Optik 127 (2016) 819–824
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Three dimensional modified theory of physical optics Yusuf Ziya Umul ∗ Electronic and Communication Department, Cankaya University, Eskis¸ehir yolu 29. km, Yenimahalle, Ankara 06810, Turkey
a r t i c l e
i n f o
Article history: Received 8 December 2014 Accepted 9 October 2015 Keywords: Physical optics Diffraction theory Edge diffraction
a b s t r a c t The three dimensional version of the modified theory of physical optics is introduced with the aid of a Green’s function that satisfies the Helmholtz equation in local spherical coordinates. The algorithm which leads to the construction of the scattering integral is given. The method is applied to the three dimensional diffraction problem of plane waves by a perfectly conducting half-plane. The comparison of the resulting field expressions with the literature shows that the modified theory of physical optics leads to the exact solution. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction The method of physical optics (PO) was first introduced by Macdonald in 1913 for the electromagnetic waves [1]. However, its scalar version had been put forth in the 19th century by Kirchhoff [2]. The main disadvantage of this method is the incorrect diffraction fields, evaluated by the edge point technique [3]. Ufimtsev tried to overcome this defect by defining additional edge currents, but he never became successful in obtaining the explicit expressions for the fringe (or non-uniform) currents as he confessed [4,5]. A good criticism of the physical theory of diffraction (PTD), which is the method of Ufimtsev, can be found in [6]. PTD only covers the highfrequency asymptotic solution of diffraction problems with the aid of the exact solutions of some canonical problems; this is far from eliminating the defect of PO. In 2004, the author introduced a new version of PO, which was leading to the exact solutions of some diffraction problems without any additional patch currents [7]. An algorithm, which was based on three mathematical and physical axioms, was introduced. The most important was the usage of a variable unit vector instead of the unit normal vector of the surface [8,9]. The new unit vector is related with the Fermat’s principle and its modified version, proposed by Keller in order to explain the edge diffraction phenomenon [10]. The incident field that hits the continuous part of the scatterer’s surfaces reflects with the angle of incidence according to the Fermat’s principle. The angles of incidence and reflection are determined according to the unit normal vector of the surface on the point of reflection. At the edge or discontinuity point, the incident ray is diffracted and many rays diverge. Thus we can define more than
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one unit vector that divides the angle between the incident and diffracted rays into two equal parts according to the Fermat’s principle of edge diffraction. The surface current is evaluated according to the unit normal vector of the surface in the classical PO. For this reason, the amplitude function of the PO integral is locked to the Fermat’s principle. The stationary phase evaluation of the PO integral leads to the correct geometrical optics (GO) reflected waves from the scatterer. However, the Fermat’s principle is not valid at the diffracting edge of the scatterer. Thus the method of PO considers the incorrect amplitude function in the edge point evaluation of the scattering integral. This choice leads to the incorrect diffracted field expressions. As a result, the reason of the PO’s defect is based on the definition of the surface current. The evaluation of this current component is performed according to the Fermat’s principle. In order to correct this problem, one must define the so-called surface current according to a more general unit vector, which will also satisfy the modified Fermat’s principle of Keller at the edge point. For this reason, the concept of the variable unit vector is proposed. These studies show that the scattering of MTPO integral yields the exact solution of the two dimensional (2D) diffraction problems of waves by a perfectly conducting half-plane [11,12]. Recently, a group of workers claimed that the theory of MTPO was wrong, because its Green’s function was not satisfying the Helmholtz equation [13]. This claim is unsubstantial as was shown in [14,15], because they use the wrong Green’s function in order to support their argument. In fact the Green’s function, used in MTPO, directly satisfies the Helmholtz equation and is not the highfrequency asymptotic expression of the zero-order second kind Hankel function. In this paper, the theory of the three dimensional (3D) case is proposed. With this aim, a Green’s function that satisfies the Helmholtz equation in local spherical coordinates was obtained. The procedure for the construction of the MTPO scattering integral
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will be introduced and the 3D method will be applied to the edge diffraction problem of plane waves by a perfectly conducting halfplane for oblique incidence. This problem is known in the literature as the 3D diffraction [16–18]. The resultant field expressions will be compared with the literature. A time factor ejωt is suppressed for ω is the angular frequency.
