Download Electromagnetic Wave Propagation in Periodic Porous Structures I

Electromagnetic Wave Propagation in Periodic
Porous Structures
I. David Abrahams1 and Gregory A. Kriegsmann2
1
School of Mathematics
University of Manchester
Oxford Road, Manchester M13 9PL, UK
2
Department of Mathematical Sciences
Center for Applied Mathematics and Statistics
New Jersey Institute of Technology
University Heights, Newark, NJ 07102
Abstract
We employ a homogenization procedure to describe the propagation of electromagnetic waves in a dielectric structure which is doubly-periodic in the X-Y plane
and of arbitrary variation in the direction of propagation, Z. The fundamental cell
is composed of an arbitrarily shaped pore filled with a dielectric and the host by
another.
Our analysis yields the structure of the electromagnetic fields at the micro level
and gives an effective medium equation at the macro level. The latter contains a
simple arithmetic average of the dielectric constants and a correction term which
involves a line integral around the pore. The integrand of this integral depends
upon the polarization of the wave and the solution to a canonical potential problem. We approximately solve this problem for small pore volumes and for large
contrasts, the pore dielectric constant being much larger than the host. We also
provide an equivalent variational formulation for the potential problem and use a
simple Raleigh-Ritz procedure to determine an approximate solution. For all these
approximations, we provide a simple macroscopic description of electromagnetic
wave propagation in our structure.
1. Introduction
In this paper we model and analyze the transmission, reflection, and propagation
of a plane electromagnetic wave through a composite slab of finite thickness. The
composite contains a host material with an index of refraction Nh2 and a periodic
arrangement of parallel dielectric cylinders with index Np2 . These cylindrical structures need not be circular and their cross sections can vary in in the direction, Z,
of propagation. The slab thickness and the wavelength of the incident microwave
2
are of the same order which is much larger than the period of the structure.
The problem considered here was originally motivated by a microwave assisted
chemical vapour process. In this setting the slab is a porous ceramic host where
Nh2 is complex and the dielectric cylinders are air filled pores with Np2 = 1. The
slab is put into an environment where a gas infuses into the pores and microwaves
are used to heat the host. The gas reacts with the heated pores, fills them, and
creates a composite material. In more realistic and complicated models of this
process [1] the slab is treated as a porous medium and linear mixing theory is used
to approximate the effective dielectric constant, or index of refraction. The periodic
porous structure studied in this paper allows a more refined approximation to this
constant and consequently will provide a basis for a more accurate model of the
heating process.
The structure we consider also has motivation in optical devices, such as quantumdot arrays [2]. Here, the host material is air with Nh2 = 1 and the cylinders are
semiconductors layered in the Z direction. In these devices the wavelength of the
incident light and the thickness of the structure are again of the same order, but
they are only a factor of three times larger than the period. Although the theory
developed in this paper requires these scales to be disparate, our results may offer
some qualitative information about these devices.
Mathematically and physically there are two approaches to formulate our problem. In the first, see e.g. [3], equations are derived for the electric and magnetic fields
in the direction of propagation, i.e. the Z direction. If the pores are circular and not
changing with Z and the fields are all proportional to eiΓZ , where Γ is the propagation constant, then applying the continuity of all tangential fields across the pore
boundary produces a doubly infinite system of algebraic equations. The solution of
these equations give the Fourier coefficients of the Z components of the electric and
magnetic fields. However, non-trivial solutions are obtained only for those values of
Γ that render a zero determinant. The complexity of this problem is exacerbated
by the fact that Γ appears in this determinant in a very nonlinear manner. This
approach was recently taken in [4] and several ad-hoc approximations were made to
determine Γ. Since this scheme required a truncation of the infinite system it produced a set of N values for the propagation constant Γi , i = 1, 2, 3...N where the
truncated system is N × N . Numerical calculations showed that the {Γi } clustered
about two distinct values of Γ, namely Γ1 and Γ2 . This suggested that there are
only two propagation constants for this problem.
The other and more fruitful approach is to compute the transverse electric and
magnetic fields and then deduce the fields in the Z direction. This is the approach
we take in this paper. We find that two potentials determine the transverse electric
and magnetic fields. These potentials satisfy homogeneous boundary conditions
across the pore contour C. In fact the potential that yields the magnetic fields has
3
two independent, elementary solutions, each giving rise to a particular polarization.
(This gives credibility to the suggestion just mentioned above.) Using this solution
along with the unknown potential for the electric fields we deduce the electric and
magnetic fields in the Z direction. From these fields and their periodicity we are
able to derive a simple macroscopic equation governing the propagation of electromagnetic waves through this structure. This equation is a second order ordinary
differential equation. It contains an effective index of refraction composed of two
parts. The first is just a mixture theory result and the second contains a contour
integral around C whose integrand depends upon a related potential function. This
function is harmonic everywhere in the cell, periodic on its boundaries, continuous across C, and has a nonhomogeneous jump in its normal derivative across this
curve. The forcing on this term is related to the polarization of the propagating
wave.
