IEEE TRhTSACTIONS ON AXTEXXAS AND PROPAGATION VOL. AP-14, so. 3 MAY, 1Ybb V. CONCLUSION REFERENCES The characteristics of the waves guided along a plane [I] P. S. Epstein,“On the possibility of electromagneticsurface waves,” Proc. Nat’l dcad. Sciences, vol. 40, pp. 1158-1165, De- interface which separates a semi-infinite region of free cember 1954. [2] T. Tamir and A. A. Oliner, “The spectrum of electromagnetic space from that of a magnetoionic medium are investi- waves guided by a plasma layer,” Proc. IEEE, vol. 51, pp. 317- gated for the case in which the static magnetic field is 332, February 1963. orientedperpendicular to the plane interface. It is [3] &I. A. Gintsburg, “Surface waves on the boundary of a plasma in magnetica field,” Rasprost. Radwvoln i Ionosf., Trudy found that surface waves exist only when w,<wp and NIZMIRAN L’SSR, no. 17(27), pp. 208-215, 1960. that also only for angular frequencieswhich lie bet\\-een [4] S. R. Seshadri and A. Hessel, “Radiation from a source near a plane interface between an isotropic and a gyrotropic dielectric,” weand 1/42times the upper hybrid resonant frequency. Canad. J. Phys., vol. 42, pp. 2153-2172, November 1964. [5] G. H. Owpang and S. R. Seshadri, “Guided waves propagating The surfacewaves propagate with a phase velocity along the magnetostatic field at a planeboundary of asemi- which is always less than the velocityof electromagnetic infinitemagnetoionic medium,” IEEE Trans. on Miomave waves in free space.The attenuation rates normal to the Tbory and Techniques, vol. MTT-14, pp. 136144, March 1966. [6] S. R. Seshadri and T. T. \Vu, “Radiation condition for a mag- interface of the surface wavefields inboth the media are netoionic medium.” to be Dublished. examined. Kumerical results of the surface wave char- acteristics are given for one typicalcase. Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media KANE S. YEE Abstracf-Maxwell’s equations are replaced by a set of finite obstacle is moderately large compared to that of an in- difEerence equations. It is shown that if one chooses the field points coming n-ave. appropriately, the set of finite difference equations is applicable for A set of finite difference equations for the system of aboundary condition involving perfectly conducting surfaces. An example is given of the scattering of an electromagnetic pulse by a partial differential equations will be introduced in the perfectly conducting cylinder. early partof this paper. We shall then show that with an appropriate choice of the points at which the various INTRODUCTION field components are to be evaluated, the set of finite OLUTIONS to thetime-dependent Maxwell’s equa- difference equations can be solved and the solution will tionsin general form are unknown except for satisfy the boundary condition. The latter part of this sa few special cases.The difficulty is due mainly to paper will specialize in two-dimensional problems, and the imposition of theboundary conditions. LT,7e shall an example illustrating scattering of an incoming pulse show in this paper how to obtain the solution numer- by a perfectly conducting square will be presented. ically when the boundary condition is that appropriate for a perfect conductor. In theory, this numerical attack AIAXTT-ELL’S EQVATIONAND THE EQUIVALENT SET can be employed for the most general case. However, OF FINITEDIFFERENCE EQUATIONS because of the limited memory capacity of present dal: llaxwell’s equations in an isotropic medium [l]are:’ computers, numerical solutions to a scattering problem aB for which the ratio of the characteristic linear dimen- -+VXE=O, sion of the obstacle tothe m-avelength is largestill at seem to be impractical. We shall show by an example that in the case of two dimensions, numerical solutions are practical even when the characteristic lengthof the Manuscript receivedAugust 24, 1965; revised January 28, 1966. D = €E, This work was performed under the auspices of the U. S. Atomic Energy Commission. The author is with the Lawrence Radiation Lab., University of California, Livermore, Calif. 1 In MKS system of units. 801 YEE: SOLUTION OF INITIAL BOUNDARY VALUE PROBLEMS 303 where J, p, and E are assumed to be given functions of space and time. In a rectangular coordinate system, (la) and (lb) are equivalent to the following system of scalar equations: dB, aE,aE, --=-.