Huygens' principle in classical electrodynamics: a distributional approach
aa r X i v : . [ m a t h - ph ] M a y Huygens’ principle in classical electrodynamics:a distributional approach
Gerald KaiserSignals & Waves, Austin, TX
November 5, 2018
Abstract
We derive Huygens’ principle for electrodynamics in terms of 4-vectorpotentials defined as distributions supported on a surface surrounding thecharge-current density. By combining the Pauli algebra with distributiontheory, a compact and conceptually simple derivation of the Stratton-Chuand Kottler-Franz equations is obtained. These are extended to freelymoving integration surfaces, so that the fields due to charge distributionsin arbitrary motion are represented. A further generalization is obtainedto multiple surfaces, which can be used to enclose clusters of transmitters,scatterers and receivers.
Contents Huygens’ principle and communication
The significance of Huygens’ principle in physics has been described by Courantand Hilbert [CH62, pp 765–766] as follows: ... our actual [3D] physical world, in which acoustic and electro-magnetic (EM) signals are the basis for communication, seems to besingled out among other mathematically conceivable models by in-trinsic simplicity and harmony.
Here is what they meant. Let P ( x , t ) be the propagator for the wave equationin n space dimensions, which is the retarded solution of (cid:3) f ( x , t ) ≡ ( c − ∂ t − ∆ n ) P ( x , t ) = δ ( x , t ) ≡ δ ( t ) δ ( x ) , x ∈ R n , where c is the propagation speed and ∆ n is the Laplacian in R n . If a pointsource δ ( x ) fixed at the origin is excited by a time signal g ( t ), then the signalreceived at x is the retarded solution of (cid:3) f ( x , t ) = g ( t ) δ ( x ), which is f ( x , t ) = Z d x ′ d t ′ P ( x − x ′ , t − t ′ ) g ( t ′ ) δ ( x ′ ) = Z d t ′ g ( t ′ ) P ( t − t ′ , x ) . The ideal communication is obtained only for n = 3 since then P ( x , t ) = δ ( t − r/c )4 πr ⇒ f ( x , t ) = g ( t − r/c )4 πr , r = | x | and g ( t ) can be recovered directly from the received wave f . For n = 1 and all even n , an impulse g ( t ) = δ ( t ) produces a wave whose value at x reverberates at times t > r/c . This can be seen in water waves ( n = 2), where the leadingripple is always followed by a train of ripples. In such a world, communicationwould require massive processing and information would generally be lost. For odd n > P depends on t through a sum of δ ( t − r/c ) and its derivatives. Thisleads to a distortion of g ( t − r/c ) by its derivatives. Thus ideal communicationis possible only in a world with three spatial dimensions, as noted by Courantand Hilbert.Huygens’ principle in R is based on the above property of the wave equation.It states that the wave emitted by a source can be represented in the exterior ofa surface S surrounding the source region as a sum of secondary waves, called Huygens wavelets, emitted by points x ∈ S . Green’s second theorem states thatthe Huygens wavelets consist of the propagators P and their normal derivativeson S . In electrodynamics, Huygens’ principle has been formulated in terms ofthe Stratton-Chu equations and the
Kottler-Franz equations.
