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

Authors: Pierre Hillion
Journal: SRX Physics
 (2010)

---

## Abstract

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.


---

## Body

## 1. Introduction

A self dual chiral medium is invariant under the duality transformations [1, 2]:(1)E⇒-H,H⇒-E,D⇒B,B⇒D,μ⇒-ε,ε⇒-μ,ξ⇒ξ
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 (ξ2<0):(2)B=μH-ξE,D=εE+ξH.
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].


## 2. Gibbs Conventional Formulation

Taking (2) into account, the Maxwell equations with ∂τ=1/c∂t(3)∇⋀E+∂τ(μH-ξE)=0,∇⋀H-∂τ(εE+ξH)=0,∇·(μH-ξE)=0,∇·(εE+ξH)=0
have the explicit form when ∂zE=0, ∂zH=0:(4a)∂yEz+∂τ(μHx-ξEx)=0,(4b)∂xEz-∂τ(μHy-ξEy)=0,(4c)∂xEy-∂yEx+∂τ(μHz-ξEz)=0,(4d)∂x(μHx-ξEx)+∂y(μHy-ξEy)+∂z(μHz-ξEz)=0,(5a)∂yEz-∂τ(εEx+ξHx)=0,(5b)∂xEz+∂τ(εEy+ξHy)=0,(5c)∂xHy-∂yHx-∂τ(εEz+ξHz)=0,(5d)∂x(εEx+ξHx)+∂y(εEy+ξHy)+∂z(εEz+ξHz)=0.
We are interested in the solutions of these equations when Ez or Hz=0.


### 2.1. Pseudo-TE Waves (Ez = 0)

Let us write (4c)(6)∂x(Ey+ξHyε)-∂y(Ex+ξHxε)-ξε(∂xHy-∂yHx)+∂τ(μHz-ξEz)=0.
Substituting (5c) into the third term of (6) with Ez=0 this equation becomes(7)∂x(εEy+ξHy)-∂y(εEx+ξHx)+(n2-ξ2)∂τHz=0,n2=εμ,
while according to (4a), (4b), Ez=0 implies(8)μHx,y-ξEx,y=0
so that (4d) is satisfied. Now writing (5a), (5b)  εEx+ξHx=∂τ-1∂yHz, εEy+ξHy=-∂τ-1∂xHz, and substituting these expressions into (7) gives the wave equation satisfied by Hz(ξ2<0):(9)[∂x2+∂y2-(n2-ξ2)∂τ2]Hz=0.
Finally, taking into account (8), (5a) and (5b) determine Ex,y in terms of Hz:(10)μ∂yHz-(n2+ξ2)∂τEx=0,μ∂xHz+(n2+ξ2)∂τEy=0
so that ∂xEx+∂yEy=0 which implies (5d) according to (8).


### 2.2. Pseudo-TM Waves (Hz = 0)

Proceeding similarly, we write (5c)(11)∂x(Hy-ξEyμ)-∂y(Hx-ξExμ)+ξμ(∂xEy-∂yEx)-∂τ(εEz+ξHz)=0.
Substituting (4c) into the third term of (11) gives with Hz=0(12)∂x(μHy-ξEy)-∂y(μHx-ξEx)+(n2-ξ2)∂τEz=0,
while according to (5a) and (5b), Hz=0 implies in agreement with (5d) that(13)εHx,y+ξHx,y=0.
Now, substituting (4a) and (4b) into (12) written as μHx-ξEx=-∂τ-1∂yEz, μHy-ξEy=∂τ-1∂xEz gives the wave equation satisfied by Ez:(14)[∂x2+∂y2-(n2-ξ2)∂τ2]Ez=0,
while according to (13), (4a) and (4b) supplying Hx,y in terms of Ez become(15)ε∂yEz+(n2+ξ2)∂τHx=0,ε∂xEz-(n2+ξ2)∂τHy=0
so that ∂xHx+∂yHy=0 imply (4d).


