terms hC
Λ
m
,
dC
Λ
n
i, we first treat the terms
for (m, n = 0) and (m, n = 1) before dealing with the general case. It
is easy to verify that hC
Λ
m
,
d
C
Λ
0
i vanish for all m and, since,
dC
Λ
1
is
a constant, we can write:
¿
C
Λ
m
,
dC
Λ
1
À
=
dC
Λ
1
Z
1
1
¡
1 ξ
2
¢
Λ
1
2
C
Λ
m
(ξ). (19)
In the particular case of m = 0, the calculation of terms hC
Λ
0
,
dC
Λ
1
i,
leads to the following relationship:
¿
C
Λ
0
,
dC
Λ
1
À
= C
Λ
0
dC
Λ
1
Z
π
0
sin
θ. (20)
The integral of the right-hand of Eq. (20) is computed by using the
following expression, which involves Bessel and Gamma functions [11]:
J
Λ
(ξ)
µ
ξ
2
Λ
=
1
π
1
2
Γ(Λ +
1
2
)
Z
π
0
cos(ξ cos θ) sin
θ. (21)
For non-negative values of Λ and for small arguments ξ (ξ
£
0 :
Λ + 1
¤
), Bessel functions have the following asymptotic form:
J
Λ
(ξ)
µ
ξ
2
Λ
'
1
Γ(Λ + 1)
. (22)
By combining Eq. (22) and Eq. (21), when ξ is closed to 0, we obtain:
Z
π
0
sin
θ ' π
1
2
Γ(Λ +
1
2
)
Γ(Λ + 1)
. (23)
24 Edee et al.
Thus hC
Λ
0
,
dC
Λ
1
i has the following analytical expression:
¿
C
Λ
0
,
dC
Λ
1
À
' π
1
2
C
Λ
0
dC
Λ
1
Γ(Λ +
1
2
)
Γ(Λ + 1)
. (24)
For m 1, by using the integral Eq. (25) which involves Gegenbauer
polynomials:
m(2Λ+m)
Z
ξ
0
¡
1y
2
¢
Λ
1
2
C
Λ
m
(y)dy = C
Λ+1
m1
(0)
¡
1ξ
2
¢
Λ+
1
2
C
Λ+1
m1
(ξ), (25)
we easily demonstrate that hC
Λ
m
,
dC
Λ
1
i = 0, for all values of m 1.
At this stage, elements hC
Λ
m
,
dC
Λ
0
i and hC
Λ
m
,
dC
Λ
1
i, are known for all
values of m. In order to calculate terms hC
Λ
m
,
dC
Λ
n
i, when n 2 and
for all values of m, we introduce the following recursive relation:
C
Λ
n
(ξ) =
1
2(n + Λ)
d
£
C
Λ
n+1
(ξ) C
Λ
n1
(ξ)
¤
, (26)
which leads to
d
C
Λ
n
(ξ) = 2(n 1 + Λ)C
Λ
n1
(ξ) +
d
C
Λ
n2
(ξ). (27)
Consequently, terms hC
Λ
m
,
d
C
Λ
n
i are obtained as:
¿
C
Λ
m
,
dC
Λ
n
À
= 2(n 1 + Λ)
¿
C
Λ
m
, C
Λ
n1
À
+
¿
C
Λ
m
,
dC
Λ
n2
À
. (28)
Finally, we will need to use the following formula
µ
dC
Λ
n
(ξ) = C
Λ+1
n1
(ξ), (29)
which is essential for the boundary conditions;, i.e., to express the
continuity of the field derivative at the interfaces ξ = 1 and ξ = 1.
From Eq. (29), we obtain:
µ
dC
Λ
n
(ξ = 1) = C
Λ+1
n1
(1),
µ
dC
Λ
n
(ξ = 1) = C
Λ+1
n1
(1) = 2Λ(1)
n1
C
Λ+1
n1
(1).
(30)
Progress In Electromagnetics Research, Vol. 133, 2012 25
4. THE PLANE WAVE IN GEGENBAUER POLYNO-
MIALS BASIS AND BOUNDARY CONDITIONS IN THE
Y DIRECTION: S-MATRIX ALGORITHM
Usually when solving problems of diffraction from lamellar gratings,
with modal methods, one follows roughly the main steps consisting
in (i) solving Maxwell’s equations through an eigenvalue problem in
the incidence, the transmittance and the grating regions, (ii) writing
the appropriate boundary conditions (TE or TM) and (iii) solving
the resulting algebraic system through the S-matrix algorithm for
example. Except for the original Fourier Modal Method (FMM) where
the solutions in the homogeneous media are given by the classical
Rayleigh expansions, for the other modal approaches, it is necessary
to solve numerically at least one eigenvalue problem [12]. This can
lead to numerical difficulties if one is dealing with normal incidence
or the Littrow configuration where some eigenvalues are degenerate
which make it difficult to associate them with the appropriate orders.