We shall now outline the remainder of this paper. Section 2 contains the description and formulation of our problem. Section 3 presents our homogenization
analysis of the Maxwell’s equations exploiting the smallness of the structure’s period to the incident wavelength. The result is the ordinary differential equation
described in the preceding paragraph. Two limiting cases are considered in Section
4; the dilute limit and the strong dielectric contrast limit. Approximate formulae
are given for these two scenarios. Section 5 contains a variational statement of the
microstructure potential problem. A simple one term Raleigh-Ritz approximation
is made to yield a simple effective index of refraction. Using the theory developed
in the preceding sections we study the scattering and transmission of a plane electromagnetic wave through a finite slab. These results are contained in Section 6.
Finally, Section 7 contains our conclusions.
2. Formulation
A plane electromagnetic wave propagates in the positive Z direction, where X,
Y , Z are Cartesian coordinates, and is normally incident upon a porous slab which
occupies the region 0 < Z < L. The wave is partially transmitted through the
structure and part is reflected. The pores are cylindrical in shape, with axes lying
parallel to the Z, have identical shape, and are distributed in a doubly-periodic
arrangement in the X-Y plane; the cross-section of the pores is arbitrary and can
vary with Z. The geometry of the problem is sketched in Figure 1 for the case of
circular pores.
In the analysis that follows we assume that the wavelength of the incident field λ
is much longer than LX and LY , the periodicity of the pore structure in the X and
Y directions respectively. However, λ is of the same order as the slab thickness L.
This naturally introduces the small parameter δ = LX /L which we shall exploit on
4
2
N h
R 0
L
Y
N
2
p
LX
Figure 1. The doubly periodic arrangement of pores indicating the fundamental cell.
developing our theory. We also introduce the cell aspect ratio l = LY /LX , which
we take to be an O(1) quantity, and without loss of generality we can henceforth
choose l ≥ 1.
The (steady-state) electromagnetic fields are governed by Maxwell’s equations,
which are given in dimensionless form by
∂E3
∂E2
−
= ikH1 ,
∂y
∂z
(1a)
∂E1
∂E3
+
= ikH2 ,
∂x
∂z
(1b)
∂E2 ∂E1
−
= ikδ 2 H3 ,
∂x
∂y
(1c)
∂H3
∂H2
−
= −ikN 2 E1 ,
∂y
∂z
(2a)
−
∂H1
∂H3
+
= −ikN 2 E2 ,
∂x
∂z
(2b)
∂H2 ∂H1
−
= −ikδ 2 N 2 E3 ,
∂x
∂y
(2c)
−
5
where the wavenumber k = 2πL/λ and N 2 is the index of refraction which takes the
value Np2 in the pore regions and the value Nh2 in the host medium. In equations
(1-2) the dimensionless independent spatial variables are defined by x = X/LX ,
y = Y /LX , and z = Z/L, and the fields by
Ej = Ej′ /E0 ,
Hj = Z0 Hj′ /E0 ,
E3 = δ −1 E3′ /E0 ,
j = 1, 2,
(3a)
H3 = δ −1 Z0 H3′ /E0 ,
(3b)
p
where Z0 = µ0 /ǫ0 , µ0 is the magnetic permeability of free space, ǫ0 is the permittivity of free space, E0 is the strength of the incident wave, and the primes denote
the dimensional electromagnetic fields. For simplicity of analysis, the pores and
host have both been taken to have the permeability of free space. We note here
that the fields in the direction of propagation have been scaled by δ, that is, they
are assumed small compared to the transverse fields in the x-y plane; we shall show
below that this scaling yields a consistent asymptotic result. Due to the periodicity
of the structure and the normal incidence of the impinging wave it is enough to find
the periodic solution to (1-2) in the fundamental cell shown in Figure 2. The region
of the cell occupied by the host is denoted Rh , occupied by the pore is RP , and the
whole cell is R ≡ Rh ∪ Rp .
3. Analyses
In the following analyses we will only be concerned with the leading order (homogenized) electromagnetic fields in the limit as δ → 0. Accordingly, we may set
δ = 0 into (1-2). From (1c) we then find that
∂E1
∂E2
−
= 0.
∂x
∂y
(4a)
Next, we take the partial derivative of (2a) with respect to x, the partial derivative
of (2b) with respect to y, and add the resulting expressions. Using (2c) with δ = 0,
this resulting expression becomes
∂ N 2 E1
∂ N 2 E2
+
= 0.
