Difference of scattering geometrical optics components and line integrals of currents in modified edge representation


Corresponding author: P. Lu, Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, 2-12-1-S3-19 O-okayama, Meguro-ku, Tokyo 152-8552, Japan. (lui@antenna.ee.titech.ac.jp)


[1] Equivalent edge currents (EECs) are widely used to asymptotically extract and express the diffraction from the periphery of the scatterers, as in the concept suggested by Young. Authors proposed novel EECs based on the unique concept named as Modified Edge Representation (MER), for surface to line integral reduction of Physical Optics (PO) radiation integral. MER is based upon not only asymptotic approximation but also Stokes' theorem, and it has remarkable accuracy even for small scatterers and is uniformly applicable at the geometrical boundaries. Moreover MER-EECs are clearly defined not only at the edges but also everywhere on the scatterer surface, with the singularity at the stationary phase point (SPP) if any. It comes that the radiation integral in general, consisting of both the diffraction and the Geometrical Optics (GO) components, could be reduced into two sets of line integration of MER-EECs along the periphery and the indentation at SPPs. In this paper, the GO reflection is numerically compared with the MER indentation integral around the reflected point for the dipole scattering from an ellipsoid; the error is empirically derived in analytical form.

1. Introduction

[2] Physical Optics (PO) [Silver, 1949] is one of the high frequency asymptotic techniques and is widely used for the diffraction analysis. In PO, the currents induced on the scatterer are approximated by Geometrical Optics (GO) and integrated over the surface to give the scattering fields. The surface integration, called radiation integral hereafter, has highly oscillating integrand in higher frequency. The surface-to-line integral reduction is important for reducing the computational load and extracting the mechanism of diffraction. To this end, in addition to a mathematical method related with Stokes' theorem, the concept of equivalent edge currents (EECs) have been established in asymptotic approaches based on the method of stationary phase (SP) for high frequency. The radiation integral could be asymptotically decomposed into two components in general, GO contributions and diffraction. It has been widely understood that the GO components are separately obtained as the contribution from stationary phase point (SPP) inside of the integration area while the diffraction is extracted from the radiation integral in terms of line integration of EECs along the periphery of the scatterer. The GO components are analytically derived using ray geometries alternatively. In the long history of high frequency diffraction analysis, various types of definitions of EECs have been proposed asymptotically and the accuracy for the surface-to-line integral reduction has been compared intensively [Michaeli, 1984; Ufimtsev, 1991; Murasaki and Ando, 1992; Johansen and Breinbjerg, 1995; Albani and Maci, 2002; Albani, 2011]. The EECs were defined along the edge and discussions were focused upon their accuracy in predicting the diffraction fields near the geometrical boundaries such as the shadow (SB) and the reflection boundaries (RB), for which inner SPP traverses the periphery of the integration area. In this background with asymptotic techniques, the EECs have been considered to have nothing to do with the GO components, the remaining contribution in the radiation integral.

[3] The Modified Edge Representation (MER) is a unique concept for defining the EECs for surface-to-line integral reduction of radiation integral [Gokan et al., 1989; Murasaki and Ando, 1992]. MER-EEC is based upon not only asymptotic approximation but also Stokes' theorem. It eliminates the singularities of fields on geometrical boundaries. Furthermore, MER-EEC is defined independently from the orientation of the actual edge and therefore, it could be available not only at the periphery but also at arbitrary points over the scatterer except the SPP where MER-EEC becomes infinite. It has been verified that the surface integral of PO currents is uniformly reduced into a MER-EEC line integration, if no inner SPP [Sakina and Ando, 2001]. The superiority to other EECs in terms of accuracy and applicability in surface-to-line integral reduction was fully demonstrated [Murasaki and Ando, 1991; Sakina and Ando, 2001; Rodriguez and Ando, 2005]. It implies that, if SPP exists inside, the surface integration is reduced into two line integrations along the periphery and the infinitesimally small indentation integration around the inner SPP, the singularity in the integrands [Miyamoto and Wolf, 1962] as in Figure 1. If the former, named MER-Periphery, is regarded as the diffraction as it was suggested byYoung [1802], it comes that the latter coming from the MER-EEC indentation integral at inner SPP should be identical with the GO constituent. This poses the new discussion about the relation between EECs and the GO component, which has never opened until the proposal of MER-EEC, which is applicable even for the small and inner integration area. The accuracy of MER-Periphery in diffraction analysis and the comparison of MER-EEC with the GO are two related but separate problems for MER-EEC. This contrasts strongly with the conventional EECs where the line integration along periphery sometimes contains not only diffraction but also the GO [Johansen and Breinbjerg, 1995; Albani and Maci, 2002].

