Modified theory of physical optics

doi:10.1364/OPEX.12.004959

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Modified theory of physical optics

Yusuf Umul »View Author Affiliations

Optics Express, Vol. 12, Issue 20, pp. 4959-4972 (2004)
http://dx.doi.org/10.1364/OPEX.12.004959

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Abstract

A new procedure for calculating the scattered fields from a perfectly conducting body is introduced. The method is defined by considering three assumptions. The reflection angle is taken as a function of integral variables, a new unit vector, dividing the angle between incident and reflected rays into two equal parts is evaluated and the perfectly conducting (PEC) surface is considered with the aperture part, together. This integral is named as Modified Theory of Physical Optics (MTPO) integral. The method is applied to the reflection and edge diffraction from a perfectly conducting half plane problem. The reflected, reflected diffracted, incident and incident diffracted fields are evaluated by stationary phase method and edge point technique, asymptotically. MTPO integral is compared with the exact solution and PO integral for the problem of scattering from a perfectly conducting half plane, numerically. It is observed that MTPO integral gives the total field that agrees with the exact solution and the result is more reliable than that of classical PO integral.

1. Introduction

The geometrical theory of diffraction (GTD), first introduced by Keller [1

J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Of America 52, 116–130 (1962). [CrossRef]

] for two canonical problems of plane wave diffraction from a perfectly conducting (PEC) half plane and cylinder [2

J. B. Keller, “Diffraction by an aperture,” J. App. Physics 28, 426–444 (1957). [CrossRef]

J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture II,” J. App. Physics 28, 570–579 (1957). [CrossRef]

], is a high frequency asymptotic ray technique. The method is developed from the known solutions of simple shapes, named as canonical problems [4

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Peter Peregrinus Ltd., London, 1976).

]. GTD has a few limitations one of which is the infinite fields which are occurring at shadow boundaries. Uniform asymptotic theory (UAT) and uniform theory of diffraction (UTD) are developed to overcome this difficulty and used with success for a wide variety of problems [5

R. C. Hansen Ed., Geometric Theory of Diffraction (IEEE Press, New York, 1981).

]. In spite of their usefulness, these theories fail near caustics and focal points. Another defect of GTD appears at the limited range of problems to which it can be applied. The ray techniques (GTD, UAT, UTD) need the appropriate diffraction coefficients, found from the exact solution of Helmholtz equation, but it is not able to find rigorous solutions to all canonical problems [6

N. D. Taket and R. E. Burge, “A physical optics version of geometrical theory diffraction,” IEEE Trans. Antennas and Propagat. 39, 719–731 (1991). [CrossRef]

]. For example a GTD formulation for the impedance half plane problem could be constructed only after the works of Senior and Maliuzhinets [8–10

T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. Roy. Soc. 213A, 436–458 (1952).

]. Some physical optics (PO) based techniques are developed to find approximated diffraction coefficients for GTD [6

N. D. Taket and R. E. Burge, “A physical optics version of geometrical theory diffraction,” IEEE Trans. Antennas and Propagat. 39, 719–731 (1991). [CrossRef]

R. E. Burge, X. C. Yuan, B. D. Caroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: A diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas and Propagat. , 47, 1515–1527 (1999). [CrossRef]

PO is a high frequency technique, which determines reflected fields and uses an approximation of the induced surface current density on a perfectly conducting surface in proportion to the tangential incident magnetic field [11

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949).

], but fails in evaluating the edge diffracted fields [4

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Peter Peregrinus Ltd., London, 1976).

,12

S. W. Lee, “Comparison of uniform asymptotic theory and Ufimtsev’s theory of electromagnetic edge diffraction,” IEEE Trans. Antennas and Propagat. 25, 162–170 (1977). [CrossRef]

]. In order to correct the PO surface field approximation, the Physical Theory of Diffraction (PTD) was developed by Ufimtsev [13

P. Ya. Ufimtsev, “Method of edge waves in the Physical Theory of Diffraction,” Air Force System Command, Foreign Tech. Div. Document ID No. FTD-HC-23-259-71, (1971).

]. PTD uses additional current components, called residual or fringe current. Michaeli developed equivalent edge currents that allow the evaluation of the far diffracted field for directions not on the Keller cone [14

A. Michaeli, “Equivalent edge currents for arbitrary aspects of observation,” IEEE Trans. Antennas and Propagat. 23, 252–258 (1984). [CrossRef]

,15

F. E. Knott, “The relationship between Mitzner’s ILDC and Michaeli’s equivalent currents,” IEEE Trans. Antennas and Propagat. 33, 112–114 (1985). [CrossRef]

]. Michaeli also introduced Extended Physical Theory of Diffraction (EPTD), as an extension to Ufimtsev’s theory to aperture integration, by formulating in terms of incremental diffraction coefficients (IDC) [16

A. Michaeli, “Incremental diffraction coefficients for the extended physical theory of diffraction,” IEEE Trans. Antennas and Propagat. 43, 732–734 (1995). [CrossRef]

]. These two methods are used frequently for the analysis of reflectors and backscattering from complex objects [17

T. Griesser and C. A. Balanis, “Backscatter analysis of dihedral corner reflectors using physical optics and physical theory of diffraction,” IEEE Trans. Antennas and Propagat. 35, 1137–1147 (1987). [CrossRef]

,18

M. Martinez-Burdalo, A. Martin, and R. Villar, “Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section,” IEEE Trans. Antennas and Propagat. 41, 1336–1339 (1993). [CrossRef]

There are two deficiencies in PO theory. First of all, only the perfectly conducting surface (or the scatter surface) is considered in forming the scattering integral and the aperture part is omitted. As a result of this negligence, the reflected and reflected diffracted fields can be evaluated, but there will be no information about incident and incident diffracted waves. A second restriction is the acceptance of the discontinuous surface as a continuous surface and taking the reflection angle equal to the incidence angle, when evaluating the PO current. This acceptance fails at the edge discontinuity.