(4b)
∂x
∂y
Introducing the potential function Φ and setting E1 = ∂Φ/∂x and E2 = ∂Φ/∂y we
find that (4a) is satisfied and (4b) yields
∇ · N 2 ∇Φ = 0,
(x, y) ∈ R,
(5)
6
FUNDAMENTAL CELL
2
N
h
R
l
r0
P
N
2
p
R
H
1
Figure 2. The non-dimensionalized fundamental cell.
where R denotes the region occupied by the fundamental cell. The gradient and
divergence operators are with respect to the transverse variables x and y only; this
notation will be followed for the remainder of this paper.
We now follow a similar path using (2c) and (1a-1c) to find that
∂H2
∂H1
−
= 0,
∂x
∂y
(6a)
∂H2
∂H1
+
= 0.
∂x
∂y
(6b)
Introducing a second potential function Ψ and setting H1 = −∂Ψ/∂y and H2 =
∂Ψ/∂x, then (6b) is satisfied and (6a) yields
∇2 Ψ = 0,
(x, y) ∈ R.
(7)
We note that, for conciseness, here and henceforth the functional dependence of Φ
and Ψ on z is implied but not written explicitly.
7
If the potentials Φ and Ψ can be found then the electric and magnetic fields in
the z direction can be deduced. This follows by substituting the potentials for the
transverse fields in (1a-b) and (2a-b); the first yields
E3 =
and the second gives
∂Φ
− ikΨ + Q(z)
∂z
(8a)
∂ 2Ψ
∂Φ
∂H3
=
− ikN 2
,
∂y
∂z ∂x
∂x
(8b)
∂2Ψ
∂Φ
∂H3
=−
+ ikN 2
,
∂x
∂z ∂y
∂y
(8c)
where Q(z) is an unknown function at this stage. Next, we deduce the boundary
conditions that Φ and Ψ must satisfy in the unit cell, R. First, the electric and
magnetic fields must be periodic in both x and y directions. Since the transverse
fields are given by the x and y derivatives of the potentials, it follows that these are
periodic too. In addition, the required periodicity of E3 implies that the right hand
side of (8a) is also periodic. Secondly, we demand that the tangential electric and
magnetic fields in the x-y plane are continuous across the boundary C of the pore.
Expressing these transverse tangential fields in terms of their potentials we deduce
(see Appendix 1), respectively, that
∂Ψ
= 0, (x, y) ∈ C,
(9a)
[Φ]C =
∂n C
where the notation [ ]C denotes the jump in the bracketed quantity across the pore
boundary C, and n is a coordinate locally normal to the curve C and measured
positive on the host material side. Thirdly, the tangential components of the electric
and magnetic fields in the longitudinal direction are also continuous across the pore
boundary. In Appendix 1 we show that this implies
2 ∂Φ
N
= 0, [Ψ]C = 0,
(x, y) ∈ C.
(9b)
∂n C
Now the solution of (7) which satisfies [Ψ]C = [∂Ψ/∂n]C = 0, with periodic
∂Ψ/∂x and ∂Ψ/∂y, is
Ψ = a(z)x + b(z)y,
(10a)
in which a(z) and b(z) are functions to be determined. This potential then gives
H1 = −b(z),
H2 = a(z),
(x, y) ∈ R,
(10b)
8
that is, the magnetic fields are constant throughout R in any plane with z constant.
For the moment we take b = 0 and consequently, H1 = 0 and H2 = a(z). Then the
periodicity of E3 given by (8a) implies
∂Φ
∂Φ
(x, 0) =
(x, l),
∂z
∂z
0 < x < 1,
∂Φ
∂Φ
(0, y) =
(1, y) − ika(z),
∂z
∂z
These can be integrated to give
Φ(x, 0) = Φ(x, l) + h(x),
(11a)
0 < y < l.
0 < x < 1,
Φ(0, y) = Φ(1, y) − ikA(z) + g(y),
(11b)
(11c)
0 < y < l,
(11d)
where dA/dz = a. The functions h(x) and g(y) are, in fact, constants. To see this
we differentiate (11c) with respect to x and obtain h′ = Φx (x, 0) − Φx (x, l). Since
Φx = E1 and this field is periodic in y, we find h′ = 0. A similar calculation holds
for g(y). We may therefore take both these constants to be zero as they have no
effect on the fields E1 and E2 .
For piecewise constant refractive index, N 2 , we can define a new potential P by
Φ = ikA{x − [N 2 ]C P }.
(12)
Inserting this into (5) and using [Φ]C = [N 2 ∂Φ/∂n]C = 0 we find that P satisfies
∇2 P = 0,
[P ]C = 0,
N
2 ∂P
∂n
(x, y) ∈ R,
=
C
∂x
,
∂n
(13a)
(x, y) ∈ C,
(13b)
and employing (11c) and (11d):
P (x, 0) = P (x, l),
0 < x < 1,
P (0, y) = P (1, y),
0 < y < l,
(13c)
that is, P is doubly periodic. If this problem can be solved then from (12) we have
∂Φ
∂P
2
E1 =
,
= ikA 1 − [N ]C
∂x
∂x
E2 = −ikA[N 2 ]C
∂P
.
