IEICE TRANS. ELECTRON., VOL.E84–C, NO.1 JANUARY 2001

74

PAPER

Mathematical Derivation of Modified Edge Representation for Reduction of Surface Radiation Integral Ken-ichi SAKINA†a) , Regular Member, Suomin CUI† , Nonmember, and Makoto ANDO† , Regular Member

SUMMARY Modified Edge Representation (MER) empirically proposed by one of the authors is the line integral representation for computing surface radiation integrals of diffraction. It has remarkable accuracy in surface to line integral reduction even for sources very close to the scatterer. It also overcomes false and true singularities in equivalent edge currents. This paper gives the mathematical derivation of MER by using Stokes’ theorem; MER is not only asymptotic but also global approximation. It proves remarkable applicability of MER, that is, to smooth curved surface, closely located sources and arbitrary currents which are irrelevant to Maxwell equations. key words: PO surface integral, modi ed edge representation,

reduction of a surface integral, Stokes' theorem

1.

Introduction

Physical optics (PO) [1] is one of the high frequency techniques, in which the total induced currents J are approximated in the sense of geometrical optics (GO). PO currents J P O thus defined are then integrated over the surface to give finite fields everywhere including geometrical boundaries and caustics in focusing systems. PO has been widely applied to the pattern analysis of reflector antennas. In Ufimtsev’s physical theory of diffraction (PTD) [2]–[4], PO currents are improved by adding another component called fringe wave currents JFW . In PO, the scattering fields are obtained by surface integrals of J P O , which are performed numerically in general. Since the reduction of the surface integral to line one provides the great saving of the numerical computation time, it has been investigated for long time by many workers. Intensive works for surface to line integral reduction have been reported in both exact [5]–[10] and asymptotic [11]–[15] manners. The exact reduction theory, derived via Kirchhoff’s surface integral, has inherent drawback in terms of applicability; it is limited to planar scatterer for which PO currents are exact locally and to the incidence satisfying Maxwell equations. On the other hand, the asymptotic reduction theory can be applied to wider class of problems, that is, to curved scatterers and to approximate inciManuscript received March 24, 2000. Manuscript revised July 21, 2000. † The authors are with the Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo, 152-8552 Japan. a) E-mail: [email protected]

dence etc. But most of them rely upon local plane wave approximation and the accuracy of existing expressions degrades rapidly as a source becomes closer to the scatterer. In order to keep the good accuracy we need at least five wave lengths for the distance from the scatterer [16], [17]. It has been reported that modified edge representation (MER) [17], [18], empirically proposed by one of the authors, is the reduction theory with the highest accuracy for the source located very close to the scatterer. This paper mathematically derives MER for the first time, which explains the remarkable accuracy of MER and extracts the conditions for the accuracy. What is significant in the derivation of MER is that Stokes’ theorem is used in order to obtain the reduction of the surface integral without the help of the image theory; it can be applied to curved surfaces, nearby sources and arbitrary incidence irrelevant to Maxwell equations. A surface integral without an inner stationary phase point is directly reduced by MER while that with an stationary phase point inside the scatterer is indirectly evaluated. The latter is replaced by the difference between the integral extending to infinity and that over the complementary surface of the scatterer without stationary phase point. The integration over the infinite surface must be conducted in general, but in most cases of the practical interest we can dispense with numerical integration by utilizing the image source or GO approximation in high frequency. 2.

Modified Edge Representation (MER)

The following is the concept of MER and shows how to calculate scattered fields by MER. MER is the line integral expression for the surface radiated integral over the scatterer S.
kη e−jkro S rˆo × (ˆ ro × J) dS E = j 4π S ro
k e−jkro +j rˆo × M dS, (1) 4π S ro where the time factor ejωt is suppressed. In the above expression of the scattered far field, J is the electric surface current, M is the magnetic surface current, ro is the distance from the integration point to the observer,

SAKINA et al.: MATHEMATICAL DERIVATION OF MODIFIED EDGE REPRESENTATION

75

Fig. 1 rˆo .

Definitions of the modified edge τˆ, unit vectors rˆi and

ro is the unit vector toward the observer as is shown in Fig. 1. Furthermore, η is the intrinsic impedance of the free space and k is the wave number. Firstly, the edge of the scatterer is replaced with the modified edge, say unit vector τˆ, satisfying the diffraction law at each point on the edge as in Fig. 1. These requirements are simply expressed as (ˆ ri + rˆo ) · τˆ = 0, n ˆ · τˆ = 0,

