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633
Multiple Reflection from Layered Heterogeneous Medium
R. Kadlec, E. Kroutilová, and P. Fiala
Department of Theoretical and Experimental Electrical Engineering
Brno University of Technology, Kolejnı́ 2906/4, Brno 612 00, Czech Republic
Abstract— The paper presents the problem of algorithm design of propagation, reflection and
refraction of the electromagnetic waves on a layered medium. The analytic solution of this issue
is very intricate and time demanding. This method is suitable for specific purposes of the detailed
analysis of the general issue. Numerical methods are more suitable for analysis of the reflection
and refraction of electromagnetic waves on layered heterogeneous medium. Fundamental law for
analysis of the reflection and the refraction of electromagnetic waves on the boundary line between
two materials are Snell’s law for electromagnetic waves. The paper deals with the problem of
complex angle of refraction in the losing medium. In non-lossy environment, interpretation of
Fresnel relations and Snell’s law is simple. For a layered heterogeneous medium, an algorithm was
prepared for the reflection on several layers in MatLab program environment. Methods described
in this paper are suitable for analysis of beam refraction to the other side from the perpendicular
line during the passage through the interface. This phenomenon is occur in metamaterial. These
materials with negative parameters constitute a group of media that possesses a negative value,
relative permittivity or relative permeability or both.
1. INTRODUCTION
Generally, inhomogeneities and regions with different parameters appear even in the cleanest materials. During the electromagnetic wave passage through a material there occur an amplitude
decrease and a wave phase shift, owing to the material characteristics such as conductivity, permittivity, or permeability. If a wave impinges on an inhomogeneity, a change of its propagation there
occurs. This change materializes in two forms, namely in reflection and refraction. In addition to
this process, polarization and interference may appear in the waves.
2. ELECTROMAGNETIC WAVES IN ISOTROPIC DIELECTRICS MATERIALS
Algorithms were generated in the Matlab program environment that simulates reflection and refraction in a lossy environment on the interface between two dielectrics. This section of the paper
is linked to the previous modelling of light applying the related geometrical laws. The reflection
and refraction is in accordance with Snell’s law for electromagnetic waves as shown in Fig. 1. The
form of Snell’s law is
p
jωµ2 · (γ2 + jωε2 )
sin θ0
k2
=
=p
,
(1)
sin θ2
k1
jωµ1 · (γ1 + jωε1 )
where k is the wave number, γ is the conductivity, ε the permittivity and µ the permeability.
Relation (1) is defining for the boundary line between the dielectrics medium. Generally, k1 and k2
are complex; then angle θ2 is also complex. An electromagnetic wave is understood as the electric
field strength and the magnetic field strength. The electric component and magnetic component
incident wave according to Fig. 1 follows the formula
Ei = E0 e−jk1 un0 ·r ,
Hi =
un0 × Ei
Zv1
(2)
where E0 is the amplitude electric field strength on the boundary line, r is the positional vector,
and un0 is the unit vector of propagation direction.
The intensity of reflection beams and the intensity of refraction beams are expressed according
to the formula
Er = E1 e−jk1 un1 ×r , Et = E2 e−jk2 un2 ×r ,
(3)
where E1 is calculated from the intensity on boundary line E0 and reflection coefficient ρE , and E2
is calculated from the intensity on boundary line E 0 and transmission factor τ E :
E1 = ρE · E0 , E2 = τE · E0 ,
(4)
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634
magnetic component is calculated from electric component and wave impedance:
H0 =
E0
,
Zv1
H1 =
E1
,
Zv1
H2 =
E2
.
Zv2
(5)
The calculation of reflection coefficient ρE and transmission factor τ E with utilization of wave
impedance Zv is according to these relations:
ρE =
E1
Zv2 cos θ1 − Zv1 cos θ2
=
,
E0
Zv2 cos θ1 + Zv1 cos θ2
τE =
E2
2Zv2 cos θ1
=
.
E0
Zv2 cos θ1 + Zv1 cos θ2
For numerical modelling, there is a suitable relation in the form of
q
µ2 k1 cos θ0 − µ1 k22 − k12 sin2 θ0
q
Er =
E0 · e−jk1 un1 ×r ,
2
2
2
µ2 k1 cos θ0 + µ1 k2 − k1 sin θ0
(6)
(7)
2µ2 k1 cos θ0
q
Et =
E0 · e−jk2 un2 ×r .
2
2
2
µ2 k1 cos θ0 + µ1 k2 − k1 sin θ0
These relations are calculated from the basic variable and they facilitate an acceleration of the
calculation process.
Interpretation of the Fresnel equations and Snell’s laws is simple in the case of the refraction
on boundary line between the dielectrics medium. In case of refraction in a lossy medium, angle
θ2 is complex. According to relation (1), angle θ2 depends on wave numbers k1 and k2 , which are
generally complex; then, in medium 2 an inhomogeneous wave is propagated.
Example of reflection and refraction on a planar boundary line in COMSOL program is shown
in Fig. 2 (at the incidence of the wave on the interface at an angle of 45◦ ).
For a layered heterogeneous medium, an algorithm is deduced for the reflection on several layers.
The reflection and refraction on a heterogonous material is solved by the help of the numerical
method. The reflection on a layered material on n layers generates n primary electromagnetic
waves, according to Fig. 4. The interpretation of propagation of electromagnetic waves on a layered
heterogeneous medium is according to relation
Erl = Eil ρEλ · e−jkl unrl ×rl ,
Etl = Eil τEλ · e−jkl untl ×rl ,
(8)
where Erl and Etl are the reflection and refraction electromagnetic waves on the boundary line
(l = 1, . . . , max) according to Fig. 4, Eil is the amplitude electric field strength on boundary line l,
Figure 1: Reflection and refraction of light [1].