∂y
(14a)
(14b)
9
We note here that the periodicity of E1 and E2 imply the same for ∂P/∂x and
∂P/∂y, respectively.
To determine the amplitude function A we return to (8b) which becomes, on
account of (10b) and a = dA/dz,
d2 A
∂Φ
∂H3
=
− ikN 2
.
2
∂y
dz
∂x
(15a)
Next, we integrate (15a) along the straight line x = x0 , 0 < y < l, where x0 is
chosen to ensure that the line is in the region Rh , i.e. it does not intersect the pore
region Rp . Since H3 must be periodic in y, the result is
d2 A
l 2 − ikNh2
dz
Z
0
l
∂Φ(x0 , y)
dy = 0.
∂x
Inserting (12) into this relationship and dividing by l we find that
d2 A
+ k 2 Nh 2
2
dz
(
1
1 − [N 2 ]C
l
Z
l
0
)
∂P
(x0 , y) dy A = 0.
∂x
(15b)
We shall now recast the differential equation (15b) into a more useful form. First,
we note that the integral in (15b) is independent of x0 . To see this we define I(x) =
Rl
∂P/∂x(x, y) dy. Since P is harmonic in Rh , it follows that dI/dx = −∂P/∂y|l0 .
0
The periodicity of ∂P/∂y yields our result. Thus, the integrand in (15b) can be
evaluated, without loss of generality, at x = x0 = 1. Next, since P and x are
harmonic functions in R we deduce ∇ · {N 2 P ∇x − N 2 x∇P } = 0. Integrating this
expression in the region Rh , and using the periodicity of P , Px , and Py , we obtain
Nh2
I
∂P +
x
ds − Nh2
∂n
I
P
+ ∂x
∂n
ds =
Nh2
Z
0
l
∂P
(1, y) dy
∂x
(15c)
H
where denotes counter clockwise integration on C, the direction derivative ∂/∂n
is defined to point normally into the host region, and the superscript + denotes
evaluation on the Rh side of the pore. We integrate the same expression in the pore
region Rp and find
−Np2
I
∂P −
ds + Np2
x
∂n
I
P−
∂x
ds = 0
∂n
(15d)
10
where the superscript − denotes evaluation on the Rp side of the pore. Adding
equations
(15c) and (15d), employing (13b), and noting (via the divergence theorem)
H
that x∂x/∂n ds = Ap , where Ap is the pore area, we arrive at
Nh2
Z
0
l
∂P
(1, y) dy = Ap − [N 2 ]C
∂x
I
P
∂x
ds.
∂n
Finally, combining this result with (15b) we find that A satisfies
I
1 22
∂x
d2 A
2
2
+ k < N > + [N ]C P
ds A = 0
dz 2
l
∂n
(15e)
(16a)
where
Ap 2
Ap 2 Ah 2
[N ]C =
N +
N ,
(16b)
l
l p
l h
Ah is the area of Rh and l is the area of the fundamental cell. Equation (16a) is an
effective medium equation for our periodic, porous material. If the pore shape is
independent of z, then the bracketed term in (16a) is the effective (constant) index
of refraction. It is the sum of a simple mixture relationship (16b) and a contour
integral, which involves the microstructure of the medium.
We note here several observations about our results to this point. First, if Ap = 0,
then there is no pore region and < N 2 >= Nh2 . Also, the line integral in (16a) is
gone so that (16c) reduces to d2 A/dz 2 + k 2 Nh2 A = 0. Furthermore, [N 2 ]C = 0,
since there is no contour, and (14) reduces to E1 = ikA and E2 = 0. Recalling that
H2 = dA/dz and H1 = 0, our results must reduce to transverse electromagnetic
(TEM) propagation through a homogeneous slab with the electric field polarized
in the x direction. This requires that Q = 0 in (8a). Our next observation is, if
[N 2 ]C = 0, then Np2 = Nh2 and the results are the same as those described in the
preceding paragraph. Out third observation is, if Ah = 0, then < N 2 >= Np2 and
the line integral in (16a) is now around the perimeter of the fundamental cell. This
integral vanishes by the periodicity of P and (16a) reduces to d2 A/dz 2 +k 2 Np2 A = 0.
The periodic solution to (13) is now P = 0, so that (14) again gives E1 = ikA and
E2 = 0. The magnetic fields are still given by H1 = 0 and H2 = dA/dz. Thus, we
again have a TEM electromagnetic wave with the electric field polarized in the x
direction.
We close this section by considering the case where Ψ = b(z)y. All of our analysis
carries over and the results are the same with a few important exceptions. First,
equation (14) is replaced by
< N 2 >= Nh2 −
E1 =
∂P
∂Φ
= −ikB[N 2 ]C
∂x
∂x
(17a)
11
∂Φ
∂P
2
E2 =
= ikB 1 − [N ]C
∂y
∂y
(17b)
where B = db/dz and P satisfies all of (13) with the exception that the second
equation in (13b) is replaced by
N
2 ∂P
∂n
=
C
∂y
,
∂n
(x, y) ∈ C.
