S-Theorem (On Regularization):Green’s Function-induced

Distributed Elementary Sources − Second KindAlireza Baghai-Wadji

Electrical Engineering DepartmentUniversity of Cape Town

Rondebosch, Western Cape, South Africaalireza.baghai-wadji@uct.ac.za

Abstract—Standard singular dyadic Green’s functions (DGFs)in computational electromagnetics are responses to idealizeddipoles - Dirac’s delta functions. The latter are generalizedsymbolic functions defined as the limit of a sequence of(η−)parametrized functions. Any member of the sequence, withnon-vanishing η, is a function in ordinary sense having finite(non-zero) or infinite support. Utilization of such distributedsource functions, rather than symbolic distributions, renormalizessingularities automatically and results in regularized DGFs. Inthis work a novel physics-inspired distributed elementary sourcefunction has been constructed for the first time. Maxwell’sequations in general media can be split into two comple-mentary systems of partial differential equations: diagonalized-and supplementary forms, D- form and S- form, respectively.Given a boundary value problem, the D-form with respect toa distinguished direction in space allows directly determiningfield components transversal to the distinguished direction. Theremaining two field components (parallel to the distinguisheddirection) can be determined a posteriori from the transversalcomponents by employing the S-form. Using the S-form, a noveldistributed elementary source has been constructed leading toself-consistently regularizing DGFs. The results have been firmlyestablished by providing the complete proof of a theorem.

I. PREPARATORY CONSIDERATIONS

A. D− and S−Forms

Consider Maxwell’s equations in an bi-anisotroic mediumand a distinguished direction in space, say, z. Then, thefollowing D− and S−Forms are equivalent with Maxwell’sequations Maxwell’s equations [1].

1) The D− Form: The form associated with theDIAGONALIZATION of Maxwell’s equations with respectto, say, z−direction is LΨ = ∂zΨ with Ψ =[E1, E2, H1, H2

].

2) The S− Form: The SUPPLEMENTARY set of differentialequations associated with Maxwell’s equations, and comple-menting the diagonalized form with respect to the assumedz−direction, is AΨ = Φ with Φ =

[E3, H3

]. The 4× 4

and 2× 4 differential operators L and A, respectively, dependon (possibly dispersive) material parameters.

II. DYADIC GREEN’S FUNCTIONS

For illustrating the trust of the method it should sufficeto consider the simplest possible three dimensional problem;

i.e., electromagnetic wave in free space induced by a dipoledirected along the x−axis: J1 = J1e1δ(x−x′, y− y′, z− z′).

Subdividing entire space by a fictitious plane z = z′ intoregions z > z′ and z < z′, using the D− Form and proceedingalong the lines outlined in [1] the components of Ψ can bedetermined. Using the S− Form the components of Φ; i.e.,E3 and H3, can be determined a posteriori. Note that thelatter field components do not play any role in satisfyingSommerfeld’s radiation conditions, nor the conditions on the‘‘fictitious’’ interface z = z′. Consequently, the fact that H3

shall satisfy the required jump discontinuity across y = y′ is atestimony for the self-consistency of the overall formulation.

III. A NOVEL PHYSICS-INSPIRED DISTRIBUTED SOURCEFUNCTION

Let H3(x|x′), being calculated a posteriori via the S−Form, denote the magnetic field response at the point x dueto an electric dipole located at x′ and oriented parallel to thex− axis. The following integral representation is valid:

H3(x|x′) =

∞∫−∞

∞∫−∞

dk12π

dk22π

{−1

2

jk2W

}× ejk1(x−x

′)ejk2(y−y′)e−W |z−z

′| (1)

with W =√k2 − k20 if k2− k20 > 0, and W = −j

√k20 − k2

if k20 −k2 > 0. Here, k0 = ω/c0, with c0 being speed of lightin free space, and ω the angular frequency. Denote H3(x|x′) inregion y−y′ > 0 (y−y′ = |y−y′|) by Hy>y′

3 (x|x′). Likewise,denote H3(x|x′) in region y − y′ < 0 (y − y′ = −|y − y′|)by Hy<y′

3 (x|x′). Then, consistency with Maxwell’s equationsrequires that the following relationship holds true:

lim|y−y′|→0

Hy>y′

3 (x|x′)− lim|y−y′|→0

Hy<y′

3 (x|x′)

= δ(x− x′, z − z′) (2)

It should be noted that the limiting process acts on the variablein the oscillating rather than the decaying exponential, thusmaking this formula exceptionally interesting and important.

Replacing |y − y′| by η and substituting the resulting ex-pressions for Hy>y′

3 (x|x′) and Hy<y′

3 (x|x′) into (2) yields the

2174978-1-4799-3540-6/14/$31.00 ©2014 IEEE AP-S 2014

desired distributed source function, a relationship the validityof which will be established in virtue of the S-Theorem:

δ(x− x′, z − z′) = limη→0

δη(x− x′, z − z′)

= limη→0

∞∫−∞

∞∫−∞

dk12π

dk22π

{−1

2

jk2W

}×ejk1(x−x

′)ejk2ηe−W |z−z′|

− limη→0

∞∫−∞

∞∫−∞

dk12π

dk22π

{−1

2

jk2W

}×ejk1(x−x

′)e−jk2ηe−W |z−z′| (3)

Remarks: (i) It is claimed that the above procedure forconstructing ‘‘physics-inspired’’ delta functions has been over-looked in literature. (ii) For η finite, however small, (3)defines the distributed source function δη(x−x′, z−z′) whichsmoothly approaches the symbolic δ(x−x′, z−z′) (refer to [2],[3] regarding intricate details of the regularization process).