This problem has been first encountered with the C-method [6] and
has been solved by simply replacing propagating plane waves by their
expressions in the new modal basis. For the sake of completeness
we give, in the following, the expressions of these waves in terms of
Gegenbauer polynomials. Let us consider the x dependence of the
function describing a plane wave:
X
m
(x) = e
ikα
m
x
. (31)
Let X
s
m
(x) be the restriction of X
m
(x) to the interval
s
= [a, b] and
ξ be the reduced variable in the interval [1, 1] defined as follow:
x =
b a
2
ξ +
b + a
2
. (32)
If we set
² =
b a
2
and δ =
b + a
2
, (33)
then
X
s
m
(ξ) = e
ikα
m
δ
e
ikα
m
²ξ
=
M
X
n=0
B
s
nm,Λ
C
Λ
n
(ξ), (34)
with
B
s
nm,Λ
=
e
ikα
m
δ
h
Λ
n
Z
1
1
(1 ξ
2
)
Λ
1
2
C
Λ
n
(ξ)e
ikα
m
ξ
. (35)
Since Fourier integrals of Gegenbauer polynomials can be expressed in
terms of Bessel functions, this integral can be finally expressed under
26 Edee et al.
the form:
B
s
nm,Λ
= Γ(Λ)
µ
2
kα
m
²
Λ
i
n
(n + Λ)J
n
(kα
m
²)e
ikα
m
δ
. (36)
Remark that, in the case of α
m
= 0, the orthogonal properties of
Gegenbauer polynomials can be used. The numerical resolution of the
wave equation, in a medium l defined by a refractive index function
ν
l
(x), gives [1 : N
max
] eigenvectors denoted by
Ψ
l
p
=
h
a
l,i
[1 : N2],p
i
i[1 : N
]
. (37)
In this nomenclature, i is referred to the subinterval
i
; i.e.:
a
l,i
[1 : N2],p
=
h
a
l,i,j=1
[1 : N2],p
a
l,i,j=2
[1 : N2],p
. . . a
l,i,j=N
i
[1 : N2],p
i
t
. (38)
The matrix of the eigenvectors Ψ
l
has consequently the following
form:
Ψ
l
=
£
Ψ
l
p
¤
p[1 : N
max
]
. (39)
The second components needed for boundary conditions in y direction,
i.e., E
x
in TM polarization and H
x
in TE polarization are represented
by the following vector Φ
l
p
:
Φ
l
p
=
·
β
l
p
η
i
a
l,i
[1 : N2], p
¸
i[1 : N
]
. (40)
According to the exp(ikβ
l
p
y) dependence and in order to satisfy the
outgoing Summerfeld condition, the root of eigenvalues β
l
p
are sorted
such that:
n
β
l
p
o
= U
+
U
(41)
with
U
+
=
n
β
l
p
β
l
p
R
+
or
³
β
l
p
C and =(β
l
p
) < 0
´o
, (42a)
U
=
n
β
l
p
β
l
p
R
or
³
β
l
p
C and =(β
l
p
) > 0
´o
. (42b)
The eigenvalues belonging to U
+
(resp U
) and their corresponding
eigenvectors are affected by the subscript + (resp ). According to
this convention, the S matrix of the interface separating the media l
and l + 1 has the following form:
S
l
=
"
Ψ
l
+
Ψ
l+1
Φ
l
+
Φ
l+1
#
1
"
Ψ
l+1
+
Ψ
l
Φ
l+1
+
Φ
l
#
. (43)
Progress In Electromagnetics Research, Vol. 133, 2012 27
5. NUMERICAL RESULTS AND DISCUSSION
In order to discuss the issue of stability of the MMGE we chose
a case known to be rather difficult for modal methods: a highly
conducting lamellar grating with the following parameters: ν
1
= 1,
ν
21
= 1 40i, ν
22
= 1, ν
3
= ν
21
, h = 0.4λ, d = 1.2361λ, f = 0.57,
and θ = arcsin(λ/2d). We give, as an example, the R
1
efficiency
computed via the classical MMGE (that we will designate MMGE1
from now on) without subdividing the two homogeneous regions
1/2
,
i.e., N
1
= N
2
= 1. For a shake of fluidity these results are given for
only one value of Λ(Λ = 0.5). Nevertheless, the conclusions deduced
from this study still valid for any value of Λ. Figure 2 summarizes
the results for both TE and TM polarizations as the total number of
polynomials N is varied.