(17c)
Secondly, the function ∂x/∂n in the line integral in (16a) is replaced by ∂y/∂n. All
of the observations made above for the limiting cases of Ap = 0, [N 2 ]C = 0, and
Ah = 0 still hold yielding a TEM electromagnetic wave polarized with the electric
field in the y direction, i.e. E1 = 0, E2 = ikB, H1 = dB/dz, and H2 = 0.
4. Further Limiting Cases
In addition to the limiting cases briefly described in the previous section, there
are two other physical scenarios where the integral in (16a) can be approximated
or easily computed. In the first, the pore is circular and its radius is r0 ≪ 1. This
is called the small pore, or low concentration, limit. If we were to rescale (13a-b)
so that the pore had unit radius, then the cell boundaries would, to leading order,
be scaled off to infinity. The periodic boundary conditions (13c) would be replaced
by the requirement that P is bounded away from the pore. The small pore limit is
often called the dilute limit, or limit of low area fraction, where the area fraction,
φ, is the ratio of the pore area to the cell area, i.e. φ = Ap /l = πr02 /l. The solution
to this limiting case, which satisfies the requisite conditions (13a-b), is

r,
cos θ 
2
P =− 2
Np + Nh2  r0 ,
r
r < r0 ,
r0 < r.
(18a)
To extend this result to incorporate the periodic boundary conditions, i.e. for higher
area fractions, would require an application of a technique such as the method of
matched asymptotic expansions or a complex variable approach (see e.g. Parnell &
Abrahams [5]). For ease of exposition we restrict attention to just the dilute pore
limit in this article. We may now compute the integral expression in (16a) using
(18a), which after simplification yields
[N 2 ]C
d2 A
2 2
A = 0.
+ k Nh 1 − 2φ 2
dz 2
Np + Nh2
(18b)
12
If the pore radius is independent of z, then the bracketed term in (18b) is the
effective index of refraction. It is a small perturbation in the area fraction φ.
In the second physical scenario we take Np2 ≫ Nh2 , that is, there is a large dielectric contrast between the pore and the host materials. Then the second boundary
condition in (13b) becomes approximately
1 ∂x
∂P −
=− 2
,
∂n
Np ∂n
(x, y) ∈ C,
(19a)
where the superscript − denotes evaluation of the normal derivative just inside the
curve C. In the pore region Rp we again have ∇ · {P ∇x − x∇P } = 0. Integrating
this expression in Rp , using the divergence theorem, we find that
I
∂x
P
ds =
∂n
I
x
∂P −
ds.
∂n
(19b)
H
Inserting (19a) into the right hand side of (19b), recalling that x∂x/∂n ds = Ap =
φl, and using this result in (16a) we obtain after some simplification
d2 A
+ k 2 Nh2 {1 + φ}A = 0.
dz 2
(19c)
If the pore fraction φ is independent of z, then the bracketed term is again the
effective index of refraction; note that it only depends on Nh . Thus, the wave speed
is essentially determined by the material with the higher propagation speed. We
observe that (19c) cannot be derived from (18b) by taking the limit Np2 /Nh2 → ∞,
that is, the limits of small area fraction and large contrast can not be interchanged.
5. A Variational Approach
The propagation of electromagnetic waves through our periodic porous medium
is governed by (16) which depends intimately on P , the solution to the boundary
value problem (13). We believe that there are no exact solutions to this problem.
Although evaluation of P can be obtained to a high degree of accuracy by a number
of numerical methods, such calculations are likely to become intensive if the pore
shape changes in z. In this section we present a variational approach, which leads
to a simple approximation.
First, we introduce the functional I(Q) defined by
ZZ
I
∂x
2
2
ds
(20)
I(Q) =
N |∇Q| dx dy + 2 Q
∂n
R
13
y
1
r
2
Np
0
r= 1/2
N2
h
x
1
Figure 3. The cell geometry for the trial function P0 .
where Q is taken from the set D of functions which are periodic on the cell R,
continuous there, and possess piecewise smooth partial derivatives. Note that the
line integral runs over the boundary of the pore. We may use standard calculus of
variations arguments to deduce that the function which minimizes I is the solution
to (13), and conversely, the solution of (13) minimizes I(Q). This allows us to
use a Raleigh-Ritz approach to reduce (13) to a finite dimensional linear algebra
problem. The procedure is straightforward and we do not present the general case
here. Rather, we shall choose a single function from D, and minimize I with respect
to it, i.e. employ a special one-dimensional Raleigh-Ritz approximation.