ut (P) = ui (P) + us (P)
(1)
where u is a component of the electromagnetic field. ui represents the incident wave. us is the scattered field and can be given by the integral
ui (Q )G(P, Q ) dl
(2)
C
for G is the Green’s function, which is an exact solution of the Helmholtz equation. k is the wavenumber. R1 and ˇ define a local polar coordinate system at the point Q. The Helmholtz equation can be written as: 1 ∂ R1 ∂R1
∂G R1 ∂R1
+
1 ∂2 G + k2 G = 0 R12 ∂ˇ2
(3)
when the singular point is excluded from the space [16]. A solution of (3) can be given by
G=
A sin
ˇ ˇ + B cos 2 2
(2)
H1/2 (kR1 )
(4) (2)
for the diverging waves. A and B are constants. H1/2 is the ½ order second kind Hankel function, which is also equal to
(2) H1/2 (kR1 )
=
2 e−jkR1 kR1
Fig. 1. 2D scattering geometry.
ui (Q )
sin
C
ˇ−˛ ˇ+˛ ∓ sin 2 2
e−jkR1
dl .
(6)
kR1
It is apparent that the Green’s function of the integral is in the form G(P, Q ) =
Before outlining the algorithm for the construction of the 3D MTPO integral, the author will review the 2D case. The scattering geometry, shown in Fig. 1, is considered. The incident field hits the scatterer with the angle ˛ at the scattering point Q. The scattered ray leaves the surface with the angle ˇ and reaches the observation point P. The PO integral also includes a virtual transmitted ray that passes through the surface as if it does not exist [19]. t is the tangent v is the variable unit vector that divides the of the curve C at Q and n angle between the incident and reflected rays into two equal parts (). R1 is the distance between P and Q for 2D geometry. The total field that occurs after the interaction of the incident wave with the scatterer can be introduced by
kej/4 2
kej/4 us (P) = √ 2
2. 3D MTPO
us (P) =
The previous studies showed that the 2D MTPO integral of the scattered fields must be as follows:
sin
ˇ−˛ ˇ+˛ ∓ sin 2 2
e−jkR1
(7)
,
kR1
which directly satisfies the Helmholtz equation, in Eq. (3), when Eq. (4) is taken into consideration. The plus and minus signs are valid for the Neumann (normal derivative of the total field is zero on the surface) and Dirichlet (total field is equal to zero on the surface) boundary conditions respectively. We can also obtain the Green’s function by using the geometry, as shown in Fig. 1. The cosine of the angle reads cos = sin
ˇ+˛ v. = sr · n 2
(8)
where sr is the unit vector in the direction of the reflected ray. Thus the 2D Green’s function can be defined by e−jkR1 v )ˇ→−ˇ ± sr · n v] G(P, Q ) = −[(sr · n kR1
(9)
in terms of the variable unit vector. The first and second terms are related to the transmitted and reflected rays, respectively. Now, MTPO will be generalized for the 3D scattering problems. The scattered field can be written as us (P) =
jk 2
ui (Q )G(P, Q ) dS
(10)
S
in a similar way with the 2D case. S is the surface of the scatterer. Because of the surface integration, the observation point is the function of two scattering angles. These are shown by ˇ and in Fig. 2. Note that ˇ varies from 0 to 2 whereas exists in the interval [0,]. For this reason, the transmitted ray is defined according to the angle ˇ. The scattering integral, in Eq. (10) has two stationary phase points, namely ˇs and −ˇs . The first one gives the reflected GO wave and the second angle represents the transmitted GO field. Thus an angle must vary between 0 and 2 to include two GO fields. In Fig. 2, (x1 , x2 , x3 ) is an orthogonal coordinate system. R1 is the projection of R to the (x1 , x2 ) plane. The Green’s function, in Eq. (10), must satisfy the Helmholtz equation given as: ∂ ∂R
R
2 ∂G
∂R
1 ∂ + sin ∂
∂G sin ∂
+
1 sin
2
∂2 G + k2 R2 G = 0, ∂2 (11)
(5)
Fig. 2. Scattering geometry for 3D problems.
Y.Z. Umul / Optik 127 (2016) 819–824
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the general solution of which can be expressed by
A sin(/2)
G=
sin
+
B cos(/2)
C sin
sin
ˇ ˇ + D cos 2 2
e−jkR R (12)
for diverging spherical waves. The procedure for the construction of the 3D MTPO scattering integral can be introduced as follows: (1) determine the variable unit vector from the relation v · si = −n v · sr , n
(13)
(2) obtain the Green’s function from the equation
G(P, Q ) = −
v ) [ (sr · n
v ] −jkR ± sr · n e R sin sin 0
Fig. 3. Geometry of the half-plane.
