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

1 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

V x £ ( x , t ) - f / i ^ ( x , 0 = 0,

(6.1)

V x W ( x , 0 - 6 ^ ( x , t ) = J ( x , 0 Also, in an isotropic conductor, the electric field satisfies Ohm's law:

It is interesting to note that the speed of wave propagation for the

elec-tric 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 = Re 1 f e + : ^ ) E(x)e~^^* I ,

l ^ ' " ^ J (6.3) W(x,0 = R e { / x - i H ( x ) e - ^ ' ^ * } ,

where a; > 0 is the frequency Summarizing these assumptions and

elimi-nating the time dependency, we arrive at the stationary (or reduced,

time-harmonic) 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 ( 1 + 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)

1 BOUNDARY-VALUE PROBLEMS IN ELECTROMAGNETICS 1 0 9

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

equa-tions let us introduce some normed spaces which are relevant for

electro-magnetic scattering With S being the boundary of a domain £), we denote

by

Ctan(5) = { a / a G C ( S ) , n a = 0 } the space of all continuous tangential fields and by

CtZiS) = { a / a e Cf''''{S), n - a = 0 } , 0 < 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 2 ( 5 ) , n a = 0 } ,

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), V« • a € C'''^{S)} , 0 < a < 1,

of uniformly H5lder continuous tangential fields with uniformly Holder

con-tinuous 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

equa-tions To this end suppose that 5 is perfectly conducting and let n

de-note the unit outward normal to 5 The scattering by a perfectly

conduct-ing obstacle is modelled by the direct electromagnetic obstacle scatterconduct-ing

problem: given Eo, HQ as an entire solution to the Maxwell equations

rep-resenting an incident electromagnetic field, find a solution E = E5 -I- EQ,

H = Hfi-f Ho to the Maxwell equations in the exterior Ds of 5 such that

the scattered field E s , ! ! ^ satisfies the Silver-Mtiller radiation condition

- ^ X Hs + E5 = o f — j as |x| -> 00, (6.8)

uniformly for all directions x / | x | , and the total electric E satisfies the

boundary conditions

n X E = 0 on 5 (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

Silver-Milller 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 ( n x E , ) , (6.11)

where A : C^^^^{S) —• C^^^^{S) is the electric to magnetic boundary

com-ponent map Note that A is a, bijective bounded operator with bounded

inverse and satisfies A^ = —J The exterior Maxwell problem with

contin-uous boundary data has also a unique solution which depends contincontin-uously

on the boundary data with respect to the uniform convergence of the

so-lution and all its derivatives on closed subsets of Dg

The interior Maxwell boundary-value problem in Di has a similar

for-mulation 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

ho-mogeneous interior Maxwell problem corresponds to a perfectly

conduct-ing 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

1 BOUNDARY-VALUE PROBLEMS IN ELECTROMAGNETICS 1 1 1

leads to an exterior boundary-value problem for Maxwell's equations with

boundary data of the form n x E — 7 n x n x H = 0 o n 5 , where 7 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 - 7 n X (n X H^) = f on 5, (6.12)

where f is a given tangential field and 7 is the given surface impedance

function

life CtlniS), 7 G C0'"(5) and R e 7 > 0 the impedance

boundary-value 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

bound-ary 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 H j , that is, n x E = n x Ei and n x H = n x H j

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, (6.13)

in Dt^t = s^i, and two transmission conditions

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

theoreti-cal arguments, these have been used to develop a C^ theory for the exterior

and interior Maxwell boundary-value problems in this setting

2 VECTOR POTENTIALS

In this section we will review the basic jump relations and regularity

prop-erties for surface potentials in the vector case Let 5 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<CaM^,S (6-17)

holds for some constant Ca depending on 5 and a

For densities a G C^ani'^)^ 0 < a < 1, the first derivatives of the vector

potential Aa can be uniformly extended in a Holder continuous fashion

from Ds into Ds and from Di into Di with boundary values

(V X A a ) i (x) = y Vx X [a(y)p(x,y,fc)]d5(y) T ^ n ( x ) x a(x), x G 5,

s (V A a ) i (x) = / Vx • [a(y)^(x, y,fc)] dS{y) T ~n(x) • a(x), xeS,

(6.18)

