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Scattering by a resistive half-plane residing between two dissimilar media
Accepted Manuscript Title: Scattering by a resistive half-plane residing between two dissimilar media Author: Yusuf Ziya Umul PII: DOI: Reference:
S0030-4026(18)32005-9 https://doi.org/10.1016/j.ijleo.2018.12.081 IJLEO 62097
To appear in: Received date: Accepted date:
25 November 2018 18 December 2018
Please cite this article as: Umul YZ, Scattering by a resistive half-plane residing between two dissimilar media, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.12.081 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Scattering by a resistive half-plane residing between two
IP T
dissimilar media
SC R
Yusuf Ziya Umul
Adress: Electronic and Communication Dept., Cankaya University, Eskisehir yolu 29. km, Etimesgut, Ankara 06790, Turkiye
U
Tel: +90 312 2331324
A
N
e-mail: [email protected]
M
Abstract: The scattering problem of plane electromagnetic waves by a resistive half-plane, located at the planar interface of two media with different
ED
electromagnetic properties, is studied. First of all, the reflection and transmission coefficients of a resistive whole plane, between two different media, are
PT
evaluated. The scattered geometrical optics waves are obtained for the problem
CC E
under consideration. They are used in order to determine the diffracted waves, which are expressed in terms of a split function that arises in the solution of the diffraction problem by a resistive half-plane. The uniform diffraction fields are
A
obtained by using the uniform theory of diffraction. The behavior of the total, total geometric optics and diffracted waves are studied numerically.
Keywords: diffraction theory; different media; refraction.
1
1. Introduction The interaction of waves by a perfect electric conductor half-plane was first investigated by Clemmow in 1953 [1]. He was studying the propagation problem of radio waves over landscapes with different electromagnetic properties. This scenario led him to obtain the
IP T
solution of a canonical problem of a half-screen between two different media. Clemmow used the method of plane wave spectrum integral [2] in order to evaluate the scattered fields
SC R
by the discontinuity of the half-plane, but he was not able to find an explicit expression for
the edge diffracted waves. Later, Coblin and Pearson [3] attacked the same problem with the aid of the method of Wiener-Hopf factorization. After the determination of the split
U
functions, they asymptotically evaluated the complex integrals and obtained high-frequency
N
asymptotic expressions for the diffraction fields. However, their solutions were not uniform
A
and the diffracted fields were approaching to infinity at the transition boundaries. du Cloux
M
studied the perfect electric conductor half-plane between different media for the illumination
ED
by a pulsed electric field from a line source [4]. Ciarkowski investigated the same problem for the incidence of three dimensional plane wave [5]. He did not give any numerical
PT
simulations in his paper. In 1990, Volakis and Collins [6] studied the scattering problem of wave by a resistive half-screen between two dielectric media. They used the Wiener-Hopf
CC E
factorization in order to evaluate the diffracted fields. The diffraction of waves by a perfect conductor half-plane, residing between two different media, was investigated by us with the aid of different approaches [7-10]. Recently we also obtained the solution of the scattering
A
problem of waves by a perfect electric conductor wedge between two dissimilar media [11]. The aim of this paper is to obtain the total, total geometric optics (GO) and diffracted
waves, excited by the interaction of plane electromagnetic waves with a resistive half-screen between two different media. We will use the method, introduced in [12]. According to this method, the diffracted fields can be determined from the scattered GO waves by using a 2
relation between them at the transition boundaries. Also the approach, introduced in [10, 11] will be considered for the modeling of the effect of the planar interface between two dissimilar media. A time factor of exp(jt) is suppressed throughout the paper. is the angular
IP T
frequency.
2. Definition of the problem
SC R
A resistive half-plane is located between two media which have different electromagnetic
properties. The half-screen and the interface is illuminated by the plane electromagnetic
(1)
A
N
Ei E0 e jk1 cos 0 ez
U
wave of
M
where E0 is the complex amplitude. ki shows the wave-number in the ith medium and 0 angle of incidence. ki has the expression of i i where i and i are the permittivity and
ED
permeability of the ith medium. The cylindrical coordinates are given by (,,z). The related
A
CC E
PT
geometry is shown in Fig. 1. P is the observation point.