(2) (3)

where rˆi is unit vectors oriented from any point on the edge of the scatterer to the source and n ˆ is the unit vector normal to the scatterer. The direction of vector τˆ is determined by the positions of observer, source and the edge point of interest; it is independent of that of the original edge tˆ, which is generally different from τˆ except the diffraction points. In the scattering problem for the disk shown in Fig. 1, the modified edge τˆ coincides with the original edge tˆ at two diffraction points D1 and D2. In MER as applied to PO ˆ × H i , M = 0), the diffracted fields have been (J = 2n empirically calculated in following steps. ˜ φ) ˜ with (a) A local spherical coordinate system (˜ r, θ, its z-axis being parallel to the modified edge τˆ, is defined at every edge point Q. The coordinate system is referred as the local modified edge one hereafter. (b) The fields H i incident on the scatterer are decomposed into the local modified edge coordinate system as Hθi and Hφi , where only the radiation terms are used in the calculation of equivalent edge currents. (c) The magnitudes of equivalent edge currents at Q on the edge are calculated for the modified edge τˆ. Since the modified edge is so defined as to satisfy the diffraction law at every edge point, we may use nonuniform Keller-type expressions which are valid only for the diffraction points. Then equivalent edge currents [17], [18] are given by   ˜i 2 sin φ i i Hφ tˆ (4) cos θ˜i Hθ − JI = jk sin θ˜i cos φ˜i + cos φ˜o

Fig. 2 Reflection and shadow regions (RB and SB). I: Reflection region, II: Region without stationary phase point, III: Shadow region.

and MI = −

2η sin φ˜o Hθi tˆ jk sin θ˜i (cos φ˜i + cos φ˜o )

(5)

which are given in the Appendix A. Furthermore, the direction of equivalent currents must be taken along the real edge tˆ and not along the modified edge τˆ. (d) The line integration of these currents along the periphery of scatterer provides the diffracted fields. In this paper we will mathematically derive generalized MER which is applicable not only to perfectly electric conducting surface (PEC) but also to that with arbitrary surface impedance. 3.

Mathematical Derivation of MER

3.1 Surface Integration without Inner Stationary Phase Point In this section we investigate the reduction of the surface integral (1) without inner stationary phase points. It corresponds to the case where GO reflection or shadowing points do not fall on the scatterer. In Fig. 2, the observer is in region II and is not in the reflection and/or shadow regions (region I and region III). (i) Phase-fixed curvilinear coordinate system on a curved surface We here define the special curvilinear coordinate system paying special attention to the phase of the integrand in (1). Let S be a smooth curved surface given by the vector equation S: r = r(σ, τ )

(6)

with two parameters σ and τ , and let the edge of S be Γ as is in depicted in Fig. 3. We denote the source, the observer and the arbitrary point on S by Pi , Po and Q, respectively. Defining two distances ri and ro as

IEICE TRANS. ELECTRON., VOL.E84–C, NO.1 JANUARY 2001

76

Fig. 3 τ -curves, σ-curves and the orthonormal frames {ˆ σ, n ˆ , τˆ} on a smooth curved surface S with the boundary Γ.

ri = Pi Q and ro = Po Q,

(7)

we can choose the curvilinear coordinate system (σ, τ ) on S so as to satisfy the following conditions: τ -curves: the family of lines on S for which ri + ro = constant, σ-curves: the family of lines orthogonal to the family of τ -curves. Let us define two unit vectors τˆ(= modified edge) and σ ˆ tangent to τ -curve and σ-curve, respectively. Note that the vector τˆ here coincides with the modified edge satisfying the diffracted law defined in (2). When we take two parameters, τ and σ, to be equal to arc lengths ∂r along τ and σ-curves, respectively, we have τˆ = ∂τ ∂r . Then the orthonormal frame {ˆ σ, n ˆ (= and σ ˆ = ∂σ τˆ × σ ˆ ), τˆ} can be set on S everywhere, as is depicted in Fig. 3. Noting the equations ∂(ri + ro ) = −(ˆ ri + rˆo ) · τˆ = 0, ∂τ ∂(ri + ro ) = −(ˆ ri + rˆo ) · σ ˆ ∂σ

(8)

We have the expression for the gradient of (ri + ro ) in the curvilinear coordinate system ∂(ri + ro ) ∂(ri + ro ) ∂(ri + ro ) ˜ i + ro ) = σ +n ˆ + τˆ ˆ ∇(r ∂σ ∂n ∂τ ∂(ri + ro ) = −(ˆ ri + rˆo ) · σ ˆσ ˆ+n ˆ ∂n ≡ ∇(ri + ro ) (9) where ∂ ∂ ∂ ˜ ≡σ ∇ ˆ +n ˆ + τˆ and ∂σ ∂n ∂τ ∂ ∂ ∂ + yˆ + zˆ . ∇≡x ˆ ∂x ∂y ∂z (ii) Assumption of surface currents We rewrite surface currents J and M as J = kJ o e−jkri and M = kM o e−jkri . It is assumed that J o and M o

are independent of k; this implies that the incident fields on the scatterer consist of only radiation terms with the k dependence of ke−jkri (see Appendix A). (iii) Derivation of line integral representation Substituting J and M into (1), we obtain the expression for the scattered field in the far region as follows:
k2 η e−jk(ri +ro ) s E =j rˆo × (ˆ ro × J o ) dS 4π S ro
k2 e−jk(ri +ro ) +j rˆo × M o dS. (10) 4π S ro ro × J o ) and rˆo × M o into We first decompose rˆo × (ˆ the sums of two parts ro × J o ) = ζ1 rˆo × (ˆ ro × τˆ) + ζ2 rˆo × τˆ, rˆo × (ˆ  rˆo × M o = ζ1 rˆo × (ˆ ro × τˆ) + ζ2 rˆo × τˆ.