It is the aim of this paper to correct the deficiencies of PO theory and obtain the exact solution of edge diffraction problems for various geometries. Three axioms are introduced with this purpose. The aperture and scatter surfaces are considered, the reflection and transmission angles are taken as variables of the scatter and aperture coordinates, and a new unit vector, which divides the angle between the incident and reflected (transmitted) rays into two equal parts, is defined. This new theory is named as Exact Theory of Physical Optics (MTPO). It is important to note that changing the scatterer coordinates is analogous to the method of Modified Edge Representation (MER), in which the edge of the scatterer is replaced with a modified one, defining a new unit vector that satisfies the diffraction law at each point [19

T. Murasaki and M. Ando, “Equivalent edge currents by the modified edge edge representation: physical optics components,” IEICE Trans. on Electronics E75-C, 617–626 (1992).

,20

K. Sakina, S. Cui, and M. Ando, “Mathematical investigation of modified edge representation,” presented at the 2000 IEEE AP-S URSI International Symposium, Salt Lake City-Utah, USA, 16–21 July 2000.

]. This method is introduced in order to overcome the false singularities in equivalent edge currents for PO and GTD. Defining a new direction for the unit vector was also offered in the literature, in order to improve PO [21

J. Goto, “Interpretation of high frequency diffraction based upon PO,” M.S. thesis (Tokyo Institute of Technology, Tokyo, 2003), Chap. 3.

Asymptotic evaluation of MTPO integral gives the exact edge diffraction coefficient. One of the original points of this paper is the expression of the phase function of the MTPO integral by a new form [22

Y. Z. Umul, E. Yengel, and A. Aydin, “Comparison of physical optics integral and exact solution for cylinder problem,” presented at Eleco’2003 International Conference, Bursa, Turkey, 3–7 Dec. 2003, http://eleco.emo.org.tr/eleco2003/ELECO2003/bsession/B5-01.pdf.

]. The new method is applied to a well known canonical problem of half plane in order to examine its validity. The exact edge diffracted fields can be found without considering additional fringe waves or equivalent currents for a wide spectrum of problems for scattering from perfectly conducting bodies with MTPO integral method. The method can also be applied to the geometries where the radius of curvature is a function of angle, like parabolic and hyperbolic reflectors.

GTD and PTD need the diffraction coefficients, found from the solution of canonical problems, in order to construct high frequency asymptotic fields for more complicated geometries. For this reason their applicability is limited with the number of solved basic problems as mentioned before. In contrary, exact edge diffracted fields can be evaluated directly from the asymptotic evaluation of the MTPO integral. Half plane problem, solved in Section 3, can be considered with this aim. GTD uses directly the edge diffraction coefficient found by Sommerfeld [23

A. Sommerfeld, Optics (Academic Press, New York, 1954).

] and can only define the behavior of edge diffracted fields with the existence of this coefficient, but exact edge diffracted fields and the related coefficient are found from the asymptotic evaluation of the MTPO integral, constructed by using the geometry.

A time factor e^jwt is assumed and suppressed throughout the paper.

2. Exact theory of physical optics

The geometry in Fig. 1 is considered. S ₁ is the perfectly conducting surface and S ₂ is the aperture part.

A general procedure will be given in order to find the total diffracted fields by taking into account these two surfaces. Three axioms can be introduced as

Scattering fields from S ₁ and S ₂ surfaces are considered. The incident waves induce a surface current on S ₁ and integration of this current gives the reflected and reflected diffracted fields as in classical PO theory, but this solution will not include information about incident diffracted fields. For this reason, S ₂ surface must be considered. Equivalent currents can be defined on the aperture according to the Equivalent Source Theorem and radiated field can be obtained by integrating the related currents on S ₂. Radiated fields contain the data about incident and incident diffracted waves. This approach is analogous to the solution of aperture antenna problems with Equivalent Source Theorem. A surface current can be defined for S ₁ as

${\vec{J}}_{es} = {\vec{n}}_{1} \times {\vec{H}}_{t} |_{S_{1}}$

(1)
where H⃗_t is the total magnetic field on the perfectly conducting surface. Equivalent Source Theorem can be applied to S ₂ and equivalent surface currents can be defined as

${\vec{J}}_{es} = {\vec{n}}_{2} \times {\vec{H}}_{i} |_{S_{2}}, {\vec{J}}_{ms} = - {\vec{n}}_{2} \times {\vec{E}}_{i} |_{S_{2}}$

(2)
for E⃗_i , H⃗_i are the incident fields on the aperture.
Fig. 1. Scattered fields from a perfectly conducting surface and an aperture continuation
Fig. 1. Scattered fields from a perfectly conducting surface and an aperture continuation
The reflection and transmission angles (β) are variables which depend on the surface (S ₁ + S ₂) coordinates.
A new unit vector (n⃗ ₁, n⃗ ₂), which divides the angle between the reflected (or transmitted) and the incident rays into two equal parts, can be defined. n⃗ ₁ can be written as

${\vec{n}}_{1} = \cos (u + α) \vec{t} + \sin (u + α) \vec{n}$

(3)
for S ₁ and

${\vec{n}}_{2} = \cos (v + α) \vec{t} - \sin (v + α) \vec{n}$

(4)
for S ₂ where α is the angle of incidence, t⃗ and n⃗ are the actual tangential and normal unit vectors of the surface, respectively. The boundary conditions in Eqs. (1) and (2) will be evaluated according these new unit vectors. u and v are equal to $\frac{π}{2} - \frac{α + β}{2}$ . The total scattered electric field can be defined as

${\vec{E}}_{t} = {\vec{E}}_{is} + {\vec{E}}_{rs}$

(5)
where

${\vec{E}}_{is} = - \frac{jω μ_{0}}{4 π} \iint_{S_{2}} {\vec{n}}_{2} \times {\vec{H}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}} dS' + \frac{1}{4 π} \iint_{S_{2}} \nabla \times ({\vec{n}}_{2} \times {\vec{E}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}}) dS'$