3
P Diffracted ray
Medium I
0 Resistive half-plane
Medium II
SC R
y
IP T
Incident ray
U
x
N
Fig. 1. The geometry of the diffraction problem of waves by a resistive half-plane between
A
two different media
M
Our aim is to find the total field and its components that will be excited when the incident wave interacts with the resistive half-screen. First of all, the effect of the media on
ED
the half-plane will be determined by solving the scattering problem of waves by a resistive whole screen between dissimilar media. Then we will find the solution of the problem, under
PT
consideration, by using the method, introduced in [12].
CC E
3. Whole resistive half-plane between two different media We will investigate the effect of two dissimilar media on the reflection and transmission
A
coefficients of the resistive surface. A resistive whole plane is located at y=0. The boundary conditions on the resistive surface can be given by n E1 E2
and
4
S
0
(2)
n n E1
S
Re n H1 H 2
(3)
S
for n is the unit normal vector of the surface [13, 14]. Re represents the resistivity of the
surface. The subscripts “1” and “2” show the fields in medium 1 and medium 2 respectively.
IP T
The incident field is given by Eq. (1). The reflected and transmitted electric fields can be expressed as
e Er e E0 e jk1 cos 0 ez
(5)
N
U
Te Et Te E0 e jk2 cos t ez
SC R
and
(4)
A
where e and Te are the reflection and transmission coefficients to be determined. The related
M
magnetic fields can be evaluated by using the Maxwell-Faraday equation. t is the angle of refraction. When the incident, reflected and transmitted electric and magnetic fields are used
ED
in Eqs. (2) and (3), t, e and Te are found to be
PT
Z1 Z 2 Re e ZZ Z 2 sin 0 Z1 sin t 1 2 Re
(6)
Z 2 sin 0 Z1 sin t
(7)
A
CC E
k1 sin 0 , k2
t sin 1
and
Te
2Z 2 sin 0 ZZ Z 2 sin 0 Z1 sin t 1 2 Re
5
(8)
respectively. Zi is the impedance of the ith medium and equal to
i / i .
4. Half-plane between two different media In this section, we will find the total field and its components (total GO and diffracted waves) by using the approach, introduced in [12]. The method can be summarized as follows; 1)
Determine the initial fields Ein by excluding the structure that causes the discontinuity in
IP T
SC R
space, 2) find the scattered GO wave from the equation
ESGO ETGO Ein ,
U
3) evaluate the f function by using the relations
(9)
(10)
A
N
sin 0 A ESGO 0 f 0
M
for f is defined in the diffracted field by the equation of
j
f ,0 e jk ez 2 cos cos 0 k
ED
e Ed
4
PT
(11)
and the operator Au means that consider only the amplitude of the part of vector function
CC E
u , omitting phase and polarization.
The reason of the edge discontinuity of the problem, under consideration, is the
A
existence of the resistive half-plane. If it is excluded from space, only the interface between two different media will exist. Thus the initial fields read
Ein Ei Er U TEtU
6
(12)
in this case. and T are the reflection and transmission coefficients of the interface and can be defined by
k1 sin 0 k 2 sin t k1 sin 0 k 2 sin t
T
2k 2 sin t k1 sin 0 k 2 sin t
(13)
(14)
SC R
respectively [10].
IP T
and
U
The total GO field can be determined from the geometry, in Fig. 1. It can be expressed
N
as
M
A
ETGO Ei Er U 1 eU 1 U Et TU 2 TeU 2 U (15)
ED
where 1 and 2 can be introduced by
0 2
(16)
PT
1 2k cos
2 2k cos
t 2
(17)
A
CC E
and
respectively. The scattered GO wave is found to be
ESGO Er e U 1 Et Te T U 2
7
(18)
from Eq. (9). This equation can be represented as
ESGO Er e 1 1 U 1 Et Te T U 2 ,
(19)
which yields the expression
(20)
IP T
ESGO Er Te T U 1 Et Te T U 2 .
