Nov 6, 1979 - B-10-4, Sendai. Aug. 1978. An Efficient Approach for Computing the Geometrical optics. Field Reflected from a Numerically Specified Surface.
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equations obtained by using the zeroth- and first-order ( m = 0 and m = 1) EBC equations as additional ones. The side width W of the square cylinder normalized by l/ko is chosen as the abscissa. In Figs. 2 and 4 the dotted curves coincide with the dashed ( [ I , = 0.707 h where h is the onesnearthefirstresonance wavelength),andthedashedcurvescoincidewiththesolid ones near the second resonance( W = 1.12 X).
v. DISCUSSION We see from Figs. 1-4 that the unstable conditions occur within a relativelynarrowfrequency-spectrumwidthinthe case of a dielectric cylinder (which is also true in the case of a conducting cylinder with TM excitation). However, it must also be noted that their ranges are not points but finite width, so that we should not use (2), or (3) and (4) without any care. Of course, there will be no problems when solutions are not sought in the near-regions of resonances, and the unstable frequency-spectrum width will be expected to shrink narrower as the precision of numerical calculation becomes higher. If the EBC equations (9) or (10) are properly used in addition to the surface equations( 2 ) , or (3) and (4), the acceptable solutions can be obtained even in the parameter ranges near resonances as shown by the dashed and dotted curves in the figures. However, it must be noted that near the second reso( m = 0) nance ( W = 1.12 X) use of onlythezeroth-order equation is not adequate to eliminate resonant solutions. This can easily beunderstood if we recognizethatthe m = 0 equation in (9) or (10)is just the same as that of (7) or ( 8 ) and that in the example treated here the coordinate originis just on a nodalline of a secondresonant TM modeintheconductingsquarehollowcylinder. On theotherhand, if the first-order ( m = 1 ) equation is used as well as the zeroth one, theacceptablesolutionscanbeobtained.Sincethehigher of fieldwith orderequations of EBC restrictthecondition angular variation (such as exp O‘ma))of the cylindrical coordinate, the situation never occurs such that the added higher order equation does not work at all to remove resonant solutions. In thissensethe m = 0 equation is the rather special no angular variation.) Furthercase. (Only this equation has more, the role of higher order equations decreases as the order m increases, which is a merit of the EBC method (while in the earlier method the role of each interior point equation does so clearly). Therefore, we will be not differ from the others able t o say that only a few lower order equations suffice for our purpose of removingresonantsolutionsthroughoutthe whole parameter range. Finally, we should remember that if we don’t want to invoke the leastsquares solution to the overdetermined equations, we could remove the same number of equations as those of the added EBC equations from among theequationswhoseobservationpointsareonthesurface boundary [ 31. ACKNOWLEDGMENT The author wishes to thank Prof.N. Kumagai of Osaka University for his continued encouragement.
REFERENCES [I] K. K.Mei and J. Van Bladel, “Low-frequency scatteringby rectangular cylinders.” IEEE Trans. Antennas Propagat.. vol. A p - I I . no. I . pp. 52-56, Jan. 1963. 121 “Scattering by perfectly-conducting rectangularcylindres.”IEEE Trans. Anfennus Propagar.,vol. AP-I I , no. 2, pp. 1 8 5 192, Mar. 1963.
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R. Mittra and C. A.Klein,“Stabilityandconvergenceofmoment method solutions,” in Numerical andAsympfotic Techniques in Ekcfromagnetics. R. Mittra, E d . Berlin, NewYork:Springer-Verlag.1975, ch. 5. T. K. Wu and L. L. Tsai. “Numerical analysisof electromagnetic fields in biological tissues,” Proc. IEEE (Letters). vol. 62, no. 8, pp. 11671168. Aug. 1974. K. K.Mei,“Unimomentmethodofsolvingantennaandscattering problems,” IEEE Trans. Antennas Propagat.. vol. AP-22, no. 6, pp. 760-766. Nov. 1974. N. Morita. “Surface integral representations for electromagnetic scattering from dielectriccylinders,” IEEE Trans. Antennas Propagat., vol. 26, no. 2, pp. 261-266. Mar. 1978. “Another method of extending the boundary condition for the problem of scattering by dielectric cylinders,” IEEE Trans. Anfennus Propagar. (Commun.).vol. AP-27, no. 1. pp. 97-99. Jan. 1979. T. K . Wu and L.L. Tsai.”Electromagneticfieldsinducedinside IEEE Trans. Microwave Theory arbitrary cylindersof biological tissue,” Tech.. vol. MlT-25, (Short Papers). pp. 6145, Jan. 1977. -. “Scattering by a r b i t r a r y cross-sectioned layered lossy dielectric cylinders,” IEEE Trans. Antennas Propagar., vol.AP-25.no. 4, pp. 518-524. Jul. 1977. C. Miller, Foundarions of rhe Marhematical Theory of Elecfromagnetic Waves. Berlin, NewYork:Springer-Verlag,1%9. V. V. Solcdukhov and E. N . Vasile’v, “Diffraction of a plane electromagnetic wave by a dielectric cylinder of arbitrary crosssection,” Sov. f h y . - T e c h . P h y s . .vol. 15, no. I , pp. 32-36, July 1970. N. Morita. “Analysis of scattering by a rectangular cylinderby means of integral equation formulation.”Trans. Insf. Elec. Commun. Eng. Japan, vol. 57-B, no. 10. pp. 72-80. Oct. 1974. J. R. Mautz and R. F. Harrington, “If-field, E-field,and combined field solutionsforbodies of revolution.”SyracuseUniv.. Syracuse, NY, Tech. Rep. TR-77-2, Feb.1977. -. “Electromagnetic scatteringfrom a homogeneous body of revolution,” SyracuseUniv.. Syracuse.NY.Tech. Rep. TR-77-10. Nov. 1977. K. Yasuura. Tech. Rep.