ˇ→−ˇ
(14) 3. Three dimensional diffraction of plane waves by a perfectly conducting half-plane
(3) write the scattering integral as jk sin 0 us (P) = − 2 ×
ui (Q ) S
v ) v (sr · n ± sr · n ˇ→−ˇ
sin sin 0
e−jkR dS . R
(15)
I will find the variable unit vector for the geometry, shown in Fig. 2, as an example. The unit vectors of the incident and reflected rays can be expressed by si = sin 0 cos ˇ0 ex1 − sin 0 sin ˇ0 ex2 − cos 0 ex3
(16)
and sr = sin cos ˇex1 + sin sin ˇex2 − cos ex3
(17)
As an application of the 3D MTPO, I will solve the scattering process of plane waves by a perfectly conducting half-plane for oblique incidence. As mentioned above, this problem is known as the three dimensional diffraction in the literature, since the incident and reflected waves are functions of besides [16–18]. The geometry of the problem is given in Fig. 3. The half-screen is located at x ∈ [0,∞), y = 0 and z ∈ (−∞,∞). The total field satisfies the Dirichlet boundary condition on the scattering surface. The incident plane wave has the following expression ui = u0 ejk(x sin 0
cos 0 +y sin 0 sin 0 +z cos 0 )
.
(22)
In order to construct the scattering integral of the 3D MTPO, the unit vectors along the incident and scattered rays must be written as si = − sin 0 cos 0 ex − sin 0 sin 0 ey − cos 0 ez
(23)
and
respectively. The variable unit vector can be written as v = Aex1 + Bex2 + C ex3 n
(18)
sr = − sin cos ˇex + sin sin ˇey − cos ez
(24)
from Fig. 3. The variable unit vector can be determined as for A, B and C are constants. The equation v = sin n
A sin 0 cos ˇ0 − B sin 0 sin ˇ0 − C cos 0 = −A sin cos ˇ − B sin sin ˇ + C cos
(19)
can be obtained from Eq. (13). The solution of Eq. (19) can be found by using the half angles. Also note that the functions of ˇ and are independent from each other, since the Green’s function, in Eq. (12), is evaluated by the method of separation of variables. As a result the variable unit vector reads v = − sin n
ˇ + ˇ0 − 0 sin ex3 2 2
(20)
Eq. (20) also yields ex2 , which is the unit normal vector of the surface at Q, at the stationary phase point = 0 , ˇ = ˇ0 . The Green’s function is found to be G(P, Q ) =
sin(( + 0 )/2)
sin sin 0
sin
ˇ − ˇ0 ˇ + ˇ0 ∓ sin 2 2
e−jkR R
ˇ + 0 − 0 sin ez 2 2
+ sin
(25)
from Eq. (13). The Green’s function reads G(P, Q ) =
sin(( + 0 )/2)
sin sin 0
ˇ − 0 ˇ + 0 ∓ sin sin 2 2
e−jkR R
(26)
according to Eq. (14). As a result the scattered field can be written by
ˇ − ˇ0 ˇ − ˇ0 − 0 − 0 cos ex1 + cos cos ex2 2 2 2 2
+ sin
ˇ − 0 ˇ − 0 − 0 − 0 cos ex + cos cos ey 2 2 2 2
(21)
from Eq. (14). It is apparent that Eq. (21) satisfies the Helmholtz equation, given in Eq. (11), according to Eq. (12).