SC R
We will take into account Eq. (20) for the determination of the diffracted wave, which can be expressed by
j
f1 U e jk1 f 2 U e jk2 2 cos cos 0 k1 cos cos t k 2
U
4
(21)
N
e Ed e z
A
for f1 and f2 are the functions to be evaluated. The relations of
M
f1 0 sin 0 Te T
ED
(22)
CC E
from Eq. (10).
PT
and
f 2 t sin t Te T
(23)
A
e is taken into account. It can be arranged as
Z 2 sin 0 Z1 sin t e
Z1 Z 2 Re
ZZ 2Z 2 sin 0 Z1 sin t Z 2 sin 0 1 2 Re
that leads to the expression of
8
,
(24)
e
sin e sin 0 sin e
(25)
where sine is defined by
Te
SC R
and the transmission coefficient reads
(26)
IP T
Z 1 Z sin e 1 sin t sin 0 1 2 Z2 Re
sin 0 sin 0 sin e
(27)
A
k1 sin 0 k 2 sin t 2k1 sin 0 k 2 sin t k1 sin 0
(28)
M
N
U
from the relation of Te=e+1. As a second step, is considered as
ED
from Eq. (13). Equation (28) can be arranged as
CC E
PT
1 k2 sin t sin 0 2 k1
1k sin 0 2 sin t sin 0 2 k1
(29)
A
that yields the expression of
sin sin 0 sin
(30)
where sin is
1k sin 2 sin t sin 0 . 2 k1 9
(31)
The transmission coefficient can be written as
T
sin 0 sin 0 sin
(32)
sin sin e sin 0 sin 0 sin e sin 0 sin
(33)
SC R
Te T
IP T
from the relation T= +1. As a result TeT becomes
according to Eqs. (27) and (32).
(34)
N
sin sin e sin 2 0 sin 0 sin e sin 0 sin
A
f1 0
U
Equation (22) is taken into consideration. f1 satisfies the relation of
M
when Eq. (33) is used in Eq. (22). Equation (34) can be arranged as
ED
1 1 sin 0 sin sin 0 sin f1 0 , sin sin e sin 0 sin sin 0 sin e
(35)
PT
which leads to the expression of
CC E
1 1 K 0 , K 0 , K 0 , e K 0 , e f1 0 sin sin e
(36)
A
by using the relation
K , K ,
sin sin sin sin
(37)
where K+ is the split function that arises in the solution of the diffraction problem of waves by a resistive half-plane [14]. The function f1 has the expression of
10
1 1 K , K 0 , K , e K 0 , e f1 sin e sin
(38)
as a result [15].
(39)
SC R
1 1 sin t sin sin t sin f 2 0 sin e sin sin t sin sin t sin e sin 0 sin t sin sin t sin sin t sin 0 sin sin 0 sin
IP T
In order to determine f2, Eq. (23) is taken into account. This equation can be written as
U
as in Eq. (35). Thus f2 can be determined as
(40)
ED
M
A
N
1 1 K 3 , K t , K , e K 0 , e f 2 sin e sin t t sin 0 cos sin cos sin 2 2 t sin 0 sin sin 0 sin cos 2
as in f1. The uniform version of the diffracted field can be given by
(41)
CC E
PT
f1 p1 f 2 p2 Ed e z U U 2 sin sin 0 2 sin sin t 2 2 2 2
A
where the functions p1 and p2 can be introduced as p1 e jk1 cos 0 sign1 F 1 e jk1 cos 0 sign1 F 1
(42)
p2 e jk2 cos t sign 2 F 2 e jk2 cos t sign 2 F 2
(43)
and
11
respectively. F[x] is the Fresnel function, which can be introduced by the integral of
F x
e
j
4
e
jv 2
(44)
dv
x
IP T
and sign(x) shows the signum function that is equal to 1 for x>0 and 1 otherwise.