-lnst.Elec. Eng. Japan. EMT-76-31,Oct. 1976. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE, vol. 53. no. 8. pp. 805-8 12. Aug. 1965. -. “scattering by dielectricobstacles.” AlfaFreq., vol. 38 (Speciale).pp. 348-352, 1969. J . Van Bladel. ElectromagneticFields. New York: McGraw-Hill, 1964, p. 373. in the integralequation N.Morita,“Resonantsolutionsinvolved approach to scattering from dielectric cylinders,”presented at Int. Symp. on Antennas and Propagat.. B-10-4, Sendai. Aug. 1978.
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An Efficient Approach for Computing the Geometrical optics Field Reflected from a Numerically Specified Surface RAJ MITTRA, FELLOW, IEEE, AND ALI RUSHDI*, STUDENT MEMBER, IEEE Abstruct-In the conventional geometrical optical analysis of smooth surfaces it is customary to search for aspecular point on the surface of the scatterer for each givencombination of source and observation points. In many situations only a numerical description of the surface is available rather than an analytical expression which lends itself more readily to a determination of the specular point.The numerically prescribed form for the surface may be locally interpolated each time a specular point is to be calculated. An alternative approach is investigated that circumvents the step of derivingthe specular point and obtains the scattered field in a more
ManuscriptreceivedDecember 10, 1978; revisedApril 13, 1979. This work was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-77-3375, and by the National Aeronautics and SpaceAdministration under Grant NASA-NAS-5-2.5062. The authors are with the Electromagnetics Laboratory, Department OfElectricalEngineering, University of Illinois, Urbana, IL 61801. * The order of the authors is arbitrary.
001 8-926X/79/1100-0871$00.75 0 1979 IEEE
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direct manner. Basically, the method begins by computing the reflected rays off the surface at the points where their coordinates, as well as the partial derivatives (or equivalently,thedirection of the normal), are numerically specified. Next, a cluster of three adjacent rays are chosen to define a “mean ray” and the divergence-factor associated with this mean ray. Finally, the amplitude, phase, and vector direction of the reflected field at a given observation point are derived by associating this point with the nearest mean ray and determining its position relativeto such aray.
SOURCE POINT
Fig. 1.
I. INTRODUCTION
NO. 6, NOVEMBER 1979 T
GIVEN INCIDENT FIELD
Reflected GO field is required at P, but specular point A is not known.
Theproblem of computingthegeometricaloptics (GO) fieldreflectedfromanarbitrarysurfacearisesfrequentlyin the ray optical analysis of scattering from structures such as shapedsubreflectorsin Cassegrainian systems.Theconventional approach [ 1 ] -[ 31 t o GO analysis is based on a search procedure for the specular pointon the surface of the scatterer. For a pair of specified source and observation points, the specular point on a surface given by z = f(x, y ) must be determined via a numerical search such that the following conditions are satisfied [ 1 1 , [ 21 :
-as= o , ax
- --
0
(1.1)
aY
s=lsll+lsgl
Fig. 2. Locating P within a pencil comprising a cluster of three adjacent rays.
(1.2)
where s1 and s2 are the distances from the source point 0 and the observation point P to the specular point A, respectively1 (see Fig. 1). Although seemingly straightforward, the problem of determining the coordinates of (x, y , z ) of A can involve extensive numerical computation for two reasons. First, for nuf ( x , y ) is notexmericallyspecifiedsurfacesthefunction plicitlyknownandmustfirstbeconstructed via atwo-dimensional interpolation technique using, for example, spline functions. Second, the nonlinear equation (1.1) must be solved via an iteration procedure, e.g., a zero search, which can also be computationally involved and hence time-consuming. In this communication we present an alternative approach which circumvents both of the above steps; viz., the surface interpolationandthespecularpointsearchprovidedthatthe slope and the location of the points on the reflecting surface arespecified-asituationthat is commonfornumerically specified surfaces. We introduce a procedure, referred to here as the “launching method,” which employs the following key steps.
