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

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1 RADIO SCIENCE, VOL. 47,, doi:0.029/20rs004899, 202 Difference of scattering geometrical optics components and line integrals of currents in modified edge representation Pengfei Lu and Makoto Ando Received 4 October 20; revised 3 February 202; accepted 22 April 202; published 3 May 202. [] 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. Citation: Lu, P., and M. Ando (202), Difference of scattering geometrical optics components and line integrals of currents in modified edge representation, Radio Sci., 47,, doi:0.029/20rs Introduction [2] Physical Optics (PO) [Silver, 949] 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 surfaceto-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 Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo, Japan. Corresponding author: P. Lu, Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, 2-2--S3-9 O-okayama, Meguro-ku, Tokyo , Japan. (lui@antenna.ee.titech.ac.jp) Copyright 202 by the American Geophysical Union /2/20RS 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, 984; Ufimtsev, 99; Murasaki and Ando, 992; Johansen and Breinbjerg, 995; Albani and Maci, 2002; Albani, 20]. 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., 989; Murasaki and Ando, 992]. 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, 200]. The superiority to other EECs in terms of accuracy and applicability in surface-to-line integral reduction was fully demonstrated [Murasaki and Ando, 99; Sakina and Ando, 200; Rodriguez and Ando, 2005]. It implies that, if SPP exists of7

2 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, 20a, 20b] 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. Figure. Surface to line integral reduction by EECs defined in MER. 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, 962] as in Figure. If the former, named MER-Periphery, is regarded as the diffraction as it was suggested by Young [802], 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, 995; Albani and Maci, 2002]. [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 2. Modified Edge Representation and Scattering Geometrical Optics Components 2.. Equivalent Edge Currents in Modified Edge Representation [6] The Modified Edge Representation (MER) is proposed empirically by Murasaki and Ando [992], 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 r 0 around it which corresponds to reflected field as shown in Figure. [7] MER line integration is shown in Figure 2, and is expressed by [Murasaki and Ando, 992], E MER ¼ j kh I 4p G þ j kh I 4p r o ½r o J MER þ M MER G Š e jkr o dl r o r o ½r o J MER þ M MER Š e jkr o dl ðþ r o where r i and r o are measured from the reflected point to source and observer respectively and the MER unit vector t is defined as: ðr i þ r o Þt ¼ 0; n t ¼ 0 ð2þ Figure 2. Parameters used in MER line integration. 2of7

3 Figure 3. Geometrical optics regions. observer location as well as the local curvature of the scattering surface as: r r ¼ ;2 2 r i þ r i þ ð8þ 2 f ;2 where ¼ sin 2 q 2 þ sin2 q f ;2 cosf a a 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2 2 q 2 cos 2 þ sin2 q 4 f a a 2 a a 2 ð9þ which is generally different from t along the actual edge. The field reflected from inner SPP is defined by the MER line integration as: 8 9 E SPP MER ¼ lim r 0 < j kh I : 4p G r o ½r o J MER þ M MER Š e jkr o r o = dl ; ð3þ [8] MER electric and magnetic line currents to be used for integration along the periphery or around the inner SPP, defined as: fr o ðr o JÞg t J MER ¼ j ðr o tþ 2 ðr i þ r o Þðn t ðr o JÞ t M MER ¼ j ðr o tþ 2 ðr i þ r o J ¼ 2n H i t Þ t Þðn tþ ð4þ [] Here q, q 2 is angle between the direction of the incident rays r i and u, u 2 which are unit vectors in two principal direction of scatterer at reflected point with principal radii of curvatures a and a 2 of the surface, as shown in Figure 4b. E i (SPP) is the incident field at the reflected point, and R is the dyadic expression of the reflection coefficient. 3. Numerical Discussions of Differences Between SGO and MER Line Integration [2] 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 H i is the radiation component of the magnetic field incident upon the point of interest from the Hertzian dipole expressed as: H i ¼ e jkr i k p r i j4pr i jkr i 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: 8 8 < I incident þ reflected ray < E r E SGO ¼ E i þ E GO II incident ray only ¼ 0 ð6þ : : III 0 E i where the GO reflected contribution from SPP is given explicitly by [Kline, 95; Balanis, 989], ð5þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E r ¼ E i r ðsppþ R r rr 2 r r þ r o r r 2 þ r o e jkr o ð7þ [0] 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 r r curvature r and r 2 of the reflected wavefront at reflected point. These parameters are related with the source and 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. 3of7

