Acoustic and electromagnetic scattering analysis using discrete sources VI the maxwell equations

Acoustic and electromagnetic scattering analysis using discrete sources VI the maxwell equations Acoustic and electromagnetic scattering analysis using discrete sources VI the maxwell equations Acoustic and electromagnetic scattering analysis using discrete sources VI the maxwell equations Acoustic and electromagnetic scattering analysis using discrete sources VI the maxwell equations Acoustic and electromagnetic scattering analysis using discrete sources VI the maxwell equations

VI THE MAXWELL EQUATIONS Up until now, we have considered the direct obstacle scattering problems for time-harmonic acoustic waves Now we want to extend these results to obstacle scattering for time-harmonic electromagnetic waves As in our analysis on acoustic scattering we begin by recalling the fundaments of the Maxwell equations After a brief discussion of the physical background of electromagnetic waves propagation, we will formulate the boundary-value problems for the Maxwell equations We then proceed to extend the jump relations and regularity properties of the vector potentials from the acoustic to the electromagnetic case The basic results of this section consist of the jump relations for vector potentials with square integrable densities As in the acoustic case we will prove some fundamental theorems which enable us to construct complete systems of vector functions on the particle surface Next, we will present the Stratton-Chu representation formulas and will establish some estimates of the solutions The last section of this chapter deals with the general null-field equations for the exterior and the transmission boundary-value problems We will discuss the existence and uniqueness of the solutions, and will show the equivalence of the null-field equations with boundary-integral equations 107 108 CHAPTER VI THE MAXWELL EQUATIONS BOUNDARY-VALUE PROBLEMS IN ELECTROMAGNETIC THEORY Let us consider the electromagnetic wave propagation in a homogeneous, isotropic medium with electric conductivity tr > 0, electric permittivity e > 0, and magnetic permeability ^ If we denote by E and W the electric and magnetic fields, respectively, and if J stands for the current density, then the Maxwell equations read as Vx£(x,t)-f/i^(x,0 = 0, (6.1) VxW(x,0-6^(x,t) = J(x,0 Also, in an isotropic conductor, the electric field satisfies Ohm's law: GE = J (6.2) It is interesting to note that the speed of wave propagation for the electric and magnetic fields is c = 1/y/ejl'^ and in free space c = l/y/eofl^ = 2.99792458 X l O ^ m s ' ^ We assume time-harmonic dependency for S and H that is, that for some time-independent vector fields E and H the following separation of variables holds £(x,0 W(x,0 = = Re f e + : ^ ) l^ '"^ E(x)e~^^* I , J (6.3) Re{/x-iH(x)e-^'^*}, where a; > is the frequency Summarizing these assumptions and eliminating the time dependency, we arrive at the stationary (or reduced, timeharmonic) Maxwell equations V X E = jfcH, V X H = -jkE (6.4) Here, the wave number k given by k^ = Ljhfi (l -f j — ) = a;'eo//o4/ir ( + J - ^ ) (6-5) is complex, its imaginary part leading to the attenuation of a propagating electromagnetic wave It is customary to write k = ko^eriJ^r (6.6) BOUNDARY-VALUE PROBLEMS IN ELECTROMAGNETICS 109 with fco = ujy/eQfx^ and the now relative complex permittivity Cr defined as so as to absorb the conductivity as its imaginary part Before we formulate the boundary-value problems for Maxwell's equations let us introduce some normed spaces which are relevant for electromagnetic scattering With S being the boundary of a domain £), we denote by Ctan(5) = { a / a G C ( S ) , n a = } the space of all continuous tangential fields and by CtZiS) = { a / a e Cf''''{S), n - a = } , < Q < l , the space of all uniformly Holder continuous tangential fields equipped with the supremum norm and the Holder norm, respectively By Cl^^{S) we denote the space of square integrable tangential fields 'C?an(5) = { a / a e L ( ) , n a = } , and clearly ^tanC*^) *s a subspace of the Hilbert space L'^{S) For proving the existence of solutions to the scattering problems by a boundary integral equation treatment it is necessary to introduce the subspace ClAS) = { a / a e C^{S), Vô a C'''^{S)} , < a < 1, of uniformly H5lder continuous tangential fields with uniformly Holder continuous surface divergence C^^^{S) is equipped with the norm I N U s = INU5+IIV.-a||„,5 Let us recall the basic boundary-value problems for the Maxwell equations To this end suppose that is perfectly conducting and let n denote the unit outward normal to The scattering by a perfectly conducting obstacle is modelled by the direct electromagnetic obstacle scattering problem: given Eo, HQ as an entire solution to the Maxwell equations representing an incident electromagnetic field, find a solution E = E5 -I- EQ, H = Hfi-f Ho to the Maxwell equations in the exterior Ds of such that the scattered field Es,!!