(11) (12)

where ro × J o )} · τˆ {ˆ ro × (ˆ (ˆ ro × J o ) · τˆ , ζ2 = . (13) 1 − (ˆ ro · τˆ)2 1 − (ˆ r0 · τˆ)2 ro × M o )} · τˆ (ˆ ro × M o ) · τˆ {ˆ ro × (ˆ ζ1 = − , ζ2 = − 1 − (ˆ ro · τˆ)2 1 − (ˆ ro · τˆ)2 (14)

ζ1 = −

Since the observer is in the far field region for which rˆo is constant over S, (10) can be written as ro × A) + jkηˆ ro × B ro × (ˆ Es ∼ = jkηˆ  + jkˆ ro × (ˆ ro × A ) + jkˆ ro × B 

(15)

where 

   A
ς1 B  ς2  −jk(r +r ) i o   = k    τˆe dS. A  4πro   S ς1   ς2 B

(16)

This is the original surface integration expression of E S in (1) for given surface currents J o , M o . Our goal is to express E S in terms of line integration of equivalent edge currents J I , M I in (4) and (5). Using (9), we have the expression ˜ −jk(ri +ro ) × n ∇e ˜ = jke−jk(ri +ro ) (rˆi + rˆo ) · σ ˆ τˆ. (17) This leads us to the following useful relation; ζ1 τˆe−jk(ri +ro ) ζ1 ˜ −jk(ri +ro ) × n ∇e = ˆ jk(ˆ ri + rˆo ) · σ ˆ

 −jk(ri +ro ) ˜ ζ1 e =∇ ×n ˆ jk(ˆ ri + rˆo ) · σ ˆ 

ζ1 ˜ ˆ. e−jk(ri +ro ) × n −∇ jk(ˆ ri + rˆo ) · σ ˆ

(18)

SAKINA et al.: MATHEMATICAL DERIVATION OF MODIFIED EDGE REPRESENTATION

77

Then A in (16) can be decomposed into the sum of two integrals as follows:


ζ1 e−jk(ri +ro ) k A= ∇ ×n ˆ dS 4πro S jk(ˆ ri + rˆo ) · σ ˆ
k e−jk(ri +ro ) n ˆ + 4πro S

 ζ1 ×∇ dS. (19) jk(ˆ ri + rˆo ) · σ ˆ

and maximum values for the extension of the scatterer surface in (σ, τ )-coordinate system. We finally have {ˆ ro × (ˆ ro × J o )} · τˆ 1 ∼ e−jk(ri +ro ) tˆdl A = j4πro Γ (1 − (ˆ ro · τˆ)2 )(ˆ ri + rˆo ) · σ ˆ 1 e−jk(ri +to ) n ˆ + 4πro k Γ (ˆ ri + rˆo ) · σ ˆ

 {ˆ ro × (ˆ ∂τ ro × J o )} · τˆ ×∇ dl. (24) 2 (1 − (ˆ ro · τˆ) )(ˆ ri + rˆo ) · σ ˆ ∂l

Since the stationary phase points are outside of S, we ˆ = 0 on S. Then Stokes’ theorem of have (ˆ ri + rˆo ) · σ the vector form below is applied to the first integral in (19).
∇f × n ˆ dS = − f tˆdl, (20)

In the above expression, the order of the second term with respect to k is higher than that of the first term by k−1 provided that J o is independent of k. We therefore obtain the approximate expression of A (B, A and B  in similar manner) for the lowest order as follows:     A JI B   M I  −jk(r +r ) 1 i o   ∼   dl, (25) A  = 4πro  JM  e Γ MM B

S

Γ

where l denotes the length of the edge Γ and tˆ is the unit tangent vector of Γ. Then we obtain 1 ζ1 e−jk(ri +ro ) ˆ A= − tdl 4πro Γ j(ˆ r + rˆo ) · σ ˆ
i 1 + e−jk(ri +ro ) n ˆ j4πro S