Figure 1.

Surface to line integral reduction by EECs defined in MER.

[4] The equivalence of MER(SPP) and Scattering Geometrical Optics (SGO) was proved rigorously for the planar surface [Rodriguez and Ando, 2005] and supported numerically for curved surfaces provided that the wavefront of the reflected ray is spherical [Rodriguez et al., 2007]. However, the difference between MER(SPP) and SGO appears in the general case where the reflected wave is non-spherical and aberration exists.

[5] This paper discusses the difference between MER(SPP) at inner SPP and SGO intensively for the ellipsoid [Lu and Ando, 2011a, 2011b] and summarizes the dependence upon the parameters, such as the ray geometry and the local shape of the scatterer. Finally the explicit and analytical expression is derived empirically. Now, the GO term is given with two independent expressions, analytical one from ray geometry and line integration of MER-EEC with analytical correction factor.

2. Modified Edge Representation and Scattering Geometrical Optics Components

2.1. Equivalent Edge Currents in Modified Edge Representation

[6] The Modified Edge Representation (MER) is proposed empirically by Murasaki and Ando [1992], which is one of the approaches for defining the EECs for PO surface integrals. It has been verified that the diffraction fields are directly reduced from PO surface integration into MER line integration along the periphery of the scatterer if not for inner SPP. As an alternative case, the additional contribution from the inner SPP should be evaluated by infinitesimally small line integration ρ′ → 0 around it which corresponds to reflected field as shown in Figure 1.

[7] MER line integration is shown in Figure 2, and is expressed by [Murasaki and Ando, 1992],

display math

where ri and ro are measured from the reflected point to source and observer respectively and the MER unit vector τ is defined as:

display math

which is generally different from math formula along the actual edge. The field reflected from inner SPP is defined by the MER line integration as:

display math
Figure 2.

Parameters used in MER line integration.

[8] MER electric and magnetic line currents to be used for integration along the periphery or around the inner SPP, defined as:

display math

Hi is the radiation component of the magnetic field incident upon the point of interest from the Hertzian dipole expressed as:

display math

2.2. Scattering Geometrical Optics Components

[9] The position of the source and the scatterer geometrically defines three regions in the space in Figure 3, where the definition of SGO field is given as:

display math

where the GO reflected contribution from SPP is given explicitly by [Kline, 1951; Balanis, 1989],

display math
Figure 3.

Geometrical optics regions.

[10] The GO theory considers an astigmatic wave leaving from the reflected point, as shown in Figure 4a. The spreading factor of the wave depends on principal radii of curvature ρ1r and ρ2r of the reflected wavefront at reflected point. These parameters are related with the source and observer location as well as the local curvature of the scattering surface as:

display math


display math
Figure 4.

Geometrical parameters of reflected ray tube and local shape (geometry) of the surface. (a) Reflected ray tube. (b) Local geometry of the surface.

[11] Here θ1, θ2 is angle between the direction of the incident rays ri and u1, u2 which are unit vectors in two principal direction of scatterer at reflected point with principal radii of curvatures a1 and a2 of the surface, as shown in Figure 4b. Ei(SPP) is the incident field at the reflected point, and math formula is the dyadic expression of the reflection coefficient.

3. Numerical Discussions of Differences Between SGO and MER Line Integration

[12] This paper discusses the difference between MER line integration in (3) and SGO components (6) in first region at reflection point for general combination of geometrical parameters, such as the radii R1, R2, R3 of ellipsoid, and the distances ri, ro as shown in Figure 5a. The source and the observer belong to the plane defined by the radii R1 and R3 of the ellipsoid. The contour for the indentation integration is defined as the intersection of the cone with the sector angle of θ and the ellipsoid as shown in Figure 5b. For making the MER line integration path Γ′ small, angle θ is made small enough. The solid line in Figure 6 presents the convergence of line integration MER(SPP) as θ decreases, while the dashed line is the result of SGO both in dB. EMERSPP/ESGO in dotted line presents the difference between these defined as:

display math
Figure 5.

Ellipsoid with an inner SPP on the axis. (a) Ellipsoid reflecting surface. (b) Integral path of MER.

Figure 6.

Convergent characteristic at SPP.

[13] In this specific geometry, MER(SPP) approaches to SGO and the difference EMERSPP/ESGO approaches to 0 dB.

[14] Numerical comparison between MER(SPP) and SGO is conducted in Figure 7, where the observation angle ϕ is changing from 0 deg to 90 deg. The radio ρ1r/ρ2r associated with the angle, is plotted at the bottom, which varies in the range from 0 to 1. As was pointed out in the previous work by Rodriguez et al. [2007], EMERSPP/ESGO goes to unity at the observation angles for which ρ1r/ρ2r takes the value of unity in (11), as are indicated by A, B and C for each geometries in Figure 7a for ri = 5λ and ro = 5000λ.

display math
Figure 7.