As can be seen, the MMGE1 witnesses instabilities as the number
of polynomials is increased regardless of the polarization. To overcome
this problem, it has been proposed [10] to subdivide the subintervals
i
into layers. The upper part of the Table 1 contains the results
when each homogeneous subinterval is subdivided into four layers
(N
1
= N
2
= 4).
The gain in stability is clearly established. Nevertheless, in the
case of MMGE1 with subdivisions, one can notice that the size of the
matrices increases too rapidly in comparison with the former case; i.e.,
0 20 40 60 80 100
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Nmax
R
TE
TM
-1
Figure 2. Minus-first order reflected efficiency R
1
of highly
conductive metal in TE and TM polarizations. Instability of the results
obtained by MMGE1 for N
1
= N
2
= 1.
28 Edee et al.
Table 1. Minus-first order reflected efficiency R
1
of highly conductive
metal in TE and TM polarizations. Comparison between the results
obtained by MMGE1 for N
1
= N
2
= 4 and those obtained by MMGE2
for N
1
= N
2
= 1.
MMGE1
N N
max
TE polarization TM polarization
4 16 0.67443 0.95306
8 48 0.62024 0.52578
12 80 0.60925 0.78212
16 112 0.60875 0.79040
20 144 0.60875 0.79057
24 176 0.60875 0.79057
MMGE2
N N
max
TE polarization TM polarization
16 28 0.61503 0.58047
32 60 0.60873 0.79048
36 68 0.60875 0.79056
40 76 0.60875 0.79057
44 84 0.60875 0.79057
48 92 0.60875 0.79057
MMGE1 without subdivisions. This can be a serious drawback since
the eigenvalue problem is the most time consuming step in the MMGE
approach. The current version of the MMGE (MMGE2) removes this
disadvantage. The results obtained by the MMGE2 are presented in
the lower part of the Table 1 and show a remarkable stability. Under
TE polarization, the forth digit of R
1
= 0.6087 is stabilized as soon
as N
max
reaches 60 while it is necessary to use N
max
= 112 in order
to have the same result with the MMGE1 (N
1
= N
2
= 4). Under
TM polarization, the same behavior is observed on R
1
= 0.7905 as
soon as N
max
reaches 76; while it is necessary to use N
max
= 144 with
the MMGE1. Thus with the new implementation of the MMGE one
not only gains in stability but lowers the computational cost by using
smaller matrices.
In order to highlight how striking the improvement of convergence
brought by the MMGE is, we present in Figures 3(a) and 3(b) a
comparison between the results obtained by the Fourier Modal Method
and MMGE2 for three arbitrary values of Λ: Λ = 0.0005, 0.5 and 1.
Progress In Electromagnetics Research, Vol. 133, 2012 29
20 40 60 80 100
9
8
7
6
5
4
3
2
1
0
Nmax
=0.0005
=0.5
=1
FMM
0 50 100 150 200
10
8
6
4
2
0
Nmax
=0.0005
=0.5
=1
FMM
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ζ
ζ
Λ
Λ
Λ
Λ
Λ
Λ
(a) (b)
Figure 3. Minus-first order reflected efficiency of highly conductive
metal obtained by MMGE2 in (a) TE and (b) TM polarizations.
Comparison with Fourier modal method.
For that purpose, we introduced the error function:
ζ(N
max
) = log
10
| 1
R
1
(N
max
P
N
i=1
N
i
)
R
1
(N
max
)
|, (44)
that measures the relative variation of the efficiency.
After this discussion on the stability, let us turn to the study of
the influence of the parameter Λ in the MMGE2 implementation. It is
of fundamental importance to notice that this parameter acts through
the weighting function w(ξ), also known as the density function, by
introducing the measure (1 ξ
2
)
Λ1/2
in the integral of the inner
product. The density increases or decreases in the vicinity of ξ = ±1
depending on the value of Λ. When Λ is close to zero, it is easy to see
that the density increases around ξ = ±1; and in this case Gegenbauer
polynomials are similar to Chebychev ones. The particular case of
Λ = 0.5, which, corresponds to Legendre polynomials, involves an
equipartition on the interval [1, 1]. This density decreases around
ξ = ±1 when Λ is greater than 0.5. All this, is from a theoretical
point of view. In practice, it is reasonable to think, that for a given
grating problem at least one optimal value of Λ may exist that allows
describing the field at best. Indeed if we look to Tables 2 and 3, we find
that the convergence is slightly improved in the case of TM polarization
for Λ = 0.45 while Λ = 0.5 seems to be the best value in the case of
TE polarization.