 ζ1 ×∇ dS (21) (ˆ ri + rˆo ) · σ ˆ Now we will show the second term is asymptotically of the higher order. Firstly, a special case is given to suggest it. When the source is far away from the ˆ and ζ1 are scatterer and the surface is planar, rˆi , rˆo , σ constant over S and the second term in (21) vanishes exactly since

 ζ1 ∇ = O. (22) (ˆ ri + rˆo ) · σ ˆ We now return to more general case where the source is at arbitrary distance and the scatterer is curved. The asymptotic treatment [19] should be applied to the second term in (21); applying the stationary phase method with respect to σ, we get


ζ1 1 −jk(ri +ro ) e n ˆ×∇ dS j4πro S (ˆ ri + rˆo ) · σ ˆ
τ max
σ(τ ) max 1 dτ e−jk(ri +ro ) n ˆ = j4πro τ min σ(τ ) min

 ζ1 ×∇ dσ (ˆ ri + rˆo ) · σ ˆ 1 1 e−jk(ri +ro ) n ˆ ≈ − 4πro k Γ (ˆ ri + rˆo ) · σ ˆ

 ζ1 ∂τ dl. ×∇ (ˆ ri + rˆo ) · σ ˆ ∂l (23) where σmin , σmax , τmin and τmax stand for minimum

where the electric and magnetic equivalent edge currents J I , M I , J M and M M are given by 



Jo

 · τˆ rˆo × rˆo × Mo JI (26) = MM j(1 − (ˆ ro · τˆ)2 )(ˆ ri + rˆo ) · σ ˆ and



 −J o

 · τˆ rˆo × Mo MI tˆ. = JM j(1 − (ˆ ro · τˆ)2 )(ˆ ri + rˆo ) · σ ˆ

(27)

This is the goal of our derivation; E S is expressed in terms of line integration. Now the diffracted electric fields can be calculated from (15) together with (25),(26) and (27). In the local modified edge coordinate system regarding τ -axis as z-axis at each point on the edge, the equivalent currents J I and M I in (26) and (27) coincide with (4) and (5), empirically proposed before [17], [18] (see Appendix B). (iv) Unique aspects of MER based upon Stokes’ theorem Although MER utilizes the asymptotic theory in part for omitting the second term in (24), the use of global tool of Stokes’ theorem provides us many unique advantages over the methods proposed before [5]–[15]. Unique aspects of MER reveal themselves as follows: (a) Equivalent edge currents in MER are flowing not along the modified edge τˆ but along the actual edge tˆ of the scatterer though they are derived in the midified edge coordinate systems. (b) Since the equivalent edge currents are derived by using Stokes’ theorem which is valid for curved surfaces, it seems natural that MER could be applied not only to planar but also smoothly curved scatterers.

IEICE TRANS. ELECTRON., VOL.E84–C, NO.1 JANUARY 2001

78

(c) The necessary conditions imposed upon incident fields or the surface currents J and M are relatively loose; local plane wave approximation is not used. MER therefore has the potential for the source considerably close to the scatterer. (d) MER can be applied not only to PO for PEC or arbitrary surface impedance, but also for aperture field integration methods[20], which are related to Maxwell equations. Furthermore, since derivation is irrelevant to Maxwell equations, the surface currents J and M do not have to be associated with real EM fields. Integration of non-physical and mathematically defined currents can also be evaluated by MER. 3.2 Surface Integration with Stationary Phase Points We here consider the surface integral (1) with stationary phase points inside the scatterer S. Observers in regions I and III in Fig. 2 correspond to these cases. The stationary phase points satisfying the equation ˆ = 0 exist inside of S. In this case, Stokes’ (ˆ ri + rˆo ) · σ theorem can not be applied to the surface S directly; alternatively, it should be applied to the complementary surface S¯ (≡ infinite surface S∞ − S), smoothly connected to S, as in Fig. 2. The discussion in 3.1 is ˆ 0 is satisfied there. now repeated since (ˆ ri + rˆo ) · σ = The scattering fields from S is given by the difference between integration over S∞ and the complimentary ¯ The edge diffracted fields yielded from S¯ are surface S. calculated by MER, whose result has the inverse sign as those from S, while the contribution from S∞ have to be evaluated numerically except the special cases. Some special but still practically familiar cases are discussed analytically for the evaluation of S∞ . If the surface is planar with surface impedance and the surface currents are defined as PO (unperturbed) part of the solution of Maxwell equations, the integration over S∞ reduces to simple GO reflection or shadowing of direct wave depending upon the observer position. The generalized PO surface currents on S are given by ˆ (28) J = (1 + α)ˆ n × H i and M = (1 − α)E i × n where α is arbitrary complex number. Special surfaces of α = 1, α = 0 and α = −1 correspond to PEC, matched (reflection-less) sheet and perfectly magnetic conductor (PMC), respectively. The field equivalence principle [21], [22] tells us that the surface radiation integral (1) of the currents in (28) over S∞ reduces to geometrical optics (GO) waves as follows.
kη e−jkro GO ≡j rˆo × (ˆ ro × J ) dS E 4π S∞ ro
e−jkro k rˆo × M dS +j 4π S∞ ro

Fig. 4

Geometry for scattering from a flat disk.