--. (24 dt ay dz dB, dE,dE, ___=__-- (2b) at az ax dB, dE,aE, (i,j,kl EY - (2c) at ay 8% aD, dH,dH, / _- Jz, (24 at ay az / X- aD, - dHz dH, Ju, (24 Fig. 1. Positions of various field components. TheE-components at dz ax are in the middle of the edges and the H-components are in the center of the faces. dD, dH,dH, Jz, (20 at ax ay BOUNDARY CONDITIONS where we have taken A = (Az,A,, A,).lye denotea grid point of the space as The boundary condition appropriate for a perfectly conducting surface is that the tangential componentsof (i,j, k) = (iAx,jay, KAz) (3) the electric field vanish.This condition also implies and for any function of space and time we put that the normal component of the magnetic field van- ishes on the surface. The conducting surface will there- F(iAx,jay, kAz, %At) = Fn(i,j, k). (4) forebe approximated by a collection of surfaces of A set of finite difference equations for (2a)-(2f) that cubes, the sides of which are parallel to the coordinate will befound convenient for perfectly conducting axes. Plane surfaces perpendicular to the x-axis will be boundary condition is as follows. chosen so asto contain points where E, and E, are For (2a) we have defined. Similarly, plane surfaces perpendicular to the other axes are chosen. BZn+l/'(i,j+ 1.2, K + 4) - Bzn-"2(i,j + 3, K + 3) At GRID SIZEAND STABILITYCRITERION The space grid size must be such that over oneincre- E,"(i,j 3, k 1) - Eyn(i,j 4, K) - + + + mentthe electromagnetic field doesnot change sig- Az nificantly. This means that, to havemeaningful results, the linear dimension of the grid must be only a fraction - E*"(i,j + 1, K + 3) - EZR(i,j,k + 3) * (5) of the wavelength. We shall choose Ax=Ay =Az. For AY computational stability, itis necessary to satisfy a rela- The finitedifference equationscorresponding to (2b) tion between the space increment and time increment and (k),respectively, can be similarly constructed. At. When E and p are variables, a rigorous stability cri- For (2d) we have terion is difficult to obtain. For constant E and p, com- D,"i + Q,j,k) - D,n-'(i + +,j,K) putational stability requires that At AX)^ + (AY)~+ (AZ)~> cAf = $/;At, (7) - Bz"-l"(i + 4,j + 3, k) - a,n-yi + +,j- 3, K) AY where c is the velocity of light. If cmsI is the maximum ayn-l/S(i+ 1 K + L) - H,n-l/2(i f 1 ' K - 4) light velocity in the region concerned, we must choose - 2,37 2 293, Az d(4~)~+ (AY)~ + (Az)~> cmaxAt. (8) Jzn-1/*(i j, K). (6) + + 3, This requirement putsa restriction on At for the chosen The equations corresponding to (2e) and (2f), respec- Ax, Ay, and Az. tively, can be similarly constructed. The grid points for the E-field and the H-field are MAXWELL'S EQCATIOXSIN Two DIMENSIOXS chosen so as to approximate the condition to be dis- To illustratethe method, me consider a scattering cussed below as accurately aspossible. The various grid problem in two dimensions. We shall assume that the positions are shown in Fig. 1. field components do not depend on thez coordinate of a so4 IEEE TRsNSACTIOh-S ON AhTEhTnTAS AND PROPAGATION BUY point. Furthermore,we take E and p to be constants and J=O. The only source of our problem is then an "in- cident" wave.This incident wave will be "scattered" after it encounters the obstacle. The obstaclewill be of a few "wavelengths"in its linear dimension. Further simplificationcan be obtained if we observe the fact that in cylindrical coordinates we can decompose any electromagnetic field into"transverse electric" and "transverse magnetic" fields if E and p are constants. The two modes of electromagnetic waves are character- ized by 1) Transverse electric wave (TE) H, = H, = 0, E, = 0, 1 AT aH, aE, aE, - - - [EZn(i,j+ 1) - ELn(i,j)] (14b) -"a,=x-z' z AY aH, aE, aa, a& E-=-) -E-> (9) 1 AT ay at ax dt - - [aqi+ 1,j) - EP(i,j)]. (14c) + Z Ax and 2) Transverse magnetic wave (TM) KUMERICALCOMPUTATIOMS FOR TXI WAVES For further numerical discussion we shall limit our- E, = & = 0, Hz = 0, selves to the TM waves. In this case we use the finite aE, aH, aH, difference equations (14a)-(14c). The values forEzo(i,j), E-=--- H,1/2(i+$,j), and HZ1l2(i,j-$) are obtained from the dt ax ay incident wave.2 Subsequent values are evaluated from ag, dE* aH, dE, the finite difference equations (14a)-(14c). The bound- p-=--, pdt=d.t-. (10) at aY ary condition is approximated by putting the boundary value of EZn(i,j) equal to zero for any n. Let C be a perfectly conducting boundary curve. ?Ye To be specific, we shall consider the diffraction of an approximate it bya polygon whose sides are parallel to incident T31 wave by a perfectly conducting square. the coordinate axes. If the griddimensions are small The dimensions of the obstacle, aswell as the profile of compared to the wavelength,we expect the approxima- the incident wave, are shown in Fig.