5. Numerical results
SC R
In this section, we will investigate the behaviors of the total field and its components
(diffracted and total GO waves) numerically. The distance of observation () is taken as 3
U
for is the wavelength. The angle of observation ( ) varies between 0o and 360o. The angle
N
of incidence (o) is equal to 60o. k1 has the value of 2/ and k2 is taken as 3k1. Z1/2Z2 and
A
CC E
PT
ED
M
A
Z1/2Re are 3 and 4 respectively. Note that the plots are independent from the wavelength.
Fig. 2. The total field excited by the incident plane wave
12
IP T SC R
U
Fig. 3. The total GO and diffracted fields
N
Figure 2 shows the variation of the total field, which is the sum of the total GO and
A
diffracted waves, versus the observation angle. The field does not have any discontinuities.
M
At 180o, a decrease in the intensity is observed. This is the location of the interface between
ED
two different media. A second decrease is seen at 240o, which is the location of the shadow boundary. The intensity of the field, before 180o is greater that the one, after 180o, since in
PT
the first region, the incident field is also added to the reflected waves. In Fig. 3, the variations of the diffracted and total GO waves with respect to the observation angle are given. The
CC E
total GO field has discontinuities at 120o and 240o, which are the locations of the reflection and shadow boundaries respectively. The diffracted field has also its maxima at these points.
A
The addition of the diffracted wave to the total GO field eliminates the discontinuities as can be seen from Fig. 2.
6. Conclusion In this paper, the scattering of electromagnetic waves by a resistive half-plane, residing between two different media, was investigated. First of all, the interaction of waves by a
13
resistive whole plane, located on the interface of two dissimilar media was analyzed and the effect of the different media on the reflection and transmission of the resistive surface was put forth. Then the problem, under consideration, was solved by using the method that enables one to determine the diffracted waves with the aid of the scattered GO fields. The
SC R
IP T
obtained field expressions were investigated numerically.
U
References
N
1. P.C. Clemmow, Radio propagation over a flat earth across a boundary separating two
A
different media, Phil. Trans. R. Soc. A 246 (1953) 1-55.
M
2. P.C. Clemmow, A method for the exact solution of two-dimensional diffraction problems, Proc. R. Soc. Lond. A 205 (1951) 286-308.
ED
3. R.D. Coblin, L.W. Pearson, A geometrical theory of diffraction for a half plane residing on the interface between dissimilar media: transverse magnetic polarized case, Radio Sci.
PT
19 (1984) 1277-1288.
CC E
4. R. du Cloux, Pulsed electromagnetic radiation from a line source in the presence of a semi-infinite screen in the plane interface of two different media, Wave Motion 6 (1984) 459-476.
A
5. A. Ciarkowski, Three-dimensional electromagnetic half-plane diffraction at the interface of different media, Radio Sci. 22 (1987) 969-975.
6. J.L. Volakis, J.D. Collins, Electromagnetic scattering from a resistive half plane on a dielectric interface, Wave Motion 12 (1990) 81-96.
14
7. Y.Z. Umul, U. Yalçın, Scattered fields of conducting half-plane between two dielectric media, Appl. Opt. 49 (2010) 4010-4017. 8. Y.Z. Umul, Diffraction of waves by a perfectly conducting half-plane between two dielectric media, Opt. Commun. 310 (2014) 64-68.
two different media, Microw. Opt. Technol. Lett. 57 (2015) 1928-1933.
IP T
9. Y.Z. Umul, Scattering theory of waves by a perfectly conducting half-screen between
SC R
10. Y.Z. Umul, Interaction of plane waves by a half-screen between two different media, Opt. Quantum Electron. 50 (2018) 84 (13 pp).
11. Y.Z. Umul, Diffraction of waves by a wedge residing between two different media,
U
Optik-Int. J. Light Electron Opt. 162 (2018) 8-18.
N
12. Y.Z. Umul, Scattering by a conductive half-screen between isorefractive media,” Appl.
A
Opt. 54 (2015) 10309-10313.