=



αE i −E i

P ∈ reflection region P ∈ shadow region

(29)

Consequently, E GO should be added to diffracted fields computed by MER if stationary phase points exist inside S. It is noted that this discussion is only true for PO calculated with J, M related to Maxwell equations. As another special case, the integration over curved or planar S∞ in high frequency, the method of stationary phase provides the reasonable approximation in the sense of GO, which will also be denoted by E GO . The special treatment above for S∞ are utilized implicitly for the observer in reflection and/or shadow regions I and III in the next section. 4.

Numerical Results

4.1 PO Scattering from Perfectly Electric Conductor (α = 1) The unique feature of (c) in Sect. 3 is highlighted. For PEC, the surface currents on the scatterer is given by J = 2ˆ n × H i and M = 0, that is, α = 1 in (28). Several numerical results of MER for PO have been already reported [17], [18]. We here show PO diffraction or scattering patterns from the disk, the rectangular plate and the parabola. The effectiveness of MER for the small distance to the source as well as the small dimensions of the scatterer is demonstrated first. Accuracy check of (15) together with (25) is demonstrated for the electric dipole wave diffracted from a flat disk as depicted in Fig. 4. Figure 5 shows the diffraction patterns for the electric dipole located on z-axis and extremely close to the scatterer at 0.1λ (λ being the wavelength). Its moment vector points to x-axis. In this case, PO-MEC1 [16] and PO-MEC2 [14] suffer from serious errors while only MER provides accurate results. It is reminded that the incidence used in MER includes only radiation term of the electric dipole fields while all the terms are included in PO surface integration (see Appendix A). Figure 6 is the geometry of the scattering of the electric dipole wave from a rectangular plate of dimensions La by Lb . Figure 7 shows the scattering

SAKINA et al.: MATHEMATICAL DERIVATION OF MODIFIED EDGE REPRESENTATION

79

(a) Eθ (φo = 0◦ ) Fig. 7 Scattered field patterns in φo = 45◦ -plane by small rectangular plate in PO. The dipole source is very close to the edge. (A plate of dimensions 0.2λ by 0.3λ, α = 1, dipole position (0.1λ, 0, 0.01λ), dipole moment (1,1,1).)

(b) Eφ (φo = 90◦ ) Fig. 5 Accuracy check of MER in diffraction from the flat disk, (a = 3λ, α = 1, source position (0,0,0.1)λ). POsurf: PO-Surface integral, PO-MEC1: The authors [16], PO-MEC2: Michaeli [14].

Fig. 6

Fig. 8

Geometry for scattering from the parabolic reflector.

the mathematical background of Stokes’ theorem behind MER. Next, the accuracy check for curved surface in terms of (b) is also conducted for the parabolic reflector in Fig. 8, in which the antenna F/D = 0.25. This geometry is a typical one demonstrating the superiority of the method of equivalent edge currents, since the diffraction points correspond to null of illumination and the simple two-point approximation such as GTD breaks down. From the Fig. 9, we can observe MER keeps the high accuracy even for curved surface and gives much more excellent approximations than the previous equivalent currents.

Geometry for scattering from the rectangular plate.

4.2 PO Scattering from Scatterer with Surface Impedance patterns from the plate of dimensions 0.2λ by 0.3λ. The dipole source with the moment (1,1,1) is located very close to the edge at (0.1λ, 0, 0.01λ). Almost perfect agreement is observed again for such an extreme example. MER maintains remarkable accuracy even for PO surface integral reduction for sources within 0.01 wavelength from the scatterer. These results suggest

One of the unique features in (d) is demonstrated next; MER is applied to arbitrary surface currents. Figure 10 shows the scattering patterns of the electric dipole wave from the square plate of dimensions 4λ by 4λ with the surface currents (28) for α = −1 or α = 0. The source with the moment (1,1,1) is located on (λ, λ, 2λ) as in

IEICE TRANS. ELECTRON., VOL.E84–C, NO.1 JANUARY 2001

80

(a) Eθ (φo = 0◦ )

Fig. 11 Scattered field patterns in φo = 45◦ -plane by square plate when the higher orders are involved in the incident fields in PO. (A plate of dimensions 4λ by 4λ, α = 1, dipole position (2λ, 0, 0.1λ), dipole moment (1,1,1).)

(b) Eφ (φo = 90◦ ) Fig. 9 Accuracy check of MER in diffraction from the parabolic reflector. (radius = 10λ, α = 1, source position (0,0,5)λ). POsurf: PO-Surface integral, PO-MEC1: The authors [16].