2 VECTOR POTENTIALS 113 where

(V X Aa)+ (x) = lim V X A(x ± /in(x)),

( V - A a ) ± ( x ) = lim V - A ( x ± / i n ( x ) ) ,

(6.19)

in the sense of uniform convergence on S and where the integrals exist as

Cauchy principal values Furthermore, we have the estimates

IIVxAall^;^^ < C a | | a | | „ _ s , f = S,i,

(6.20)

for some constant Ca depending on 5 and a (cf Colton and Kress [32],

[35])

The divergence of a vector potential Ao with continuous tangential

density a possessing a continuous surface divergence Vs a can be expressed

in the form of a single-layer potential

V Aa(x) = y V , a(y)^(x, y,A:) d5(y), x e R^ - 5 (6.21)

Then, using the identity V x V x E = ~ A E 4- V (V • E) which holds for

twice continuously differentiable vector fields E, we find that

Consequently, the double curl of a vector potential Aa with tangential

density a € C^^ d{S)^ 0 < a < 1, can be uniformly extended in a H5lder

continuous fashion from Dg into Ds and from Di into Dt with boundary

The estimates

||V X V X Aa|U,-o, < Cc,\MaAS^^ = ^^h (6.24) hold for some constant Ca depending on 5 and a

The following jump relations are valid for vector potentials with

con-tinuous tangential densities:

(a) lim n(x) x [V x Aa(x ± hn{x))]

= y n(x) X {Vx X [a(y)^(x,y,A:)]}d5(y) ± i a ( x ) , x € 5,

s

(b) lim ii(x) X {V X V X [Aa(x -•- /in(x))

- A a ( x - / i n ( x ) ) ] } = 0, x e 5 ,

(c) ^lim n(x) x | V x V x [ A » ( x ± /in(x)) - A*„(xT hn{x.))\ |

= n(x) X I V X V X y a(y) (^(x, y.fc,) - g{x, y.fci)] d5(y) I , x G 5,

(6.25) where

A'/{x) = Ja(y)p(x,y,k,,i) dS{y), x e R ' - 5 (6.26)

The operator M is compact in Ctan(S'), C^iniS) and C^;,^^^{S), 0 < a < 1,

and map Ctan('S') into Ctan(5') (cf., e.g Colton and Kress [32] and

Mar-tin and Ola [100]) We will also use the principal value singular integral operator P, called the electric dipole operator, defined by

(Pa)(x) = n(x) X V x V x /a(y)^(x,y,fc)d5(y) , X G 5 (6.28)

2 VECTOR POTENTIALS 1 1 5

The operator Vs - Vi is compact in Ctan(5') and C^^ni^)^ 0 < a < 1, and

map Ctan(S') into C^an ('5) Excepting a multiplication constant, the operator

V is related to the well-known operator A/" in [32] by ATa = V{n x a)

The adjoint of M with respect to the L^ bilinear form

{a,b> = | ^ a b d S ' (6.29)

s

is defined by

(A<a,b) = {a,A<'b) (6.30) for any a,b € Ctan(5') By interchanging the order of integration we see

that

M'a = n X M(n x a) (6.31) Similarly, the operator P ' designates the adjoint of V with respect to the

pairing (a, b ) Using the identity

Af{n X a) = P (n X (n X a)) = -Pa for all a G Cf^^ d('^)^ ^^^ ^^^ ^^^^ ^^^^ -^ ^^ '^^^^ adjoint we have

(Pa, b) = - (n X a,'P (n x b)), (6.32) whence

7^'a = n x ' P ( n x a ) (6.33) Following Kersten [79], [80] we can extend the jump relations for con-

tinuous densities to square integrable densities by making use of the Lax

theorem [90] Analogously to the scalar case we have the following theorem

THEOREM 2.1: Assume that S is a closed surface of class C^ and let

Aa be the vector potential with square integrable tangential density a Then

-n X I V X V X y a(y) [gi.^yA) - gi^vA)]d5(y)

2,S

0

(6.34)

Proof: The proof is similar to that of theorem 2.1 given in Chapter

Our further completeness analysis relies on the results of the following theorems