Figure 2: Reflection and refraction on a planar
boundary line [2].
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Figure 3: Perpendicular incidence of the wave
on layered material.
635
Figure 4: Reflection and refraction on a layered heterogenous material.
ρEl and τ El are the reflection coefficient and transmission factor on boundary line l, kl is the wave
number of layer, rl is the electromagnetic wave positional vector on boundary line l, u ntl and u nrl
are the unit vectors of propagation direction.
Special case is perpendicular incidence of the electromagnetic wave on the interface according
to Fig. 3. The interpretation of perpendicular incidence for incident wave ẼA and reflection wave
←
EA is according to relation
ẼA =
ejk2 ·unA ×r + ρ12 ρ23 · e−jk2 ·unA ×r
ẼD ,
τ12 τ23
←
EA =
ρ12 ejk2 ·unA ×r + ρ23 e−jk2 ·unA ×r
ẼD ,
τ12 τ23
(9)
where ρ12 is reflection coefficient of wave in external medium which is reflective on boundary line
1, ρ21 is reflection coefficient of wave in internal medium which is reflective on boundary line 1.
Transmission factors τ are indexed analogically. k2 is the wave number internal medium and unA ×r
is distance between boundary lines.
With some redefinitions, the formalism of transfer matrices and wave impedances for perpendicular incidence translates almost like to the case of oblique incidence. By separating the fields into
transverse and longitudinal components with respect to the direction the dielectrics are stacked
(the z-direction), we show that the transverse components satisfy the identical transfer matrix
relationships as in the case of perpendicular incidence. The transverse components of the electric
fields are defined differently in the two polarization cases. We recall from that an obliquely-moving
wave will have, in general, both TM and TE components, then we use generally electric component
ET with both polarizations, in this paper [5].
Using the matching and propagation matrices for transverse fields we derive here the layer
recursions for multiple dielectric slabs at oblique incidence [5]. Fig. 4 shows such a multilayer
structure. The layer recursions relate the various field quantities, such as the electric fields and the
reflection waves, on the top of each interface.
We assume that there are no incident fields from the down side of the structure. The reflection/refraction angles in each medium are related to each other by Snel’s law applied to each of
the M + 1 interfaces:
k1 · sin θ0 = k2 · sin θ2 = kl · sin θl ,
l = 1, 2, . . . , M
(10)
To obtain the layer recursions for the electric fields, we apply the propagation matrix [5] to the
fields on the top of interface l + 1 and propagate them to the down of the interface l, and then,
apply a matching matrix [5] to pass to the left of that interface:
#
"
#
·
¸ · jk ·u ×r
¸"
ẼT,l+1
ẼTl
1
1 ρT l
e l nl
.
(11)
=
←
←
e−jkl ·unl ×r
τ T l ρT l 1
ET,l+1
ETl
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Multiplying the matrix factors, we obtain:
"
#
#
·
¸"
ẼTl
ẼT,l+1
1
ejkl ·unl ×r
ρT l e−jkl ·unl ×r
=
.
←
←
e−jkl ·unl ×r
τT l ρT l ejkl ·unl ×r
ETl
ET,l+1
636
(12)
The recursion is initialized on the to of the (M +1)st interface by performing an additional matching
to pass to the down of that interface:
#
"
·
¸· 0
¸
ẼT,M +1
1
1
ρT,M +1
ẼT,M +1
=
.
(13)
←
1
0
τT l,M +1 ρT,M +1
ET,M +1
Similarly, we obtain the following recursions for the total transverse electric and magnetic fields at
each interface (they are continuous across each interface):
¸ ·
¸·
¸
·
cos(kl · unl ×r)
j · Zvl sin(kl · unl ×r)
ET,l+1
ETl
=
.
(14)
−1
HTl
HT,l+1
j · Zvl
sin(kl · unl ×r)
cos(kl · unl ×r)
3. CONCLUSIONS
For simple cases (such as a planar interface), the behaviour of an impinging wave can be calculated analytically by the help of Snell’s refraction/reflection law and the Fresnel equations, which
are fundamental law for analysis of the reflection and the refraction of electromagnetic waves on
boundary line between two materials. Problem of algorithm design of propagation, reflection and
refraction of the electromagnetic waves on a layered medium is very intricate and time demanding.
This method is suitable for specific purposes of detail analysis of general issue. Therefore, numerical methods are applied to facilitate the calculation process, and a wide range of programs like
ANSYS, Comsol, or Matlab can be utilized in the realization of numerical modelling.
Methods describe in this paper are suitable for analysis of beam refraction to the other side
from the perpendicular line during the passage through the interface. This phenomenon is occur
in metamaterials.
Algorithms created in the Matlab environment are verified by the help of programs based on
the finite element method, namely programs such as Comsol and ANSYS.
ACKNOWLEDGMENT
The research described in the paper was financially supported by the research program MSM
0021630516 and research plan MSM 0021630513, Ministry of Defence of the CR, Ministry of Industry and Trade of the CR (Diagnostics of Superfast Objects for Safety Testing, FR-TI1/368), Czech
Science Foundation (102/09/0314) and project of the BUT Grant Agency FEKT-S-11-5.
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