# Inner stationary phase point contribution of physical optic in terms of the modified edge representation line integrals (curved surfaces)

## Abstract

[1] The unique concept of the modified edge representation (MER) was proposed for the surface to the line integral reduction of the physical optic (PO). The equivalence between the MER line and the PO surface integration was analytically derived by using the Stokes theorem relations as well as asymptotic treatments, for the smooth scattering surfaces without inner stationary phase points (SPP). Later on, for the planar surface, the MER line integration around the inner SPP was investigated, and it was identified as the scattering geometrical optics (SGO). In this paper, findings related with the MER line integration around the inner SPP are extended to curved surfaces. The accuracy and the applicability of the SGO extraction in terms of the MER line integration are numerically investigated for different radii of curvatures of the scattering surfaces. Authors introduce a geometrical criterion for the applicability of the method. The MER line integration provides an alternative way to the stationary phase method or the classical geometrical optics for calculating SGO. In addition, this numerical result indirectly identifies the entity of the MER line integration along the periphery of the scattering illuminated region, irrespective of the position of observer, as not other than diffraction.

## 1. Introduction

[2] The physical optics (PO) is widely used for estimating the high frequency scattering of electromagnetic fields [see Harrington, 1961]. PO predicts the scattering fields from conducting surfaces by two well-defined steps. Firstly, the induced currents at the point of interest in the illuminated region are approximated in the sense of the geometrical optics (GO) by replacing the scattering surface with its tangential infinite plane. Secondly, the scattering fields are obtained by integrating the PO currents over the illuminated region of the scattering surface. The resulting PO fields are free of discontinuities and singularities on the geometrical boundaries and the caustics where GO and the geometrical theory of diffraction (GTD) fail as shown by Yamashita [1990, and references therein].

[3] The PO surface radiation integral suffers of difficulties in the physical interpretation. Theoretical works by Miyamoto and Wolf [1962], Rubinowicz [1965] and Asvestas [1986], asymptotics by Ando [1984], Michaeli [1986], Gokan et al. [1989], Murasaki and Ando [1991], Sakina et al. [2001], Rodriguez and Ando [2004a, 2004b], and Shijo et al. [2004] and exacts by Johansen and Breinbjerg [1995] and Albani and Maci [2002] on the surface-to-line integral reduction developments have been presented. Some of these works have provided important and effective tools for the extraction and corrections of the PO errors [see, e.g., Ufimtsev, 1971; Ando et al., 1991; Oodo and Ando, 1994].

[4] For the surface-to-line radiation integral reduction the equivalent edge currents (EEC) method is introduced with two principal streams in the study. The works in the first category are associated with the asymptotic discussions based upon the principle of stationary phase; where the line integration of the equivalent edge currents around the periphery of the scattering surface is regarded as the edge contributions or the diffraction components as shown by Ando [1984], Michaeli [1986] and Oodo and Ando [1994]. This concept is harmonizing with the work by Young [1802] and was demonstrated rigorously by Rubinowicz [1965] for scalar potentials. In the second category the line integration is derived rigorously based upon the field equivalence theorem by Asvestas [1986], Johansen and Breinbjerg [1995] and Albani and Maci [2002] where the integral reduction for the planar surface was conducted rigorously with the help of the image theory.

[5] The modified edge representation (MER) concept was proposed for defining the equivalent edge currents on the periphery of the scattering surface, reported by Ando [1984]. MER is conceptually different from the general EECs in its derivation and have two important advantages, the wider applicability to curved surfaces and for the source close to the scattering surface [Rodriguez and Ando, 2007]. The line integration of the MER currents along the periphery of the scattering illuminated region reproduces the PO diffracted fields with remarkably high accuracy [see Gokan et al., 1989; Murasaki and Ando, 1991].

[6] For the integration region having no stationary phase point (SPP), Sakina et al. [2001] mathematically showed, by applying Stoke theorem identities and the high frequency approximation, the equivalence of the MER line integration around the periphery of the scattering surface to the PO surface integration, reproducing the diffraction components. On the other hand, if the SPP is located in the surface integration area the entity of the MER line integration along the periphery, whether it is only diffraction or it is scattering is not clear, as it is also the case with the general EECs.

[7] This paper presents novel findings obtained as the by-product in the above study of MER for the case where SPP is inside of the integration area. The MER line integration along the infinitesimally small contour at SPP is calculated numerically, showing a good approximation to the scattering geometrical optics (SGO) for large variety of surface curvatures.

