On The Concept of Vector (Polarization) Electromagnetic Inverse Boundary
Conditions for the Perfectly and Imperfectly Conducting Cases and its
Applications: Why is renewed interest in EM-IBC forthcoming?
Wolfgang-Martin Boerner, Professor Emeritus & Director
UIC-ECE Communications, Sensing & Navigation Laboratory
900 W. Taylor St., SEL (607) W-4210, M/C 154, CHICAGO IL/USA-60607-7018
T&F: + [1] (312) 996-5480, Email: boerner@ece.uic.edu

Abstract: The inverse problem of electromagnetic scattering pertains to the problem of recovering the size, shape,material surface and interior constituents of an unknown scatterer, given the incident and the scattered fields every
where, together with the laws governing the interaction. In the direct problems of scattering and diffraction, the electric
size, the shape and the material surface and interior constituents are known a priori together with the pre-specified
incident field in terms of a computational coordinate system so that their unknown vector electric and vector magnetic
fields can be derived from Maxwell’s equations by incorporating the known parameters into the boundary conditions
which are well established. In contrary, in the inverse problem such local boundary conditions must be sought which inparticular do not depend on either the size, shape or the surface normal and material properties of the scattering bodyand its enclosing surface ,but must allow to specify these parameters uniquely from the recovered near fields.
Here two separate cases are considered, a perfectly and an imperfectly conducting closed-shaped smooth body forwhich the total complex vector electric fields

E = Einc + E scat and H = H inc + H scat are assumed to be given

everywhere and expressed in terms of a computational coordinate system, where for a
1. Perfectly conducting closed-shaped, convex and smooth body, the following inverse boundary conditions hold

E H=0and

Einc = E scat

E × E* = 0

necessary and sufficient(1)

necessary and not locally sufficient(2)

2. Imperfectly conducting closed-shaped, convex and smooth body, for which the following inverse boundaryconditions are derived from the Leontovich direct boundary condition

E × nˆ = η nˆ × (H × nˆ )(3)Where n̂ denotes outward local surface normal, η defines the local normal Leontovich surface impedance
By inverting this equation (3), two characteristic orthogonal vector quantities, A and B, are obtained which aretangential to the local surface and independent of its local normal

A = E × E* − ηη *H × H* andB = η E* × H − η *E × H*

(4)

Satisfying the following three necessary but not locally yet globally sufficient conditions

A B=0

orthogonality condition

(5a)

A =B

normality condition

(5b)

22

nˆ A = 0 and nˆ B = 0

tangentiality condition(5c)

These two sets of boundary conditions are tested for the case of cylinders and spheres of different radii and for twoslightly different frequencies for verifying that these conditions are indeed satisfied for (i) the perfectly conducting and(ii) the imperfectly conducting cases explaining why these conditions are exact, and why all of these conditions arenecessary but some of the conditions are not locally but only globally sufficient.
Although attempts had been made soon after these vector inverse electromagnetic boundary conditions were first
discovered forty years ago, verifying them with experimental data was not possible at that time due to the lack ofmeasurement capabilities and accuracy. However, within the past decade this has changed and sufficiently accurate
measurement data for near and far field measurements are now becoming available for testing them which is of
considerable relevance to non-destructive material testing and also for advancing vector electromagnetic inverse
scattering theory and techniques.

References:
Boerner, W-M. (V.H. Weston), A Bistatic Electromagnetic Inverse Scattering Technique, Can. J. Physics, 47(11), pp.
1177-1184, 1969.
Boerner, W-M. and H.P.S. Ahluwalia, On a Continuous Wave Electromagnetic Inverse Boundary Condition, Canadian
Journal of Physics, 50(23), pp. 3023-3061, 1972.
Boerner, W-M. (H.P.S. Ahluwalia), Application of a set of electromagnetic inverse boundary conditions to the profile
characteristics inversion of imperfectly conducting cylindrical shapes, IEEE Trans. AP 21 (5), 1973, pp. 663-672.
Boerner, W-M (H.P.S. Ahluwalia), Profile characteristics inversion of spherical shapes, IEEE Trans. AP 22 (5), 1974,
pp. 673-682.
Boerner, W-M., O.A. Aboul-Atta, On the concept of electromagnetic inverse boundary conditions, 1975 Int. IEEE/APS Symposium Proc., 2-5 June 1975, Urbana, IL, USA, 130-135.
Boerner, W-M. (O.A. Aboul-Atta), Vectorial impedance identity for the natural dependence of harmonic fields inclosed boundaries, Can. Journal of Phys. 53, 1975, 1404-1407.

Nagasaki University, EEE –ISL, Colloquium Seminar, 070703, prepared 070507=070528