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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 20 — Oct. 4, 2004
  • pp: 4959–4972
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Modified theory of physical optics

Yusuf Z. 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.

© 2004 Optical Society of America

1. Introduction

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

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

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

,3

3. 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

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

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

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

8. 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

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

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]

].

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

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

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

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

,12

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

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

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

,15

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

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

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

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

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

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

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

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.

A time factor ejwt 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

  1. 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

    Jes=n1×Ht|S1
    (1)

    where H⃗t 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

    Jes=n2×Hi|S2,Jms=n2×Ei|S2
    (2)

    for E⃗i , H⃗i 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
  2. The reflection and transmission angles (β) are variables which depend on the surface (S 1 + S 2) coordinates.
  3. 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

    n1=cos(u+α)t+sin(u+α)n
    (3)

    for S 1 and

    n2=cos(v+α)tsin(v+α)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 π2α+β2. The total scattered electric field can be defined as

    Et=Eis+Ers
    (5)

    where

    Eis=μ04πS2n2×Hi|S2ejkR2R2dS'+14πS2×(n2×Ei|S2ejkR2R2)dS'
    (6)

    and

    Ers=μ04πS1n1×Ht|S1ejkR1R1dS'
    (7)

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

    Ht=His+Hrs
    (8)

    where

    His=14πS2×(n2×Hi|S2ejkR2R2)dS'+ε4πS2n2×Ei|S2ejkR2R2dS'
    (9)

    and

    Hrs=14πS1×(n1×Ht|S1ejkR1R1)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 E⃗rs (H⃗rs ) and E⃗is (H⃗is ) 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

Hi=EiZ0(exsinϕ0eycosϕ0)ejk(xcosϕ0+ysinϕ0)
(11)

Er=ezErejk(xcosβysinβ)
(12)

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

Er=Eiejkx'(cosϕ0cosβ)
(13)

by using the boundary condition of n⃗ 1×(E⃗i +E⃗r )|S1 = 0, where E⃗i =e⃗z Eie jk(xcosϕ0+ysinϕ0) and n⃗ 1 =cos(u + ϕ 0)e⃗x + sin(u + ϕ 0)e⃗y . For this problem, u is equal to π2ϕ0+β2. The MTPO surface current can be evaluated as

JMTPO=EiZ0[cosucos(u+β+ϕ0)]ejkx'cosϕ0ez
(14)

which is equal to n⃗ 1 × (H⃗i + H⃗r )|S2 = J⃗ETPO at the perfectly conducting half plane. The classical PO surface current component is equal to

JPO=ez2EiZ0sinϕ0ejkx'cosϕ0
(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

JMTPO=2EiZ0sin(β+ϕ02)ejkx'cosϕ0ez
(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

Ers=ezjkEi2πx'=0z'=ejkx'cosϕ0ejkR1R1sin(β+ϕ02)dx'dz'
(17)

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

R1=(xx')2+y2+(zz')2.
(18)

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

cejkchα=πjH0(2)(kR)
(19)

by using the variable change of (z-z') = R shα where R is equal to (xx')2+y2. As a result one obtains

Ers=ezkEi2x'=0ejkx'cosϕ0H0(2)(kR)sin(β+ϕ02)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

Eieq=ezEiejk(xcosϕ0ysinϕ0)
(21)

and magnetic field is found to be

Hieq=EiZ0(exsinϕ0+eycosϕ0)ejk(xcosϕ0ysinϕ0)
(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

JMTPO=2EiZ0sin(β+ϕ02)ejkx'cosϕ0ez
(23)

where n⃗ 2 is equal to cos(v + ϕ 0)e⃗x - sin(v + ϕ 0)e⃗y for the aperture surface and v=π2ϕ0+β2. The operations from Eqs. (17) to (20) are also valid for this case. As a result one obtains

EisezkEi2πejπ4x'=0ejkx'cosϕ0ejkR1kR1sinϕ0+β2dx'
(24)

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

Es=Ers+Eis
(25)

for

ErsezkEi2πejπ4x'=0ejkx'cosϕ0ejkR1kR1sin(β+ϕ02)dx'
(26)

denoting the reflected scattered field found from Eq. (20) by using the Debye asymptotic expansion of H0(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 ϕπ 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

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

,23

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

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

EtezkEi2πejπ4(x'=0ejkx'cosϕ0ejkR1kR1sinϕ0+β2dx'x'=0ejkx'cosϕ0ejkR1kR1sin(β+ϕ02)dx')
(27)

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

g(x')=ρcosγ+x'(cosβcosϕ0)
(28)

