- Email: [email protected]
Diffraction of waves by a wedge residing between two different media
Optik 162 (2018) 8–18
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Diffraction of waves by a wedge residing between two different media Yusuf Ziya Umul Electronic and Communication Dept., Cankaya University, Eskisehir yolu 29. km, Etimesgut, Ankara, 06790, Turkiye
a r t i c l e
i n f o
Article history: Received 21 January 2018 Accepted 20 February 2018 Keywords: Diffraction theory Wedge Refraction
a b s t r a c t The scattering problem of plane waves by a perfectly electric conducting wedge, residing at the planar interface of two media with different electromagnetic properties, is investigated. The structure of the scattered geometrical optics waves is 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 obtained with the aid of uniform theory of diffraction. The behavior of the total, total geometrical optics and diffracted waves are studied numerically. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction The diffraction phenomenon of waves by a wedge is a canonical problem that finds many areas of applications in optics. The high-frequency expressions of the diffracted fields by soft and hard wedges underline the foundations of the geometrical theory of diffraction [1] and uniform theory of diffraction (UTD) [2]. However, a more fundamental problem, related with wedge, has not been investigated in the literature yet, to our knowledge. This problem describes a wedge, residing between two media with different permittivitiy and permeability. In fact the scattering of waves by a wedge in a single medium is a special case of this scenario. Similar scattering problems were studied for a perfectly conducting [3–6] and resistive [7] halfplanes. However, the diffraction field expressions, obtained by these studies contain spurs. We put forth alternative solutions, which do not have any spurious fields, for the perfectly conducting half-screen problem [8–10], but these expressions are approximate. The aim of this paper is to investigate the diffraction problem of plane waves by a soft wedge, located on a planar junction between two different media. First of all the initial fields, which occur on a planar whole interface between two different media, will be obtained. Then the total geometric optics (GO) waves will be determined for the problem under consideration. The scattered GO fields will be evaluated by subtracting the initial fields from the total GO waves. The diffracted fields, excited by the wedge, will be obtained with the aid of a relation, put forth by us [11–13]. The uniform diffraction waves will be derived by using UTD [2,14]. The behaviors of the total, total GO and diffracted waves will be analyzed numerically. A time factor of exp(jωt) is assumed and suppressed throughout the paper. ω is the angular frequency.
E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2018.02.059 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
Y.Z. Umul / Optik 162 (2018) 8–18
9
2. Introduction of the method We suppose an incident wave that has the time harmonic electric field component of ui interacts with an obstacle on its path of propagation. The wave satisfies the Helmholtz equation
∇ 2 ui + k2 ui = 0
(1)
where k is the wave-number. The total field of uT = ui + uS
(2)
is excited when the incident wave interacts with an obstacle on its path of propagation [15]. ui is the wave distribution when there is not any scatterer in the medium. uS is the scattered field and shows the effect of the obstacle on the incident wave. In some problems, like junction diffraction or canonical objects between different media, it is more suitable to redefine Eq. (2) as uT = uin + uS
(3)
for uin represents the initial fields that exist in space when the scattering object is excluded [11–13]. Eq. (3) can be further expressed by uTGO + uTd = uin + uSGO + uSd
(4)
in terms of the diffracted and GO waves. Eq. (4) can be decomposed into two parts as uTGO = uin + uSGO
(5)
and uTd = uSd
(6)
since the initial fields do not contain diffracted wave. We will show the diffracted field by ud from now on. Our previous studies showed a relation between the scattered GO fields and diffracted waves [11–13]. Once the scattered GO field is determined, the diffracted field can be constructed by using suitable functions. Generally, the relation of
K+ ˛, ˇ K+ ∓ ˛, ˇ =
sin ˛ sin ˇ sin ˛ + sin ˇ
(7)
is encountered in the scattering problems by non-perfectly conducting surfaces. K+ (˛,ˇ) is the split function that arises in the solution of the resistive half-screen problem [16]. It can be introduced by the equation of
K+ ˛, ˇ =
×
where
(x)
⎧ ⎪ ⎨
4
sin ˇ sin
/2 − ˛ + ˇ √ 1 + 2 cos 2
2
−˛+ˇ
⎪ ⎩
3
2
2
˛ 2
3/2 − ˛ − ˇ √ 1 + 2 cos 2
⎫2 ⎬ −˛−ˇ ⎪
2
(8)
⎪ ⎭
is the Maliuzhinets function [17] that has the expression
⎡
(x) = exp ⎣−
1 8
x
√ sin v − 2 2 sin v/2 + 2v cos v
⎤
dv⎦ .
(9)
0
First of all, the scattered GO wave must be determined from the equation uSGO = uSGO − uin
(10)
as a second step, the diffracted wave will be constructed by using the relation between the scattered GO and diffracted waves. Now we will derive this relation for the problem of wedge diffraction. A plane wave of ui = u0 ejk cos(ϕ−ϕ0 )
(11)
hits a soft wedge (total field is equal to zero on the surface) with an angle of incident of ϕ0 . The wedge is located in a single medium. u0 is the complex amplitude. The cylindrical coordinates are given by (,ϕ,z). The geometry of the problem is shown in Fig. 1. The outer angle of the wedge is equal to . P is the observation point.
10
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 1. The geometry of the wedge in a single medium.
The diffracted field by the wedge can be given by
sin n e−j 4 ud = u0 √ n 2
1 cos
n
ϕ−ϕ0 n
− cos
−
1 cos
n
− cos
ϕ+ϕ0 n
e−jk
(12)
k
for n has the expression of / [1]. The total GO field can be written as
uTGO = ui U −0− − ur0 U −0+
(13)
when only face-0 is illuminated. uTGO will become
uTGO = ui − ur0 U −0+ − urn U n+
(14)
if the incident field hits both of the faces. ur 0 and urn are the reflected waves from face-0 and face-n, which can be defined by ur0 = u0 ejk cos(ϕ+ϕ0 )
(15)
and urn = u0 ejk cos[ϕ+ϕ0 −2(n−1)]
(16)
respectively. U(x) is the unit step function, which is equal to one for x>0 and zero otherwise. 0± and n± are the detour parameters, which can be introduced by
0± = − and
2k cos
n± = −
2k cos
ϕ ± ϕ0 2
(17)
ϕ ± ϕ0 − 2 (n − 1) 2
(18)
respectively [14]. The scattered GO waves read
uSGO = −ur0 U −0+ − ui U 0− and
uSGO = −ur0 U −0+ − urn U n+
(19)
(20)
for the cases, given by Eqs. (13) and (14) when the incident field is ui . The diffracted field, in Eq. (12), can be rewritten as
ud = u0
sin n e−j 4 √ n 2 cos
2 sin n
− cos
ϕ−ϕ0 n
ϕ
ϕ0 n cos n
e−jk
sin
n
− cos
ϕ+ϕ0 n
.