M
13. T.B.A. Senior, Half plane edge diffraction, Radio Sci. 10 (1975) 645-650.
A
CC E
PT
IEE, London, 1995.
ED
14. T.B.A. Senior, J.L. Volakis, Approximate Boundary Conditions in Electromagnetics,
15
S0030-4026(18)32005-9 https://doi.org/10.1016/j.ijleo.2018.12.081 IJLEO 62097
To appear in: Received date: Accepted date:
25 November 2018 18 December 2018
Please cite this article as: Umul YZ, Scattering by a resistive half-plane residing between two dissimilar media, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.12.081 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Scattering by a resistive half-plane residing between two
IP T
dissimilar media
SC R
Yusuf Ziya Umul
Adress: Electronic and Communication Dept., Cankaya University, Eskisehir yolu 29. km, Etimesgut, Ankara 06790, Turkiye
U
Tel: +90 312 2331324
A
N
e-mail: [email protected]
M
Abstract: The scattering problem of plane electromagnetic waves by a resistive half-plane, located at the planar interface of two media with different
ED
electromagnetic properties, is studied. First of all, the reflection and transmission coefficients of a resistive whole plane, between two different media, are
PT
evaluated. The scattered geometrical optics waves are obtained for the problem
CC E
under consideration. They are used in order to determine the diffracted waves, which are expressed in terms of a split function that arises in the solution of the diffraction problem by a resistive half-plane. The uniform diffraction fields are
A
obtained by using the uniform theory of diffraction. The behavior of the total, total geometric optics and diffracted waves are studied numerically.
Keywords: diffraction theory; different media; refraction.
1
1. Introduction The interaction of waves by a perfect electric conductor half-plane was first investigated by Clemmow in 1953 [1]. He was studying the propagation problem of radio waves over landscapes with different electromagnetic properties. This scenario led him to obtain the
IP T
solution of a canonical problem of a half-screen between two different media. Clemmow used the method of plane wave spectrum integral [2] in order to evaluate the scattered fields
SC R
by the discontinuity of the half-plane, but he was not able to find an explicit expression for
the edge diffracted waves. Later, Coblin and Pearson [3] attacked the same problem with the aid of the method of Wiener-Hopf factorization. After the determination of the split
U
functions, they asymptotically evaluated the complex integrals and obtained high-frequency
N
asymptotic expressions for the diffraction fields. However, their solutions were not uniform
A
and the diffracted fields were approaching to infinity at the transition boundaries. du Cloux
M
studied the perfect electric conductor half-plane between different media for the illumination
ED
by a pulsed electric field from a line source [4]. Ciarkowski investigated the same problem for the incidence of three dimensional plane wave [5]. He did not give any numerical
PT
simulations in his paper. In 1990, Volakis and Collins [6] studied the scattering problem of wave by a resistive half-screen between two dielectric media. They used the Wiener-Hopf
CC E
factorization in order to evaluate the diffracted fields. The diffraction of waves by a perfect conductor half-plane, residing between two different media, was investigated by us with the aid of different approaches [7-10]. Recently we also obtained the solution of the scattering
A
problem of waves by a perfect electric conductor wedge between two dissimilar media [11]. The aim of this paper is to obtain the total, total geometric optics (GO) and diffracted
waves, excited by the interaction of plane electromagnetic waves with a resistive half-screen between two different media. We will use the method, introduced in [12]. According to this method, the diffracted fields can be determined from the scattered GO waves by using a 2
relation between them at the transition boundaries. Also the approach, introduced in [10, 11] will be considered for the modeling of the effect of the planar interface between two dissimilar media. A time factor of exp(jt) is suppressed throughout the paper. is the angular
IP T
frequency.
2. Definition of the problem
SC R
A resistive half-plane is located between two media which have different electromagnetic
properties. The half-screen and the interface is illuminated by the plane electromagnetic
(1)
A
N
Ei E0 e jk1 cos 0 ez
U
wave of
M
where E0 is the complex amplitude. ki shows the wave-number in the ith medium and 0 angle of incidence. ki has the expression of i i where i and i are the permittivity and
ED
permeability of the ith medium. The cylindrical coordinates are given by (,,z). The related
A
CC E
PT
geometry is shown in Fig. 1. P is the observation point.