(6)
and

${\vec{E}}_{rs} = - \frac{jω μ_{0}}{4 π} \iint_{S_{1}} {\vec{n}}_{1} \times {\vec{H}}_{t} |_{S_{1}} \frac{e^{- jk R_{1}}}{R_{1}} dS'$

(7)
for an electric polarized incident wave. The total scattered magnetic field can be written as

${\vec{H}}_{t} = {\vec{H}}_{is} + {\vec{H}}_{rs}$

(8)
where

${\vec{H}}_{is} = \frac{1}{4 π} \iint_{S_{2}} \nabla \times ({\vec{n}}_{2} \times {\vec{H}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}}) dS' + \frac{jω ε}{4 π} \iint_{S_{2}} {\vec{n}}_{2} \times {\vec{E}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}} dS'$

(9)
and

${\vec{H}}_{rs} = \frac{1}{4 π} \iint_{S_{1}} \nabla \times ({\vec{n}}_{1} \times {\vec{H}}_{t} |_{S_{1}} \frac{e^{- jk R_{1}}}{R_{1}}) dS'$

(10)
for a magnetic polarized incident wave. It is important to note that the rotational operation in Eqs. (6), (9) and (10) is applied according to the coordinates of the observation point. Here E⃗_rs (H⃗_rs ) and E⃗_is (H⃗_is ) denote the reflected scattered and incident scattered fields, respectively.

3. Scattering from a perfectly conducting half plane: MTPO approach

The geometry in Fig. 2 is considered. An electric polarized (electrical field is parallel to the surface) plane wave is illuminating the half plane.

Fig. 2 Reflection geometry from a perfectly conducting half plane

Exact Theory of Physical Optics will be applied to this problem in order to evaluate scattered fields. The magnetic field of the plane wave can be written as

{\vec{H}}_{i} = - \frac{E_{i}}{Z_{0}} ({\vec{e}}_{x} \sin ϕ_{0} - {\vec{e}}_{y} \cos ϕ_{0}) e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})}

(11)

where ϕ ₀ is the angle of incidence. The method, given in the previous part, will be applied to this problem. As can be seen from Fig. 2, there are two scattered ray paths. One belongs to the reflected (for 0≤ϕ≤π-ϕ ₀) and reflected diffracted (for 0≤ϕ≤2π) rays and the second is the incident (for π≤ϕ π + ϕ ₀) and the incident diffracted (for 0≤ϕ≤2π) rays. The integrals of Eqs. (6) and (7) will be considered. Reflected plane wave can be written as

{\vec{E}}_{r} = {\vec{e}}_{z} E_{r} e^{jk (x \cos β - y \sin β)}

(12)

where β is a variable angle which is the function of surface coordinates. The amplitude of the electric field can be found as

E_{r} = - E_{i} e^{jkx' (\cos ϕ_{0} - \cos β)}

(13)

by using the boundary condition of n⃗ ₁×(E⃗_i +E⃗_r )|_{S
₁} = 0, where E⃗_i =e⃗_z E_ie ^{jk(xcosϕ
₀+ysinϕ
₀)} and n⃗ ₁ =cos(u + ϕ ₀)e⃗_x + sin(u + ϕ ₀)e⃗_y . For this problem, u is equal to

\frac{π}{2} - \frac{ϕ_{0} + β}{2}

. The MTPO surface current can be evaluated as

{\vec{J}}_{MTPO} = \frac{E_{i}}{Z_{0}} [\cos u - \cos (u + β + ϕ_{0})] e^{jkx' \cos ϕ_{0}} {\vec{e}}_{z}

(14)

which is equal to n⃗ ₁ × (H⃗_i + H⃗_r )|_{S
₂} = J⃗_ETPO at the perfectly conducting half plane. The classical PO surface current component is equal to

{\vec{J}}_{PO} = {\vec{e}}_{z} 2 \frac{E_{i}}{Z_{0}} \sin ϕ_{0} e^{jkx' \cos ϕ_{0}}

(15)

for this problem. The trigonometric expression in Eq. (14) can be evaluated by using the geometry in Fig. 2. As a result one obtains

{\vec{J}}_{MTPO} = \frac{2 E_{i}}{Z_{0}} \sin (\frac{β + ϕ_{0}}{2}) e^{jkx' \cos ϕ_{0}} {\vec{e}}_{z}

(16)

for the MTPO surface current. The current that will flow on the surface for a magnetic polarized incident wave is calculated and compared with PO surface current in the Appendix section. The scattered electrical field can be written as

{\vec{E}}_{rs} = - {\vec{e}}_{z} \frac{jk E_{i}}{2 π} \int_{x' = 0}^{\infty} \int_{z' = - \infty}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{R_{1}} \sin (\frac{β + ϕ_{0}}{2}) dx' dz'

(17)

by using Eq. (7). Here R ₁ is equal to

R_{1} = \sqrt{{(x - x')}^{2} + y^{2} + {(z - z')}^{2}} .

(18)

z' part of Eq. (17) gives a Hankel function a

\int_{c} e^{- jkchα} dα = \frac{π}{j} H_{0}^{(2)} (kR)

(19)

by using the variable change of (z-z') = R shα where R is equal to

\sqrt{{(x - x')}^{2} + y^{2}}

. As a result one obtains

{\vec{E}}_{rs} = - {\vec{e}}_{z} \frac{k E_{i}}{2} \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (kR) \sin (\frac{β + ϕ_{0}}{2}) dx'

(20)

for the reflected scattered wave. The incident scattered field can be found by considering Fig. 3 and Eq. (7). The image of the incident field is considered. The reflected waves for x'∈ (- ∞,0] is equal to the transmitted incident field for this region. The equivalent image field can be written as

{\vec{E}}_{ieq} = - {\vec{e}}_{z} E_{i} e^{jk (x \cos ϕ_{0} - y \sin ϕ_{0})}

(21)

and magnetic field is found to be

{\vec{H}}_{ieq} = - \frac{E_{i}}{Z_{0}} ({\vec{e}}_{x} \sin ϕ_{0} + {\vec{e}}_{y} \cos ϕ_{0}) e^{jk (x \cos ϕ_{0} - y \sin ϕ_{0})}

(22)

for the plane waves illuminating the upper part of the plane.