(21)
k
The relation of fn ( ∓ ϕ0 , ϕ0 ) = −2 sin
ϕ0 ∓ ϕ0 sin A[uSGO ]ϕ=∓ϕ0 n n
(22)
can be written for the case when only face-0 is illuminated. If the incident wave hits both of face-0 and face-n, the relations of fn ( − ϕ0 , ϕ0 ) = −2 sin
− ϕ0 ϕ0 sin A[uSGO ]ϕ=−ϕ0 n n
(23)
Y.Z. Umul / Optik 162 (2018) 8–18
11
Fig. 2. The geometry of the wedge for the illumination of single face.
and fn (2 − − ϕ0 , ϕ0 ) = 2 sin
+ ϕ0 ϕ0 sin A[uSGO ]ϕ=2−−ϕ0 n n
(24)
can be defined. The function fn can be introduced by the equation of
ud = u0
sin n e−j 4 √ n 2 cos
n
− cos
fn (ϕ, ϕ0 ) ϕ−ϕ0 cos n n
e−jk
− cos
ϕ+ϕ0 n
(25)
k
The operator A[z] means, “only consider the amplitude of z by eliminating phase”. In order to construct the diffracted field, Eq. (25) can be considered as a tool. The function fn can be obtained by using the relations, in Eqs. (22)–(24) and Eq. (7). Once the value of fn is found, the diffracted field becomes determined. 3. The solution of the problem A soft wedge, which is located on a planar interface between two different media, is interacting with an incident wave that has the expression of ui = u0 ejk1 cos(ϕ−ϕ1 )
(26)
for k1 and ϕ1 are the wave-number and angle of incidence in the first medium. The two media have different permittivities and permeabilities. We will take into consideration two cases. In the first case, the incident field interacts only with face-0 of the wedge. In the second one, the interaction of the wave with both of the wedge faces will be analyzed. 3.1. Incident field hits face-0 The geometry of the problem is given in Fig. 2. The incident wave illuminates only face-0 of the wedge. ϕ2 is the angle of refraction, which can be defined by ϕ2 = cos−1
k
1
k2
cos ϕ1
(27)
where k2 is the wave-number in the second medium. The cause of the diffraction process is the existence of the wedge in Fig. 2. If it is excluded from the geometry, only an interface between two media exists. The initial field reads uin = (ui + Rur0 ) U ( − ϕ) + Tut U (ϕ − )
(28)
in this case. R and T are the reflection and transmission coefficients of the interface, which can be introduced as R=
k1 sin ϕ1 − k2 sin ϕ2 k1 sin ϕ1 + k2 sin ϕ2
(29)
2k1 sin ϕ1 k1 sin ϕ1 + k2 sin ϕ2
(30)
and T=
respectively [10]. ur 0 is the reflected field from face-0 and can be expressed by ur0 = u0 ejk1 cos(ϕ+ϕ1 ) ,
(31)
12
Y.Z. Umul / Optik 162 (2018) 8–18
and it has the same expression with the reflected wave from the interface (except the reflection coefficient). ut can be written as ut = u0 ejk2 cos(ϕ−ϕ2 ) ,
(32)
and represents the refracted wave. We will define the transmission coefficient as T = T1 U ( − ϕ) + T2 U (ϕ − )
(33)
for our purposes. T1 and T2 have the expressions of
T1 = T11 U 1 + T12 U 2 and
(34)
T2 = T21 U 1 + T22 U 2
(35)
where T11 , T12 , T21 , T22 can be defined by T11 = T12 =
sin ϕ1 sin 1 1 , sin 1 sin ϕ1 + sin 1 sin ϕ1 + sin 2
sin ϕ1 sin ϕ2 + sin 2
sin ϕ sin sin ϕ2 1 2 2+ ,
T21 = and T22 =
(36)
sin ϕ1 sin ϕ2 + sin 1
(37)
sin ϕ1 + sin 2
sin 2
sin ϕ2 sin 1 sin ϕ2 sin 1 sin ϕ1 + sin 1 sin ϕ2 + sin 1
1
+
sin 2
2 sin ϕ2
(38)
sin ϕ sin 2 2
(39)
sin ϕ2 + sin 2
respectively. The coefficients Tij are derived from the transmission coefficient and defined for different media and different values of j . sin 1 and sin 2 can be represented as sin 1 =
1 2
and sin 2 =
1 2
k
2
k1
k
1
k2
sin ϕ2 − sin ϕ1
sin ϕ1 − sin ϕ2
(40)
(41)
respectively. The total GO field can be expressed by
uTGO = ui + Rur0 U 10+ − ur0 U −10+
U ( − ϕ) + Tut U −20− U (ϕ − )
(42)
from the geometry, in Fig. 2. The parameters 10± and 20± read
10± = − and
20± = −
2k1 cos
ϕ − ϕ1 2
(43)
2k2 cos
ϕ − ϕ2 2
(44)
respectively. The scattered GO wave is found to be
uSGO = −Tur0 U −10+ − Tut U 20−
(45)
from Eq. (10). The unknown diffracted field is written as ud = ud1 U ( − ϕ) + ud2 U (ϕ − )
(46)
for ud1 and ud2 can be expressed by
ud1 = u0
sin n e−j 4 √ n 2 cos
n
− cos
fn1 (ϕ, ϕ1 ) ϕ−ϕ1 cos n n
− cos
fn2 (ϕ, ϕ2 ) ϕ−ϕ2 cos n n
e−jk1
− cos
ϕ+ϕ1 n
− cos
ϕ+ϕ2 n
(47)
k1
and
ud2 = u0
sin n e−j 4 √ n 2 cos
n
e−jk2
k2
(48)
Y.Z. Umul / Optik 162 (2018) 8–18
13
respectively. The relations of fn1 ( − ϕ1 , ϕ1 ) = 2T1 sin
− ϕ1 ϕ1 sin n n
(49)
ϕ2 + ϕ2 sin n n
(50)
and fn2 ( + ϕ2 , ϕ2 ) = 2T2 sin
can be written from Eq. (22). The coefficients T11 , T12 , T21 and T22 can be arranged as
T11 = T12 =
K+ ϕ1 , 1 K+ − ϕ1 , 1 sin 1 sin ϕ1 + sin 2
sin ϕ1 sin ϕ2 + sin 2
and T22 =
,
sin ϕ2 sin 1 sin ϕ1 + sin 1
sin 2
2 sin ϕ2
+
(52)
sin 2
sin ϕ1 sin ϕ2 + sin 1
1
(51)
sin ϕ2 2+ K+ ϕ1 , 2 K+ − ϕ1 , 2 .