The former givethe exterior field in terms of the fields on S , while the latter give it in termsof the tangential fields on S . These equations have become a standard tool foranalyzing EM scattering problems. 2 generalization of Huygens’ principle for scalar waves was derived in [HK9] byletting S be a sphere of radius R and analytically continuing in R . This resultedin the deformation of the Huygens wavelets emitted by the points x ∈ S topulsed-beam wavelets emitted by disks D tangent to S . The representation ofradiation and scattering fields as superpositions of such pulsed beams has someattractive practical features. For example, the beams missing a given observercan be ignored without incurring a large error, and this gives an efficient methodfor numerical computation. A similar generalization was obtained in [HK11] ofHuygens’ principle in electrodynamics, where the exterior field is represented asa sum of EM pulsed-beam wavelets. An added degree of numerical efficiencywas gained by surrounding both the emitting and receiving sources by spheresand then analytically continuing in both radii.I believe it is useful to study Huygens’ principle in electrodynamics from freshpoints of view in order to see how the above developments can be best ap-plied and extended. In Section 2 we derive the
Pauli algebra as the associativecompletion of the vector algebra in R . This allows a well-known compact for-mulation of electrodynamics, as reviewed in Section 3. In Section 4 we develop ageneralized EM Huygens principle by synthesizing a global field from arbitrarilyspecified interior and exterior fields of a closed surface S . Distribution theorygives the required sources on S radiating the given fields. In Section 5, this isextended to arbitrarily moving surfaces. In Section 6 we further generalize thisscheme by synthesizing an EM field from its values in an arbitrary number of cells ˜ E k partitioning spacetime, with the appropriate sources on the interfacesbetween cells. This includes the two-sphere scheme in [HK11], whose cells con-sist of the interiors of the emission and reception spheres and the region betweenthe two spheres. The equations of electrodynamics will be greatly simplified by using an associa-tive algebra called
Pauli algebra which can be regarded as a simple extension ofthe usual (non-associative) vector algebra in R . Two vectors A , B ∈ R definea scalar product A · B and a vector product A × B . We look for a product AB consisting of a linear combination of these two bilinear expressions: AB = A · B + λ A × B , (1)where λ is to be chosen so that the new product is associative: ( AB ) C = A ( BC ) ≡ ABC . (2) Scalar (acoustic) and electromagnetic wavelets were introduced in [K11] by analyticallycontinuing solutions of the wave equation and Maxwell’s equations to complex spacetime. AB ) C = λ ( A × B ) · C + ( A · B ) C + λ ( A · C ) B − λ ( B · C ) AA ( BC ) = λ ( A × B ) · C + ( B · C ) A + λ ( A · C ) B − λ ( A · B ) C . Thus (2) is satisfied if and only if λ = −
1. We arbitrarily choose λ = i anddefine the Pauli product of A and B as AB = A · B + i A × B . (3)The other choice λ = − i is obtained by complex conjugation. For real vectors A , B , C , AB is the sum of a real scalar and an imaginary axial vector h AB i s = A · B , h AB i v = i A × B , (4)while ( AB ) C is the sum of an imaginary scalar and a real vector. It followsthat a general element of the Pauli algebra is the sum of a complex scalar anda complex vector, which we denote by A = A + A with A ∈ C , A ∈ C . (5)The product of two such elements is then A B = ( A B + A · B ) + ( A B + B A + i A × B ) , (6)and (2) implies that this, too, is associative:( A B ) C = A ( B C ) ≡ A B C . (7)A concrete representation of the algebra is given in terms of 2 × A ↔ (cid:20) A + A A + iA A − iA A − A (cid:21) , A ∈ C , A = ( A , A , A ) ∈ C , (8)with A B represented by the matrix product. Under this correspondence, thePauli algebra is therefore isomorphic to the algebra GL (2 , C ) of all complex2 × Consider electrodynamics in vacuum with Heaviside-Lorentz units ( ε = µ = 1)and c = 1. Define the spacetime differential operators D = ∂ t − ∇ and ¯ D = ∂ t + ∇ , (9) Along with complex numbers and quaternions, the Pauli algebra is one of the simplestexamples of