Fig. 3. Transmission geometry for the modified theory of physical optics

The reflected electric and magnetic fields can be evaluated by following the steps between Eqs. (12) and (13). The equivalent surface current is found to be

{\vec{J}}_{MTPO} = - \frac{2 E_{i}}{Z_{0}} \sin (\frac{β + ϕ_{0}}{2}) e^{jkx' \cos ϕ_{0}} {\vec{e}}_{z}

(23)

where n⃗ ₂ is equal to cos(v + ϕ ₀)e⃗_x - sin(v + ϕ ₀)e⃗_y for the aperture surface and

v = \frac{π}{2} - \frac{ϕ_{0} + β}{2}

. The operations from Eqs. (17) to (20) are also valid for this case. As a result one obtains

{\vec{E}}_{is} \approx {\vec{e}}_{z} \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin \frac{ϕ_{0} + β}{2} dx'

(24)

for the incident scattered wave. The total scattered electric field can be written as

{\vec{E}}_{s} = {\vec{E}}_{rs} + {\vec{E}}_{is}

(25)

for

{\vec{E}}_{rs} \approx - {\vec{e}}_{z} \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin (\frac{β + ϕ_{0}}{2}) dx'

(26)

denoting the reflected scattered field found from Eq. (20) by using the Debye asymptotic expansion of

H_{0}^{(2)}

(kR ₁). The integrals of Eqs. (24) and (26) denote the scattered fields from the half plane. Eq. (26) gives the reflected and reflected diffracted waves. Eq. (24) consists of incident field for ϕ≥π and incident diffracted wave for ϕ∈[0, 2π].

4. Asymptotic evaluation of scattering integrals

The incident, reflected and diffracted fields will be evaluated asymptotically by the method of stationary phase and edge point technique [4

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Peter Peregrinus Ltd., London, 1976).

,23

A. Sommerfeld, Optics (Academic Press, New York, 1954).

] for k→∞. The total electric field can be obtained as

{\vec{E}}_{t} \approx {\vec{e}}_{z} \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} (\int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin \frac{ϕ_{0} + β}{2} dx' - \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin (\frac{β + ϕ_{0}}{2}) dx')

(27)

by combining Eqs. (24) and (26). The phase functions of the integrals in Eq. (27) can be written as

g (x') = ρ \cos γ + x' (\cos β - \cos ϕ_{0})

(28)

for the geometry in Figs. 2 and 3. The first derivative of the Eq. (28) can be expressed as

\frac{\partial g}{\partial x'} = - ρ \sin γ \frac{dγ}{dx'} - x' \sin β \frac{dβ}{dx'} + \cos β - \cos ϕ_{0}

(29)

where γ is equal to

γ = π - β \mp ϕ

(30)

according to the geometry in Figs. 2 and 3. One obtains

ρ \sin γ \frac{dγ}{dx'} = - x' \sin β \frac{dβ}{dx'}

(31)

by using sine relations and the derivative of Eq. (30) according to x'. As a result one obtains

β_{s} = ϕ_{0}

(32)

which denotes the reflection law by considering

\frac{\partial g}{\partial x'} ∣_{s} = \cos β_{s} - \cos ϕ_{0} = 0

(33)

at the stationary phase point. The second derivative of Eq. (28) can be evaluated as

\frac{\partial^{2} g}{\partial {x'}^{2}} |_{s} = \frac{\sin^{2} ϕ_{0}}{l}

(34)

where l can be expressed as

l = R_{1} |_{s} = ρ \cos γ_{s}

(35)

denoting the ray path of the reflected and incident fields. γ_s is equal to π-ϕ ₀+ϕ for the incident wave and π-ϕ ₀-ϕ) for the reflected wave. The phase function of Eq. (28) can be expanded for the first three terms as

g (x') \approx l + \frac{1}{2} \frac{\sin^{2} ϕ_{0}}{l} {(x' - {x'}_{s})}^{2}

(36)

and the amplitude function is evaluated for the first term of Taylor series as

f ({x'}_{s}) \approx \pm \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \frac{\sin ϕ_{0}}{\sqrt{kl}}

(37)

where it can be seen that the trigonometric parts of the MTPO current reduces to the PO current. The integrals in Eqs. (24) and (26) can be written as

E_{r, i} \approx \pm \frac{k E_{i} \sin ϕ_{0}}{\sqrt{2 π}} e^{j \frac{π}{4}} \frac{e^{- jkl}}{\sqrt{kl}} \int_{- \infty}^{\infty} e^{- jk \frac{\sin^{2} ϕ_{0}}{l} (x' - {x'}^{2})} dx'

(38)

and can be evaluated easily by considering the integral of

\int_{- \infty}^{\infty} e^{- \frac{y^{2}}{2}} dy = \sqrt{2 π} .

(39)

As a result, one obtains

E_{r} \approx - E_{i} e^{jkρ \cos (ϕ + ϕ_{0})}

(40)

for the reflected fields from the half plane. The radiated wave from the aperture can be written as

E_{i} \approx E_{i} e^{jkρ \cos (ϕ - ϕ_{0})}

(41)

from the stationary point contribution. The edge diffracted fields can be evaluated by using the formula

{\vec{E}}_{d} \approx \mp {\vec{e}}_{z} \frac{1}{jk} \frac{f (0)}{g' (0)} e^{- jkg (0)}

(42)

where f(0) and g(0) denote the values of the amplitude and the phase functions of Eqs. (24) and (26) at the edge point, respectively [4

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Peter Peregrinus Ltd., London, 1976).