jku0 sin 0 us (P) = 2 ×
∞
x =0
sin(( + 0 )/2)
sin sin 0
∞
ejk(x
sin 0 cos 0 +z cos 0 )
z =−∞
sin
ˇ − 0 ˇ + 0 − sin 2 2
e−jkR dz dx R (27)
for the Dirichlet boundary conditions. R is equal to
(x − x )2 + y2 + (z − z )2 . The total field, in Eq. (1), can also be expressed as
ut (P) = uis (P) + urs (P)
(28)
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Y.Z. Umul / Optik 127 (2016) 819–824
as a sum of the incident and reflected scattered waves that can be defined by
jku0 sin 0 uis (P) = ui (P) + 2
∞
e x =0
jk(
x
sin 0 cos 0
+z
cos 0 )
sin sin 0
k sin 0 [R1 − x cos 0 + x cos 0 + y sin 0 ]
(29)
(40)
is introduced for the integral ∞
ejkx
M=
sin 0 cos 0
sin
0
ˇ − 0 e−jkR1 sin 0 dx 2 kR1 sin 0
(41)
in Eq. (39). can also be expressed as
and jku0 sin 0 urs (P) = − 2
∞
∞
e x =0
=
jk(x sin 0 cos 0 +z cos 0 )
z =−∞
ˇ + 0 e−jkR × sin dz dx 2 R sin sin 0 sin(( + 0 )/2)
(30)
The integral, in Eq. (29), is taken into account. The z part of the integral can be written as
∞
ejkz
cos 0
−∞
sin(( + 0 )/2) e−jkR dz R sin sin
(31)
∞
I=
ejk(
cos 0 −R) sin(( + 0 )/2)
z
dz
(32)
when therelation R1 = R sin is considered from Fig. 3. R1 is also
equal to
(x
=−
− x )2
+ y2 . The variable transform
2 R − R1 sin 0 − (z − z) cos 0 sin 0
R1
(33)
RR1 sin 0
sin(( + 0 )/2)
(34)
d
by noting the relation (35)
from Fig. 3. Then I becomes
∞
sin 0 )
e−jk(
2 /2)R 1
sin 0
d
(36)
−∞
which can be easily evaluated by using the relation [7] ∞
e
−(y2 /2)
dy =
√
2.
(37)
As a result the z part of the integral is found to be cos 0
e−jkR1
sin 0
.
∞
ejkx 0
d
(44)
(45)
∞
2
e−j d
(46)
x
and − has the expression
− = −
2k sin 0 cos
− 0 2
(47)
Eq. (47) can be derived by evaluating Eq. (40) at x = 0. At this point, R1 and ˇ are equal to and − , respectively [7]. As a result, the incident scattered field is obtained as cos 0 +y sin 0 sin 0 +z cos 0 )
{1 − F[−− ]}
(48)
by using Eq. (45) in Eq. (39). Eq. (48) can be arranged as uis (P) = u0 ejk(x sin 0
ej/4 1 − F[−− ] = √
cos 0 +y sin 0 sin 0 +z cos 0 )
F[− ]
(49)
sin 0 cos 0
sin
−−
2
e−j d = F[− ].
(50)
−∞
Now the reflected scattered wave will be expressed in terms of the Fresnel function by using Eq. (30). The z part of the integral can be directly evaluated as in the previous sub-section. The remaining part can be written as urs (P) = −
kej/4 u0 sin 0 jkz e √ 2
∞
ejkx
×
cos 0
ˇ − 0 e−jkR1 sin 0 dx 2 kR1 sin 0
Note that the variable transform → − is used in the integral, in Eq. (50), in order to obtain the Fresnel function at the right-hand side.
(38)
kR1 sin 0
kej/4 u0 sin 0 jkz uis (P) = ui (P) + e √ 2 ×
2R1
k sin 0 sin((ˇ − 0 )/2)
The integral M is found to be √ 2e−j/4 M = −ejk sin 0 (x cos 0 +sin 0 ) F[−− ] k sin 0
The incident scattered field can be written by
3.2. Reflected scattered wave
−∞
√ I = e−j(/4) 2ejkz
in Fig. 3. dx reads
(43)
by using the relation
z − z cos = R
I = ejk(z cos 0 −R1
(42)
y x − x = cos ˇ sin ˇ
uis (P) = u0 ejk(x sin 0
is introduced. dz reads dz =
R1 =
ej/4 F[x] = √
RR1 sin 0
−∞
ˇ + 0 2
for F[x] is the Fresnel integral, which can be defined as
0
which can also be represented by
2kR1 sin 0 sin
dx = −
3.1. Incident scattered wave
I=
by using the sine relations
respectively. Now I will evaluate both of these integrals exactly in order to compare them with the literature.