T H E O R E M 2.2: Consider Di a bounded domain of class C^ with ary S Let the vector potential Aa with density a e ^?an('S') satisfy

almost everywhere on S The integral equation (6.38) is a Predholm

inte-gral equation of the second kind According to Mikhfin [101] we find that

a ~ ao G Ctan(S') Then, since M maps Ctan(5') into C^l^i^) ^^ see that

ao G Ctan('S') Going further, since any solution to the integral equation

(6.38) which belongs to Cfan(5) automatically belongs to C^^^di'^) ^^ ^ ^ tain ao € C^andi*^)' From to the jump relation for the double curl of the

vector potential AQQ we conclude that n x 5^_ = 0 Since ao € Cfand('^)

we see that £, H G C^'^{Ds) solve the homogeneous exterior Maxwell boundary-value problem and therefore S = H = 0 in Dg Finally, applica- tion of the jump relation n x H+ — n x H- = ao = 0 finishes the proof of

the theorem

2 VECTOR POTENTIALS 1 1 7

T H E O R E M 2.3: Consider Di a bounded domain of class C^ with

bound-ary S and exterior Dg- Assume that k is not an eigenvalue of the interior

Maocwell problem and let the vector potential AQ with density a G £?an(*5')

satisfy

V xV xAa = 0 in Ds (6.39) Then a ~ 0 on S,

Proof: The proof is similarly to that of the previous theorem

Con-structing the fields £ and H according to (6.36) we obtain a Predholm

integral equation of the second kind for the surface density a Using the

same regularity arguments we find that a ^ ao G C^^n,d(^)' ^^^e jump

re-lations lead to n X 5 - = 0 In this case £, W € C^'^{Di) solve the

ho-mogeneous interior Maxwell boundary-value problem and since k is not

an irregular frequency we conclude that £ ^ H = 0 in Di Then, from

n X H^ — n X H- = ao = 0, we obtain a '>^ 0 on 5

T H E O R E M 2.4: Consider Di a bounded domain of class C^ with

bound-ary S Let the vector potential Aa with density a G C'^g^ni'^) satisfy

V X A a - A - V X V X Anxa = 0 in D^ for Re\ > 0 (6.40)

k Then a ~ 0 on S

Proof: The proof is similar to that of theorem 2.6 given in Chapter

2 Let us define the electromagnetic field

where H and G are the mean curvature and the Gaussian curvature of

the surface, respectively Consider now a spherical surface SR of radius R

enclosing Di Simple calculations show that £ and H satisfy the weak form

of the radiation condition, that is

^lim^ f\nx n-£\^dS = ^lim^ / {|W x n|^ + \£\^

SR SR (6.45)

~ 2 R e [ ( n x f ) W * ] } d 5 = 0

Application of Gauss' theorem in the region DhR , bounded by the surface

Sh and the spherical surface SR gives

If Re(A) > 0 the conclusion a '^ 0 on 5 follows immediately If Re(A) = 0

and Im(A;) > 0 we obtain £ = 0 in Z>5 Finally, if Re(A) = 0 and Im(fc) = 0

we get /^^ |5|^d5 - • 0 as i? -^ oo, whence 5 = 0 in JD^ follows The jump

relations (6.42) may now be used to conclude

It is noted that similar arguments can be employed to prove the

the-orem 2.2 For instance, let £ and H be given by (6.36) The null-field

condition (6.35) gives 5 = W = 0 in £>i Therefore,

lim ||nx£(.+/in(.))||2,5 = 0,

/I—•U4

lim ||nxW(.+/in(-))-a|l2,s = «•

(6.50)

For a psirallel exterior surface Sh — {y/ y = x + /in(x), x e 5, /i > 0} and

sufficiently small h we have

/ ( n X Syn'dS = f{nx[nx S{.+hn(.))]} • [n x n*{.+hn{.))]

Sh S

X (1 - 2hH + h^K)dS

(6.51) and therefore

T H E O R E M 2.5: Consider Di a hounded domain of class C^ with

bound-ary S and exterior D3 Let the vector potential Aa with density a G ^tan(*^)

satisfy

V X Aa + A7-V X V X Anxa = 0 in D^ for ReX > 0 (6.54)

k Then a ~ 0 on S

Proof: Let us define the electromagnetic field