For simplicity, we take the cell to be square, i.e. l = 1 and the pore to be circular
with radius r0 and center (1/2, 1/2). We then introduce a larger circle with radius
1/2 and the same center as the pore; see Figure 3. As a trial function we take

1

 (1 − 2 )r,
r < r0 ,


4r0

2
4r0 cos θ
P0 = 2
(21a)
1
r0 < r < 1/2,
(r − ),
Np + Nh2 + 4r02 [N 2 ]C 


4r


0,
elsewhere.
This function is harmonic within the larger circle, satisfies the boundary conditions
14
(13b), vanishes on the circumference of the larger circle and in the corners of the
remaining region. We now take Q = αP0 and choose α to minimize I(Q). Inserting
this value of Q into (20) and setting its derivative with respect to α to zero yields
∂x
ds
∂n
α = − RR
,
N 2 |∇P0 |2 dx dy
R
H
P0
(21b)
where the contour integral is taken around the circular pore. A straightforward
calculation of the integrals in (21b) yields α = 1. Then, when P0 is inserted into
the integral in (16a) the approximate amplitude equation becomes
d2 A
+ k 2 Ω2 A = 0,
2
dz
I
∂x
2
2
2 2
Ω =< N > +[N ]C P0
ds.
∂n
(21c)
(21d)
Finally, we carry out the integration required in (21d) and find
Ω2 = Nh2
(
2φ[N 2 ]C
1−
Np2 + Nh2 + 4 πφ [N 2 ]C
)
(21e)
where now φ = Ap = πr02 , since l = 1, and r0 may depend upon z.
In Figure 4 we have plotted Ω2 /Nh2 as a function of Np2 /Nh2 for three different
values of r0 . Each curve passes through the point (1, 1); that is when Np = Nh , Ω2 =
Nh2 as expected. As r0 increases, the effect of the pore becomes more pronounced.
The opposite is true as r0 decreases. In each case, Ω2 → γ 2 (r0 ) as Np /Nh → ∞,
where γ 2 = 1 + 2Ap /(1 − 4r02 ). This behavior is qualitatively the same as given by
(19c), but the asymptote differs by the factor 2/(1 − 4r02 ). This error is a result
of employing the simple test function P0 . Finally we note that Ω2 given by (21e)
reduces to the dilute limit (18b) as φ → 0.
6. Propagation and Reflection
Up until now we have been concerned with developing a theory for electromagnetic wave propagation through a periodic, porous medium. In this section the
problem of propagation through, and reflection from, a finite slab of this material
is analyzed and simple formulae for the reflection and transmission coefficients will
be derived.
15
3
r0 = 2/5
r0 = 1/3
r0 = 1/6
2
2
Ω /N h
2
1
0
0
1
2
2
N p/N
3
2
h
Figure 4. The effective wavenumber against contrast of
refractive indices.
We begin by considering an incident plane electromagnetic wave whose electric
field is polarized in the x direction (i.e. E2 = 0, H1 = 0). In the fundamental cell,
but external to the porous slab, the electric field E1 and the magnetic field H2 may
be written, respectively, as the eigenfunction expansions
X
E1 = eikz + ρe−ikz +
e1nm ψnm eknm z ,
(22a)
n,m
H2 = eikz − ρe−ikz +
X
h2nm ψnm eknm z ,
(22b)
n,m
where ρ is the reflection coefficient,
p the infinite sums omit the n = m = 0 term,
2πi(nx+my/l)
ψnm = e
, and knm = 4π 2 (n2 + m2 /l2 )/δ 2 − k 2 assuming the waves
propagate in a vacuum (i.e. N = 1). We note that the first two terms, i.e. the
m = n = 0 terms, in (22) represent the incident and reflected plane waves respectively. The form of these fields (22) follows from the fact all the components of the
electromagnetic fields satisfy the same Helmholtz equation, namely
Exx + Eyy + δ 2 (Ezz + k 2 N 2 E) = 0,
as may be derived from equations (1) and (2). We note here that the infinite sums
contain only evanescent modes, which decay rapidly away from the interface between
16
p
n2 + m2 /l2 ≫ 1. The other
the slab and free-space. This is because knm ∼ 2π
δ
components of the electric and magnetic fields, i.e. E2 , E3 , H1 , and H3 , contain
only evanescent terms.
Now in the porous slab, we take Ψ = xdA/dz and accordingly H2 = dA/dz
and E1 is given by (14a). Since E1 and H2 are tangential fields at the interface
z = 0, they are continuous across it. Thus, we equate them to (22a) and (22b),
respectively, at z=0 and find
X
∂P
2
,
(23a)
1+ρ+
e1nm ψnm = ikA(0) 1 − [N ]C
∂x
n,m
1−ρ+
X
h2nm ψnm =
n,m
dA
(0).
dz
(23b)
Next, these equations are integrated over the region R at z = 0 to give, respectively
ZZ
2
l(1 + ρ) = ikA(0) l − [N ]C
∂P/∂x dx dy ,
(24a)
R
l(1 − ρ) = l
dA
(0).
dz
(24b)
RR
Equation (24a) simplifies to (1 + ρ) = ikA(0), since R ∂P/∂x dx dy = 0. This fact
is readily deduced from integrating the quantity ∇ · (P ∇x) over the region R, using
the divergence theorem and the continuity of P across the pore interface. Finally,
combining the simplified version of (24a):
ρ = ikA(0) − 1
(25a)
with (24b) we deduce
dA
(0) + ikA(0) = 2.