Other numerical investigations (not reported here) concerning the
filling factor, the dielectric permittivity and the angle of incidence did
not allow us to draw a general rule about the choice of Λ. All we can
30 Edee et al.
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
groove width ( m)
minus first order efficiency
µ
Figure 4. Minus-first order reflected efficiency of a metallic
lamellar grating with respect to the groove width in TM polarization.
Numerical parameters: θ = 30
, ν
3
= ν
22
= 10i, ν
1
= ν
21
= 1,
d = h = 0.5 µm, λ = 0.6328 µm, N
max
= 20.
Table 2. Minus-first order reflected efficiency obtained by MMGE2 of
highly conductive metal in TE polarization. Influence of the parameter
Λ in the convergence of the results.
TE polarization
N
max
N Λ = 5e 4 Λ = 0.15 Λ = 0.25 Λ = 0.45 Λ = 0.5
50 27 0.60854 0.60858 0.60863 0.60877 0.60882
54 29 0.60865 0.60866 0.60868 0.60875 0.60877
58 31 0.60870 0.60871 0.60871 0.60874 0.60875
62 33 0.60873 0.60873 0.60873 0.60874 0.60875
66 35 0.60874 0.60874 0.60874 0.60874 0.60875
70 37 0.60874 0.60874 0.60874 0.60874 0.60875
74 39 0.60874 0.60874 0.60874 0.60874 0.60875
78 41 0.60874 0.60874 0.60874 0.60875 0.60875
82 43 0.60874 0.60874 0.60874 0.60875 0.60875
86 45 0.60874 0.60874 0.60874 0.60875 0.60875
90 47 0.60875 0.60875 0.60875 0.60875 0.60875
94 49 0.60875 0.60875 0.60875 0.60875 0.60875
Progress In Electromagnetics Research, Vol. 133, 2012 31
Table 3. Minus-first order reflected efficiency obtained by MMGE2 of
highly conductive metal in TM polarization. Influence of the parameter
Λ in the convergence of the results.
TM polarization
N
max
N Λ = 5e 4 Λ = 0.15 Λ = 0.25 Λ = 0.45 Λ = 0.5
50 27 0.79273 0.79229 7.9178 0.79023 0.78972
54 29 0.79164 0.79149 7.9128 0.79054 0.79028
58 31 0.79107 0.79102 7.9094 0.79060 0.79048
62 33 0.79079 0.79078 7.9075 0.79060 0.79054
66 35 0.79067 0.79067 7.9065 0.79059 0.79056
70 37 0.79061 0.79061 7.9061 0.79058 0.79057
74 39 0.79059 0.79059 7.9059 0.79058 0.79057
78 41 0.79058 0.79058 7.9058 0.79058 0.79057
82 43 0.79058 0.79058 7.9058 0.79058 0.79057
86 45 0.79058 0.79058 7.9058 0.79058 0.79057
90 47 0.79058 0.79058 7.9058 0.79058 0.79058
94 49 0.79058 0.79058 7.9058 0.79058 0.79058
assert is that the extreme values of the interval [0, 0.5] are far from
being the optimal in certain cases.
Finally, we test the stability of our approach over a grating
configuration that posed numerical problems to the FMM [13] and
for which solutions have been proposed by some authors [14, 15]. It
consists of a grating with a relative dielectric permittivity equal to
100 and all the other parameters are given in Figure 4.
The minus-first reflected order is drawn versus the groove width.
It is remarkable to see that this curve is obtained by N
max
as small as
20 without any instability.
6. CONCLUSION
In the present work, the modal method by Gegenbauer polynomials
expansions has been improved by getting rid of undesirable
instabilities. These ones have been shown to be directly linked to
the computation of the Gegenbauer polynomials through a series sum.
The introduction of a suitable weighting function in the inner product
gives an additional degree of freedom that allows for stable and efficient
evaluation of this latter. Thus the current version of the MMGE is
very stable, accurate and converges very rapidly. It clearly extends the
domain of action of modal methods with outstanding performances.