Fig. 10 Scattered field patterns in φo = 0◦ -plane by square plate for α = −1 and α = 0. (A plate of dimensions 4λ by 4λ, dipole position (λ, λ, 2λ), dipole moment (1,1,1).)

Fig. 6. MER is showing fine agreement with POexact almost everywhere though for α = 0 (matching surface) the field strength in the reflection region is very low and degradations are observable.

4.3 Effects of Higher Order Terms of the Incidence MER is derived by using the Stokes’ theorem as well as asymptotic technique on the assumption that the induced currents has the k dependence of ke−jkri which correspond to radiation terms as was mentioned in the feature (c). It is suggested that the inclusion of higher order terms into the incidence in constructing the equivalent edge currents in MER even degrades the agreement between the surface and line integral expressions. On the other hand, for observer in region I, the evaluation of S∞ based upon GO is only valid for the incidence with the higher order terms which satisfies the Maxwell equations. This situation is explained numerically. Figure 11 compares two types of incidence used in MER for the dipole wave scattering by the square plate; one incidence consists of radiation terms only while the other is exact and with all the higher order terms with respect to k. The observer is in region I with stationary phase points on the scatterer. The serious degradation is observed for MER with the higher order incidence. On the contrary, PO surface integration must use all the higher order terms and if not for them, PO is seriously deviating from original PO. It is numerically shown in this case that MER with the radiation terms only and PO with the exact incidence agrees almost perfectly. Figure 12 conducts the similar comparison for two typical observation points in regions I and II, as functions of frequency. For region I where evaluation of S∞ is necessary, PO with radiation term only suffers the large error while the MER with higher order terms deviates as the frequency becomes very low. PO with exact incidence and MER are in

SAKINA et al.: MATHEMATICAL DERIVATION OF MODIFIED EDGE REPRESENTATION

81

Fig. 12 Diffracted field patterns in φo = 0◦ -plane by square plate as functions of frequency. (A plate of dimension 4λ by 4λ, dipole position (2,2,5), dipole moment (1,1,1).)

Fig. 13 Accuracy of MER for the artificial integrand rˆo × x ˆ(x+ 1)(y + 1)e−jk(ri +ro ) . (A plate of dimensions 2λ by 2λ, source position (λ, λ, 5λ), φo = 45◦ -plane.)



(x + 1)(y + 1)(ˆ r × (ˆ r×x ˆ)) · τˆ 2) jkˆ r · σ ˆ (1 − (ˆ r · τ ˆ ) Γ · e−jk(ri +ro ) tˆdl + O(k−2 ),

+ rˆ × good agreement. For region II, on the other hand, PO with radiation term only also gives good agreement. The above results suggest the important advantage of MER in practical application; MER is more reliable than PO if only the radiation fields are known numerically or experimentally for the source, as is often the case with practical antennas such as horns with subreflectors in Cassegrain antennas, microstrip antennas with ground planes, mobile phone antennas with handset etc. 4.4 Non-Physical and Mathematical Distribution of Surface Currents Important flexibility of MER mentioned in (d) is that the currents J and M in the integrand are not necessarily satisfying Maxwell equations. To demonstrate this, we consider the surface integral for the plate S of dimensions 2λ by 2λ with the currents irrelevant to Maxwell equations as follows.
rˆo × x ˆ(x + 1)(y + 1)e−jk(ri +ro ) dS, (30) I= S

where (x, y) is Cartesian coordinate system on S. Note that the incidence associated with (30) satisfies the assumption of (ii) in 3.1 but is irrelevant to Maxwell equation. Applying the method proposed in this paper to (30), we have the reduced line integral representation as   −jk(ri +ro ) j2π e Λ ˆ I ∼ = −    rˆ × x k f f − f 2  xx yy



+ rˆ× rˆ× Γ

xy

(31)

where f (x, y) = ri + ro , QS is the stationary phase point of e−jkf (x,y) and

1 if QS is inside S Λ= (32) 0 if QS is outside S The source position or the origin of ri is at (λ, λ, 5λ). A critical observation plane of φo = 45◦ is selected where the edge stationary phase points fall on the corner. Figure 13 shows the numerical results as to (30) and (31). In reflection region, the integration over S∞ is calculated by the method of stationary phase [23]. The almost perfect agreement is observed except around θo = 20◦ and 160◦ , that is, the critical situation for which the stationary phase point of (30) traverses the scatterer corner. 5.

Conclusion

This paper provides mathematical foundation of MER for the first time. This is accomplished through the application of Stokes’ theorem to the radiation integral without using the image theory and the field equivalence principle. Various advantages of MER over conventional surface-to-line integral reduction methods are explained mathematically. MER is applicable to the source close to the scatterer, to the smooth curved surfaces and to arbitrary currents irrelevant to Maxewell equations. The remarkable accuracy of MER is numerically confirmed for various examples. References

QS

(x + 1)(y + 1)(ˆ r×x ˆ) · τˆ −jk(ri +ro ) ˆ e tdl jkˆ r·σ ˆ (1 − (ˆ r · τˆ)2 )

[1] S. Silver, Microwave antenna theory and design, pp.148– 158, Dover, 1947.

