Perfect electromagnetic conductor

arXiv:physics/0503232v1 [physics.class-ph] 31 Mar 2005

Abstract
In dierential-form representation, the Maxwell equations are represented by simple differential relations between the electromagnetic two-forms and source three-forms while the electromagnetic medium is dened through a constitutive relation between the twoforms. The simplest of such relations expresses the electromagnetic two-forms as scalar multiples of one another. Because of its strange properties, the corresponding medium has been considered as nonphysical. In this study such a medium is interpreted in terms of the classical Gibbsian vectors as a bi-isotropic medium with innite values for its four medium parameters. It is shown that the medium is a generalization of both PEC (perfect electric conductor) and PMC (perfect magnetic conductor) media, with similar properties. This is why the medium is labeled as PEMC (perfect electromagnetic conductor). Dening a certain class of duality transformations, PEMC medium can be transformed to PEC or PMC media. As an application, plane-wave reection from a planar interface of air and PEMC medium is studied. It is shown that, in general, the reected wave has a cross-polarized component, which is a manifestly nonreciprocal eect.

Lindell, Sihvola:

Perfect electromagnetic conductor

Introduction

Dierential-form calculus is a branch of mathematics based on the algebra of multivectors (elements of space Ep ) and dual multivectors (elements of space Fp ) over an n-dimensional space of vectors [1, 2, 3]1 . Its application to electromagnetic theory instead of the classical Gibbsian vector analysis is suggested by the simplicity and elegance obtained in writing the basic Maxwell equations as d = m, (1) Here, d is the four-dimensional dierential operator and the exterior product, while = B + E d, = D H d d = e.

(2)

(3)

represent the four-dimensional electromagnetic two-forms (elements of the dual bivector space F2 ) in terms of three-dimensional two-forms B, D and one-forms E, H. The electric and magnetic source three-forms e , m F3 are combinations of three-dimensional charge three-forms e , m and current two-forms Je , Jm as e = e Je d, m = m Jm d. (4)

Here stands for the normalized time = ct. Denoting by | the scalar product between multivectors and dual multivectors of the same grade [2, 3], the most general linear relation between the two electromagnetic two-forms has the form = M|, (5)

where M is the medium dyadic involving 36 scalar parameters [3]. Inserting (3) in (5) and separating the spatial and temporal components, the medium equations can be represented in terms of a set of three-dimensional dyadics , , , by D = |B + |E, The dyadic is in general not the same as in the alternative representation D = |E + |H, B = |E + |H, (8) (9) H = 1 | B + | E . (6) (7)

corresponding to the same four-dimensional medium dyadic . In the Gibbsian representation we can apply medium equations similar to (8), (9), D = E + H, B = E + H, (10) (11)

for the vector-valued elds D, B, E, H E1 , in terms of the medium dyadics , , , . One must note that the eld and medium quantities in (8), (9) and (10), (11) are denoted by the same symbols even if they are elements of dierent spaces, because they represent the same physical quantities.
1

The notation applied in the present paper coincides with that of [3].

to appear in JEMWA

19(7) 861-869 (2005)

Simple isotropic medium

Obviously, the simplest electromagnetic medium as dened by (5) is obtained when the M dyadic is a scalar factor M so that the relation becomes = M . (12)

In terms of three-dimensional one- and two-forms or Gibbsian vectors the relation (12) has the form D = M B, H = M E. (13) From the viewpoint of four-dimensional dierential forms (12) appears to dene the only possible isotropic medium in the sense that it is invariant in all possible ane transformations [3]. This means that the medium has no special spatial direction in the fourdimensional space and it appears the same for all observers moving with constant velocity. In contrast, it is known that change of motion of the observer changes the medium known as isotropic in the Gibbsian representation (10), (11) to a more general bi-anisotropic medium. While being mathematically simple, the medium dened by (12) appears physically very strange, because for constant M a contradiction arises in the Maxwell equations (1) and (2), unless the sources satisfy the special relation e = M m . In this special source system the two Maxwell equations become the same and the solution is not unique. This is why such a medium has been labeled as nonphysical [4, 5]. It appears that the problem cannot be posed in such a simple way but one should consider the process of creating the sources in the medium more carefully. On the other hand, everywhere outside the source region in such a medium the Poynting two-form E H and energy-density three-form E D + H B vanish, which means that there cannot exist any energy or transmission of electromagnetic power in such a medium. Since the simple isotropic medium dened by (12) arises so naturally in the dierentialform formalism, let us consider its representation in the Gibbsian vector formalism more closely. It turns out that the medium can be represented as a bi-isotropic medium dened in terms of four scalar medium parameters , , , as D = E + H, B = E + H. (14) (15)