,24

L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waves (IEEE Press, New York, 1994). [CrossRef]

]. The plus sign in Eq. (42) is used when the edge point is the lower value of the integral, and minus sign is used for the upper limit. g'(0) is the value of the first derivative of the phase function at the edge point. The related phase and amplitude quantities for Eq. (26) can be written as

g (0) = ρ,

(43)

g' (0) = - (\cos ϕ + \cos ϕ_{0})

(44)

and

f (0) = - \frac{k E_{i}}{2 \sqrt{2 π}} e^{j \frac{π}{4}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2})}{\sqrt{kρ}}

(45)

for x' _e = 0, β_e = π - ϕ as the edge coordinates for the half plane. One obtains

{\vec{E}}_{rd} = {\vec{e}}_{z} \frac{E_{i}}{2 \sqrt{2 π}} \frac{e^{- jkρ}}{\sqrt{kρ}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}} e^{- j \frac{π}{4}}

(46)

for the reflected edge diffracted field. Reflected edge diffraction coefficient can be written as

D_{rd} = \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}}

(47)

for the MTPO integral. The phase and the amplitude functions of Eq. (24) can be expressed as

g (0) = ρ,

(48)

g' (0) = - (\cos ϕ + \cos ϕ_{0})

(49)

and

f (0) = - \frac{k E_{i}}{2 \sqrt{2 π}} e^{j \frac{π}{4}} \frac{\cos (\frac{ϕ + ϕ_{0}}{2})}{\sqrt{kρ}}

(50)

for x' _e = 0, β_e =ϕ-π as the edge coordinates for the aperture integral. The incident edge diffracted field can be evaluated as

{\vec{E}}_{id} = - {\vec{e}}_{z} \frac{E_{i}}{2 \sqrt{2 π}} \frac{e^{- jkρ}}{\sqrt{kρ}} \frac{\cos (\frac{ϕ + ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}} e^{- j \frac{π}{4}}

(51)

by using Eqs. (48), (49) and (50) in Eq. (42). The incident edge diffraction coefficient can be written as

D_{id} = - \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} \frac{\cos (\frac{ϕ + ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}}

(52)

and one obtains

D_{t} = D_{id} + D_{rd} = \frac{e^{- j \frac{π}{4}}}{\sqrt{2 π}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2}) - \cos (\frac{ϕ + ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}}

(53)

for the total edge diffraction coefficient of MTPO integral.

5. Numerical results

MTPO integral for scattering from a perfectly conducting half plane will be compared with the classical PO approach, MTPO integral for β = ϕ ₀ and exact solution. Equation (27) is valid for this case and it represents the total scattered field (incident, reflected and edge diffracted). The total field, calculated by the physical optics theory, is given by

E_{TPO} \approx E_{i} e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})} - \frac{k E_{i}}{2} \sin ϕ_{0} \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (k R_{1}) dx'

(54)

for a plane wave illuminated perfectly conducting half plane [12

S. W. Lee, “Comparison of uniform asymptotic theory and Ufimtsev’s theory of electromagnetic edge diffraction,” IEEE Trans. Antennas and Propagat. 25, 162–170 (1977). [CrossRef]

]. The case of β = ϕ ₀ will also be plotted and compared numerically as

E_{MTPO} |_{β = ϕ_{0}} \approx \frac{k E_{i}}{2} (\int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (k R_{1}) \sin ϕ_{0} dx' - \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (k R_{1}) \sin ϕ_{0} dx')

(55)

with the series expression of Eq. (58). The first integral in Eq. (55) represents the radiated field and the second one is the physical optics integral. The incident field will be added to the PO integral for 0≤ϕ≤2π and the geometrical optics fields (incident and reflected) will be considered for the radiated wave in the interval of ϕ ∈ [0, π]. The total fields, obtained by the mentioned procedure, must be divided by two, because the value of the field is doubled for all values of ϕ. The total field can be written as

E_{TMTPO} = \frac{1}{2} [E_{i} (e^{jkρ \cos (ϕ - ϕ_{0})} - e^{jkρ \cos (ϕ + ϕ_{0})}) u (π - ϕ) + E_{TPO} + E_{R}]

(56)

where E_R is the radiated field. The unit step function in Eq. (56) can be defined as

u (π - ϕ) = {\begin{matrix} 1, ϕ \leq π \\ 0, ϕ \geq π \end{matrix}

(57)

in order to express the addition of total geometrical optics fields (reflected and incident) for ϕ∈[0,π].

The exact solution of a half plane problem can be given as

E_{t} = 2 E_{i} \sum_{m = 1}^{\infty} e^{jm \frac{π}{4}} J_{\frac{m}{2}} (kρ) \sin \frac{m}{2} ϕ \sin \frac{m}{2} ϕ_{0}

(58)

for an electric polarized incident plane wave [25

W.L. Stutzman and G. A. Thiele, Antenna Theory and Design (John Wiley & Sons, New York, 1988).

]. The MTPO integral can be written as

E_{MTPO} \approx \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} (\int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin \frac{ϕ_{0} + β}{2} dx' - \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin (\frac{β + ϕ_{0}}{2}) dx')

(59)

for the half plane problem. The first integral represents the incident field for π ≤ ϕ ≤ π + ϕ ₀ and incident diffracted waves for 0 ≤ ϕ ≤ 2π. The second integral in Eq. (59) consists of the reflected and reflected diffracted fields.

Fig. 4. Regions for scattered fields in a perfectly conducting half plane

The half plane geometry is divided into three regions as shown in Fig. 4. The geometrical optics and edge diffracted fields are plotted by considering this geometry. In Region I, it is apparent that there will be incident (u_i ), reflected (u_r ) and diffracted fields (u_di +u_dr ). In Region II, there are incident and diffracted fields. There are only diffracted fields in Region III. Since PO integral consists of reflected and reflected diffracted fields, the incident field is added to Eq. (54) for all values of ϕ. The incident field is added to the integrals in Eq. (59) for ϕ≤π. The figures are plotted for

ϕ_{0} = \frac{π}{4}

and ρ = 6λ.