^ satisfies the Silver-Mtiller radiation condition - ^ X Hs + E5 = o f — j as |x| -> 00, X (6.8) 110 CHAPTER VI THE MAXWELL EQUATIONS uniformly for all directions x / | x | , and the total electric E satisfies the boundary conditions n X E = on (6.9) The direct electromagnetic scattering problem is a particular case of the following Maxwell problem Exterior Maxwell boundary-value problem Find a solution E^, H, G C^{Ds) nC{Dg) to the Maxwell equations in Ds satisfying the SilverMilller radiation condition at infinity and the boundary condition n X E^ = f on 5, (6.10) where f is a given tangential field If f G C^^ ^{S) then there exists an unique solution to the exterior Maxwell boundary-value problem (cf Colton and Kress [32]) Imposing that the boundary data belongs to €^^^^{8) we guarantee that E^ and Hg both belong to C^'^{D8) and depend continuously on f in the norm of C^an dC*^)* ^^ particular, the tangential component of the magnetic field n X H^ belongs to C^^^^^{S) and n x H , =>t(nxE,), (6.11) where A : C^^^^{S) —• C^^^^{S) is the electric to magnetic boundary component map Note that A is a, bijective bounded operator with bounded inverse and satisfies A^ = —J The exterior Maxwell problem with continuous boundary data has also a unique solution which depends continuously on the boundary data with respect to the uniform convergence of the solution and all its derivatives on closed subsets of Dg The interior Maxwell boundary-value problem in Di has a similar formulation but with the radiation condition excluded If this problem has a unique solution, we say that k is not an eigenvalue of the interior Maxwell problem All such eigenvalues are known to be real Physically, the homogeneous interior Maxwell problem corresponds to a perfectly conducting cavity resonator The spectrum of eigenvalues of the interior Maxwell boundary-value problem in Di will be denoted by cr{Di) For the scattering problem, the boundary values are the restriction of an analytic field EQ, HQ to the boundary and therefore they are as smooth as the boundary In our analysis we will assume that the surface S is of class C^ such that the solution E^,!!^ G C^''*(Ds) The mathematical formulation of the scattering by a body which is not perfectly conducting but which also does not allow the total electric field E and the total magnetic field H to penetrate deeply into the body BOUNDARY-VALUE PROBLEMS IN ELECTROMAGNETICS 111 leads to an exterior boundary-value problem for Maxwell's equations with boundary data of the form n x E — n x n x H = o n , where is the surface impedance The impedance boundary-value problem for Maxwell's equations can be formulated as follows Impedance boundary value-problem Find a solution Eg,Us G C^(jDs)nC(jDs) to the Maxwell equations in Dg satisfying the Silver-Muller radiation condition at infinity and the boundary conditions n X E3 - 7n X (n X H^) = f on 5, (6.12) where f is a given tangential field and is the given surface impedance function life CtlniS), G C0'"(5) and Re7 > the impedance boundaryvalue problem possesses an unique solution (cf Colton and Kress [31]) For dielectric obstacles it_is more convenient to make the change of variables: E< = Et/y/et and Ht = H t / y ^ , where the material constants €t and fi^ are now the relative electric permittivity and relative magnetic permeability of the domain Dt, i.e kt = ko^/e^ and f = 5, i The boundary conditions consist of the continuity of the tangential components ^f the total electric and magnetic fields in the surrounding medium E and H, and the tangential components of the^electric and magnetic^ fields inside the particle Ei and Hj, that is, n x E = n x Ei and n x H = n x Hj Scattering of electromagnetic waves by dielectric obstacles is modelled by the following boundary-value problem Transmission boundary-value problem Given Eo,Ho as an entire solution to the Maxwell equations representing an incident electromagnetic field^ find the vector fields, Es.Hs € C^{Ds)r\C(Ds) and Ei,H, £ C^{Di)n C{Di) satisfying the Maxwell equations V xEt = jfcoMtfit, V X Ht = -jkoetEt, in Dt^t = s^i, and two transmission nxEi-nxEs (6.13) conditions = n x Eo, (6.14) nxHj-nxHs = n x HQ, on S, In addition, the scattered field Eg ,Hs must satisfy the Silver-Miiller radiation condition r—: X y7i7H5 - v ^ E ^ = o f r r j as |x| ^00, (6.15) 112 CHAPTER VI THE MAXWELL EQUATIONS uniformly for all directions x / | x | The transmission boundary-value problem possesses a unique solution (of MUller [114]) Imposing n x EQ, n x HQ E C^^n^di^)^ we have n x E^, n X H , € C^^n^iS) and n X Ei, n X Hi G C^^^AS)- Consequently, E^, H , G CO'"(D,) and Ei.Hi G C^'^CA) We note here that one can pose the boundary-value problems for the Maxwell equations in a weak formulation for Lipschitz boundaries and boundary data in >C?an('S^)' ^^ ^^^^ context, Rellich-type identities that are relevant for the Maxwell equations on arbitrary Lipschitz domains in R^ have been devised by Mitrea [111] Together with certain spectral theoretical arguments, these have been used to develop a C^ theory for the exterior and interior Maxwell boundary-value problems in this setting VECTOR POTENTIALS In this section we will review the basic jump relations and regularity properties for surface potentials in the vector case Let be a boundary of class C^ and let Aa stand for the vector potential with integrable surface density a, Aa(x) = J a(y)^(x, y,fc) d5(y), x G R ' - S (6.16) The vector potential Aa with continuous density a (not necessarily tangential) is uniformly Holder continuous throughout R^ and the estimate \\AaL,RS