Actually, (12) is a special case of such a medium and the four parameters can be expressed in terms of just two parameters M and q as = Mq, = q, = q, = q/M. In fact, (14) and (15) now become D = q (M E + H ), B = q (E + H/M ). (17) (16)

This shows us that one of the equations, D = M B, is satised for any values of q and M while the other equation H = M E requires that the parameter q must become innite: q . (18)

Lindell, Sihvola:

Perfect electromagnetic conductor

From (16) we see that, this being the case, all four medium parameters actually become innite. Their relations can be expressed as = = = q, / = M. (19) This kind of a medium can be characterized as a special Tellegen medium (nonchiral and nonreciprocal bi-isotropic medium) [6].

PEMC medium

To have some insight in the medium dened by (16), let us perform a duality transformation which is known to transform a set of elds and sources to another set and the medium to another one. In its most general form, the duality transformation can be dened as a linear relation between the electromagnetic elds, and represented in terms of four scalar parameters A, B, C, D as [7] E H =
d

A B C D

E H

AD BC = 0.

(20)

Requiring that the Maxwell equations be transformed to Maxwell equations for the dual elds, the other two eld vectors must be transformed as D B and the medium parameters as 1 = AD BC d =
d

D C B A

D B

(21)

D 2 CD CD C 2 BD AD BC AC BD BC AD AC B2 AB AB A2

(22)

It is easy to see that for = we also have d = d , which means that a Tellegen medium is transformed to another Tellegen medium. Now let us study whether there exist possible transformations leading to vanishing magnetoelectric parameters d = d = 0 for a given medium (16). This requirement leads to the condition BDM + AD + BC AC/M = (A BM )(D C/M ) = 0 (23)

for the transformation parameters. Thus, there are two possible transformations denoted by subscripts 1 and 2 and dened by the conditions A1 = B1 M, D2 = C2 /M. (24)

Inserted in (22), the two duality transformations lead to the respective two sets of transformed medium parameters with vanishing d and d . Remarkably, it appears that, in each case, three of the four parameters are transformed to zero: q (MD1 C1 )/B1 M 0 , = (25) 0 0 1d

to appear in JEMWA 0 0 = 0 q (A2 B2 M )/D2 M 2d

19(7) 861-869 (2005)

(26)

From the assumption AD BC = 0 we must also have MD1 C1 = 0 and A2 B2 M = 0. Thus, for q the nonzero transformed medium elements become innite in both cases. This shows us that it is possible to nd a duality transformation which transforms the bi-anisotropic medium dened by (16) and q to either a PEC (perfect electric conductor) medium satisfying [8] 1d , 1d = 1d = 1d = 0, (27)

or to a PMC (perfect magnetic conductor) medium satisfying 2d = 2d = 2d = 0, 2d . (28)

Actually, PEC and PMC are special cases of the medium (5). In fact, because in the PMC medium we have D = E + H = 0 and H = (B E)/ = 0 in the Gibbsian vector representation, these correspond to vanishing of the two-form = 0. Thus, the PMC condition can be characterized by the special parameter value M = 0 in (5). Similarly, the PEC medium corresponds to = 0 and M = . This gives us reason to call the more general medium dened by (5) as the perfect electromagnetic conductor (PEMC). It is a one-parameter class of media. Because the Poynting vector and energy density are transformed as Ed Hd = (AD BC )E H, (29) Ed Dd + Hd Bd = (AD BC )(E D + H B), (30)

in the general duality transformation and since these quantities vanish for PEC and PMC media, they also vanish in the PEMC medium. Because the duality transformation can be inverted, the medium (5) can also be dened as one obtained from a PEC or a PMC medium through a special duality transformation. The eect of the duality transformation can be made more transparent by the following special choice of transformation parameters: A = D = cos(/2), B= 1 sin(/2), M C = M sin(/2). (31)

In this case the PEMC medium parameters are transformed to those of another PEMC medium through the parameter as d = qM (1 + sin ), d = d = q cos , d = q (1 sin ). M (32)

It is seen that varying in the interval /2 /2 reduces the PEMC medium to PMC and PEC at the end points and the original medium is obtained at = 0. For all values of the medium satises the condition d d = d d .