Fig. 5. Reflected and diffracted fields from perfectly conducting half plane (PO and exact solution)

Fig. 6. Reflected and diffracted fields from perfectly conducting half plane [MTPO (β = ϕ ₀) and exact solution]

Fig. 7. Reflected and diffracted fields from perfectly conducting half plane (MTPO and exact solution)

Figure 5 shows the variation of Eqs. (54) and (58) versus observation angle. It can be seen that PO integral deviates from the exact asymptotic solution after ϕ = π + ϕ ₀, since the edge diffraction field, found from the PO phase contribution, is not the exact field. Incident field is added to Eq. (54) for all values of ϕ.

Figure 6 depicts the variation of MTPO (β = ϕ ₀) field and the exact scattered fields in Eqs. (56) and Eq. (58) versus the observation angle. It is observed that the MTPO integral, written for a constant reflection and transmission angle, is harmonious with the exact waves for all values of ϕ.

In Fig. 7, the integral in Eq. (59) is compared with Eq. (58). It can be seen that the fields are compatible.

7. Conclusion

In this work, a new approach to Physical Optics concept is introduced by defining three axioms. This method is named as Exact Theory of Physical Optics (MTPO). The introduced procedure is applied to a well known problem of perfectly conducting half plane and exact scattered fields (reflected and edge diffracted) and also incident field for π ≤ ϕ ≤ π + ϕ ₀ is evaluated by using asymptotic methods. Numerical results show that MTPO integral gives the exact fields harmonious with the exact series solution of Helmholtz equation. It is important to note that the integral, related with the reflected field, gives the reflected diffracted fields and the incident diffracted waves are evaluated from the integral, written for the aperture.

8. Appendix

MTPO current for a magnetic polarized incident wave will be evaluated. Figure 2 can also be considered for a magnetic polarized illumination. Magnetic field of the incident wave can be written as

{\vec{H}}_{i} = {\vec{e}}_{z} H_{i} e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})}

(60)

and electric field can be found as

{\vec{E}}_{i} = Z_{0} H_{i} (\sin ϕ_{0} {\vec{e}}_{x} + \cos ϕ_{0} {\vec{e}}_{y}) e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})}

(61)

by using the Maxwell-Ampere equation. Reflected plane wave can be written as

{\vec{H}}_{r} = {\vec{e}}_{z} H_{r} e^{jk (x \cos β - y \sin β)}

(62)

where β is a variable angle which is the function of surface coordinates, as in Eq. (12). The reflected electric field can be found as

{\vec{E}}_{r} = - Z_{0} H_{r} (\sin β {\vec{e}}_{x} + \cos β {\vec{e}}_{y}) e^{jk (x \cos β - y \sin β)}

(63)

by considering Maxwell-ampere equation. One obtains

H_{r} e^{jkx' \cos β} = H_{i} e^{jkx' \cos ϕ_{0}}

(64)

as a result of the boundary condition of n⃗ ₁ ×(E⃗_i × E⃗_r )|_{S
₁} = 0, with Eqs. (61) and (63) for n⃗ ₁ = cos(u + ϕ ₀)e⃗_x + sin(u + ϕ ₀)e⃗_y . The MTPO surface current can be evaluated as

{\vec{J}}_{MTPO} = 2 H_{i} [{\vec{e}}_{x} \cos \frac{β - ϕ_{0}}{2} - {\vec{e}}_{y} \sin \frac{β - ϕ_{0}}{2}] e^{jkx' \cos ϕ_{0}}

(65)

which is equal to n⃗ ₁ × (H⃗_i + H⃗_r )|_{S
₂} = J⃗_MTPO on the perfectly conducting half plane. The classical PO surface current component is equal to

{\vec{J}}_{PO} = {\vec{e}}_{x} 2 H_{i} e^{jkx' \cos ϕ_{0}}

(66)

for this problem. It is apparent that, the current, given in Eq. (65) is equal to J⃗_PO for β = ϕ ₀.

References and links

1.	J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Of America 52, 116–130 (1962). [CrossRef]
2.	J. B. Keller, “Diffraction by an aperture,” J. App. Physics 28, 426–444 (1957). [CrossRef]
3.	J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture II,” J. App. Physics 28, 570–579 (1957). [CrossRef]
4.	G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Peter Peregrinus Ltd., London, 1976).
5.	R. C. Hansen Ed., Geometric Theory of Diffraction (IEEE Press, New York, 1981).
6.	N. D. Taket and R. E. Burge, “A physical optics version of geometrical theory diffraction,” IEEE Trans. Antennas and Propagat. 39, 719–731 (1991). [CrossRef]
7.	R. E. Burge, X. C. Yuan, B. D. Caroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: A diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas and Propagat. , 47, 1515–1527 (1999). [CrossRef]
8.	T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. Roy. Soc. 213A, 436–458 (1952).
9.	G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).
10.	J. L. Volakis, “A uniform geometrical theory of diffraction for an imperfectly conducting half-plane,” IEEE Trans. Antennas and Propagat. 34, 172–180 (1986). [CrossRef]
11.	S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949).
12.	S. W. Lee, “Comparison of uniform asymptotic theory and Ufimtsev’s theory of electromagnetic edge diffraction,” IEEE Trans. Antennas and Propagat. 25, 162–170 (1977). [CrossRef]
13.	P. Ya. Ufimtsev, “Method of edge waves in the Physical Theory of Diffraction,” Air Force System Command, Foreign Tech. Div. Document ID No. FTD-HC-23-259-71, (1971).
14.	A. Michaeli, “Equivalent edge currents for arbitrary aspects of observation,” IEEE Trans. Antennas and Propagat. 23, 252–258 (1984). [CrossRef]
15.	F. E. Knott, “The relationship between Mitzner’s ILDC and Michaeli’s equivalent currents,” IEEE Trans. Antennas and Propagat. 33, 112–114 (1985). [CrossRef]
16.	A. Michaeli, “Incremental diffraction coefficients for the extended physical theory of diffraction,” IEEE Trans. Antennas and Propagat. 43, 732–734 (1995). [CrossRef]
17.	T. Griesser and C. A. Balanis, “Backscatter analysis of dihedral corner reflectors using physical optics and physical theory of diffraction,” IEEE Trans. Antennas and Propagat. 35, 1137–1147 (1987). [CrossRef]
18.	M. Martinez-Burdalo, A. Martin, and R. Villar, “Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section,” IEEE Trans. Antennas and Propagat. 41, 1336–1339 (1993). [CrossRef]
19.	T. Murasaki and M. Ando, “Equivalent edge currents by the modified edge edge representation: physical optics components,” IEICE Trans. on Electronics E75-C, 617–626 (1992).
20.	K. Sakina, S. Cui, and M. Ando, “Mathematical investigation of modified edge representation,” presented at the 2000 IEEE AP-S URSI International Symposium, Salt Lake City-Utah, USA, 16–21 July 2000.
21.	J. Goto, “Interpretation of high frequency diffraction based upon PO,” M.S. thesis (Tokyo Institute of Technology, Tokyo, 2003), Chap. 3.
22.	Y. Z. Umul, E. Yengel, and A. Aydin, “Comparison of physical optics integral and exact solution for cylinder problem,” presented at Eleco’2003 International Conference, Bursa, Turkey, 3–7 Dec. 2003, http://eleco.emo.org.tr/eleco2003/ELECO2003/bsession/B5-01.pdf.
23.	A. Sommerfeld, Optics (Academic Press, New York, 1954).
24.	L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waves (IEEE Press, New York, 1994). [CrossRef]
25.	W.L. Stutzman and G. A. Thiele, Antenna Theory and Design (John Wiley & Sons, New York, 1988).
26.	A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, New Jersey, 1991).

