Pseudo-TE/TM Waves in a Self-Dual Chiral Medium

Using successively the conventional Gibbs and the differential-form formulations of Maxwell’s equations, we analyze the electromagnetic fields in a self-dual chiral material (invariance under the exchanges [(E⇔ −H), (D⇔B), (ε ⇔ −μ)]. We prove that such a medium supports pseudo-TE/TM waves (only a component is null with two relations between four of the five other components). An interesting example is supplied by the Courant-Hilbert quasi-undistorted progressing waves: their form does not change along propagation; only their amplitude could change.


Introduction
A self dual chiral medium is invariant under the duality transformations [1,2]: in which E, B, D, and H are the components of the electromagnetic field and ε • μ • ξ constant permittivity, permeability, and chirality parameters.A simple illustration is supplied by the Post constitutive relations [3] used in this work, ε • μ are real and ξ pure imaginary (ξ 2 < 0): We analyze the behaviour of electromagnetic fields in this self-dual medium using successively the Gibbs conventional and the differential-form formulations of Maxwell's equations.These last ones, in a isotropic homogeneous medium, have TE and TM waves as particular solutions when the fields do not depend on one coordinate x, y, z.We prove here the existence of pseudo-TE/TM waves, pseudo meaning that only one of their components is null while two relations depending on ε, μ, ξ exist between four of the five other components.An example is supplied by quasi-undistorted free fields of the Courant-Hilbert type [4].

Gibbs Conventional Formulation
Taking (2) into account, the Maxwell equations with have the explicit form when ∂ z E = 0, ∂ z H = 0: We are interested in the solutions of these equations when E z or H z = 0.

Pseudo-TM Waves (H z = 0)
. Proceeding similarly, we write (5c) Substituting (4c) into the third term of (11) gives with H z = 0 while according to (5a) and (5b), H z = 0 implies in agreement with (5d) that Now, substituting (4a) and (4b) into (12) written as while according to ( 13), (4a) and (4b) supplying H x,y in terms of E z become 2.3.Quasi Distortion Free Solutions.Harmonic plane waves are the most elementary solutions of the wave equations ( 9) and ( 14) but the Courant-Hilbert quasi-undistorted progressing waves [4] are of some interest.The form of these waves, defined on characteristic surfaces (wave fronts), does not change along their propagation but their amplitude is not constant.To prove that ( 9) and ( 14) have such solutions, we introduce the variables and, these wave equations become with quasi-undistorted solutions [5,6]: and, it is easily noticed that the wave front is a solution of the characteristic partial differential equation (ξ 2 < 0): Substituting ( 18) into (16) gives {E x , E y } in terms of H z from which are obtained {H x , H y } according to (8).A similar result follows from (18) substituted into (15), just exchanging {μ, ξ} into {ε, −ξ) (see (13)) and the roles of E x,y , H x,y in the previous statement.Now according to (16 , and Similar expressions are obtained from (15) for the components H x,y of the pseudo-TM waves.The roles of x, y may be changed giving with The fields (18) and ( 22) represent 1D-modulated Gaussian beams [7].

Differential-Form Formulation
The three-dimensional differential-form formulation of Maxwell's equations is [1], in absence of charge and current, with the exterior derivative operator d = dx∂ x + dy∂ y + dz∂ z and τ = ct: In these equations E, H are the 1-forms: and B, D the 2-forms: Now, let * h be the Hodge star operator, then * h dx, dy, dz =⇒ dy ∧ dz, dz ∧ dx, dx ∧ dy , from which we get the permittivity, permeability, and chirality operators: so that in a self-dual chiral medium, the 2-forms D, B become and the coefficients of the differentials in (24b) are Substituting (24a) and (24b) into (23), this set of Maxwell's equations becomes Finally, a simple calculation gives for the second set (23) of Maxwell's equations: 3.1.Wave Equations for Fields.We get according to (23) and ( 27) and, eliminating H from (31) supplies the differential-form formulation of the wave equation W e E = 0 for the E-field.A simple calculation gives after multiplication by The first two terms of this expression become using the Hodge star operators ε, ξ and its inverse μ −1 : In the last term of (32), we have, still using the inverse Hodge star operator μ −1 , so that Δ and ∇. are the Laplacian and the divergence operators, and we get Then, substituting (33) and ( 35), multiplied by μ into (32) gives with and, substituting (29a) into (36), we get finally Similarly, eliminating E from (31) gives the wave equation These equations will be used by imposing that one of the three terms in (37) and (38) is null, that is, one component of the E and H fields is solution of a wave equation, the other two components being obtained from Maxwell's equations.

Pseudo-TE/TM Fields.
To get the pseudo-TM fields we suppose that the coefficient of the dx ∧ dy term in (37) is null which gives In addition, these fields must satisfy the conditions and then, (39) reduces to the wave equation ( 14) satisfied by E z .To get the other components, we further assume D x = D y = 0 so that Then, the first two terms of the Maxwell equations (29b) are null and, we are left with while we get from (29a), still taking into accont (40),( 41) The set (42a), (42b) supplies two equations to determine the components H x , H y in terms of the solutions E z of the wave equation ( 14).For the pseudo-TE fields, making null the coefficient of dx ∧ dy term in (38) null and imposing (40) with E z = 0 instead of H z = 0, supply the wave equation ( 9) satisfied by H z .We further assume B x,y = 0 so that Then the first two terms of (29a) are null and one is left with while we get from (29b) Thus, we obtain a set of two equations (44), (45) to determine E x,y once known the solution H z of the wave equation (9).The integration of these 2-forms has to be performed on 2D-manifolds with a suitable numerical technique such as finite elements [7] using a judicious choice of test functions among which the Whitney forms [8] have a particular interest.Many works have been devoted to this integration problem [8][9][10] where further references can be found.

Discussion
Two topics emerge from this work.The existence of pseudo-TE/TM waves in self media allows to make a comparison between the conventional and the differential-form formulations of Maxwell's equations.Let us limit this discussion to TE waves (similar conclusions hold valid for pseudo-TM waves).In both formalisms, we start with the component H z satisfying in some domain of R 3 the wave equation ( 9) with proper boundary conditions.Then, since, according to (8), H x,y are obtained at once from E x,y , we are left to get these last two components in terms of H z .
In the conventional formalism, this requirement leads, according to (21), to perform the numerical integration of the expressions: which is a rather ordinary business.
On the other hand, in the differential-form formalism,we have to cope with the differential equations (44), (45) whose integration on a 2D-manifold M requires an important work as just mentioned at the end of Section 3. So, an interesting question is to investigate when a formalism outpaces the other one, taking into account accuracy and computation time.
The second topics concern the existence in self-dual media of quasi-undistorted progressing waves.These fields carry on an infinite energy but, using the finite aperture approximation for diffraction, we may obtain such fields with a finite energy, able, in principle, to be launched in the physical space [11].Then, they could be used in communications [12] and to generate beams of directed energy (electromagnetic bullets) of interest in laser and radar technologies [13].