3
P Diffracted ray
Medium I
0 Resistive half-plane
Medium II
SC R
y
IP T
Incident ray
U
x
N
Fig. 1. The geometry of the diffraction problem of waves by a resistive half-plane between
A
two different media
M
Our aim is to find the total field and its components that will be excited when the incident wave interacts with the resistive half-screen. First of all, the effect of the media on
ED
the half-plane will be determined by solving the scattering problem of waves by a resistive whole screen between dissimilar media. Then we will find the solution of the problem, under
PT
consideration, by using the method, introduced in [12].
CC E
3. Whole resistive half-plane between two different media We will investigate the effect of two dissimilar media on the reflection and transmission
A
coefficients of the resistive surface. A resistive whole plane is located at y=0. The boundary conditions on the resistive surface can be given by n E1 E2
and
4
S
0
(2)
n n E1
S
Re n H1 H 2
(3)
S
for n is the unit normal vector of the surface [13, 14]. Re represents the resistivity of the
surface. The subscripts “1” and “2” show the fields in medium 1 and medium 2 respectively.
IP T
The incident field is given by Eq. (1). The reflected and transmitted electric fields can be expressed as
e Er e E0 e jk1 cos 0 ez
(5)
N
U
Te Et Te E0 e jk2 cos t ez
SC R
and
(4)
A
where e and Te are the reflection and transmission coefficients to be determined. The related
M
magnetic fields can be evaluated by using the Maxwell-Faraday equation. t is the angle of refraction. When the incident, reflected and transmitted electric and magnetic fields are used
ED
in Eqs. (2) and (3), t, e and Te are found to be
PT
Z1 Z 2 Re e ZZ Z 2 sin 0 Z1 sin t 1 2 Re
(6)
Z 2 sin 0 Z1 sin t
(7)
A
CC E
k1 sin 0 , k2
t sin 1
and
Te
2Z 2 sin 0 ZZ Z 2 sin 0 Z1 sin t 1 2 Re
5
(8)
respectively. Zi is the impedance of the ith medium and equal to
i / i .
4. Half-plane between two different media In this section, we will find the total field and its components (total GO and diffracted waves) by using the approach, introduced in [12]. The method can be summarized as follows; 1)
Determine the initial fields Ein by excluding the structure that causes the discontinuity in
IP T
SC R
space, 2) find the scattered GO wave from the equation
ESGO ETGO Ein ,
U
3) evaluate the f function by using the relations
(9)
(10)
A
N
sin 0 A ESGO 0 f 0
M
for f is defined in the diffracted field by the equation of
j
f ,0 e jk ez 2 cos cos 0 k
ED
e Ed
4
PT
(11)
and the operator Au means that consider only the amplitude of the part of vector function
CC E
u , omitting phase and polarization.
The reason of the edge discontinuity of the problem, under consideration, is the
A
existence of the resistive half-plane. If it is excluded from space, only the interface between two different media will exist. Thus the initial fields read
Ein Ei Er U TEtU
6
(12)
in this case. and T are the reflection and transmission coefficients of the interface and can be defined by
k1 sin 0 k 2 sin t k1 sin 0 k 2 sin t
T
2k 2 sin t k1 sin 0 k 2 sin t
(13)
(14)
SC R
respectively [10].
IP T
and
U
The total GO field can be determined from the geometry, in Fig. 1. It can be expressed
N
as
M
A
ETGO Ei Er U 1 eU 1 U Et TU 2 TeU 2 U (15)
ED
where 1 and 2 can be introduced by
0 2
(16)
PT
1 2k cos
2 2k cos
t 2
(17)
A
CC E
and
respectively. The scattered GO wave is found to be
ESGO Er e U 1 Et Te T U 2
7
(18)
from Eq. (9). This equation can be represented as
ESGO Er e 1 1 U 1 Et Te T U 2 ,
(19)
which yields the expression
(20)
IP T
ESGO Er Te T U 1 Et Te T U 2 .