OCIS Codes
(260.0260) Physical optics : Physical optics
(260.1960) Physical optics : Diffraction theory
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Research Papers

History
Original Manuscript: August 2, 2004
Revised Manuscript: September 26, 2004
Published: October 4, 2004

Citation
Yusuf Umul, "Modified theory of physical optics," Opt. Express 12, 4959-4972 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4959

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References

J. B. Keller, �??Geometrical theory of diffraction,�?? J. Opt. Soc. Of America 52, 116-130 (1962) [CrossRef]
J. B. Keller, �??Diffraction by an aperture,�?? J. App. Physics 28, 426-444 (1957) [CrossRef]
J. B. Keller, R. M. Lewis and B. D. Seckler, �??Diffraction by an aperture II,�?? J. App. Physics 28, 570-579 (1957) [CrossRef]
G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Peter Peregrinus Ltd., London, 1976)
R. C. Hansen Ed., Geometric Theory of Diffraction (IEEE Press, New York, 1981)
N. D. Taket and R. E. Burge, �??A physical optics version of geometrical theory diffraction,�?? IEEE Trans. Antennas and Propagat. 39, 719-731 (1991) [CrossRef]
R. E. Burge, X. C. Yuan, B. D. Caroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket and C. J. Oliver, �??Microwave scattering from dielectric wedges with planar surfaces: A diffraction coefficient based on a physical optics version of GTD,�?? IEEE Trans. Antennas and Propagat., 47, 1515-1527 (1999). [CrossRef]
T. B. A. Senior, �??Diffraction by a semi-infinite metallic sheet,�?? Proc. Roy. Soc. 213A, 436-458 (1952)
G. D. Maliuzhinets, �??Excitation, reflection and emission of surface waves from a wedge with given face impedances,�?? Sov. Phys. Dokl. 3, 752-755 (1958)
J. L. Volakis, �??A uniform geometrical theory of diffraction for an imperfectly conducting half-plane,�?? IEEE Trans. Antennas and Propagat. 34, 172-180 (1986) [CrossRef]
S. Silver, "Microwave Antenna Theory and Design", (McGraw-Hill, New York, 1949)
S. W. Lee, �??Comparison of uniform asymptotic theory and Ufimtsev�??s theory of electromagnetic edge diffraction,�?? IEEE Trans. Antennas and Propagat. 25, 162-170 (1977) [CrossRef]
P. Ya. Ufimtsev, �??Method of edge waves in the Physical Theory of Diffraction,�?? Air Force System Command, Foreign Tech. Div. Document ID No. FTD-HC-23-259-71, (1971).
A. Michaeli, �??Equivalent edge currents for arbitrary aspects of observation,�?? IEEE Trans. Antennas and Propagat. 23, 252-258 (1984) [CrossRef]
F. E. Knott, �??The relationship between Mitzner�??s ILDC and Michaeli�??s equivalent currents,�?? IEEE Trans. Antennas and Propagat. 33, 112-114 (1985) [CrossRef]
A. Michaeli, �??Incremental diffraction coefficients for the extended physical theory of diffraction,�?? IEEE Trans. Antennas and Propagat. 43, 732-734 (1995) [CrossRef]
T. Griesser and C. A. Balanis, �??Backscatter analysis of dihedral corner reflectors using physical optics and physical theory of diffraction,�?? IEEE Trans. Antennas and Propagat. 35, 1137-1147 (1987) [CrossRef]
M. Martinez-Burdalo, A. Martin and R. Villar, �??Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section,�?? IEEE Trans. Antennas and Propagat. 41, 1336-1339 (1993) [CrossRef]
T. Murasaki and M. Ando, �??Equivalent edge currents by the modified edge edge representation: physical optics components,�?? IEICE Trans. on Electronics E75-C, 617-626 (1992)
K. Sakina, S. Cui and M. Ando, �??Mathematical investigation of modified edge representation,�?? presented at the 2000 IEEE AP-S URSI International Symposium, Salt Lake City-Utah, USA, 16-21 July 2000
J. Goto, �??Interpretation of high frequency diffraction based upon PO,�?? M.S. thesis (Tokyo Institute of Technology, Tokyo, 2003), Chap. 3.
Y. Z. Umul, E. Yengel, and A. Aydýn, �??Comparison of physical optics integral and exact solution for cylinder problem,�?? presented at Eleco�??2003 International Conference, Bursa, Turkey, 3-7 Dec. 2003, <a href="http://eleco.emo.org.tr/eleco2003/ELECO2003/bsession/B5-01.pdf">http://eleco.emo.org.tr/eleco2003/ELECO2003/bsession/B5-01.pdf</a>
A. Sommerfeld, Optics (Academic Press, New York, 1954)
L. B. Felsen and N. Marcuwitz, "Radiation and Scattering of Waves", (IEEE Press, New York, 1994) [CrossRef]
W.L. Stutzman and G. A. Thiele, "Antenna Theory and Design", (John Wiley & Sons, New York, 1988)
A. Ishimaru, "Electromagnetic Wave Propagation, Radiation and Scattering", (Prentice Hall, New Jersey, 1991).