A. S−Theorem

The relationships in (3) are valid.Proof: Symmetry considerations in (1) followed by taking

the lim|y−y′|→0

gives

lim|y−y′|→0

H3(x|x′) =1

2π2lim

|y−y′|→0

∞∫0

∞∫0

dk1dk2

{k2W

}× cos[k1(x− x′)] sin[k2(y − y′)]e−W |z−z

′| (4)

Observe that the term lim|y−y′|→0

sin[k2(y − y′)] is non-zero

only for k2 → ∞ (and thus for k =√k21 + k22 → ∞).

Consequently, with limk→∞

W ∝ k, (4) is equivalent with

lim|y−y′|→0

H3(x|x′) =1

2π2lim

|y−y′|→0

∞∫0

∞∫0

dk1dk2 (5)

× cos[k1(x− x′)] {k2 sin[k2(y − y′)]}

{e−k|z−z

′|

k

}

With k2 sin[k2(y − y′)] = −∂/∂y cos[k2(y − y′)] ande−k|z−z

′|/k = −sgn(z − z′)∫dze−k|z−z

′| (5) takes the form

lim|y−y′|→0

H3(x|x′) =1

2π2lim

|y−y′|→0

∞∫0

∞∫0

dk1dk2

× cos[k1(x− x′)]{∂

∂ycos[k2(y − y′)]

}×{

sgn(z − z′)∫dze−k|z−z

′|}

(6)

Exchanging the order of integral- and differential operators

∞∫0

∞∫0

dk1dk2∂∂y

∫dz =⇒ ∂

∂y

∫dz∞∫0

∞∫0

dk1dk2 yields:

lim|y−y′|→0

H3(x|x′) = lim|y−y′|→0

∂

∂ysgn(z − z′)

∫dz

× 1

2π2

∞∫0

∞∫0

dk1dk2 cos[k1(x− x′)] cos[k2(y − y′)]

×e−k|z−z′| (7)

The double integral can be calculated in closed form:

lim|y−y′|→0

H3(x|x′) = lim|y−y′|→0

∂

∂ysgn(z − z′)

∫dz

× 1

4π

|z − z′|[(x− x′)2 + (y − y′)2 + (z − z′)2]3/2

(8)

Absorbing sgn(z − z′) into |z − z′| yields:

lim|y−y′|→0

H3(x|x′) = lim|y−y′|→0

∂

∂y(9)

× 1

4π

∫dz

z − z′

[(x− x′)2 + (y − y′)2 + (z − z′)2]3/2

The following calculation is a delicate interplay of terms,essentially replacing z−z′ in the numerator in (9) by |y−y′|,which is crucially important for further arguments.

The integral in (9) can also be calculated in closed form:

lim|y−y′|→0

H3(x|x′) = lim|y−y′|→0

∂

∂y(10)

× 1

4π

{−[(x− x′)2 + (y − y′)2 + (z − z′)2

]−1/2}Carrying out the differentiation with respect to y and using

y−y′ = |y−y′| and y−y′ = −|y−y′| for y > y′ and y < y′,respectively, in the numerators of the resulting expressions,result in:

lim|y−y′|→0

Hy>y′

3 (x|x′)− lim|y−y′|→0

Hy<y′

3 (x|x′) (11)

= lim|y−y′|→0

1

2π

|y − y′|[(x− x′)2 + (y − y′)2 + (z − z′)2]

3/2

Identifying the limits at the RHS as δ(x− x′, z − z′), (seereferences in [3]) the claim in the Theorem is immediate(validity of (2), respectively, (3) ). 4

ACKNOWLEDGEMENTSThis work is based on the research supported in part by the

National Research Foundation (UID: 85889). The initial stageof the work was carried out under an Australian ResearchCouncil (ARC) Linkage Grant: LP0775463.

REFERENCES

[1] A. R. Baghai-Wadji, ‘‘Theory and Applications of Green’s Functions,’’Selected Topics in Electronics and Systems,Vol. 20: Advances in SurfaceAcoustic Wave Technology, Systems and Applications, Editors: C. Ruppeland T. Fjeldly, World Scientific, vol. 2, pp. 83–149, 2001.

[2] A. R. Baghai-Wadji, ‘‘Self-consistent Physics-based δη−RegularizedGreens Function for 2D Poissons Equation in Anisotropic DielectricMedia,’’ Proceedings ACES, Florida, USA March, 2014 (submitted).

[3] A. R. Baghai-Wadji, ‘‘Self-consistent Physics-based δη−RegularizedGreens Function for 3D Poissons Equation in Anisotropic DielectricMedia,’’ Proceedings ACES, Florida, USA March, 2014 (submitted).

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