T21 =
K+ 2 − ϕ2 , 1 K+ − ϕ2 , 1
K+ 2 − ϕ2 , 2 K+ − ϕ2 , 2
(53)
(54)
by using Eq. (7). fn1 and fn2 can be determined as fn1 (ϕ, ϕ1 ) = 2T1 sin
ϕ1 ϕ sin n n
(55)
ϕ ϕ2 sin n n
(56)
and fn2 (ϕ, ϕ2 ) = 2T2 sin
where the coefficients T11 , T12 , T21 and T22 read
T11 = T12 =
T21 = and T22 =
K+ − ϕ, 1 K+ − ϕ1 , 1 sin 1 2 sin 2 sin
ϕ 2
ϕ 2
sin
ϕ1 2
sin ϕ1 2
sin ϕ2 + sin 2
sin ϕ1 2 sin 2 sin
ϕ 2
sin
ϕ2 2
+ sin 2
ϕ 2
sin
ϕ2 2
,
sin ϕ2 2+ K+ − ϕ, 2 K+ − ϕ1 , 2 , sin 2
+ sin 1
sin 1 sin ϕ1 + sin 1
2 1 + ϕ sin 2 2 sin 2 sin
ϕ2 2
(57)
K+ 3 − ϕ, 1 K+ − ϕ2 , 1
K+ 3 − ϕ, 2 K+ − ϕ2 , 2
(58)
(59)
(60)
respectively. The terms cos[(ϕ−ϕ1 )/2] and cos[(ϕ+ϕ2 )/2] are chosen in order to satisfy the principle of reciprocity. Thus the solution of the problem consists of the functions fn1 and fn2 , given in Eqs. (55) and (56). 3.2. Incident field hits face-0 and face-n The geometry, in Fig. 3, is taken into account. The two faces of the wedge are illuminated in this case. The incident wave, given in Eq. (26), hits face-0. Face-n is interacts with the refracted field. The initial velocity potential, given in Eq. (28), is also valid for this case. The angle , in Fig. 3, is equal to +ϕ0 −. The total field can be written as
uTGO = ui + Rur0 U 10+ − ur0 U −10+
U ( − ϕ) + T ut − urn U 2n+
U (ϕ − )
(61)
from the geometry, in Fig. 3. urn is the reflected wave from face-n and has the expression of urn = u0 ejk2 cos(ϕ+ϕ2 ) .
(62)
The parameter n1,2± can be defined as
n1,2± = −
2k1,2 cos
ϕ ± ϕ1,2 − 2 (n − 1) . 2
(63)
14
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 3. The geometry of the wedge for the illumination of both faces.
The scattered GO field reads
uTGO = −Tur0 U −10+ − Turn U 2n+
(64)
according to Eq. (10). The relations of fn1 ( − ϕ1 , ϕ1 ) = 2T1 sin
ϕ1 − ϕ1 sin n n
(65)
and fn2 (2 − − ϕ2 , ϕ2 ) = −2T2 sin
+ ϕ2 ϕ2 sin n n
(66)
can be written from Eqs. (23) and (24). The coefficients T11 and T12 are the same with the ones, in Eqs. (57) and (58). However, T21 and T22 are defined as
T21 = and T22 =
sin ϕ1 2 sin
2−ϕ 2
sin
ϕ2 2
sin ϕ2 sin 1 sin ϕ1 + sin 1
1 + sin 2 2 sin
2 2−ϕ 2
sin
+ sin 1 K+ 2 − − ϕ, 1 K+ − ϕ2 , 1 K+ 3 − ϕ, 1
ϕ2 2
K+ 4 − 2 + ϕ2 , 2
K+ 2 − − ϕ, 2 K+ − ϕ2 , 2 K+ 3 − ϕ, 2 K+ 4 − 2 + ϕ2 , 2
(67)
(68)
respectively. The functions fn1 and fn2 can be represented by fn1 (ϕ, ϕ1 ) = 2T1 sin
ϕ ϕ1 sin n n
(69)
ϕ ϕ2 sin n n
(70)
and fn2 (ϕ, ϕ2 ) = 2T2 sin
respectively. It is important to note that the only difference between the previous case, is in the definitions of T21 and T22 .
Y.Z. Umul / Optik 162 (2018) 8–18
15
4. The uniform diffracted field The expression of the diffracted field from a soft wedge, between different media, can be expressed as
ud = u0
−j n e√ 4 n 2
sin
+ u0
−j n e√ 4 n 2
sin
e−jk1 fn1 (ϕ, ϕ1 ) U ( − ϕ) ϕ − ϕ1 ϕ + ϕ1 k1 cos − cos cos − cos n n n n
fn2 (ϕ, ϕ2 ) U (ϕ − ) ϕ − ϕ2 ϕ + ϕ2 cos − cos cos − cos n n n n
(71)
e−jk2
k2
for fn1 and fn2 were determined in Sections 3.1 and 3.2. It is apparent that Eq. (71) has asymptotes at −ϕ1 , +ϕ2 or 2(n−1) −ϕ2 according to the problem. In order to obtain a finite expression, we will use UTD for wedge. The details and application of this theory to wedge diffraction can be found in the works of Kouyoumjian & Pathak [2] and Umul [14]. In order to outline the main theme, we take into consideration Eq. (12). The diffracted field can be written as ud = −
u0 (ud0− + ud0+ + udn− + udn+ ) n
(72)
where ud0± and udn ± can be introduced by
−(ϕ±ϕ
0) e−j 4 cos e−jk 2n ud0± = ∓ √ 0) 2 2 sin −(ϕ±ϕ k 2n
(73)
and
+(ϕ±ϕ
0) e−jk e−j 4 cos 2n udn± = ∓ √ +(ϕ±ϕ0 ) 2 2 sin k 2n
(74)
respectively. The approximate relation of sign (x) F [|x|] ≈
2 e−j( 4 +x ) √ 2 x
(75)
is considered for x»1 [18]. sign(x) is the signum function, which is equal to 1 for x>0 and −1 for x<0. F[x] shows the Fresnel function, which can be defined by the integral of
ej 4 F [x] = √
∞ 2
e−jt dt.
(76)
x
Eqs. (73) and (74) can be arranged as
−(ϕ±ϕ
0) cos e−j 4 cos 2n ud0± = ∓ √ −(ϕ±ϕ0 ) cos 2 2 sin 2n
ϕ±ϕ0 2 ϕ±ϕ0 2
e−jk
(77)
k
and
+(ϕ±ϕ
0) cos e−j 4 cos 2n udn± = ∓ √ +(ϕ±ϕ0 ) 2 2 sin cos 2n
ϕ±ϕ0 −2(n−1) 2 ϕ±ϕ0 −2(n−1) 2
e−jk
(78)
k
respectively. As a result, the uniform functions are found to be ud0± = ±
cos sin
−(ϕ±ϕ0 ) 2n −(ϕ±ϕ0 ) 2n
cos
ϕ ± ϕ0 jk cos(ϕ±ϕ0 ) e sign 0± F 0± 2
(79)
+(ϕ±ϕ0 ) 2n +(ϕ±ϕ0 ) 2n
cos
ϕ ± ϕ0 − 2 (n − 1) jk cos(ϕ±ϕ0 −2(n−1)) e sign −n± F n± 2
(80)
and udn± = ∓
cos sin
when Eq. (75) is taken into account. The uniform expression of the diffracted wave, in Eq. (71), can be obtained by using the above steps. The diffracted field is found to be ud = −
u0 [T1 (I01− + I01+ + In1− + In1+ ) U ( − ϕ) + T2 (I02− + I02+ + In2− + In2+ ) U (ϕ − )] n
(81)
16
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 4. The variation of total field when face-0 is illuminated.