### 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(16)u=(n2-ξ2)-1/2τ+y,v=(n2-ξ2)-1/2τ-y,
and, these wave equations become(17)(∂x2-4∂u∂v)Ψ(u,v,x)=0
with quasi-undistorted solutions [5, 6]:(18)Ψ(u,v,x)=(u+ia)-1/2exp[ikv-ikx2(u+ia)-1],
and, it is easily noticed that the wave front(19)Ω=v-x2(u+ia)-1
is a solution of the characteristic partial differential equation (ξ2<0):(20)(∂xΩ)2+(∂yΩ)2-(n2-ξ2)(∂τΩ)2=0.
Substituting (18) into (16) gives {Ex,Ey} in terms of Hz from which are  obtained {Hx,Hy} 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 Ex,y, Hx,y in the previous statement.Now according to (16) ∂y=∂u-∂v so that (10) with m=(n2-ξ2)-1/2, p2=(n2+ξ2)-1, and Hz=Ψ gives(21)Ex=μ∫-∞∞dτp2(∂u-∂v)Ψ,Ey=-μ∫-∞∞dτp2∂xΨ.
Similar expressions are obtained from (15) for the components Hx,y of the pseudo-TM waves. The roles of  x, y may be changed giving with u′=(n2-ξ2)-1/2τ+x, v′=(n2-ξ2)-1/2τ-x  the quasi-undistorted waves:(22)Ψ(u′,v′,y)=(u′+ia)-1/2exp[ikv′-iky2(u′+ia)-1].
The fields (18) and (22) represent 1D-modulated Gaussian beams [7].


## 2.1. Pseudo-TE Waves (Ez = 0)

Let us write (4c)(6)∂x(Ey+ξHyε)-∂y(Ex+ξHxε)-ξε(∂xHy-∂yHx)+∂τ(μHz-ξEz)=0.
Substituting (5c) into the third term of (6) with Ez=0 this equation becomes(7)∂x(εEy+ξHy)-∂y(εEx+ξHx)+(n2-ξ2)∂τHz=0,n2=εμ,
while according to (4a), (4b), Ez=0 implies(8)μHx,y-ξEx,y=0
so that (4d) is satisfied. Now writing (5a), (5b)  εEx+ξHx=∂τ-1∂yHz, εEy+ξHy=-∂τ-1∂xHz, and substituting these expressions into (7) gives the wave equation satisfied by Hz(ξ2<0):(9)[∂x2+∂y2-(n2-ξ2)∂τ2]Hz=0.
Finally, taking into account (8), (5a) and (5b) determine Ex,y in terms of Hz:(10)μ∂yHz-(n2+ξ2)∂τEx=0,μ∂xHz+(n2+ξ2)∂τEy=0
so that ∂xEx+∂yEy=0 which implies (5d) according to (8).


## 2.2. Pseudo-TM Waves (Hz = 0)

Proceeding similarly, we write (5c)(11)∂x(Hy-ξEyμ)-∂y(Hx-ξExμ)+ξμ(∂xEy-∂yEx)-∂τ(εEz+ξHz)=0.
Substituting (4c) into the third term of (11) gives with Hz=0(12)∂x(μHy-ξEy)-∂y(μHx-ξEx)+(n2-ξ2)∂τEz=0,
while according to (5a) and (5b), Hz=0 implies in agreement with (5d) that(13)εHx,y+ξHx,y=0.
Now, substituting (4a) and (4b) into (12) written as μHx-ξEx=-∂τ-1∂yEz, μHy-ξEy=∂τ-1∂xEz gives the wave equation satisfied by Ez:(14)[∂x2+∂y2-(n2-ξ2)∂τ2]Ez=0,
while according to (13), (4a) and (4b) supplying Hx,y in terms of Ez become(15)ε∂yEz+(n2+ξ2)∂τHx=0,ε∂xEz-(n2+ξ2)∂τHy=0
so that ∂xHx+∂yHy=0 imply (4d).