4 [4] Numerical comparison between MER(SPP) and SGO is conducted in Figure 7, where the observation angle f is changing from 0 deg to 90 deg. The radio r r /r r 2 associated with the angle, is plotted at the bottom, which varies in the range from 0 to. As was pointed out in the previous work by Rodriguez et al. [2007], E SPP MER /E SGO goes to unity at the observation angles for which r r /r r 2 takes the value of unity in (), as are indicated by A, B and C for each geometries in Figure 7a for r i =5l and r o = 5000l. r r =rr 2 ¼ ðþ [5] For the ellipsoid, the observation angle which satisfies () is explicitly given by [Rodriguez et al., 2007], rffiffiffiffi a 2 f ¼ arccos a ð2þ [6] Another case of far source located at the distance r i = l and far field observer r o = l, is shown in Figure 7b, as special cases with small errors. E SPP MER /E SGO is almost 0 db at every observation angle from 0 deg to 90 deg, even if the radio r r /r r 2 is not unity. [7] In Figure 8, a reciprocal property of difference E SPP MER / E SGO 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 Figure 5. Ellipsoid with an inner SPP on the axis. (a) Ellipsoid reflecting surface. (b) Integral path of MER. parameters, such as the radii R,R 2,R 3 of ellipsoid, and the distances r i,r o as shown in Figure 5a. The source and the observer belong to the plane defined by the radii R and R 3 of the ellipsoid. The contour for the indentation integration is defined as the intersection of the cone with the sector angle of q and the ellipsoid as shown in Figure 5b. For making the MER line integration path G small, angle q is made small enough. The solid line in Figure 6 presents the convergence of line integration MER(SPP) as q decreases, while the dashed line is the result of SGO both in db. E SPP MER /E SGO in dotted line presents the difference between these defined as: MER SGO ¼ 20lgE SPP MER 20lgE SGO ¼ 20lg ESPP MER j j E SGO ð0þ [3] In this specific geometry, MER(SPP) approaches to SGO and the difference E SPP MER /E SGO approaches to 0 db. 4of7 Figure 6. Convergent characteristic at SPP.

5 Figure 7. Comparison between MER and SGO and the ratio of the two principal radii of reflected ray r r /r r 2. (a) Ellipsoid reflecting surface. (b) Integral path of MER line integration. incident distance r i is changed from 0 to 00l for the fixed observer distance r o = 0000l. Thedashedlinesarethe 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. 4. Analytical Expression of Difference [8] 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. [9] Here we introduce the image field E image 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: E image ¼ e jkr 3 kh j4pr jkr þ 3 k 2 r 2 p ¼ pþ2nðnpþ r i ¼ r i 2nðn r i Þ r ¼ r i þ r o p r i r i þ þ jkr k 2 r 2 p ð3þ [20] The relation E SPP MER /E image between MER(SPP) in (3) and the image field is shown in Figure 0 for various shape of the ellipsoid as functions of the observation angle, where the image source moves as is schematized in Figure. In Figure 0, noteworthy is that the angular dependence disappears; the ratio E SPP MER /E image is constant for the change in observation angle f, as is expressed by: E SPP MER ¼ A E image ð4þ 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 R,R 2, R 3 0, the coefficient A =, shown by the dotted line. [2] For deriving the analytical expression for E SPP MER / E SGO, we have applied the relation (4) and equation (7), then the correction term in (0) is derived as: D ¼ ESPP MER E SGO ¼ A E image E SGO r i ¼ A r i þ r o s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r þ r o r r 2 þ r o r r rr 2 ð5þ 5of7

6 Figure 0. Linear relation between MER line integration and reflection field from plane. Figure 8. Reciprocity in the difference MER-SGO. [22] From one of the sufficient condition D = if r r = r 2 r = r, obtained in the previous section in Figure 7a, the parameter A is specified as: A ¼ r i þ r o r i r ð6þ r þ r o r r r, r 2 is changing with the observation angle f. The r is defined as: r ¼ 2 r i þ r i 2 þ sin 2 q 2 þ sin2 q cosf a a 2 ð7þ where q, q 2 and f is given by q, q 2 and f which are r r obtained from r = r 2 in (8) and (9), the parameters are shown in Figure 4b. [23] Finally, equations (5) and (6) lead us to the final analytical expression of the correction term for D as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ ESPP MER ¼ r r r þ r o r r 2 þ r o E SGO r þ r o r r rr 2 ð8þ [24] It is clear from (8) that D = for the case r r = r r 2 = r and also if r i or r o goes to 0 or, the condition D = holds. The expression (8) is checked numerically in Figure 2, where E SPP MER /D is compared with E SGO for the ellipsoid as Figure 9. source. Tangential plane at inner SPP and the image Figure. Position of source and observer for ellipsoid and tangential plane. 6of7