Fig. 5. The total GO and diffracted fields for the illumination of face-0.
for I01,2± and In1,2± can be introduced as I01,2± = ±
cos sin
−(ϕ±ϕ1,2 ) 2n −(ϕ±ϕ1,2 ) 2n
cos
ϕ ± ϕ1,2 jk cos(ϕ±ϕ ) 01,2± 1,2 sign e 1,2 01,2± F 2
(82)
+(ϕ±ϕ1,2 ) 2n +(ϕ±ϕ1,2 ) 2n
cos
ϕ ± ϕ1,2 − 2 (n − 1) jk cos(ϕ±ϕ −2(n−1)) 1,2 e 1,2 sign n1,2± F n1,2± 2
(83)
and In1,2± = ±
cos sin
respectively. These expressions are valid for both of the illumination cases, mentioned in Section 3. Only the coefficients T21 and T22 differ. 5. Numerical results In this section, we will analyze the behavior of the fields numerically. The distance of observation () is taken as where
is the wavelength. The angle of incidence (ϕ1 ), in the first medium, is equal to 60◦ . ϕ2 is 80.4◦ in this case. The angle of observation (ϕ) varies from 0◦ to . k2 will be taken as 3k1 . Fig. 4 shows the variation of the total field versus the observation angle when only face-0 is illuminated by the incident field. is equal to 315◦ for this case. In medium I, the incident field interferes with the reflected waves from the face-0 of wedge and the interface between two media. Since the reflected field from the interface has smaller amplitude from the incident wave, the maximum interference intensity is equal to 1.5. However the field reflects with the same amplitude from the surface of the wedge. For this reason, the maximum amplitude is equal to 2 in the left side of the plot. After 180◦ , only the transmitted GO field and the diffracted wave exist. In the second medium, their interference is seen. Fig. 5 shows the components of the total field, which are the total GO and diffracted waves. It can be observed from the figure that the maxima of the diffracted field are in the same locations with the discontinuities of the total GO wave.
Y.Z. Umul / Optik 162 (2018) 8–18
17
Fig. 6. The total field for the illumination of wedge’s both faces.
Fig. 7. The total GO and diffracted fields when both of the wedge faces are illuminated.
Fig. 8. The comparison of this work with the exact solution for k2 is equal to k1 .
In Fig. 6, the variation of the total field with respect to the observation angle is seen for both of the faces of the wedge are illuminated by the incident wave. has the value of 225◦ . The behavior of the fields is similar to the previous case in the first medium. However, the intensity of the interference field at the second medium seems to increase. The reason is the interference of two GO waves in medium II. Fig. 7 shows the total GO and diffracted fields the sum of which gives the total velocity potential, in Fig. 6.
18
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 8 gives the comparison of the total field, obtained in this paper, versus the exact solution of the wedge diffraction problem in a single medium. k2 is equal to k1 for our solution. The exact total field for the wedge problem in a single medium can be expressed by 4 jvm e 2 Jvm (k) sin vm ϕ sin vm ϕ0 n ∞
uT =
(84)
m=1
where vm is equal to m/n [19]. It can be clearly seen from the figure that our solution exactly reduces to the rigorous solution of the wedge diffraction problem in a single medium when k2 is equal to k1 . 6. Conclusions In this paper, we obtained a solution of the diffraction problem of waves by a wedge, located between two different media. The diffracted fields are expressed in terms of the split functions, which occur in the diffraction problem by a resistive halfplane. Two cases are investigated according to the illumination of wedge’s faces. The numerical results show that the solution, obtained in this paper, reduces to the exact total field of the wedge diffraction problems in a single medium when k2 is equal to k1 . References [1] J.B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Am. 52 (1962) 116–130. [2] R.G. Kouyoumjian, P.H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. IEEE 62 (1974) 1448–1461. [3] P.C. Clemmow, Radio propagation over a flat earth across a boundary separating two different media, Philos. Trans. R. Soc. Lond. A 246 (1952) 1–55. [4] 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. 19 (1984) 1277–1288. [5] 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. [6] A. Ciarkowski, Three-dimensional electromagnetic half-plane diffraction at the interface of different media, Radio Sci. 22 (1987) 969–975. [7] J.L. Volakis, J.D. Collins, Electromagnetic scattering from a resistive half plane on a dielectric interface, Wave Motion 12 (1990) 81–96. [8] Y.Z. Umul, U. Yalcin, Scattered fields of conducting half-plane between two dielectric media, Appl. Opt. 49 (2010) 4010–4017. [9] Y.Z. Umul, Diffraction of waves by a perfectly conducting half-plane between two dielectric media, Opt. Commun. 310 (2014) 64–68. [10] Y.Z. Umul, Scattering theory of waves by a perfectly conducting half-screen between two different media, Microw. Opt. Technol. Lett. 57 (2015) 1928–1933. [11] Y.Z. Umul, Scattering by a conductive half-screen between isorefractive media, Appl. Opt. 54 (2015) 10309–10313. [12] Y.Z. Umul, Wave diffraction by the junction between resistive and soft-hard half-screens, Optik Int. J. Light Electron. Opt. 130 (2017) 750–756. [13] Y.Z. Umul, Interaction of plane waves by a half-screen between two different media, Opt. Quantum Electron. 50 (2018) 84. [14] Y.Z. Umul, Effect of shadow geometry in wedge diffraction, J. Opt. Soc. Am. A 26 (2009) 1926–1931. [15] S.-W. Lee, Comparison of uniform asymptotic theory and Ufimtsev’s theory of electromagnetic edge diffraction, IEEE Trans. Antennas Propag. 25 (1977) 162–170. [16] T.B.A. Senior, Half plane edge diffraction, Radio Sci. 10 (1975) 645–650. [17] G.D. Maliuzhinets, Das Sommerfeldsche Integral und die lösung von Beugungsaufgaben in Wimkelgebieten, Ann. Phys. (Leipzig) 461 (1960) 107–112. [18] Y.Z. Umul, Equivalent functions for the Fresnelfresnel integral, Opt. Express 13 (2005) 8469–8482. [19] C. Balanis, Advanced Engineering Electromagnetics, Wiley, NJ, 2012.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Diffraction of waves by a wedge residing between two different media Yusuf Ziya Umul Electronic and Communication Dept., Cankaya University, Eskisehir yolu 29. km, Etimesgut, Ankara, 06790, Turkiye
a r t i c l e
i n f o
Article history: Received 21 January 2018 Accepted 20 February 2018 Keywords: Diffraction theory Wedge Refraction