32 Edee et al.
APPENDIX A. AN EXAMPLE OF THE MATRICES
CONSTRUCTION IN THE CASE OF TWO MEDIA
1
AND
2
For the both media, the resolution of the wave equation leads to:
"
L
1
[N2]×[N]
0
0 L
2
[N2]×[N]
#"
a
1
[1 : N],p
a
2
[1 : N],p
#
= β
2
p
"
G
1
[N2]×[N]
0
0 G
2
[N2]×[N]
#"
a
1
[1 : N],p
a
2
[1 : N],p
#
. (A1)
By using the boundary conditions of Eqs. (4), (5) and (6), the
expansion coefficients of highest orders a
1
(N1),p
, a
1
N,p
, a
2
(N1),p
and
a
2
N,p
are expressed in terms of the coefficients of lower orders. For that
purpose let us set:
T
i
[n]
(x
0
, κ) = κ
h
b
i
[n]
(x
0
)
i
"
db
i
[n]
dx
#
x
0
. (A2)
Eqs. (4), (5) and (6) lead to
a
1
[N1,N],p
a
2
[N1,N],p
= T
a
1
[1 : N2],p
a
2
[1 : N2],p
, (A3)
where the matrix T is defined as follows:
T =
T
1
[N1,N]
(0, 1) T
2
[N1,N]
(0, 1)
T
1
[N1,N]
(fd d, τ) T
2
[N1,N]
(fd, 1)
1
T
1
[1 : N2]
(0, 1) T
2
[1 : N2]
(0, 1)
T
1
[1 : N2]
(fd d, τ) T
2
[1 : N2]
(fd, 1)
. (A4)
τ = e
ik sin θd
is the pseudo-periodic factor and f denoted the
filling factor of the lamellar grating. Therefore the vector
h
a
1
[1 : N],p
, a
2
[1 : N],p
i
t
of Eq. (A1) can be expressed in term of the
vector
h
a
1
[1 : N2],p
, a
2
[1 : N2],p
i
t
with the following matrix relation:
"
a
1
[1 : N],p
a
2
[1 : N],p
#
= C
"
a
1
[1 : N2],p
a
2
[1 : N2],p
#
. (A5)
Progress In Electromagnetics Research, Vol. 133, 2012 33
The matrix C can be written in the following concise form:
C =
·
C
11
C
12
C
21
C
22
¸
. (A6)
The matrices C
ii
of N × (N 2) size, describe the coupling between
higher and lower coefficients of a same subinterval namely C
11
in the
subinterval
1
and C
22
in
2
:
C
11
=
1 0 . . . 0
0 1
.
.
.
0
.
.
. . . .
.
.
.
.
.
.
0 0 . . . 1
T
1,1
T
1,2
. . . T
1,N2
T
2,1
T
2,2
. . . T
2,N2
, (A7)
C
22
=
1 0 . . . 0
0 1
.
.
.
0
.
.
. . . .
.
.
.
.
.
.
0 0 . . . 1
T
3,N2+1
T
3,N2+2
. . . T
3,2(N2)
T
4,N2+1
T
4,N2+2
. . . T
4,2(N2)
, (A8)
whereas C
ij
, i 6= j of N × (N 2) size take onto account the
interconnection between the subintervals:
C
12
=
0 0 . . . 0
0 0
.
.
.
0
.
.
. . . .
.
.
.
.
.
.
0 0 . . . 0
T
1,N2+1
T
1,N2+2
. . . T
1,2(N2)
T
2,N2+1
T
2,N2+2
. . . T
2,2(N2)
, (A9)
C
21
=
0 0 . . . 0
0 0
.
.
.
0
.
.
. . . .
.
.
.
.
.
.
0 0 . . . 0
T
3,1
T
3,2
. . . T
3,N2
T
4,1
T
4,2
. . . T
4,N2
. (A10)
34 Edee et al.
It is easy to show that
"
G
1
[N2]×[N]
0
0 G
2
[N2]×[N]
#
·
C
11
C
12
C
21
C
22
¸
=
"
G
1
[N2]×[N2]
0
0 G
2
[N2]×[N2]
#
, (A11)
consequently, Eq. (A1) is written as follows:
"
L
1
[N2]×[N]
0
0 L
2
[N2]×[N]
#
·
C
11
C
12
C
21
C
22
¸
"
a
1
[1 : N2],p
a
2
[1 : N2],p
#
= β
2
p
"
G
1
[N2]×[N2]
0
0 G
2
[N2]×[N2]
#"
a
1
[1 : N2],p
a
2
[1 : N2],p
#
. (A12)
Finally in the case of two media, we are led to the compu-
tation of the eigenvalues β
2
p
and their associated eigenvectors
h
a
1
[1 : N2],p
, a
2
[1 : N2],p
i
t
of a matrix with dimension N
max
= (2(N
2)) × (2(N 2)).