Clifford algebra, also known as geometric algebra [H66, DL3]. Although its firstapplication to physics was in quantum mechanics, it has also turned out to be useful in otherfields, in particular classical electrodynamics [B99]. The Pauli matrices σ k corresponds to thevectors ˆ x k = ∇ x k . k = 1 , , A ( x ) = A ( x , t ) on spacetime R by D A = ( ∂ t − ∇ )( A + A ) = ( ∂ t A − ∇ · A ) + ( ∂ t A − ∇ A − i ∇ × A ) (10)¯ D A = ( ∂ t + ∇ )( A + A ) = ( ∂ t A + ∇ · A ) + ( ∂ t A + ∇ A + i ∇ × A ) . Then D ¯ D = ¯ D D = ∂ t − ∇ ≡ (cid:3) is the scalar wave operator. Now consider the scalar wave equation (cid:3) f ( x ) = g ( x ) (11)where g ( x ) is a given source function which, for convenience, is assumed to be adistribution of compact support. The wave radiated by g is the unique causal solution f ( x ) = Z R d x ′ P ( x − x ′ ) g ( x ′ ) = P ∗ g ( x ) , (12)where ∗ denotes spacetime convolution and P is the retarded propagator, whichis the wave radiated by g ( x ) = δ ( x ) ≡ δ ( x ) δ ( t ): P ( x ) = P ( x , t ) = δ ( t − | x | )4 π | x | , (cid:3) P ( x ) = δ ( x ) . (13)Hence the wave operator is invertible on the space of such fields, with (cid:3) − = P ∗ . Since (cid:3) is a scalar operator, it operates on Pauli fields A ( x ) by (cid:3) A ( x ) = (cid:3) A ( x ) + (cid:3) A ( x ) . Thus we may extend the wave equation (11) to Pauli fields as (cid:3) F ( x ) = G ( x ) (14)where G ( x ) is a Pauli-valued distribution with compact support. The uniquecausal solution is F ( x ) = P ∗ G ( x ) = Z R d x ′ P ( x − x ′ ) G ( x ′ ) . (15)We now apply the Pauli algebra to classical electrodynamics, more or less fol-lowing [B99]. An EM field in free space consists of two vector fields E ( x ) , B ( x )satisfying Maxwell’s equations ∂ t E − ∇ × B = − J ∇ · E = ρ (16) ∂ t B + ∇ × E = ∇ · B = 0 (17) In this context, causality simply means that f is supported in the future region of g . If g vanishes at t = −∞ as assumed here, it suffices to take the ‘initial condition’ f ( x , −∞ ) = 0. ρ, J ) is a given charge-current density. The obvious symmetry of theseequations suggest combining the two fields into a single complex field F ( x ) = E ( x ) + i B ( x ) , (18)for which Maxwell’s equations reduce to ∂ t F + i ∇ × F = − J ∇ · F = ρ. (19)Now interpret F ( x ) as a Pauli field with vanishing scalar component. Then (10)shows that (19) further reduces to the single equation¯ D F = ρ − J ≡ J . (20)The homogeneous equations (17) state that the source J is real, but it will beuseful to allow J to be complex: J = J e + i J m J e = ρ e − J e J m = ρ m − J m (21)where J e and J m represent electric and magnetic sources, respectively. Al-though Maxwell’s equations require J m = 0, virtual magnetic sources will beneeded in the general formulation of Huygens’ principle. As we shall see, thiswill not violate the prohibition of magnetic sources in nature.To solve (20) for F , apply D : (cid:3) F = D ¯ D F = D J ≡ G (22)where G = ( ∂ t − ∇ )( ρ − J ) = ( ∂ t ρ + ∇ · J ) + ( i ∇ × J − ∇ ρ − ∂ t J ) . (23)Since the left side of (22) is a pure vector field, the scalar component of theright side of (23) must vanish. This gives the continuity equation h G i s = ∂ t ρ + ∇ · J = 0 (24) ρ = ρ e + iρ m , J = J e + i J m , whose real and imaginary parts state that electric and magnetic charge areconserved. Assuming the initial condition F ( x , −∞ ) = , we obtain the