a b s t r a c t The scattering problem of plane waves by a perfectly electric conducting wedge, residing at the planar interface of two media with different electromagnetic properties, is investigated. The structure of the scattered geometrical optics waves is 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 obtained with the aid of uniform theory of diffraction. The behavior of the total, total geometrical optics and diffracted waves are studied numerically. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction The diffraction phenomenon of waves by a wedge is a canonical problem that finds many areas of applications in optics. The high-frequency expressions of the diffracted fields by soft and hard wedges underline the foundations of the geometrical theory of diffraction [1] and uniform theory of diffraction (UTD) [2]. However, a more fundamental problem, related with wedge, has not been investigated in the literature yet, to our knowledge. This problem describes a wedge, residing between two media with different permittivitiy and permeability. In fact the scattering of waves by a wedge in a single medium is a special case of this scenario. Similar scattering problems were studied for a perfectly conducting [3–6] and resistive [7] halfplanes. However, the diffraction field expressions, obtained by these studies contain spurs. We put forth alternative solutions, which do not have any spurious fields, for the perfectly conducting half-screen problem [8–10], but these expressions are approximate. The aim of this paper is to investigate the diffraction problem of plane waves by a soft wedge, located on a planar junction between two different media. First of all the initial fields, which occur on a planar whole interface between two different media, will be obtained. Then the total geometric optics (GO) waves will be determined for the problem under consideration. The scattered GO fields will be evaluated by subtracting the initial fields from the total GO waves. The diffracted fields, excited by the wedge, will be obtained with the aid of a relation, put forth by us [11–13]. The uniform diffraction waves will be derived by using UTD [2,14]. The behaviors of the total, total GO and diffracted waves will be analyzed numerically. A time factor of exp(jωt) is assumed and suppressed throughout the paper. ω is the angular frequency.
E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2018.02.059 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
Y.Z. Umul / Optik 162 (2018) 8–18
9
2. Introduction of the method We suppose an incident wave that has the time harmonic electric field component of ui interacts with an obstacle on its path of propagation. The wave satisfies the Helmholtz equation
∇ 2 ui + k2 ui = 0
(1)
where k is the wave-number. The total field of uT = ui + uS
(2)
is excited when the incident wave interacts with an obstacle on its path of propagation [15]. ui is the wave distribution when there is not any scatterer in the medium. uS is the scattered field and shows the effect of the obstacle on the incident wave. In some problems, like junction diffraction or canonical objects between different media, it is more suitable to redefine Eq. (2) as uT = uin + uS
(3)
for uin represents the initial fields that exist in space when the scattering object is excluded [11–13]. Eq. (3) can be further expressed by uTGO + uTd = uin + uSGO + uSd
(4)
in terms of the diffracted and GO waves. Eq. (4) can be decomposed into two parts as uTGO = uin + uSGO
(5)
and uTd = uSd
(6)
since the initial fields do not contain diffracted wave. We will show the diffracted field by ud from now on. Our previous studies showed a relation between the scattered GO fields and diffracted waves [11–13]. Once the scattered GO field is determined, the diffracted field can be constructed by using suitable functions. Generally, the relation of
K+ ˛, ˇ K+ ∓ ˛, ˇ =
sin ˛ sin ˇ sin ˛ + sin ˇ
(7)
is encountered in the scattering problems by non-perfectly conducting surfaces. K+ (˛,ˇ) is the split function that arises in the solution of the resistive half-screen problem [16]. It can be introduced by the equation of
K+ ˛, ˇ =
×
where
(x)
⎧ ⎪ ⎨
4
sin ˇ sin
/2 − ˛ + ˇ √ 1 + 2 cos 2
2
−˛+ˇ
⎪ ⎩
3
2
2
˛ 2
3/2 − ˛ − ˇ √ 1 + 2 cos 2
⎫2 ⎬ −˛−ˇ ⎪
2
(8)
⎪ ⎭
is the Maliuzhinets function [17] that has the expression
⎡
(x) = exp ⎣−
1 8
x
√ sin v − 2 2 sin v/2 + 2v cos v
⎤
dv⎦ .
(9)
0
First of all, the scattered GO wave must be determined from the equation uSGO = uSGO − uin
(10)
as a second step, the diffracted wave will be constructed by using the relation between the scattered GO and diffracted waves. Now we will derive this relation for the problem of wedge diffraction. A plane wave of ui = u0 ejk cos(ϕ−ϕ0 )
(11)
hits a soft wedge (total field is equal to zero on the surface) with an angle of incident of ϕ0 . The wedge is located in a single medium. u0 is the complex amplitude. The cylindrical coordinates are given by (,ϕ,z). The geometry of the problem is shown in Fig. 1. The outer angle of the wedge is equal to . P is the observation point.
10
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 1. The geometry of the wedge in a single medium.
The diffracted field by the wedge can be given by
sin n e−j 4 ud = u0 √ n 2
1 cos
n
ϕ−ϕ0 n
− cos
−
1 cos
n
− cos
ϕ+ϕ0 n
e−jk
(12)
k
for n has the expression of / [1]. The total GO field can be written as
uTGO = ui U −0− − ur0 U −0+
(13)
when only face-0 is illuminated. uTGO will become
uTGO = ui − ur0 U −0+ − urn U n+
(14)
if the incident field hits both of the faces. ur 0 and urn are the reflected waves from face-0 and face-n, which can be defined by ur0 = u0 ejk cos(ϕ+ϕ0 )
(15)
and urn = u0 ejk cos[ϕ+ϕ0 −2(n−1)]
(16)
respectively. U(x) is the unit step function, which is equal to one for x>0 and zero otherwise. 0± and n± are the detour parameters, which can be introduced by
0± = − and
2k cos
n± = −
2k cos
ϕ ± ϕ0 2
(17)
ϕ ± ϕ0 − 2 (n − 1) 2
(18)
respectively [14]. The scattered GO waves read
uSGO = −ur0 U −0+ − ui U 0− and
uSGO = −ur0 U −0+ − urn U n+
(19)
(20)
for the cases, given by Eqs. (13) and (14) when the incident field is ui . The diffracted field, in Eq. (12), can be rewritten as
ud = u0
sin n e−j 4 √ n 2 cos
2 sin n
− cos
ϕ−ϕ0 n
ϕ
ϕ0 n cos n
e−jk
sin
n
− cos
ϕ+ϕ0 n
.