SC R
We will take into account Eq. (20) for the determination of the diffracted wave, which can be expressed by
j
f1 U e jk1 f 2 U e jk2 2 cos cos 0 k1 cos cos t k 2
U
4
(21)
N
e Ed e z
A
for f1 and f2 are the functions to be evaluated. The relations of
M
f1 0 sin 0 Te T
ED
(22)
CC E
from Eq. (10).
PT
and
f 2 t sin t Te T
(23)
A
e is taken into account. It can be arranged as
Z 2 sin 0 Z1 sin t e
Z1 Z 2 Re
ZZ 2Z 2 sin 0 Z1 sin t Z 2 sin 0 1 2 Re
that leads to the expression of
8
,
(24)
e
sin e sin 0 sin e
(25)
where sine is defined by
Te
SC R
and the transmission coefficient reads
(26)
IP T
Z 1 Z sin e 1 sin t sin 0 1 2 Z2 Re
sin 0 sin 0 sin e
(27)
A
k1 sin 0 k 2 sin t 2k1 sin 0 k 2 sin t k1 sin 0
(28)
M
N
U
from the relation of Te=e+1. As a second step, is considered as
ED
from Eq. (13). Equation (28) can be arranged as
CC E
PT
1 k2 sin t sin 0 2 k1
1k sin 0 2 sin t sin 0 2 k1
(29)
A
that yields the expression of
sin sin 0 sin
(30)
where sin is
1k sin 2 sin t sin 0 . 2 k1 9
(31)
The transmission coefficient can be written as
T
sin 0 sin 0 sin
(32)
sin sin e sin 0 sin 0 sin e sin 0 sin
(33)
SC R
Te T
IP T
from the relation T= +1. As a result TeT becomes
according to Eqs. (27) and (32).
(34)
N
sin sin e sin 2 0 sin 0 sin e sin 0 sin
A
f1 0
U
Equation (22) is taken into consideration. f1 satisfies the relation of
M
when Eq. (33) is used in Eq. (22). Equation (34) can be arranged as
ED
1 1 sin 0 sin sin 0 sin f1 0 , sin sin e sin 0 sin sin 0 sin e
(35)
PT
which leads to the expression of
CC E
1 1 K 0 , K 0 , K 0 , e K 0 , e f1 0 sin sin e
(36)
A
by using the relation
K , K ,
sin sin sin sin
(37)
where K+ is the split function that arises in the solution of the diffraction problem of waves by a resistive half-plane [14]. The function f1 has the expression of
10
1 1 K , K 0 , K , e K 0 , e f1 sin e sin
(38)
as a result [15].
(39)
SC R
1 1 sin t sin sin t sin f 2 0 sin e sin sin t sin sin t sin e sin 0 sin t sin sin t sin sin t sin 0 sin sin 0 sin
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In order to determine f2, Eq. (23) is taken into account. This equation can be written as
U
as in Eq. (35). Thus f2 can be determined as
(40)
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M
A
N
1 1 K 3 , K t , K , e K 0 , e f 2 sin e sin t t sin 0 cos sin cos sin 2 2 t sin 0 sin sin 0 sin cos 2
as in f1. The uniform version of the diffracted field can be given by
(41)
CC E
PT
f1 p1 f 2 p2 Ed e z U U 2 sin sin 0 2 sin sin t 2 2 2 2
A
where the functions p1 and p2 can be introduced as p1 e jk1 cos 0 sign1 F 1 e jk1 cos 0 sign1 F 1
(42)
p2 e jk2 cos t sign 2 F 2 e jk2 cos t sign 2 F 2
(43)
and
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respectively. F[x] is the Fresnel function, which can be introduced by the integral of
F x
e
j
4
e
jv 2
(44)
dv
x
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and sign(x) shows the signum function that is equal to 1 for x>0 and 1 otherwise.