=
z =−∞
ˇ − 0 e−jkR sin dz dx 2 R
sin (( + 0 )/2)
×
∞
when Eq. (38) is taken into account. The variable transform
sin 0 cos 0
0
(39)
cos 0
sin
ˇ + 0 e−jkR1 sin 0 dx . 2 kR1 sin 0
(51)
The variable transform
=−
k sin 0 [R1 − x cos 0 + x cos 0 − y sin 0 ]
(52)
Y.Z. Umul / Optik 127 (2016) 819–824
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will be used for the integral. can also be expressed as
=−
2kR1 sin 0 sin
dx is equal to
dx =
ˇ − 0 . 2
2R1
k sin 0 sin((ˇ + 0 )/2)
(53)
d
(54)
As a result the reflected scattered wave becomes urs (P) = −u0 ejk(x sin 0
cos 0 −y sin 0 sin 0 +z cos 0 ) F[+ ]
(55)
where + is
+ = −
2k sin 0 cos
+ 0 , 2
(56)
Fig. 5. The total scattered field for different values of 0 .
which can be found from Eq. (53) from its edge point value. The total field reads
ut (P) = u0 ejk(x sin 0
cos 0 +y sin 0 sin 0 +z cos 0 )
−ejk(x sin 0
F[− ]
cos 0 −y sin 0 sin 0 +z cos 0 )
F[+ ] ,
(57)
for a half-plane that has the Dirichlet boundary condition. It can be seen that Eq. (57) is the exact scattered wave by a half-plane for oblique incidence when it is compared with the exact field expressions, given in the literature [16,17]. The only difference is a sin 0 term, in the scattered fields of the literature. This term comes from the polarization of the incident wave. Since the scalar problem is studied, the sine term does not exist. 4. Numerical results In this section, the scattered fields by the half-plane will be investigated numerically. Before examining the behavior of the fields, the diffracted and GO waves must be defined. The total GO and diffracted waves can be written as utGO (P) = u0 ejkz
cos 0
ejk
−ejk sin 0 and utd (P) = u0 ejkz
cos 0
−ejk sin 0
ejk
sin 0 cos(−0 ) cos(+0 )
U(−+ )
sin 0 cos(−0 )
cos(+0 )
U(−− ) (58)
sign(− )F[|− |]
sign(+ )F[|+ |]
(59)
in the cylindrical coordinates respectively. Fig. 4 shows the variation of the total scattered, diffracted and GO waves versus the observation angle . The distance of observation is taken at 6 for is the wavelength. z is equal to 5 . 0 and 0 are 60◦ , respectively. The GO field has two discontinuities, located at 120◦ and 240◦ . These points are the reflection and shadow boundaries. The reflected GO wave goes to zero at 120◦ . Before
Fig. 6. The variation of the total diffracted wave versus 0 .
this point, the total GO field is the interference of the incident and reflected GO waves as can be seen from the figure. Between 120◦ and 240◦ , only the incident GO field exist. Since it is a plane wave, the intensity is constant, as shown in Fig. 4. The total diffracted wave has two maxima, which are located at the reflection and shadow boundaries respectively. This behavior compensates the discontinuity of the total GO field and the scattered field is continuous in the whole space. Fig. 5 plots the variation of the total scattered wave with respect to the observation angle for different values of 0 . The values of the other parameters are the same with the ones, shown in Fig. 4. The scattered field shows strong fluctuations according to amplitude and phase. The important point is that the field does not vary at the reflection and shadow boundaries. The reason is the constant value of 0 . The locations of the transition regions are determined according to this angle. Fig. 6 depicts the variation of the total diffracted wave versus the observation angle for different values of 0 . In this plot, the constant locations of the transition regions and the fluctuations of the field are seen clearly. It can be observed that, the intensity of the field increases with the decreasing values of 0 . 5. Conclusions
Fig. 4. The behaviors of the total scattered, diffracted and GO fields.
In this paper, the 3D MTPO was introduced for the first time in the literature. The procedure of the method is explained in Section 2. The 3D diffraction problem of plane waves by a soft half-screen is investigated as an example. It is shown that MTPO leads to the exact scattered waves. The field expressions are examined numerically. This paper is also a rigorous response to the flawed claims of the authors of [5]. The Green’s function of MTPO satisfies the Helmholtz equation and the 2D Green’s function is not the asymptotic expansion of the zero order first kind Hankel function, but can be exactly obtained from the 3D Green’s function, given in Eq. (26). This paper also shows that it is possible to obtain the rigorous solutions of the diffraction problems with PO and Kirchhoff
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