1 ) Launchraysfromthepointsource 0 t o all theprescribed grid points of the reflecting surface. Typically, thecoordinates as well as thepartialderivatives(or equivalently,thedirection of thesurfacenormal)are specified at these points. 2) Reflect these incident rays off the surface (Section11). 3) Define a mean ray associated with each pencil comprising a cluster of three adjacent rays (Section111). 4) Findthepencil, if any,withinwhichtheobservation point P falls (see Fig. 2). If P is not within any of the pencils, then it is in the shadow region (SectionIV). 5) Compute the magnitude of the field at P in terms of the divergence factor of the associated pencil (SectionV). 6) Use a quadratic phase approximation to obtain the phase of the field at P in terms of its relative location with re1 Note that vectors are indicated in the text by boldface letters and in the figures by arrows over the letters.
spect to the associated mean ray and to its three bouAding rays (Section VI). 7) Obtain the direction of the vector field at P in terms of thelocation of P withinitsassociatedpencil(Section VII). It will become clear as we develop the material and provide details of the procedure that the launching method does not an interpolarequire either a search for the specular point or tion of the reflector surface which can be rather arbitrary. It relies,instead, on the fundamental aspects of the theory of geometrical optics which allows the field at a given observation point to be expressed in terms of those in the immediate neighborhood via a quadratic interpolation formula based on GO. Since the interpolation formula is analytical rather than numerical, the field computation can be carried out very efficientlywithoutsacrificingtheaccuracy.Insupport of this statement,timeandaccuracycomparisonswiththeconventional method are presented in Section VIII. 11. THE REFLECTED FIELDAT A SPECIFIED REFLECTOR POINT
The first step in our procedureis to launch a bundleof rays from the point source 0 to all of the prescribed grid points on the reflecting surface. The surface normals at these points are presumedknownandastraightforwardapplication of the principles of GO allows us to compute the field at any observation point in space, as long as this point is located on one of the reflected rays. The problem of computing the reflected field when the observation point is not located on the reflected rays is described in this communication. Let 0 be the source point and A be an arbitrary point on the reflector. Also let the incident field atA be given by
E’(A) = Z,H’(A) X i
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where Z, = @ is the free-space wave impedance, f(b) is the primary pattern of the source, and
s'=IA-Ol,
i = (A
-
(2.3)
O)/s',
(2.4)
= unit vector along the incident ray.
Noting that a ray field is a locally plane wave and that the reflector has a smooth locally flat perfectly conducting surface, then by application of Snell's law and the boundary conditions [ 4 ] we get
fi = g
-
2(j
.$)i
(2.5)
Hr(A) = H'(A) - 2(Hi(A)
*
$6
(2.6) E. E,
Er(A) = -Ei(A)
+ 2(Ei(A) 33
(2.7)
where fi is the unit normal to the surface at point A, a n d 2 is the unit vector along the reflected ray. The indices i , r denote incident and reflected fields, respectively. Since the reflected field is a ray field, the E and H fields at any observation point are related by
Er(A) = ZoHr(A) X 2.
Fig. 3.
Illustrating choice of a mean ray.
The reflected mean ray can also be defined in a similar manner by first introducing a set of three points D,, E,, and Gz along the reflected rays
(2.8)
E, = El + L A 2
(3.8)
G2 = G1 + L i 3
(3.9)
111. THE MEAN RAY The key to deriving the interpolation formula that would allow us to compute the fields at an arbitrary point in space not directly located on any of the reflected rays is to introduce the concept of a "mean" ray. We do this in the following manner. Consider a cluster of three rays corresponding t o a set of three adjacent points D, E, and G on the reflector (see Fig. 3). We construct the reflected rays off the surface, along the direction I?,, k z , and k 3 .Without loss of generality, let us assume that
Next we introduce a new set definition
of points D l , E l ,
GI via
the
(3.2)
E, = E
(3.3)
G, = G + (I OEI- I OG 11fi~.
(3.4)
Our motivation for introducing these new points is to construct a triangle DIEIGl of equiphase vertices in the vicinity of DEG as a preliminary t o defining the mean ray. Let MI be the pointof intersection of the mediansof the triangleDl El G I , 1.e.,
The mean ray for the incident pencil is then defined as OM,, and the cluster of three rays corresponding to the points D, E, and G is hereafter associated with this mean ray. An arbitrary point M along the mean rayis given by
-O),
O