5. Numerical results
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In this section, we will investigate the behaviors of the total field and its components
(diffracted and total GO waves) numerically. The distance of observation () is taken as 3
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for is the wavelength. The angle of observation ( ) varies between 0o and 360o. The angle
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of incidence (o) is equal to 60o. k1 has the value of 2/ and k2 is taken as 3k1. Z1/2Z2 and
A
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PT
ED
M
A
Z1/2Re are 3 and 4 respectively. Note that the plots are independent from the wavelength.
Fig. 2. The total field excited by the incident plane wave
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IP T SC R
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Fig. 3. The total GO and diffracted fields
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Figure 2 shows the variation of the total field, which is the sum of the total GO and
A
diffracted waves, versus the observation angle. The field does not have any discontinuities.
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At 180o, a decrease in the intensity is observed. This is the location of the interface between
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two different media. A second decrease is seen at 240o, which is the location of the shadow boundary. The intensity of the field, before 180o is greater that the one, after 180o, since in
PT
the first region, the incident field is also added to the reflected waves. In Fig. 3, the variations of the diffracted and total GO waves with respect to the observation angle are given. The
CC E
total GO field has discontinuities at 120o and 240o, which are the locations of the reflection and shadow boundaries respectively. The diffracted field has also its maxima at these points.
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The addition of the diffracted wave to the total GO field eliminates the discontinuities as can be seen from Fig. 2.
6. Conclusion In this paper, the scattering of electromagnetic waves by a resistive half-plane, residing between two different media, was investigated. First of all, the interaction of waves by a
13
resistive whole plane, located on the interface of two dissimilar media was analyzed and the effect of the different media on the reflection and transmission of the resistive surface was put forth. Then the problem, under consideration, was solved by using the method that enables one to determine the diffracted waves with the aid of the scattered GO fields. The
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obtained field expressions were investigated numerically.
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References
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1. P.C. Clemmow, Radio propagation over a flat earth across a boundary separating two
A
different media, Phil. Trans. R. Soc. A 246 (1953) 1-55.
M
2. P.C. Clemmow, A method for the exact solution of two-dimensional diffraction problems, Proc. R. Soc. Lond. A 205 (1951) 286-308.
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3. R.D. Coblin, L.W. Pearson, A geometrical theory of diffraction for a half plane residing on the interface between dissimilar media: transverse magnetic polarized case, Radio Sci.
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19 (1984) 1277-1288.
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4. R. du Cloux, Pulsed electromagnetic radiation from a line source in the presence of a semi-infinite screen in the plane interface of two different media, Wave Motion 6 (1984) 459-476.
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5. A. Ciarkowski, Three-dimensional electromagnetic half-plane diffraction at the interface of different media, Radio Sci. 22 (1987) 969-975.
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7. Y.Z. Umul, U. Yalçın, Scattered fields of conducting half-plane between two dielectric media, Appl. Opt. 49 (2010) 4010-4017. 8. Y.Z. Umul, Diffraction of waves by a perfectly conducting half-plane between two dielectric media, Opt. Commun. 310 (2014) 64-68.
two different media, Microw. Opt. Technol. Lett. 57 (2015) 1928-1933.
IP T
9. Y.Z. Umul, Scattering theory of waves by a perfectly conducting half-screen between
SC R
10. Y.Z. Umul, Interaction of plane waves by a half-screen between two different media, Opt. Quantum Electron. 50 (2018) 84 (13 pp).
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Optik-Int. J. Light Electron Opt. 162 (2018) 8-18.
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12. Y.Z. Umul, Scattering by a conductive half-screen between isorefractive media,” Appl.
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Opt. 54 (2015) 10309-10313.
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13. T.B.A. Senior, Half plane edge diffraction, Radio Sci. 10 (1975) 645-650.
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IEE, London, 1995.
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14. T.B.A. Senior, J.L. Volakis, Approximate Boundary Conditions in Electromagnetics,
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