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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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