Lindell, Sihvola:

Perfect electromagnetic conductor

Reection from a PEMC boundary

Because the PEMC medium does not allow electromagnetic energy to enter, an interface of such a medium serves as an ideal boundary to the electromagnetic eld. Let us consider the boundary of PEMC medium and air with unit normal vector n. Because tangential components of the E and H elds are continuous at any interface of two media, one of the boundary conditions for the medium in the air side is n (H + M E ) = 0 , (33)

The latter condition can also be written as

because a similar term vanishes in the PEMC-medium side. The other condition is based on the continuity of the normal component of the D and B elds which gives another boundary condition as n (D M B ) = 0 . (34) n (o E Mo H) = 0. (35)

Put together, these can be expressed as the vector condition

o =

o /o .

(36)

As a check, for we obtain the PEC conditions n E = 0 and n H = 0.

As an application, let us consider plane-wave reection from a PEMC boundary plane at z = 0. For simplicity, the incident and reected plane waves in the region z < 0 are assumed polarized parallel to the boundary. The total elds are E(z ) = Ei ejko z + Er ejko z , o H(z ) = uz Eiejko z uz Er ejko z . Mo (Ei + Er ) = uz (Ei Er ), or (uz I Mo I) Er = (uz I + Mo I) Ei , (40) (37) (38)

Inserted in the boundary conditions (36) at z = 0 with n = uz we have (39)

with I = ux ux + uy uy . Multiplying this by the dyadic (uz I + Mo I), the reected eld is obtained as 1 2 [(1 + M 2 o )Ei + 2Mo uz Ei ]. (41) Er = 2 1 + M 2 o This means that, for a linearly polarized incident eld (real Ei ), the eld reected from such a boundary has both a co-polarized component (multiple of Ei ) and a cross-polarized component (multiple of uz Ei ) in the general case. For the PMC and PEC special cases (M = 0 and M = , respectively,) the cross-polarized component vanishes. For the special PEMC case M = 1/o , we have Ei = uz Ei , (42)

which means that the reected eld appears totally cross-polarized. Thus, the boundary acts as a twist polarizer which is a nonreciprocal device ([6], p.84).

to appear in JEMWA

19(7) 861-869 (2005)

Conclusion

In this study, a class of electromagnetic media, dened in the simplest possible manner as the isotropic medium when using the four-dimensional dierential-form formalism, was given an interpretation in terms of the corresponding Gibbsian formalism. It turned out that the class can be dened as that of certain bi-isotropic media whose all four scalar parameters have innite values and no power propagation in the medium is possible. Since this kind of one-parameter class of media could be shown to be a generalization of both PEC and PMC media, it can be called the class of PEMC media. Since a PEMC medium acts as an ideal boundary, plane-wave reection was considered from a planar boundary. In the general case, the reected wave has both a co-polarized and a cross-polarized component.

References
[1] H. Flanders, Dierential Forms, New York: Academic Press, 1963. [2] G.A. Deschamps, Electromagnetics and dierential forms, Proc. IEEE, vol.69, no.6, pp.676696, June 1981. [3] I.V. Lindell, Dierential Forms in Electromagnetics, New York: Wiley and IEEE Press, 2004. [4] F.W. Hehl, Yu.N. Obukhov, Foundations for Classical Electrodynamics Boston: Birkh auser 2003, p.246. [5] I.V. Lindell, Ane transformations and bi-anisotropic media in dierential-form approach, J. Electromag. Waves and Appl., vol.18, no.9, pp.1259-1273, 2004. [6] I.V. Lindell, A.H. Sihvola, S.A. Tretyakov, A.J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media, Boston: Artech House, 1994. [7] I.V. Lindell, L.H. Ruotanen, Duality transformations and Green dyadics for bianisotropic media, J. Electromag. Waves and Appl., vol.12, pp.1131-1152, 1998. [8] A.F. Stevenson, Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer)/wavelength, J. Appl. Phys., vol.24, no.9, pp.11341142, 1953.