uniquecausal solution F = P ∗ G = P ∗ ( D J ) = D ( P ∗ J ) , (25)where the last equality follows because (cid:3) commutes with ¯ D and D , and G = −∇ ρ − ∂ t J + i ∇ × J (26)by (23) and (24). Thus we can obtain F in two ways: by propagating the source G , or by using the right side of (25): F ( x ) = D A ( x ) (27)6here the Pauli field A = P ∗ J = Φ − A , with Φ = P ∗ ρ and A = P ∗ J , (28)representing the 4-potential, is the causal solution of the wave equation (cid:3) A = ¯ D D A = ¯ D F = J . (29)In fact, since F is a pure vector field, F = D A = ( ∂ t Φ + ∇ · A ) − ∇ Φ − ∂ t A + i ∇ × A (30)shows that A satisfies the Lorenz gauge condition ∂ t Φ + ∇ · A = 0 . (31)If J is complex as in (21), then so is A : A = A e + i A m A e = Φ e − A e A m = Φ m − A m . (32)The Maxwell field with electric and magnetic sources is then given by E = Re F = −∇ Φ e − ∂ t A e − ∇ × A m (33) B = Im F = −∇ Φ m − ∂ t A m + ∇ × A e . Of course, the homogeneous Maxwell equations require J m = A m = 0. Butthe expressions (33) with a virtual magnetic 4-potential A m will be used toformulate Huygens’ principle. The assumption that J ( x ) is compactly supported was made for convenienceand can be relaxed. While it is reasonable to assume that the spatial supportof J is bounded at any time, we want to allow sources persisting in time, forexample a set of charged particles following world lines or extended chargedsystems evolving in time. This includes, among other things, time-harmonicsystems. The above results remain valid provided the integrals converge.Let the sources be spatially bounded. To simplify the analysis, assume that thespatial support of J ( x , t ) is contained in the interior of a closed surface S ⊂ R at all times t . We assume that S is a smooth manifold, at least of class C . Due to L V Lorenz and not H A Lorentz; see [B99]. Evidently the Lorenz gauge isselected by causality, although non-causal gauges like the Coulomb gauge are admissible sincethe potentials are themselves unobservable in classical electrodynamics. This will be generalized to sources in arbitrary motion in Section 5 by allowing λ todepend on time. Here we assume a fixed surface S , as is commony done in the derivation ofHuygens’ principle. S by E and its interior by I . Both E and I are taken tobe open sets, so that R is the disjoint union R = E ∪ S ∪ I. Let λ ( x ) be a C function such that x ∈ E ⇒ λ ( x ) > x ∈ S ⇒ λ = 0 and |∇ λ | = 1 x ∈ I ⇒ λ ( x ) < . The characteristic functions χ E and χ I of E and I may be written in terms ofthe Heaviside step function H as χ E ( x ) = H ( λ ( x )) = ( , x ∈ E , x ∈ I χ I ( x ) = H ( − λ ( x )) = ( , x ∈ E , x ∈ I. Define the distributional vector field N ( x ) ≡ ∇ χ E ( x ) = −∇ χ I ( x ) = δ S ( x ) n ( x ) (35)where n ( x ) = ∇ λ ( x ) and δ S ( x ) = H ′ ( λ ( x )) = δ ( λ ( x )) . Thus n is the outward unit normal on S and d x δ S ( x ) is the 2D area measureon S , regarded as a singular 3D measure:d x δ S ( x ) = d S ( x ) . (36) Remark.
Since the characteristic function χ E ( x ) does not depend on the choiceof λ , neither does the distributional field N = ∇ χ E . The introduction of λ ismerely a convenience which helps clarify the concepts by using the relation H ′ = δ . Similar remarks will apply when λ ( x , t ) is time-dependent, allowing formoving boundaries.Let F ′ be an interior field whose source is supported in E at all times, i.e., J ′ ≡ ¯ D F ′ , supp x J ′ ( x , t ) ⊂ E ∀ t, (37) An example of a function with these properties is λ ( x ) = d ( x ) , x ∈ E , x ∈ S − d ( x ) , x ∈ I, where d ( x ) is the shortest distance from x to S . For x ∈ S , we define χ E ( x ) = χ I ( x ) = 1 /