IEICE TRANS. ELECTRON., VOL.E84–C, NO.1 JANUARY 2001

82

[2] P.Y. Ufimtsev, “Method of edge waves in the physical theory of diffraction,” Prepared by the U.S. Air Force Foreign Technology Division Wright Patterson AFB, OHIO, 1971. [3] M. Ando, “Radiation pattern analysis of reflector antennas,” IEICE Trans., vol.J67-B, no.8, pp.853–860, Aug. 1984. [4] A. Michaeli, “Elimination of infinities in equivalent edge currents, Pt.1: Fringe current components,” IEEE Trans. Antennas & Propag., vol.AP-34, no.7, pp.912–918, July 1986. [5] F. Kottler, “Diffraction at a black screen, Part II: Electromagnetic theory,” Progress in Optics, vol.6, pp.331–377, 1967. [6] J.S. Asvestas, “The physical optics fields of an aperture on a perfectly conducting screen in terms of line integrals,” IEEE Trans. Antennas & Propag, vol.AP-34, no.9, pp.1155–1159, Sept. 1986. [7] J.S. Asvestas, “Line integrals and physical optics. Part I: The transformation of the solid-angle surface integral to a line integral,” J. Opt. Soc. Am. A., vol.2, no.6, pp.891–895, June 1985. [8] J.S. Asvestas, “Line integrals and physical optics. Part II: The conversion of the Kirchhoff surface integral to a line integral,” J. Opt. Soc. Am. A., vol.2, no.6, pp.896–902, June 1985. [9] P.M. Johansen and O. Breinbjerg, “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., vol.43, no.7, pp.689–696, July 1995. [10] E. Martini, G. Pelosi, and G. Toso, “The physical optics fields of a penetrable planar structure in terms of line integrals,” JINA 98-Nice-International Symposium on Antennas, pp.218–221, France, Nov. 1998. [11] W.V.T. Rusch, “Physical-optics diffraction coefficients for a parabolic,” Electron. Lett., vol.10, no.17, pp.358–360, Aug. 1974. [12] C.H. Knopp, “An extension of Rusch’s asymptotic physical optics diffraction theory of a paraboloid antenna,” IEEE Trans. Antennas & Propag., vol.AP-23, no.5, pp.741–743, Sept. 1975. [13] M. Safak, “Calculation of radiation patterns of paraboloidal reflectors by high-frequency asymptotic techniques,” Electron. Lett., vol.12, pp.229–231, April 1976. [14] A. Michaeli, “Eimination of infinities in equivalent edge currents, Pt.2: Physical optics components,” IEEE Trans. Antennas & Propag., vol.AP-34, pp.1034–1037, Aug. 1986. [15] O. Breinbjerg, Y. Rahmat-Sami, and J. Appel-Hansen, “A theoretical examination of the physical theory of diffraction and related equivalent currents,” Technical Report of Electromagnetics, Institute of Technical University Denmark, May 1987. [16] M. Ando, T. Murasaki, and T. Kinoshita, “Elimination of false singularities in GTD equivalent edge currents,” IEE Proc., Part H., vol.138, no.4, pp.289–296, Aug. 1991. [17] T. Murasaki and M. Ando, “Equivalent edge currents by the modified edge representation: Physical optics components,” IEICE Trans. Electron., vol.E75-C, no.5, pp.617–626, May 1992. [18] T. Murasaki, M. Sato, Y. Inasawa, and M. Ando, “Equivalent edge currents for modified edge representation of flat plates: Fringe wave components,” IEICE Trans. Electron., vol.E76-C, no.9, pp.1412–1419, Sept. 1993. [19] L.B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, p.387, IEEE Press, 1994. [20] A.D. Yaghjian, “Equivalence of surface current and aperture field integrations for reflector antenna,” IEEE Trans.

Antennas & Propagat., vol.AP-32, pp.1355–1358, Dec. 1984. [21] J.A. Stratton, Electromagnetic Theory, pp.464–468, McGraw-Hill, 1941. [22] P.Y. Ufimtsev, “Diffraction of electromagnetic waves at blackbodies and semitransparent plates,” Radiophysics and Quantum Electronics, vol.11, no.6, pp.527–538, 1968. [23] M. Born and E. Wolf, Principles of Optics, Appendix III, Pergamon, 1980.