Comparison between MER and SGO and the ratio of the two principal radii of reflected ray ρ1r/ρ2r. (a) Ellipsoid reflecting surface. (b) Integral path of MER line integration.

[15] For the ellipsoid, the observation angle which satisfies (11) is explicitly given by [Rodriguez et al., 2007],

display math

[16] Another case of far source located at the distance ri = 500000λ and far field observer ro = 50000000λ, is shown in Figure 7b, as special cases with small errors. EMERSPP/ESGO is almost 0 dB at every observation angle from 0 deg to 90 deg, even if the radio ρ1r/ρ2r is not unity.

[17] In Figure 8, a reciprocal property of difference EMERSPP/ESGO is investigated by changing the positions of observer and source for four sets of observation angle 0 deg, 30 deg, 53 deg and 70 deg. The solid lines show the difference when incident distance ri is changed from 0 to 100λ for the fixed observer distance ro = 10000λ. The dashed lines are the reciprocal cases where observer moves for the fixed position of the source. They are indistinguishable with each other and are showing the reciprocity. Moreover the error is decreasing as both of the distance increase as was suggested in Figure 7b.

Figure 8.

Reciprocity in the difference MER-SGO.

4. Analytical Expression of Difference

[18] The difference depends upon both geometry of the surface and angle of observation. Before going into the empirical derivation of the analytical expression for the difference, the dependence upon the observation angle is extracted as a preliminary step.

[19] Here we introduce the image field Eimage with respect to the tangential plane at the point of reflection on the ellipsoid, together with the image source p′ as shown in Figure 9. The electromagnetic fields at observer by this image is expressed by:

display math
Figure 9.

Tangential plane at inner SPP and the image source.

[20] The relation EMERSPP/Eimage between MER(SPP) in (3) and the image field is shown in Figure 10 for various shape of the ellipsoid as functions of the observation angle, where the image source moves as is schematized in Figure 11. In Figure 10, noteworthy is that the angular dependence disappears; the ratio EMERSPP/Eimage is constant for the change in observation angle ϕ, as is expressed by:

display math

where A is a function of the geometrical parameters and is independent of the angle of observation and source. If local curvature of the ellipsoid vanishes R1, R2 → , R3 → 0, the coefficient A = 1, shown by the dotted line.

Figure 10.

Linear relation between MER line integration and reflection field from plane.

Figure 11.

Position of source and observer for ellipsoid and tangential plane.

[21] For deriving the analytical expression for EMERSPP/ESGO, we have applied the relation (14) and equation (7), then the correction term in (10) is derived as:

display math

[22] From one of the sufficient condition Δ = 1 if ρ1r = ρ2r = ρ′, obtained in the previous section in Figure 7a, the parameter A is specified as:

display math

ρ1r, ρ2r is changing with the observation angle ϕ. The ρ′ is defined as:

display math

where θ1′, θ2′ and ϕ′ is given by θ1, θ2 and ϕ which are obtained from ρ1r = ρ2r in (8) and (9), the parameters are shown in Figure 4b.

[23] Finally, equations (15) and (16) lead us to the final analytical expression of the correction term for Δ as:

display math

[24] It is clear from (18) that Δ = 1 for the case ρ1r = ρ2r = ρ′ and also if ri or ro goes to 0 or , the condition Δ = 1 holds. The expression (18) is checked numerically in Figure 12, where EMERSPP/Δ is compared with ESGO for the ellipsoid as one of more general tests. They agree perfectly for various shapes of the ellipsoid and the analytical form of the difference Δ in (18) is confirmed numerically.

Figure 12.

Agreement of EMERSPP/Δ and SGO.

5. Conclusions

[25] The MER was proposed for the reduction of the PO surface integration to the line one. First, the accuracy and the applicability of the SGO extraction in terms of MER indentation line integration around the SPP was numerically investigated in ellipsoid case. The line integral of MER has high accuracy with SGO when reflected wave is spherical on condition ρ1r = ρ2r, or the distances from reflection point to source and observer both are large. Second, the relation between MER line integration and reflected field from tangential plane surface has been found. The correction term has been explicitly expressed, the MER line integration corrected by this correction term perfectly matched with SGO numerically.


[26] This work is in part supported by the Research and Development Project for Expansion of Radio Spectrum Resources of the Ministry of Internal Affairs and Communications and JSPS Grant-in-Aid for Scientific Research (22656086).