2; but this singular case will not be neededsince it does not affect χ E and χ I as distributions. x denotes spatial support. Since E is open and the support of J ′ isby definition closed, it must actually be contained in some closed set V ⊂ E .Hence F ′ is defined and sourceless in an open neighborhood of S as well asin the interior region I . Thus both F and F ′ are defined and sourceless on aneighborhood of S .We shall construct a field F S whose sources are concentrated on S at all timesand which coincides with the given field F in E and with F ′ in I . These twopartial fields are ‘glued’ into a single field defined by F S ( x ) = χ E ( x ) F ( x ) + χ I ( x ) F ′ ( x ) , x = ( x , t ) ∈ R , (38)and the source of F S is defined by applying ¯ D in a distributional sense: J S = ρ S − J S ≡ ¯ D F S . (39)Note that ¯ D χ E = ∇ χ E = N and¯ D ( χ E F ) = ( ¯ D χ E ) F + χ E ¯ D F = N F + χ E ¯ D F . Similarly, since ¯ D χ I = ∇ χ I = −∇ χ E = − N ,¯ D ( χ I F ′ ) = − NF + χ I ¯ D F ′ . Therefore J S = N F j + χ E J + χ I J ′ (40)where F j = F − F ′ = E j + i B j E j = E − E ′ , B j = B − B ′ (41)is the jump discontinuity across S . Since J is supported in I and J ′ is supportedin E , we have the global identities χ E J ≡ χ I J ′ ≡ . Hence J S = N F j = N · F j + i N × F j = δ S ( n · F j + i n × F j ) (42)is a distributional charge-current density supported spatially on S , with a surfacecharge-current density ( σ, K ) given by ρ S = δ S σ where σ = n · F j (43) J S = δ S K where K = − i n × F j . Like J , J S satisfies the distributional continuity equation h D J S i s = ∂ t ρ S + ∇ · J S = 0 , (44)9hich states that charge, now restricted to flow on S , is conserved.Note that even though J is real, J S is in general complex, consisting of electricand magnetic sources on S : J S = J S e + i J S m J S e = ρ S e − J S e J S m = ρ S m − J S m (45)with ρ S e = δ S σ e , σ e = n · E j J S e = δ S K e , K e = n × B j (46) ρ S m = δ S σ m , σ m = n · B j J S m = δ S K m , K m = − n × E j . (47)If we wish to construct a physically realizable surface source J S , then the absenceof magnetic monopoles requires it to be real: J S m = 0 ⇔ n · B j = 0 and n × E j = on S. (48)That is, the normal component of B S and tangential components of E S mustbe continuous across S . It can be shown that the scalar condition follows fromthe vector condition and Maxwell’s homogeneous vector equation. Since we arefree to choose any sourceless interior field F ′ , (46) and (48) can be viewed as aset of boundary conditions for ( E ′ , B ′ ) with ( E , B ) given. Thus we look for aninterior field F ′ = E ′ + i B ′ such that ∂ t F ′ + i ∇ × F ′ = in I n × E ′ = n × E and n · B ′ = n · B on S. (49)(Recall that F and F ′ extend as sourceless fields to a neighborhood of S .) Thisboundary-value problem has a unique solution if F is continuous in an openneighborhood of S , which will be the case if J is continuous in time. (Recallthat we have also assumed S to be of class C .) For this unique interior field,(46) and (48) are the jump conditions on the interface between the interior andexterior regions [J99, pp 16–18].If the interior field does not satisfy (49), then the Huygens representations weare developing, although useful mathematically for expressing the given ‘real’field F ( i.e., with J m = 0) by a surface integral, cannot be realized physically by actual surface sources. This is what was meant by saying that the magneticsource J S m is virtual. In either case, we now derive the Huygens representations.By (28) and (36), the 4-potential A S = P ∗ J S for F S is given by A S ( x ) = Z R d x ′ P ( x − x ′ ) J S ( x ′ ) = Z d x ′ [ J S ]4 πr (50) This follows in the frequency domain from Equation (6.38) in [CK92]. This is a sufficient but not necessary condition, as follows from the properties of thepropagator (13). Due to the factor δ ( t − r ), the spread of J in both time and space tends tosmooth F . r = | x − x ′ | and [ J S ]( x, x ′ ) = J S ( x ′ , t − r )denotes the retarded source. HenceΦ S ( x ) = Z d S ( x ′ ) ˆ n · [ F j ]4 πr (51) A S ( x ) = − i Z d S ( x ′ ) n × [ F j ]4 πr . Explicitly, the electric and magnetic 4-potentials areΦ S e ( x ) = Z d S ( x ′ ) ˆ n · [ E j ]4 πr (52) A S e ( x ) = Z d S ( x ′ ) n × [ B j ]4 πr Φ S m ( x ) = Z d S ( x ′ ) ˆ n · [ B j ]4 πr A S m ( x ) = − Z d S ( x ′ ) n × [ E j ]4 πr . Although the integrations are formally over R , they reduces to surface integralsover S by (36). We therefore have the following result. Theorem 1