[8] This methodology provides a new and alternative way for calculating SGO emanating from curved surfaces, which differs from the stationary phase method and the classical geometrical optics. For the special case of the planar scattering surface, this numerical result was also proved mathematically by Rodriguez et al. [2005] and different types of line integral expressions of GO were independently derived as part of the development by Yukimasa et al. [2006].

[9] In addition, this numerical result indirectly identifies the entity of the MER line integration along the periphery for the scattering surface with inner SPP and answers aforementioned open question; the MER line integration along the periphery of the scattering surface corresponds only to diffraction irrespectively of the observer position.

## 2. Physical Optics Surface Radiation Integral and SGO Definitions

[10] As a general definition, the total field consists of the scattering field added to the incident field. The incident field Ei radiated by the source is assumed to be given and the scattering field Es, radiated by the induced currents on the scattering surface is expressed by the following surface radiation integral:

where ro is the distance from the integration point to the observer and o is the unit vector toward the observer. In PO, the surface induced electric current J is approximated by JPO = 2 × Hi on the illuminated region of the scattering surface and JPO = 0 in its complement by Harrington [1961]. The surface normal unit vector is and Hi is the unperturbed magnetic field incident on the scattering surface. Then the radiation integral (1) is denoted by EPOS hereafter. The unperturbed incident Hi field in the PO integration must be the exact unless the source is far enough from the scattering surface where higher order terms other than the radiation terms are dispensable.

[11] The source and the scattering surface define geometrically three observation regions in the space which are related with the GO definition, Figure 1. The Region I and Region III are the reflection and the shadow region respectively, Region II is defined as diffraction region. When the stationary phase method is applied to the PO surface integration in high frequency (1), two important contributions are found, one is the diffraction and the other is SGO. The diffraction is the contribution from the boundary of the scattering surface and SGO is related with the inner SPP. The inner SPP is a point on the scattering surface representing the reflection point for Region I or the intersection of the scattering surface and the straight line between the source and the observer in Region III. In Region I, the SPP contributions reproduce the reflected wave (GO approximation). In Region III, the SPP reproduces the cancelling wave, creating the GO shadow behind the scattering surface. Hereafter, these two components, reflection and cancelling waves are named under the terminology of SGO.

[12] By the use of the Kline [1951] ansatz, the general expression of the reflected geometrical optic (GO) terms is obtained as:

with

GO considers an astigmatic ray tube which leaves from the reflection point (SPP), as shown in Figure 2. The spreading factor of the ray tube depends on principal radii of curvature ρ1r and ρ2r of the reflected ray tube, as well as, ro measured from the stationary phase point. The definitions of ρ1r and ρ2r are related with the source and observer location, as well as, the curvatures of the scattering surface. EGOi is the first term of the Luneberg-Kline series of the incident ray evaluated at the specular point and is the dyadic expression of the reflection coefficients.

[13] The SGO definition given here has been reproduced numerically by the modified edge representation line integrals around SPP by Rodriguez and Ando [2004c] and Rodriguez et al. [2005], for specific curved and planar scattering surfaces. The applicability and the accuracy criterion for the case of the curved surfaces will be discussed in this article.

## 3. Modified Edge Representation

[14] In the original definition introduced by Gokan et al. [1989], MER is a vector which satisfies the diffraction law at any point on the periphery of the scattering surface as is shown in Figure 3. The MER unit vector () is defined as:

The MER direction is generally different from the real edge of the scattering surface; the vector always satisfies the diffraction law but the vector , tangential to the real edge only at some specific points. For the points, where the vector MER and the vector have the same direction (D1 and D2 in Figure 3), the phenomenon of the diffraction occurs.

[15] By use of the MER vector , the equivalent MER line currents are defined by Ando [1984]. The line integration of the MER currents along the periphery of the scattering surface provides the PO diffracted fields accurately for Region II, without stationary phase points on the illuminated surface region as showed by Murasaki and Ando [1991]. Given the good numerical accuracy between the PO diffracted fields and the line integration of the MER currents even for planar and smooth surfaces, Sakina et al. [2001] conducted the analytical derivation for Region II.