(21)
k
The relation of fn ( ∓ ϕ0 , ϕ0 ) = −2 sin
ϕ0 ∓ ϕ0 sin A[uSGO ]ϕ=∓ϕ0 n n
(22)
can be written for the case when only face-0 is illuminated. If the incident wave hits both of face-0 and face-n, the relations of fn ( − ϕ0 , ϕ0 ) = −2 sin
− ϕ0 ϕ0 sin A[uSGO ]ϕ=−ϕ0 n n
(23)
Y.Z. Umul / Optik 162 (2018) 8–18
11
Fig. 2. The geometry of the wedge for the illumination of single face.
and fn (2 − − ϕ0 , ϕ0 ) = 2 sin
+ ϕ0 ϕ0 sin A[uSGO ]ϕ=2−−ϕ0 n n
(24)
can be defined. The function fn can be introduced by the equation of
ud = u0
sin n e−j 4 √ n 2 cos
n
− cos
fn (ϕ, ϕ0 ) ϕ−ϕ0 cos n n
e−jk
− cos
ϕ+ϕ0 n
(25)
k
The operator A[z] means, “only consider the amplitude of z by eliminating phase”. In order to construct the diffracted field, Eq. (25) can be considered as a tool. The function fn can be obtained by using the relations, in Eqs. (22)–(24) and Eq. (7). Once the value of fn is found, the diffracted field becomes determined. 3. The solution of the problem A soft wedge, which is located on a planar interface between two different media, is interacting with an incident wave that has the expression of ui = u0 ejk1 cos(ϕ−ϕ1 )
(26)
for k1 and ϕ1 are the wave-number and angle of incidence in the first medium. The two media have different permittivities and permeabilities. We will take into consideration two cases. In the first case, the incident field interacts only with face-0 of the wedge. In the second one, the interaction of the wave with both of the wedge faces will be analyzed. 3.1. Incident field hits face-0 The geometry of the problem is given in Fig. 2. The incident wave illuminates only face-0 of the wedge. ϕ2 is the angle of refraction, which can be defined by ϕ2 = cos−1
k
1
k2
cos ϕ1
(27)
where k2 is the wave-number in the second medium. The cause of the diffraction process is the existence of the wedge in Fig. 2. If it is excluded from the geometry, only an interface between two media exists. The initial field reads uin = (ui + Rur0 ) U ( − ϕ) + Tut U (ϕ − )
(28)
in this case. R and T are the reflection and transmission coefficients of the interface, which can be introduced as R=
k1 sin ϕ1 − k2 sin ϕ2 k1 sin ϕ1 + k2 sin ϕ2
(29)
2k1 sin ϕ1 k1 sin ϕ1 + k2 sin ϕ2
(30)
and T=
respectively [10]. ur 0 is the reflected field from face-0 and can be expressed by ur0 = u0 ejk1 cos(ϕ+ϕ1 ) ,
(31)
12
Y.Z. Umul / Optik 162 (2018) 8–18
and it has the same expression with the reflected wave from the interface (except the reflection coefficient). ut can be written as ut = u0 ejk2 cos(ϕ−ϕ2 ) ,
(32)
and represents the refracted wave. We will define the transmission coefficient as T = T1 U ( − ϕ) + T2 U (ϕ − )
(33)
for our purposes. T1 and T2 have the expressions of
T1 = T11 U 1 + T12 U 2 and
(34)
T2 = T21 U 1 + T22 U 2
(35)
where T11 , T12 , T21 , T22 can be defined by T11 = T12 =
sin ϕ1 sin 1 1 , sin 1 sin ϕ1 + sin 1 sin ϕ1 + sin 2
sin ϕ1 sin ϕ2 + sin 2
sin ϕ sin sin ϕ2 1 2 2+ ,
T21 = and T22 =
(36)
sin ϕ1 sin ϕ2 + sin 1
(37)
sin ϕ1 + sin 2
sin 2
sin ϕ2 sin 1 sin ϕ2 sin 1 sin ϕ1 + sin 1 sin ϕ2 + sin 1
1
+
sin 2
2 sin ϕ2
(38)
sin ϕ sin 2 2
(39)
sin ϕ2 + sin 2
respectively. The coefficients Tij are derived from the transmission coefficient and defined for different media and different values of j . sin 1 and sin 2 can be represented as sin 1 =
1 2
and sin 2 =
1 2
k
2
k1
k
1
k2
sin ϕ2 − sin ϕ1
sin ϕ1 − sin ϕ2
(40)
(41)
respectively. The total GO field can be expressed by
uTGO = ui + Rur0 U 10+ − ur0 U −10+
U ( − ϕ) + Tut U −20− U (ϕ − )
(42)
from the geometry, in Fig. 2. The parameters 10± and 20± read
10± = − and
20± = −
2k1 cos
ϕ − ϕ1 2
(43)
2k2 cos
ϕ − ϕ2 2
(44)
respectively. The scattered GO wave is found to be
uSGO = −Tur0 U −10+ − Tut U 20−
(45)
from Eq. (10). The unknown diffracted field is written as ud = ud1 U ( − ϕ) + ud2 U (ϕ − )
(46)
for ud1 and ud2 can be expressed by
ud1 = u0
sin n e−j 4 √ n 2 cos
n
− cos
fn1 (ϕ, ϕ1 ) ϕ−ϕ1 cos n n
− cos
fn2 (ϕ, ϕ2 ) ϕ−ϕ2 cos n n
e−jk1
− cos
ϕ+ϕ1 n
− cos
ϕ+ϕ2 n
(47)
k1
and
ud2 = u0
sin n e−j 4 √ n 2 cos
n
e−jk2
k2
(48)
Y.Z. Umul / Optik 162 (2018) 8–18
13
respectively. The relations of fn1 ( − ϕ1 , ϕ1 ) = 2T1 sin
− ϕ1 ϕ1 sin n n
(49)
ϕ2 + ϕ2 sin n n
(50)
and fn2 ( + ϕ2 , ϕ2 ) = 2T2 sin
can be written from Eq. (22). The coefficients T11 , T12 , T21 and T22 can be arranged as
T11 = T12 =
K+ ϕ1 , 1 K+ − ϕ1 , 1 sin 1 sin ϕ1 + sin 2
sin ϕ1 sin ϕ2 + sin 2
and T22 =
,
sin ϕ2 sin 1 sin ϕ1 + sin 1
sin 2
2 sin ϕ2
+
(52)
sin 2
sin ϕ1 sin ϕ2 + sin 1
1
(51)
sin ϕ2 2+ K+ ϕ1 , 2 K+ − ϕ1 , 2 .