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

gx'=ρsinγdx'x'sinβdx'+cosβcosϕ0
(29)

where γ is equal to

γ=πβϕ
(30)

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

ρsinγdx'=x'sinβdx'
(31)

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

βs=ϕ0
(32)

which denotes the reflection law by considering

gx's=cosβscosϕ0=0
(33)

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

2gx'2|s=sin2ϕ0l
(34)

where l can be expressed as

l=R1|s=ρcosγs
(35)

denoting the ray path of the reflected and incident fields. γs 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')l+12sin2ϕ0l(x'x's)2
(36)

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

f(x's)±kEi2πejπ4sinϕ0kl
(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

Er,i±kEisinϕ02πejπ4ejklklejksin2ϕ0l(x'x'2)dx'
(38)

and can be evaluated easily by considering the integral of

ey22dy=2π.
(39)

As a result, one obtains

ErEiejkρcos(ϕ+ϕ0)
(40)

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

EiEiejkρcos(ϕϕ0)
(41)

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

Edez1jkf(0)g'(0)ejkg(0)
(42)

g(0)=ρ,
(43)
g'(0)=(cosϕ+cosϕ0)
(44)

and

f(0)=kEi22πejπ4cos(ϕϕ02)
(45)

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

Erd=ezEi22πejkρcos(ϕϕ02)cosϕ+cosϕ0ejπ4
(46)

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

Drd=ejπ422πcos(ϕϕ02)cosϕ+cosϕ0
(47)

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

g(0)=ρ,
(48)
g'(0)=(cosϕ+cosϕ0)
(49)

and

f(0)=kEi22πejπ4cos(ϕ+ϕ02)
(50)

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

Eid=ezEi22πejkρcos(ϕ+ϕ02)cosϕ+cosϕ0ejπ4
(51)

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

Did=ejπ422πcos(ϕ+ϕ02)cosϕ+cosϕ0
(52)

and one obtains

Dt=Did+Drd=ejπ42πcos(ϕϕ02)cos(ϕ+ϕ02)cosϕ+cosϕ0
(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

ETPOEiejk(xcosϕ0+ysinϕ0)kEi2sinϕ00ejkx'cosϕ0H0(2)(kR1)dx'
(54)

for a plane wave illuminated perfectly conducting half plane [12

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

EMTPO|β=ϕ0kEi2(x'=0ejkx'cosϕ0H0(2)(kR1)sinϕ0dx'0ejkx'cosϕ0H0(2)(kR1)sinϕ0dx')
(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

ETMTPO=12[Ei(ejkρcos(ϕϕ0)ejkρcos(ϕ+ϕ0))u(πϕ)+ETPO+ER]
(56)

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

u(πϕ)={1,ϕπ0,ϕπ
(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

Et=2Eim=1ejmπ4Jm2()sinm2ϕsinm2ϕ0
(58)

for an electric polarized incident plane wave [25

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

EMTPOkEi2πejπ4(x'=0ejkx'cosϕ0ejkR1kR1sinϕ0+β2dx'x'=0ejkx'cosϕ0ejkR1kR1sin(β+ϕ02)dx')
(59)

for the half plane problem. The first integral represents the incident field for πϕπ + 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 (ui ), reflected (ur ) and diffracted fields (udi +udr ). 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 ϕπ. The figures are plotted for ϕ0=π4 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

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

Hi=ezHiejk(xcosϕ0+ysinϕ0)
(60)

and electric field can be found as

Ei=Z0Hi(sinϕ0ex+cosϕ0ey)ejk(xcosϕ0+ysinϕ0)
(61)

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

Hr=ezHrejk(xcosβysinβ)
(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

Er=Z0Hr(sinβex+cosβey)ejk(xcosβysinβ)
(63)

by considering Maxwell-ampere equation. One obtains

Hrejkx'cosβ=Hiejkx'cosϕ0
(64)

as a result of the boundary condition of n⃗ 1 ×(E⃗i × E⃗r )|S1 = 0, with Eqs. (61) and (63) for n⃗ 1 = cos(u + ϕ 0)e⃗x + sin(u + ϕ 0)e⃗y . The MTPO surface current can be evaluated as

JMTPO=2Hi[excosβϕ02eysinβϕ02]ejkx'cosϕ0
(65)

which is equal to n⃗ 1 × (H⃗i + H⃗r )|S2 = J⃗MTPO on the perfectly conducting half plane. The classical PO surface current component is equal to

JPO=ex2Hiejkx'cosϕ0
(66)

for this problem. It is apparent that, the current, given in Eq. (65) is equal to J⃗PO 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

  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. 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>
  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).

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