[16] For the observation positions remaining in the Regions I and III, the inner SPP exists in the former integration region and the MER singularity is identified. Due to this particular singularity, the application of the Stokes theorem is not possible and the development by Sakina et al. [2001] needs to be modified to include this contribution in the MER theory.

## 4. MER and EEC Line Integrals Interpretation of the SGO

### 4.1. PO Surface to MER Line Integral Reduction for Regions I and III

[17] Following Sakina et al. [2001], for the observer into the distance far from the scattering surface, the PO surface integral (1) is written as:

where

For the case of A (B, A′ and B′ are similar):

The PO currents JO necessary for the MER current definition include only the radiation terms of the incident fields at the illuminated region of the scattering surface.For the observer in Region II, the integrants of (8) are well defined and free of singularities. By direct application of the Stokes theorem identities, the first surface integral is expressed by the line integration of the MER currents along the boundary of S. The second surface integral is reduced to a line integral by the stationary phase method, showing to be of higher asymptotic order.

[18] Then, the PO surface integration in (1) is defined by the MER line integration as:

The expression in (9) is regarded as diffraction for the observer in Region II and it is named as EMERdiff hereafter. The equivalent electric and magnetic line currents along the actual edge are defined using the modified edge vector (4) as:

For the observer located in the Regions I and III, SPP is inner to the PO integration area S. By definition, the scattering field consists of both, the diffracted and the SGO terms. For these regions, the surface integrals in (8) suffer from a singularity at SPP due to (i + o) · ( × ) → 0 and the procedure based upon the Stokes Theorem should be reformulated as follows:

[19] According to Figure 4, the PO radiation integral (1) is decomposed into two integrals:

with

In (11), S′ is the small area which contains SPP and So its complement. The exclusion of SPP from the region So guarantees the applicability of the results developed by Sakina et al. [2001]. Therefore, the surface integral So is rewritten by two MER line integrals as:

The field EPOS in (1) or (11) is now expressed as the sum of two contour MER line integrals and a PO surface integral. Taking the limit ρ′ → 0 in (13) around the SPP, then (1) is expressed as:

The first MER line integral is evaluated taken around the infinitesimal contour Γ′ of SPP and the second is along the periphery Γ of the illuminated surface, clockwise and counterclockwise respectively. The PO surface integral over the tiny portion of the surface S′ containing the SPP vanishes as ρ′ → 0 since its integrand is well-defined. The MER line integration along Γ′ does not necessarily vanish as ρ′ → 0 since the MER currents are singular as the evaluation is approaching to SPP. The surface radiation integral (1) is finally expressed only by the MER line integration as:

where

The expression EMERPer is found to be mathematically the same than the obtained by Sakina et al. [2001] for the diffracted waves in the Region II while EMERSPP is new one. This last expression is associated with SPP and the relation to the SGO defined in (2) and (3) for the smooth surfaces is the main discussion of this paper.

### 4.2. Interpretation of the MER Line Integration Around the Stationary Phase Point

[20] The objective in this section is to investigate the MER line integration around the inner SPP (16) and its relation with SGO. Then, the line integration is evaluated and compared numerically with the classical SGO. The results for the observer in the reflected and shadow regions (Regions I and III) are included. Since the behavior as well as the integration of EECs has been discussed only at the periphery of the illuminated scattering surface region in the history, this discussion is quite new and full of interest.

[21] Figure 5 shows the results of the EMERSPP as ρ′ → 0, normalized by the SGO (cancelling wave). The distance to SPP of the source and the observer is 5 λ respectively. The observer is in the Region III and the scattering surfaces are spheres with various radii of curvature. For ρ′ smaller than 0.1 λ, the fast convergence of the EMERSPP to the cancelling wave −Ei, is observed for variety of surface curvatures. The results, in agreement with the field equivalent principle, suggest the creation of the GO shadow. For the observer in Region I, EMERSPP and SGO in (2) are also calculated and presented in Figure 6. The amplitudes are normalized by the image of the dipole Eimag. The amplitude of SGO (reflected field in Region I) varies with the curvature of the scattering surfaces. As the scattering surface radii of curvature become infinite (planar scattering surface), the SGO approaches to the image surface (results for R = 100 λ in the figure). For ρ′< 0.01 λ, EMERSPP shows good convergence to SGO.