T21 =
K+ 2 − ϕ2 , 1 K+ − ϕ2 , 1
K+ 2 − ϕ2 , 2 K+ − ϕ2 , 2
(53)
(54)
by using Eq. (7). fn1 and fn2 can be determined as fn1 (ϕ, ϕ1 ) = 2T1 sin
ϕ1 ϕ sin n n
(55)
ϕ ϕ2 sin n n
(56)
and fn2 (ϕ, ϕ2 ) = 2T2 sin
where the coefficients T11 , T12 , T21 and T22 read
T11 = T12 =
T21 = and T22 =
K+ − ϕ, 1 K+ − ϕ1 , 1 sin 1 2 sin 2 sin
ϕ 2
ϕ 2
sin
ϕ1 2
sin ϕ1 2
sin ϕ2 + sin 2
sin ϕ1 2 sin 2 sin
ϕ 2
sin
ϕ2 2
+ sin 2
ϕ 2
sin
ϕ2 2
,
sin ϕ2 2+ K+ − ϕ, 2 K+ − ϕ1 , 2 , sin 2
+ sin 1
sin 1 sin ϕ1 + sin 1
2 1 + ϕ sin 2 2 sin 2 sin
ϕ2 2
(57)
K+ 3 − ϕ, 1 K+ − ϕ2 , 1
K+ 3 − ϕ, 2 K+ − ϕ2 , 2
(58)
(59)
(60)
respectively. The terms cos[(ϕ−ϕ1 )/2] and cos[(ϕ+ϕ2 )/2] are chosen in order to satisfy the principle of reciprocity. Thus the solution of the problem consists of the functions fn1 and fn2 , given in Eqs. (55) and (56). 3.2. Incident field hits face-0 and face-n The geometry, in Fig. 3, is taken into account. The two faces of the wedge are illuminated in this case. The incident wave, given in Eq. (26), hits face-0. Face-n is interacts with the refracted field. The initial velocity potential, given in Eq. (28), is also valid for this case. The angle , in Fig. 3, is equal to +ϕ0 −. The total field can be written as
uTGO = ui + Rur0 U 10+ − ur0 U −10+
U ( − ϕ) + T ut − urn U 2n+
U (ϕ − )
(61)
from the geometry, in Fig. 3. urn is the reflected wave from face-n and has the expression of urn = u0 ejk2 cos(ϕ+ϕ2 ) .
(62)
The parameter n1,2± can be defined as
n1,2± = −
2k1,2 cos
ϕ ± ϕ1,2 − 2 (n − 1) . 2
(63)
14
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 3. The geometry of the wedge for the illumination of both faces.
The scattered GO field reads
uTGO = −Tur0 U −10+ − Turn U 2n+
(64)
according to Eq. (10). The relations of fn1 ( − ϕ1 , ϕ1 ) = 2T1 sin
ϕ1 − ϕ1 sin n n
(65)
and fn2 (2 − − ϕ2 , ϕ2 ) = −2T2 sin
+ ϕ2 ϕ2 sin n n
(66)
can be written from Eqs. (23) and (24). The coefficients T11 and T12 are the same with the ones, in Eqs. (57) and (58). However, T21 and T22 are defined as
T21 = and T22 =
sin ϕ1 2 sin
2−ϕ 2
sin
ϕ2 2
sin ϕ2 sin 1 sin ϕ1 + sin 1
1 + sin 2 2 sin
2 2−ϕ 2
sin
+ sin 1 K+ 2 − − ϕ, 1 K+ − ϕ2 , 1 K+ 3 − ϕ, 1
ϕ2 2
K+ 4 − 2 + ϕ2 , 2
K+ 2 − − ϕ, 2 K+ − ϕ2 , 2 K+ 3 − ϕ, 2 K+ 4 − 2 + ϕ2 , 2
(67)
(68)
respectively. The functions fn1 and fn2 can be represented by fn1 (ϕ, ϕ1 ) = 2T1 sin
ϕ ϕ1 sin n n
(69)
ϕ ϕ2 sin n n
(70)
and fn2 (ϕ, ϕ2 ) = 2T2 sin
respectively. It is important to note that the only difference between the previous case, is in the definitions of T21 and T22 .
Y.Z. Umul / Optik 162 (2018) 8–18
15
4. The uniform diffracted field The expression of the diffracted field from a soft wedge, between different media, can be expressed as
ud = u0
−j n e√ 4 n 2
sin
+ u0
−j n e√ 4 n 2
sin
e−jk1 fn1 (ϕ, ϕ1 ) U ( − ϕ) ϕ − ϕ1 ϕ + ϕ1 k1 cos − cos cos − cos n n n n
fn2 (ϕ, ϕ2 ) U (ϕ − ) ϕ − ϕ2 ϕ + ϕ2 cos − cos cos − cos n n n n
(71)
e−jk2
k2
for fn1 and fn2 were determined in Sections 3.1 and 3.2. It is apparent that Eq. (71) has asymptotes at −ϕ1 , +ϕ2 or 2(n−1) −ϕ2 according to the problem. In order to obtain a finite expression, we will use UTD for wedge. The details and application of this theory to wedge diffraction can be found in the works of Kouyoumjian & Pathak [2] and Umul [14]. In order to outline the main theme, we take into consideration Eq. (12). The diffracted field can be written as ud = −
u0 (ud0− + ud0+ + udn− + udn+ ) n
(72)
where ud0± and udn ± can be introduced by
−(ϕ±ϕ
0) e−j 4 cos e−jk 2n ud0± = ∓ √ 0) 2 2 sin −(ϕ±ϕ k 2n
(73)
and
+(ϕ±ϕ
0) e−jk e−j 4 cos 2n udn± = ∓ √ +(ϕ±ϕ0 ) 2 2 sin k 2n
(74)
respectively. The approximate relation of sign (x) F [|x|] ≈
2 e−j( 4 +x ) √ 2 x
(75)
is considered for x»1 [18]. sign(x) is the signum function, which is equal to 1 for x>0 and −1 for x<0. F[x] shows the Fresnel function, which can be defined by the integral of
ej 4 F [x] = √
∞ 2
e−jt dt.
(76)
x
Eqs. (73) and (74) can be arranged as
−(ϕ±ϕ
0) cos e−j 4 cos 2n ud0± = ∓ √ −(ϕ±ϕ0 ) cos 2 2 sin 2n
ϕ±ϕ0 2 ϕ±ϕ0 2
e−jk
(77)
k
and
+(ϕ±ϕ
0) cos e−j 4 cos 2n udn± = ∓ √ +(ϕ±ϕ0 ) 2 2 sin cos 2n
ϕ±ϕ0 −2(n−1) 2 ϕ±ϕ0 −2(n−1) 2
e−jk
(78)
k
respectively. As a result, the uniform functions are found to be ud0± = ±
cos sin
−(ϕ±ϕ0 ) 2n −(ϕ±ϕ0 ) 2n
cos
ϕ ± ϕ0 jk cos(ϕ±ϕ0 ) e sign 0± F 0± 2
(79)
+(ϕ±ϕ0 ) 2n +(ϕ±ϕ0 ) 2n
cos
ϕ ± ϕ0 − 2 (n − 1) jk cos(ϕ±ϕ0 −2(n−1)) e sign −n± F n± 2
(80)
and udn± = ∓
cos sin
when Eq. (75) is taken into account. The uniform expression of the diffracted wave, in Eq. (71), can be obtained by using the above steps. The diffracted field is found to be ud = −
u0 [T1 (I01− + I01+ + In1− + In1+ ) U ( − ϕ) + T2 (I02− + I02+ + In2− + In2+ ) U (ϕ − )] n
(81)
16
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 4. The variation of total field when face-0 is illuminated.