The field F S = E S + i B S radiated by the surface charge-currentdensity J S = ρ S − J S on S is given by surface integral F S = −∇ Φ S − ∂ t A S + i ∇ × A S (53) or E S = −∇ Φ S e − ∂ t A S e − ∇ × A S m (54) B S = −∇ Φ S m − ∂ t A S m + ∇ × A S e where the surface potentials are given by (52) in terms of the retarded jumpdiscontinuities ( E j , B j ) between the exterior and interior fields across S . Thisrepresentation is valid both in the exterior region E , where F S = F , and in theinterior region I , where F S = F ′ . The
Stratton-Chu equations [HY99, page 32] are a special case of (54) obtainedby choosing F ′ = and assuming that x ∈ E . Since F ′ = does generally notsatisfy the boundary conditions (49), the Stratton-Chu formulation of Huygens’principle requires virtual magnetic sources on S . However, if F ′ is chosen to bethe unique solution of (49), the Stratton-Chu equations reduce to the simplerexpressions E S = −∇ Φ S e − ∂ t A S e B S = ∇ × A S e . (55)11eturning to the general case (51) and (53), note that A S involves only the tan-gential components of F j on S while Φ S involves only the normal components.The latter can be eliminated as follows. Begin with ∂ t F S = − i ∇ × F S − J S = − i ∇ × ( i ∇ × A S − ∇ Φ S − ∂ t A S ) − J S = ∇ × ∇ × A S + i ∇ × ∂ t A S + i N × F j . This involves only the tangential component n × F j of F j on S , and it can beintegrated using the initial condition A S ( x , −∞ ) = to obtain F S = ∇ × ∇ × ∂ − t A S + i ∇ × A S + i N × ∂ − t F j (56)where ∂ − t A S ( x , t ) = Z t −∞ d t ′ A S ( x , t ′ ) . This is a generalization of
Kottler-Franz equations [HY99, page 34], obtainedby choosing F ′ = and assuming that x ∈ E : E S = ∇ × ∇ × ∂ − t A S e − ∇ × A S m (57) B S = ∇ × ∇ × ∂ − t A S m + ∇ × A S e . Like the Stratton-Chu equations, (56) and (57) involve virtual magnetic sourceson S . If we assume that the interior field satisfies the physical boundary condi-tions (49), then (57) simplify to E S = ∇ × ∇ × ∂ − t A S e B S = ∇ × A S e . (58) Remark.
Equations (56) and (53), unlike the Stratton-Chu and Kottler-Franzequations, are global.
They remain valid when x ∈ I (where F S = F ′ ) and, ina distributional sense, even when x ∈ S , as indicated by the last term in (56)which is missing in (57). Consequently, they also solve the interior problem, where we are given a source J ′ with spatial support in E and required to findits field in I in terms of an equivalent source on S . This is useful for describing reception. The above can be generalized to sources in arbitrary motion simply by letting λ depend on time, so that the 2D surface S t = { x : λ ( x , t ) = 0 } ⊂ R enclosing the source J ( x , t ) at time t is time-dependent. The 3D hypersurface˜ S = { x = ( x , t ) : λ ( x ) = 0 } ⊂ R history of S t . It is the oriented boundary separating theexterior and interior spacetime regions:˜ E = { x : λ ( x ) > } , ˜ I = { x : λ ( x ) < } , ˜ S = ∂ ˜ I = − ∂ ˜ E. Define the Huygens field F ˜ S ( x ) = χ ˜ E ( x ) F ( x ) + χ ˜ I ( x ) F ′ ( x ) (59)where χ ˜ E ( x ) = H ( λ ( x )) and χ ˜ I ( x ) = H ( − λ ( x )) = 1 − χ ˜ E ( x )are the characteristic functions of ˜ E and ˜ I in spacetime. Then the same argu-ments as above give J ˜ S ≡ ¯ D F ˜ S = ( ¯ D χ ˜ E ) F j = δ ˜ S ( ˙ λ F j + n · F j + i n × F j ) (60)with charge- and current distributions ρ ˜ S = δ ˜ S n · F j , J ˜ S = − δ ˜ S ( ˙ λ F j + i n × F j ) , (61)where ˙ λ = ∂ t λ and δ ˜ S ( x ) = δ ( λ ( x )) (62)is the distribution supported on ˜ S representing the measured x δ ˜ S ( x ) = d t d x δ ( λ ( x , t )) = d t d S t ( x ) . The term − δ ˜ S ˙ λ F j in (61) is a drag current generated by the motion of S t . Since − ˙ λ F j = ˙ λ n × ( n × F j ) − ˙ λ n ( n · F j ) , it has tangential and normal components. Equations (51) generalize toΦ ˜ S ( x ) = Z R d x ′ δ ˜ S ( x ′ ) P ( x − x ′ ) n ( x ′ ) · F j ( x ′ ) (63) A ˜ S ( x ) = − Z R d x ′ δ ˜ S ( x ′ ) P ( x − x ′ ) n ˙ λ ( x ′ ) F j ( x ′ ) + i n ( x ′ ) × F j ( x ′ ) o The electric and magnetic 4-potentials are the real and imaginary parts, andthe Stratton-Chu equations for a moving surface are obtained exactly as in (54).Again, the source J ˜ S is virtual in general. To make it real, hence realizableas a physical surface charge-current density, we must enforce the boundaryconditions n · B j = 0 and ˙ λ B j + n × E j = 0 . (64)Note that the vector condition implies the scalar condition if ˙ λ = 0. For ˙ λ = 0,this can be proved by letting S t acquire a small velocity ˙ λ at x and then takingthe limit ˙ λ →
0. 13
Partitions of unity and energy flow
The expression (59) defines a global field F ˜ S using a partition of spacetime intothe exterior and interior regions separated by the interface ˜ S : R = ˜ E ∪ ˜ S ∪ ˜ I. When the sources (transmitters, scatterers, and receivers) form several clusters, it is useful to surround each cluster by a closed surface. Let us therefore startwith a finite (or even infinite) partition of spacetime into open cells ˜ E k withcharacteristic functions χ k ( x ) = , x ∈ E k , x ∈ ∂ ˜ E k , x ∈ E l l = k which implies X k χ k ( x ) ≡ . (65)This can be used to define a global field F from local fields F k by F ( x ) = X k χ k ( x ) F k ( x ) , (66)where the superscript ˜ S has been dropped. We assume that F k is sourcelessin an open spacetime region O k containing the closure of ˜ E k , thus extendingbeyond its boundary. (The values of F k outside of O k don’t matter due to thefactor χ k in (66).) Since χ k ¯ D F k = 0 ∀ k, the source of F is the distribution J ≡ ¯ D F = X k ( ¯ D χ k ) F k = X k {∇ χ k · F k + ˙ χ k F k + i ∇ χ k × F k } , (67)giving the surface charge-current density ρ = X k ∇ χ k · F k and J = − X k { ˙ χ k F k + i ∇ χ k × F k } . (68) In mathematics, a set of functions with the property (65) is called a partition of unity, al-though the functions are usually assumed to be differentiable. That the characteristic functionsare discontinuous is not a problem, as we have seen, provided we view them as distributionswhen applying derivatives. The values of χ k on ∂ ˜ E k don’t actually matter since (65) stillholds almost everywhere. D χ k is supported on ∂ ˜ E k , J is supported on the boundary ˜ S = [ k ∂ ˜ E k . (69)Furthermore, since x ∈ ˜ E k ∪ ˜ E l ∪ ( ∂ ˜ E k ∩ ∂ ˜ E l ) ⇒ ¯ D ( χ k + χ l ) = 0 , (70) J depends only on the jump fields F jkl = F k − F l , x ∈ ˜ E k ∩ ˜ E l across the interfaces between adjoining regions ˜ E k , ˜ E l . Thus F = D A where A = P ∗ J (71)gives a generalized Huygens representation of F in terms of sources supportedon ˜ S . Furthermore, the projection property χ l ( x ) χ m ( x ) = δ lm χ l ( x ) (72)implies that quadratic expressions in F have similar partitions. For example,the scalar Lorentz invariant F = F · F = E − B + 2 i E · B has the local partition F = X k χ k F k , and the EM energy-momentum density S ≡ F F ∗ = 12 { F · F ∗ + i F × F ∗ } = U + S (73) U = 12 (cid:0) E + B (cid:1) , S = E × B , Since χ k decreases from 1 to 0 as we leave ˜ E k , ∂ ˜ E k is oriented by the unit normal pointinginto its interior. Therefore each interface ∂ ˜ E k ∩ ∂ ˜ E l between adjoining regions occurs twice in (69), with opposite orientations. Hence the oriented sum ( chain ) P k ∂ ˜ E k vanishes but theset-theoretic union ˜ S does not. Equation (72) fails numerically on ∂ ˜ E k ∩ ∂ ˜ E l where χ k ( x ) = χ l ( x ) = 1 /