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... These cases often require a high number of Fourier basis functions to converge which increases the computational times. To remedy this situation, one can replace the FMM by the very efficient and well established Polynomial Modal Method (PMM) [30][31][32][33][34] equipped with Perfectly Matched Layers (PMLs); an approach that proves to be more efficient and robust even though at the price of a more complex numerical implementation. ...
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The classical adjoint-based topology optimization (TO) method, based on the use of a random continuous dielectric function as an adjoint variable distribution, is known to be one of the most efficient optimization methods that enable the design of optical devices with outstanding performances. However, the strategy for selecting the optimal solution requires a very fine pixelation of the permittivity function of the profile under optimization. Typically, at least 28 pixels are needed while optimizing a one wavelength wide 1D metagrating. This makes it very difficult to extend TO methods to large-scale optimization problems. In this paper, we introduce a new concept of adjoint-based topology optimization that enables fast and efficient geometry based design of both periodic and aperiodic metasurfaces. The structures are built from nano-rods whose widths and positions are to be adjusted. Our new approach requires a very low number of design parameters, thus leading to a drastic reduction in the computational time: about an order of magnitude. Hence, this concept makes it possible to address the optimization of large-scale structures in record time. As a proof-of-concept we apply this method to the design of (i) a periodic metagrating, optimized to have a specific response into a particular direction, and (ii) a dielectric metalens (aperiodic metasurface), enabling a high energy focusing into a well-defined focal spot.
... The polynomial modal method [2] is one such method. It was revisited in [3], [4] and [5]. It was shown to outperform all other numerical modal methods for dielectric metallic structures. ...
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We propose a simple semi analytical model that allows to compute the transmittance and reflectance of a one dimensional subwavelength graphene strip grating under an external static magnetic field. In this model graphene is treated as an anisotropic layer with atomic thickness and a frequency dependent complex permittivity tensor. The model is based on an effective medium approach (EMA) and a rigorous phase correction. The scattering matrix approach is also used to take into account the different resonant phenomena occurring in the structure. The approach is validated against the Polynomial Modal Method (PMM) through numerical examples.
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The classical adjoint-based topology optimization (TO) method, based on the use of a 1 random continuous dielectric function as design variable distribution is known to be one of the timely efficient and fast optimization methods enable a very high performance functional optical devices. It relies on the computation of the gradient of a figure of merit (FOM) with respect to the design parameters. The gradient of the figure of merit (FOM) may then be used to update the design vector element in several scenarios. One of the most common use scenario consists in 6 updating simultaneously all the design parameter vector elements. In a linear problem case involving a simply convex FOM-function shape, utilizing the gradient information, it is a relatively easy to reach an optimal solution. In the case of constrained and non linear problems stated in an infinite and indeterminate design space, the conventional TO, a local optimizer, may require multiple restarts, with multiple initial points and multiple runs. The algorithm strongly depends on the initial conditions. We report in the paper, a global-like optimizer inspired by a wolves-pack hunting, enabling efficient design of metasurfaces through their geometrical parameters. We apply the method to design a non periodic metasurface consisting of plasmonic metalenses, enabling a high energy flow focusing into a well-defined 2D focus spot. Numerical results show that the proposed inverse design method has a low sensitivity to initial conditions. In our design method of metalens, we optimize the full micro device at once, and demonstrate that the proposed method may provide both symmetric and more creative unexpected asymmetric on-axis metalenses even though under a normal illumination.
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A rigorous modal theory describing the diffraction properties of a dielectric lamellar grating is presented. The numerical implementation is shown to be suited to modelling the behaviour of high refractive index gratings. This suggests that an approach of this type may be successfully applied to the problem of lossy metallic lamellar gratings.
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This paper details to an electromagnetic modeling of an extreme ultraviolet (EUV) lithography mask. For that purpose, a modal method based on a spline nodal expansion (MMSNE) is presented. The results obtained using first, and second-order splines as basis functions are compared with those obtained using other modal methods, such as modal method by Fourier expansion (MMFE). The agreement between the results obtained using different methods is very good, and a convergence test is also performed. The spline nodal basis function implemented in this paper is the first step toward the realization of a multiresolution scheme that is expected to perform much more efficiently than conventional schemes.
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