Fig. 5. The total GO and diffracted fields for the illumination of face-0.
for I01,2± and In1,2± can be introduced as I01,2± = ±
cos sin
−(ϕ±ϕ1,2 ) 2n −(ϕ±ϕ1,2 ) 2n
cos
ϕ ± ϕ1,2 jk cos(ϕ±ϕ ) 01,2± 1,2 sign e 1,2 01,2± F 2
(82)
+(ϕ±ϕ1,2 ) 2n +(ϕ±ϕ1,2 ) 2n
cos
ϕ ± ϕ1,2 − 2 (n − 1) jk cos(ϕ±ϕ −2(n−1)) 1,2 e 1,2 sign n1,2± F n1,2± 2
(83)
and In1,2± = ±
cos sin
respectively. These expressions are valid for both of the illumination cases, mentioned in Section 3. Only the coefficients T21 and T22 differ. 5. Numerical results In this section, we will analyze the behavior of the fields numerically. The distance of observation () is taken as where
is the wavelength. The angle of incidence (ϕ1 ), in the first medium, is equal to 60◦ . ϕ2 is 80.4◦ in this case. The angle of observation (ϕ) varies from 0◦ to . k2 will be taken as 3k1 . Fig. 4 shows the variation of the total field versus the observation angle when only face-0 is illuminated by the incident field. is equal to 315◦ for this case. In medium I, the incident field interferes with the reflected waves from the face-0 of wedge and the interface between two media. Since the reflected field from the interface has smaller amplitude from the incident wave, the maximum interference intensity is equal to 1.5. However the field reflects with the same amplitude from the surface of the wedge. For this reason, the maximum amplitude is equal to 2 in the left side of the plot. After 180◦ , only the transmitted GO field and the diffracted wave exist. In the second medium, their interference is seen. Fig. 5 shows the components of the total field, which are the total GO and diffracted waves. It can be observed from the figure that the maxima of the diffracted field are in the same locations with the discontinuities of the total GO wave.
Y.Z. Umul / Optik 162 (2018) 8–18
17
Fig. 6. The total field for the illumination of wedge’s both faces.
Fig. 7. The total GO and diffracted fields when both of the wedge faces are illuminated.
Fig. 8. The comparison of this work with the exact solution for k2 is equal to k1 .
In Fig. 6, the variation of the total field with respect to the observation angle is seen for both of the faces of the wedge are illuminated by the incident wave. has the value of 225◦ . The behavior of the fields is similar to the previous case in the first medium. However, the intensity of the interference field at the second medium seems to increase. The reason is the interference of two GO waves in medium II. Fig. 7 shows the total GO and diffracted fields the sum of which gives the total velocity potential, in Fig. 6.
18
Y.Z. Umul / Optik 162 (2018) 8–18
Fig. 8 gives the comparison of the total field, obtained in this paper, versus the exact solution of the wedge diffraction problem in a single medium. k2 is equal to k1 for our solution. The exact total field for the wedge problem in a single medium can be expressed by 4 jvm e 2 Jvm (k) sin vm ϕ sin vm ϕ0 n ∞
uT =
(84)
m=1
where vm is equal to m/n [19]. It can be clearly seen from the figure that our solution exactly reduces to the rigorous solution of the wedge diffraction problem in a single medium when k2 is equal to k1 . 6. Conclusions In this paper, we obtained a solution of the diffraction problem of waves by a wedge, located between two different media. The diffracted fields are expressed in terms of the split functions, which occur in the diffraction problem by a resistive halfplane. Two cases are investigated according to the illumination of wedge’s faces. The numerical results show that the solution, obtained in this paper, reduces to the exact total field of the wedge diffraction problems in a single medium when k2 is equal to k1 . References [1] J.B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Am. 52 (1962) 116–130. [2] R.G. Kouyoumjian, P.H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. IEEE 62 (1974) 1448–1461. [3] P.C. Clemmow, Radio propagation over a flat earth across a boundary separating two different media, Philos. Trans. R. Soc. Lond. A 246 (1952) 1–55. [4] 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. 19 (1984) 1277–1288. [5] 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. [6] A. Ciarkowski, Three-dimensional electromagnetic half-plane diffraction at the interface of different media, Radio Sci. 22 (1987) 969–975. [7] J.L. Volakis, J.D. Collins, Electromagnetic scattering from a resistive half plane on a dielectric interface, Wave Motion 12 (1990) 81–96. [8] Y.Z. Umul, U. Yalcin, Scattered fields of conducting half-plane between two dielectric media, Appl. Opt. 49 (2010) 4010–4017. [9] Y.Z. Umul, Diffraction of waves by a perfectly conducting half-plane between two dielectric media, Opt. Commun. 310 (2014) 64–68. [10] Y.Z. Umul, Scattering theory of waves by a perfectly conducting half-screen between two different media, Microw. Opt. Technol. Lett. 57 (2015) 1928–1933. [11] Y.Z. Umul, Scattering by a conductive half-screen between isorefractive media, Appl. Opt. 54 (2015) 10309–10313. [12] Y.Z. Umul, Wave diffraction by the junction between resistive and soft-hard half-screens, Optik Int. J. Light Electron. Opt. 130 (2017) 750–756. [13] Y.Z. Umul, Interaction of plane waves by a half-screen between two different media, Opt. Quantum Electron. 50 (2018) 84. [14] Y.Z. Umul, Effect of shadow geometry in wedge diffraction, J. Opt. Soc. Am. A 26 (2009) 1926–1931. [15] S.-W. Lee, Comparison of uniform asymptotic theory and Ufimtsev’s theory of electromagnetic edge diffraction, IEEE Trans. Antennas Propag. 25 (1977) 162–170. [16] T.B.A. Senior, Half plane edge diffraction, Radio Sci. 10 (1975) 645–650. [17] G.D. Maliuzhinets, Das Sommerfeldsche Integral und die lösung von Beugungsaufgaben in Wimkelgebieten, Ann. Phys. (Leipzig) 461 (1960) 107–112. [18] Y.Z. Umul, Equivalent functions for the Fresnelfresnel integral, Opt. Express 13 (2005) 8469–8482. [19] C. Balanis, Advanced Engineering Electromagnetics, Wiley, NJ, 2012.