(25b)
dz
We can repeat the same analysis at z = 1, the other interface between the
porous medium and free space. In free-space beyond the slab, z > 1, the outgoing/evanescent transverse fields may be written as
E1 = τ eikz +
X
ē1nm ψnm eknm z ,
(26a)
X
h̄2nm ψnm eknm z ,
(26b)
n,m
H2 = τ eikz +
n,m
17
where τ is the transmission coefficient; as before, the electric field is polarized in
the x direction. Again integrating these over R, and equating on z = 1 with the
field inside the slab we deduce
dA
(1) − ikA(1) = 0,
dz
(27a)
τ = ike−ik A(1).
(27b)
Now, equations (25b) and (27a) are the boundary conditions required by the
amplitude equation (16a). This is a standard two-point boundary value problem. If
the pore shape is independent of z, and the refractive indices are constant throughout the slab, then it can be solved exactly; omitting details we find in this uniform
case
(K + k)eiK(z−1) + (K − k)eiK(1−z)
A(z) =
(K 2 + k 2 ) sin K + 2iKk cos K
in which
I
∂x
1 22
2
ds ,
K = k < N > + [N ]C P
l
∂n
2
2
with < N 2 > given in (16b) and the reflection and transmission coefficients are
ρ=
(k 2 − K 2 ) sin K
,
(K 2 + k 2 ) sin K + 2iKk cos K
τ=
2iKke−iK
.
(K 2 + k 2 ) sin K + 2iKk cos K
Note that, if K = k, then the slab has the same effective wavenumber as that in
free-space, and we find ρ = 0, τ = 1 as expected. If the pore shape depends on z
then, in general, numerical methods must be employed to obtain an approximate
solution to A(z). In either case, the reflection and transmission coefficients are
determined uniquely from the boundary conditions at 0 and 1.
7. Conclusions
We have presented a leading order homogenization analysis of electromagnetic
wave propagation through a composite slab of finite thickness. Our analysis gives
an explicit description of the electromagnetic fields within the slab. Specifically,
the field components are proportional to either the amplitude A(z) or its derivative
dA
. This amplitude satisfies a second order differential equation with an effective
dz
18
index of refraction, composed of two parts. The first is a linear combination of the
host and pore indices of refraction, weighted by the relative areas of the host and
pore, respectively. This is just a simple mixture term. The second part contains a
line integral whose integrand depends upon the wave polarization and a harmonic
function P . This potential function is doubly periodic in the plane z = constant, as
are its derivatives, and satisfies an inhomogeneous jump condition across the pore
interface. This term describes the microstructure effect on the effective index of
refraction.
We have considered three approximations of this potential function, each yielding
an approximate, effective index. In the first case, the pore area was assumed to be
small compared to that of the host, i.e. the dilute concentration limit. The effective
index in this case is Nh2 with a small correction due to the pores. In the second case,
the index of the pore was assumed to be much larger than that of the host. The
effective index in this case is Nh2 [1 +φ] where φ is the relative area of the pore to the
unit cell. In the third case we made use of an equivalent variational statement of our
potential problem. A simple one-term Raleigh-Ritz approximation was employed to
produce an approximate effective index of refraction. We are presently investigating
various numerical methods to better approximate our potential function, and hence
the effective index of refraction.
Finally, we note here that our analysis can easily be extended to handle a smooth
2
N . In this case Φ still satisfies (5), but now P is defined by Φ = ikA{x − P } for
the x polarization, where, as in Section 3, A is a function of z only. It follows that
P now satisfies
∂ 2
N
(28)
∇ · (N 2 ∇P ) =
∂x
and the same periodic boundary conditions. The derivation of the ordinary differential equation for A follows along a parallel path. The exception is that the
∂
identity ∇ · (P N 2 ∇x − xN 2 ∇P ) = (P − x) N 2 is now integrated around the
∂x
entire fundamental cell. Omitting the details it is easy to demonstrate that A
satisfies
ZZ
∂ 2
1
d2 A
2
2
P
+k < N >+
N dx dy A = 0,
(29a)
dz 2
l
∂x
R
where
1
< N >=
l
2
ZZ
N 2 dx dy,
(29b)
R
where R denotes the fundamental cell. Again the effective index of refraction contains an average, or mixture, term and a part that depends upon the microstructure
of the electric field.