Appendix A The electric dipole with dipole moment p radiates the electric and magnetic fields 

 e−jkri 3 3 Ei = kη −1 − + 2 2 p · rˆi rˆi j4πri jkri k ri    1 1 + 1+ − 2 2 p (A· 1) jkri k ri and Hi =

  e−jkri 1 k 1+ p × rˆi , j4πri jkri

(A· 2)

where rˆi is the unit vector toward the dipole source from the scatterer [9]. The first terms in (A· 1) and (A· 2) are the radiation terms while the remaining terms are the higher order ones. In this case the following relation holds for the radiation terms. ri × H i E i = ηˆ

(A· 3)

Appendix B ˜ φ) ˜ in When we define the spherical coordinates (˜ r, θ, the orthonormal frame {ˆ σ, n ˆ , τˆ} on, the unit vectors rˆi and rˆo are rˆi ≡ (sin θ˜i cos φ˜i , sin θ˜i sin φ˜i , cos θ˜i ) rˆo ≡ (sin θ˜o cos φ˜o , sin θ˜o sin φ˜o , cos θ˜o )

(A· 4) (A· 5)

where the relation θ˜o = π − θ˜i holds at arbitrary point on S, and subscripts i and o denote quantities with respect to the source and observer, respectively. The incident fields in the orthonormal frame are expressed as E i = (Eσi , Eni , Eτi ) = (cos θ˜i cos φ˜i Eφi − sin φ˜i Eφi , cos θ˜i sin φ˜i Eθi + cos φ˜i E i , − sin θ˜i E i ) (A· 6) φ

θ

and H i = (Hσi , Hni , Hτi ) = (cos θ˜i cos φ˜i Hφi − sin φ˜i Hφi , cos θ˜i sin φ˜i Hθi + cos φ˜i H i , − sin θ˜i H i ) (A· 7) φ

θ

Noting σ ˆ = (1, 0, 0) n ˆ = (0, 1, 0), τˆ = (0, 0, 1), J o =

SAKINA et al.: MATHEMATICAL DERIVATION OF MODIFIED EDGE REPRESENTATION

83

2ˆ n × H i and M o = 0, we have the following relations n × H i )} · τˆ = 2 sin2 θ˜i sin φ˜o Hθi , 2{ˆ ro × (ˆ ro × (˜ n × H i ))} · τˆ 2{ˆ ro × (ˆ ˜ i = 2 sin2 θ˜i cos θ˜i (cos φ˜i + cos φ)H

(A· 8)

− 2 sin2 θ˜i sin φ˜i Hφi , ˆ = sin θ˜i (cos φ˜i + cos φ˜o ) (ˆ ri + rˆo ) · σ

(A· 9) (A· 10)

rˆo · τˆ = − cos θ˜i

(A· 11)

θ

and

Then Eqs. (26) and (27) reduce to the equivalent currents in the local modified edge coordinate system as (4) and (5) [17].

Ken-ichi Sakina was born in Hokkaido, Japan, on October 12, 1949. He received the B.S. degree from Kanto Gakuin University in 1972 and the M.S. degree from Chiba University in 1975, in electrical engineering. He studied General Relativity and Spinor in space-time as a Graduate Research Student from 1976 to 1980 at Tohoku University. Since 1998, he has been pursuing the D.E. degree at Tokyo Institute of Technology. His research interests include the scattering of electromagnetic wave and electromagnetic theory.

Suomin Cui was born in Shaanxi, China, on November 11, 1967. He received the B.S. degree in physics from Shaanxi Normal University in 1989, the M.S. degree in radio physics in 1992, and D.E. degree in electrical engineering in 1995 from Xidian University, China, respectively. From 1995 to 1997, he worked in Nanjing University of Science and Technology as a postdoctoral fellow, and he was appointed as associate professor. Currently, he is postdoctoral fellow of the Japan Society for the Promotion of Science. His primary fields of interest include the high frequency diffraction analysis, computational electromagnetic and RCS computation of complex objects.

Makoto Ando was born in Hokkaido, Japan, on February 16, 1952. He received the B.S., M.S. and D.E. degrees in electrical engineering from Tokyo Institute of Technology, Tokyo, Japan in 1974, 1976, 1979, respectively. From 1979 to 1983, he worked at Yokosuka Electrical Communication Laboratory, NTT, and was engaged in development of antennas for satellite communication. He was a Research Associate at Tokyo Institute of Technology from 1983 to 1985, and is currently a Professor there. His main interests have included high-frequency diffraction theory such as physical optics and geometrical theory of diffraction, the design of reflector antennas and waveguide planar arrays for direct broadcast from satellite (DBS) and very small aperture terminal (VSAT), and the design of high-gain millimeter-wave antennas. Dr. Ando received the Young Engineers Award of IEICE Japan in 1981 and the Achievement Award and the Paper Award from IEICE Japan in 1993. He also received the fifth Telecom Systems Award in 1990 and the eighth Inoue Prize for Science in 1992.