. Opt. Soc. Of America

J. B. Keller, �??Geometrical theory of diffraction,�?? J. Opt. Soc. Of America 52, 116-130 (1962) [CrossRef]

IEEE Trans. Antennas and Propagat.

N. D. Taket and R. E. Burge, �??A physical optics version of geometrical theory diffraction,�?? IEEE Trans. Antennas and Propagat. 39, 719-731 (1991) [CrossRef]
A. Michaeli, �??Incremental diffraction coefficients for the extended physical theory of diffraction,�?? IEEE Trans. Antennas and Propagat. 43, 732-734 (1995) [CrossRef]
T. Griesser and C. A. Balanis, �??Backscatter analysis of dihedral corner reflectors using physical optics and physical theory of diffraction,�?? IEEE Trans. Antennas and Propagat. 35, 1137-1147 (1987) [CrossRef]
M. Martinez-Burdalo, A. Martin and R. Villar, �??Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section,�?? IEEE Trans. Antennas and Propagat. 41, 1336-1339 (1993) [CrossRef]
J. L. Volakis, �??A uniform geometrical theory of diffraction for an imperfectly conducting half-plane,�?? IEEE Trans. Antennas and Propagat. 34, 172-180 (1986) [CrossRef]
S. W. Lee, �??Comparison of uniform asymptotic theory and Ufimtsev�??s theory of electromagnetic edge diffraction,�?? IEEE Trans. Antennas and Propagat. 25, 162-170 (1977) [CrossRef]
A. Michaeli, �??Equivalent edge currents for arbitrary aspects of observation,�?? IEEE Trans. Antennas and Propagat. 23, 252-258 (1984) [CrossRef]

IEICE Trans. on Electronics

T. Murasaki and M. Ando, �??Equivalent edge currents by the modified edge edge representation: physical optics components,�?? IEICE Trans. on Electronics E75-C, 617-626 (1992)

J. App. Physics

J. B. Keller, �??Diffraction by an aperture,�?? J. App. Physics 28, 426-444 (1957) [CrossRef]
J. B. Keller, R. M. Lewis and B. D. Seckler, �??Diffraction by an aperture II,�?? J. App. Physics 28, 570-579 (1957) [CrossRef]

Proc. Roy. Soc.

T. B. A. Senior, �??Diffraction by a semi-infinite metallic sheet,�?? Proc. Roy. Soc. 213A, 436-458 (1952)

Sov. Phys. Dokl.

G. D. Maliuzhinets, �??Excitation, reflection and emission of surface waves from a wedge with given face impedances,�?? Sov. Phys. Dokl. 3, 752-755 (1958)

The relationship between Mitzner???s ILDC

F. E. Knott, �??The relationship between Mitzner�??s ILDC and Michaeli�??s equivalent currents,�?? IEEE Trans. Antennas and Propagat. 33, 112-114 (1985) [CrossRef]

Trans. Antennas and Propagat.

R. E. Burge, X. C. Yuan, B. D. Caroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket and C. J. Oliver, �??Microwave scattering from dielectric wedges with planar surfaces: A diffraction coefficient based on a physical optics version of GTD,�?? IEEE Trans. Antennas and Propagat., 47, 1515-1527 (1999). [CrossRef]

Other

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE Peter Peregrinus Ltd., London, 1976)
R. C. Hansen Ed., Geometric Theory of Diffraction (IEEE Press, New York, 1981)
K. Sakina, S. Cui and M. Ando, �??Mathematical investigation of modified edge representation,�?? presented at the 2000 IEEE AP-S URSI International Symposium, Salt Lake City-Utah, USA, 16-21 July 2000
J. Goto, �??Interpretation of high frequency diffraction based upon PO,�?? M.S. thesis (Tokyo Institute of Technology, Tokyo, 2003), Chap. 3.
Y. Z. Umul, E. Yengel, and A. Aydýn, �??Comparison of physical optics integral and exact solution for cylinder problem,�?? presented at Eleco�??2003 International Conference, Bursa, Turkey, 3-7 Dec. 2003, <a href="http://eleco.emo.org.tr/eleco2003/ELECO2003/bsession/B5-01.pdf">http://eleco.emo.org.tr/eleco2003/ELECO2003/bsession/B5-01.pdf</a>
A. Sommerfeld, Optics (Academic Press, New York, 1954)
L. B. Felsen and N. Marcuwitz, "Radiation and Scattering of Waves", (IEEE Press, New York, 1994) [CrossRef]
W.L. Stutzman and G. A. Thiele, "Antenna Theory and Design", (John Wiley & Sons, New York, 1988)
A. Ishimaru, "Electromagnetic Wave Propagation, Radiation and Scattering", (Prentice Hall, New Jersey, 1991).
P. Ya. Ufimtsev, �??Method of edge waves in the Physical Theory of Diffraction,�?? Air Force System Command, Foreign Tech. Div. Document ID No. FTD-HC-23-259-71, (1971).
S. Silver, "Microwave Antenna Theory and Design", (McGraw-Hill, New York, 1949)

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