19
Acknowledgments
The work of G. A. Kriegsmann was sponsored by the Department of Energy under
grant number DE-FG02-04ER25654.
Appendix 1.
In this Appendix we derive the boundary conditions given in equations (9a) and
(9b) in the text. To begin we describe our pore surface by the dimensional equation
XP = LX r0 (θ, Z/L)n̂ + Z k̂,
(A1)
where n̂ = (cos θ, sin θ, 0) and r0 is a smooth function of its arguments and is 2π
periodic in θ. This is a very general pore surface, which reduces to a circular
structure when r0 is independent of θ. We observe that the intersection of the
surface with the plane z = Z/L = constant is the curve C shown in Figure 2 (in
the case when r0 is independent of θ).
The unit tangents to our pore surface are given by
r0θ n̂ + r0 t̂
,
T̂1 = p 2
2
r0 + r0θ
δr0z n̂ + k̂
,
T̂2 = p
2
1 + δ 2 r0z
(A2)
(A3)
where t̂ = (− sin θ, cos θ, 0) and r0θ , r0z indicate partial derivatives of r0 with respect
to the specified variable. We note that the vector T̂1 lies in the plane z = constant
and is tangent to the curve C. The small parameter, δ = Lx /L, in T̂2 implies that
the pore shape is slowly changing in z. Taking the curl of these two tangent vectors
gives the unit normal to the pore surface
p
r 2 + r 2 n̂1 − δr0 r0z k̂
,
(A4)
N̂ = p0 2 0θ2
2 δ2
r0 + r0θ + r02 r0z
p
2 ) is the unit normal to the curve C, i.e. n̂ · T̂ =
where n̂1 = (−r0θ t̂+r0 n̂)/ r02 + r0θ
1 1
0.
We now have the prerequisite geometry to derive the boundary conditions (9).
Recall from (3) that the dimensional electric field is given by E′ = E0 {E1 î + E2 ĵ +
δE3 k̂}. Expressing the Ei in terms of the potentials Φ and Ψ we have
E′ = E0 {
∂Φ
∂Φ
î +
ĵ + δ(Φz − ikΨ)k̂}.
∂x
∂y
(A5)
20
From Maxwell’s equations one can deduce [3] that the tangential component of the
electric field along T̂1 must be continuous across the surface. Taking the dot product
of E′ with this unit vector gives [∇Φ · T̂1 ]C = 0, where ∇ is the two-dimensional
gradient operator given in terms of x and y only. But this is just the directional
derivative of Φ along the curve C. From this observation we deduce that [Φ]C =
a constant which, without loss of generality, we take to be zero. This is the first
boundary condition in (9a).
Similarly from (3) the dimensionless magnetic field is given in terms of the potential Ψ and H3 by
H′ = H0 {−Ψy î + Ψx ĵ + δH3 k̂}.
(A6)
Its component along T̂1 must also be continuous across the pore surface, see [4].
Taking the dot product of H′ and T̂1, noting
the definition of n̂1 , and using the
∂Ψ
= 0. This is the second boundary
same reasoning as above we deduce
∂n C
condition in (9a) where the subscript denotes the normal derivative of Ψ across the
curve C.
To obtain the two remaining boundary conditions in (9b) we can proceed along
two paths. The first would be to demand the continuity of the electric and magnetic
field components along T̂2 across the surface. A second and equivalent approach,
see [4], is to require that N 2 (E′ · N̂ ) and H′ · N̂ are continuous across the pore
surface. We choose the latter for ease of presentation and calculation. We find by
taking the dot product of (A5) with (A4) and multiplying the result by N 2 that
∂Φ
+ O(δ 2 ),
(A7)
N 2 (E′ · N̂ ) = N 2
∂n
where the O(δ 2 ) involves the component of E′ along k̂. Since we are only concerned
with the leading order approximations of all the fields, we neglect the O(δ 2 ) term
in (A7). Then the continuity of the resulting expression across C gives the first
boundary condition in (9b). Performing the same analysis on H′ · N̂ gives the
second.
References
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(2) X. Mei, M. Blumin, M. Sun, D. Kim, Z.H. Wu, and H.E. Ruda, “Highly Ordered GaAs/AlGaAs Quantum Dot Arrays on GaAs(01) Substrates”, Applied Physics Letters 82 (2003), pp. 967-969.
21
(3) D. S. Jones, Acoustic and Electromagnetic Scattering, Oxford Science Publications, Clarendon Press, Oxford (1989).
(4) G. A. Kriegsmann, “Electromagnetic Propagation in Periodic Porous Structures”, Wave Motion, 36 (2002) pp. 457-472.
(5) W. J. Parnell and I. D. Abrahams, “Dynamic Homogenization in Periodic
Fibre Reinforced Media: Quasi-Static Limit for SH Waves”, Wave Motion,
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