### 4.3. MER and Other EECs Line Integrals Convergence to SGO and the Dependence Upon the Integration Contour

[22] On the basis of the asymptotic approximations, the general EEC techniques assume the diffracted waves derivation from the line integration along the illuminated periphery of the scattering surface [see Michaeli, 1986]. From the discussion in the previous section, this treatment implies that EEC line integration around the inner SPP should also approach to the SGO, if EEC is accurate in surface to line integral reduction (diffraction component). By the use of MER, which is also one of the EECs, Rodriguez and Ando [2004a] has reproduced the SGO terms with high accuracy. The SGO extraction by the general EECs was carefully introduced by Yukimasa et al. [2006]. In this section, the EECs line integrations proposed for various authors are calculated around SPP and also the convergence to SGO depending on the integration path is compared.

[23] Figure 7 shows the line integration around SPP for three general EECs [Murasaki and Ando, 1991; Michaeli, 1986] and MER. The scattering surface is a sphere of radio 100 λ, the source is located at 5 λ from SPP. The far field results for the reflection and the shadow regions are shown according to the incident angle θi. The results obtained by using the circular and rectangular integration path are also shown. Different values of convergence are observed for different shape of the path for each EECs. Only the MER shows the accurate results independent of the shape of the contour.

## 5. Necessary Condition for the MER to SGO Convergence

[24] In this section, the necessary condition to guarantee the MER to the SGO convergence is discussed from numerical data. Figure 8 presents the comparison of the MER line integration around the SPP (EMERSPP) and the SGO for different ellipsoids. The study is based in the variation of the scattering surface parameters R1, R2 and R3. The integration path is Γ′ and the radio ρ′ is small enough to obtain good convergence. The accuracy is in direct relation with the radii of curvature ρ1r and ρ2r of the reflected ray tube. Errors are notable if the ray tube is astigmatic as it was suggested by Albani et al. [2006], where the discontinuity on the shadow boundary is removing by introducing an astigmatic correction factor. In the figure, the astigmatic ratio defined by ρ1r/ρ2r is also plotted for each observation angles. High accuracy between EMERSPP and SGO is observed for the cases when the ratio ρ1r/ρ2r is close to unity reproducing a spherical reflected ray tube. From certain angle toward normal aspects for the incident reflected rays at the specular point the reconstruction of the reflected wave-front is accurate and it is explained as follows:

[25] For the incident wave from a dipole, the principal radii of curvatures ρ1r and ρ2r of the astigmatic reflected ray tubes are calculated as:

where ri is the distance from the source to SPP and f1,2 are the focal distances:

the plus sign is associated with and the minus sign with while 1 and 2 are the angles between the direction of incidence and the principal directions of the surface, a1 and a2 are the surface radii of curvature as are defined by McNamara et al. [1990]. For the source in the plane of R1 and R3, with θ2 = , θ1 = + θi (Figure 8) and the corresponding curvatures of the spherical reflected ray tube, the necessary condition for the convergence of the MER line integration to the SGO is found to be:

Expression (20) describes the relation between the incident angle at SPP and the geometry of the scattering surface for which the reflected ray tube becomes spherical. With the knowledge of the local scattering surface radii of curvature at SPP, those incident angles θi for which EMERSPP is almost perfectly approaching to SGO are calculated by (20).

[26] On the basis of results in Figure 8, numerical calculation of (20) is shown in Figure 9. The radii of the ellipsoid R1 and R3 are constant while R2 varies. The source is a dipole and it is located at 5 λ from the SPP (the top of the ellipsoids). The angle of incidence in (20) for the agreement EMERSPP and SGO is plotted by the solid line for different radii of curvature. The angles for which the agreement are observed in Figure 8 are depicted into Figure 9 by A–E. The angle θi with a1 and a2 as implicit function of the scattering surface parameters R1, R2 and R3, are correctly predicted in Figure 9 according to variations of R2.

## 6. Conclusions

[27] The accuracy and the applicability of the SGO extraction in terms of the MER line integration was numerically investigated for smooth surfaces with different radii of curvatures. A new criterion has been introduced, which satisfies the convergence of the MER line integrations around SPP to SGO. The calculation of various EECs line integrations around SPP have also been presented; only MER provides the stable limiting values approaching to SGO. The MER as one of the EECs has probed to be an alternative way to the stationary phase method and the classical geometrical optics for calculating the SGO contributions. In addition, this numerical result and the criterion given, indirectly identifies the entity of MER line integration along the periphery of the illuminated region as the diffraction only, irrespective of the position of observer.