7 [26] Acknowledgments. 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 ( ). Figure 2. Agreement of E SPP MER /D and SGO. one of more general tests. They agree perfectly for various shapes of the ellipsoid and the analytical form of the difference D in (8) is confirmed numerically. 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 r r = r r 2, 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. References Albani, M. (20), Boundary diffracted wave and incremental geometrical optics: A numerically efficient and physically appealing line-integral representation of radiation integrals. Aperture scalar case, IEEE Trans. Antennas Propag., 59(2), , doi:0.09/tap Albani, M., and S. Maci (2002), An exact line integral representation of the PO radiation integral from a flat perfectly conducting surfaces illuminated by elementary electric or magnetic dipoles, Turk. J. Electr. Eng., 0(2), Balanis, C. A. (989), Geometrical theory of diffraction, in Advanced Engineering Electromagnetics, pp , John Wiley, New York. Gokan T., M. Ando, and T. Kinoshita (989), A new definition of equivalent edge currents in a diffraction analysis, IEICE Tech. Rep., AP89-64, Inst. of Electron., Inf., and Commun. Eng., Tokyo. Johansen, P. M., and O. Breinbjerg (995), An exact line integral representation of the physical optics scattered field: The case of a perfectly conducting polyhedral structure illuminated by electric Herzian dipoles, IEEE Trans. Antennas Propag., 43(7), , doi:0.09/ Kline, M. (95), An asymptotic solution of Maxwell s equations, Commun. Pure Appl. Math., 4, , doi:0.002/cpa Lu, P., and M. Ando (20a), Expressions of difference between scattering geometrical optics and indentation line integral around the pole of modified edge representation, IEICE Tech. Rep., AP200-9, pp. 07 2, Inst. of Electron., Inf., and Commun. Eng., Tokyo. Lu, P., and M. Ando (20b), Differences between scattering geometrical optics and line integral of modified edge representation, paper presented at Antennas and Propagation and USNC/URSI National Radio Science Meeting, Inst. of Electr. and Electron. Eng., Spokane, Wash. Michaeli, A. (984), Equivalent edge currents for arbitrary aspects of observation, IEEE Trans. Antennas Propag., 32, , doi:0.09/ TAP Miyamoto, K., and E. Wolf (962), Generalization of the Maggi- Rubinowicz theory of the boundary diffraction wave. Part I, J. Opt. Soc. Am., 52, , doi:0.364/josa Murasaki, T., and M. Ando (99), Elimination of false singularities in GTD equivalent edge currents, IEE Proc., Part H, 38(4), Murasaki, T., and M. Ando (992), Equivalent edge currents by the modified edge representation: Physical optics components, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 75-C(5), Rodriguez, L., and M. Ando (2005), Direct and analytical derivation of the vectorial geometrical optics from the modified edge representation line integrals for the physical optics, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 88-C(2), Rodriguez, L., K. Yukimasa, T. Shijo, and M. Ando (2007), Inner stationary phase point contribution of physical optic in terms of the modified edge representation line integrals (curved surfaces), Radio Sci., 42, RS6S24, doi:0.029/2007rs Sakina, K., and M. Ando (200), Mathematical derivation of modified edge representation for reduction of surface radiation integral, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 84-C(), Silver, S. (949), Microwave Antenna Theory and Design, McGraw-Hill, New York. Ufimtsev, P. Y. (99), Elementary edge waves and the physical theory of diffraction, Electromagnetics, (2), 25 60, doi:0.080/ Young, T. (802), The Bakerian lecture: On the theory of light and colours, Philos. Trans. R. Soc. London, 20, of7

Partial reduction of the physical optics surface integral to the modified edge representation line integral

RADIO CIENCE, VOL. 44,, doi:1.129/28r3992, 29 Partial reduction of the physical optics surface integral to the modified edge representation line integral L. Rodriguez, 1,2 Y. Katakai, 1 M. Oishi, 1 and