2, but it holds weakly, in the sense of distributions, i.e., Z d x χ l ( x ) χ m ( x ) f ( x ) = δ lm Z d x χ l ( x ) f ( x )for any continuous function f with compact support or rapid decay (needed when ˜ E k or ˜ E l are unbounded). In [K11a] it was shown that | F | is related to the electromagnetic inertia and the reactiveenergy of the field. F ∗ is the ordinary complex conjugate of F ∈ C ) has the local partition S = X k χ k S k U = X k χ k U k S = X k χ k S k . Hence the local power density (rate of increase of energy density) is h ¯ D S i s = ˙ U + ∇ · S = X l { ˙ χ k U k + ∇ χ k · S k } + X k χ k n ˙ U k + ∇ · S k o . Since F k is sourceless in O k , it follows from Poynting’s theorem that˙ U k + ∇ · S k = 0 in O k and thus ˙ U + ∇ · S = X k { ˙ χ k U k + ∇ χ k · S k } . (74)Here ˙ χ k U k is the rate of increase in the energy density coming into ˜ E k duethe motion of the boundary ∂ ˜ E k , and ∇ χ k · S k is that due to the incomingmomentum flowing through ∂ ˜ E k . By (70), the right side of (74) involves onlythe differences U jkl = U k − U l and S jkl = S k − S l on ∂ ˜ E k ∩ ∂ ˜ E l . Since the general partition (66) allows arbitrary choices of sourceless fields F k in domains O k containing the closure of ˜ E k , these differences need not vanish.Physically, this means that energy must be pumped in or out of the boundary˜ S to maintain these fields. By enforcing boundary conditions on any interface ∂ ˜ E k ∩ ∂ ˜ E l , the corresponding terms can be made to vanish. But then thatinterface can be removed, thus merging the two cells into one.It is instructive to confirm (74) using the expression (68) for the surface current J . Recall that J is generally complex, including magnetic as well as electricsources: J = J e + i J m . The generalized Poynting theorem for a complex surface current density is de-rived by applying the distributional Maxwell equations ∂ t F + i ∇ × F = − J ∂ t F ∗ − i ∇ × F ∗ = − J ∗ to ∂ t U + ∇ · S = 12 { ∂ t F · F ∗ + F · ∂ t F ∗ + i ∇ × F · F ∗ − i F · ∇ × F ∗ } , which gives ∂ t U + ∇ · S = −
12 ( J · F ∗ + J ∗ · F ) = − J e · E − J m · B . (75)16he right side is, like J , a distribution supported on ˜ S . The partitions − J = X k { ˙ χ k F k + i ∇ χ k × F k } F = X l χ l F l give − J · F ∗ = X kl { χ l ˙ χ k F k · F ∗ l + iχ l ∇ χ k × F k · F ∗ l } . Using ∇ χ k × F k · F ∗ l = ∇ χ k · F k × F ∗ l , (75) gives ∂ t U + ∇ · S = 12 X kl { ( χ l ˙ χ k + χ k ˙ χ l ) F k · F ∗ l + i ( χ l ∇ χ k + χ k ∇ χ l ) · F k × F ∗ l } . But the projection property (72) implies the distributional identities χ l ˙ χ k + χ k ˙ χ l = δ kl ˙ χ k and χ l ∇ χ k + χ k ∇ χ l = δ kl ∇ χ k , therefore ∂ t U + ∇ · S = X k { ˙ χ k U k + ∇ χ k · S k } in agreement with (74). This confirms the consistency of our computationsinvolving bilinear distributional expressions. We have derived a concise generalization of Huygens’ principle for EM fields bycombining the Pauli algebra with distribution theory. Given a closed surface S ,we computed the surface source J S on S required to radiate arbitrarily givenexterior and interior fields F and F ′ . Then (50) gives the 4-vector potentialsof F and F ′ as surface integrals over S . The expressions of the fields in termsof these potentials generalize the Stratton-Chu and Kottler-Franz equations.This idea was extended to a time-dependent surface, required for sources ingeneral motion (Section 5), and to multiple surfaces (Section 6). The latter canbe applied, for example, when any number of sources, including transmitters,scatterers, and receivers, form multiple clusters in spacetime. Acknowledgements
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