## 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(16)u=(n2-ξ2)-1/2τ+y,v=(n2-ξ2)-1/2τ-y,
and, these wave equations become(17)(∂x2-4∂u∂v)Ψ(u,v,x)=0
with quasi-undistorted solutions [5, 6]:(18)Ψ(u,v,x)=(u+ia)-1/2exp[ikv-ikx2(u+ia)-1],
and, it is easily noticed that the wave front(19)Ω=v-x2(u+ia)-1
is a solution of the characteristic partial differential equation (ξ2<0):(20)(∂xΩ)2+(∂yΩ)2-(n2-ξ2)(∂τΩ)2=0.
Substituting (18) into (16) gives {Ex,Ey} in terms of Hz from which are  obtained {Hx,Hy} 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 Ex,y, Hx,y in the previous statement.Now according to (16) ∂y=∂u-∂v so that (10) with m=(n2-ξ2)-1/2, p2=(n2+ξ2)-1, and Hz=Ψ gives(21)Ex=μ∫-∞∞dτp2(∂u-∂v)Ψ,Ey=-μ∫-∞∞dτp2∂xΨ.
Similar expressions are obtained from (15) for the components Hx,y of the pseudo-TM waves. The roles of  x, y may be changed giving with u′=(n2-ξ2)-1/2τ+x, v′=(n2-ξ2)-1/2τ-x  the quasi-undistorted waves:(22)Ψ(u′,v′,y)=(u′+ia)-1/2exp[ikv′-iky2(u′+ia)-1].
The fields (18) and (22) represent 1D-modulated Gaussian beams [7].


## 3. 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:(23)(a)d⋀E̲+∂τB̲=0,(b)d⋀B̲=0,(a)d⋀H̲-∂τD̲=0,(b)d⋀D̲=0.
In these equations E̲, H̲ are the 1-forms:(24a){E̲,H̲}={Ex,Hx}dx+{Ey,Hy}dy+{Ez,Hz}dz,
and B̲, D̲ the 2-forms:(24b){B̲,D̲}={Bx,Dx}(dy⋀dz)+{By,Dy}(dz⋀dx)+{Bz,Dz}(dx⋀dy)+.
Now, let h* be the Hodge star operator, then(25)h*(dx,dy,dz)⇒(dy⋀dz,dz⋀dx,dx⋀dy),
from which we get the permittivity, permeability, and chirality operators:(26a)ε=εh*,μ=μh*,ξ=ξh*,
so that in a self-dual chiral medium, the 2-forms D̲, B̲ become(27)D̲=εE̲+ξH̲,B̲=μH̲-ξE̲,
and the coefficients of the differentials in (24b) are(28a)Dx,y,z=εEx,y,z+ξHx,y,z,Bx,y,z=μHx,y,z-ξEx,y,z.
Substituting (24a) and (24b) into (23), this set of Maxwell’s equations becomes(29a)(∂yEz-∂zEy+∂τBx)(dy⋀dz)+(∂zEx-∂xEz+∂τBy)(dz⋀dx)+(∂xEy-∂yEx+∂τBz)(dx⋀dy)=0,(29b)(∂yHz-∂zHy-∂τDx)(dy⋀dz)+(∂zHx-∂xHz-∂τDy)(dz⋀dx)+(∂xHy-∂yHx-∂τDz)(dx⋀dy)=0.
Finally, a simple calculation gives for the second set (23) of Maxwell’s equations:(30)[∂x{Bx,Dx}+∂y{By,Dy}+∂z{Bz,Dz}](dx⋀dy⋀dz)=0.


### 3.1. Wave Equations for Fields

We get according to (23) and (27)(31)∂τ-1(d⋀E̲)+μH̲-ξE̲=0,∂τ-1(d⋀H̲)-εE̲+ξH̲=0,
and, eliminating H̲ from (31) supplies the differential-form formulation of the wave equation WeE̲=0 for the E-field. A simple calculation gives after multiplication by ∂τ2(32)WeE̲=(ε+ξ2μ-1)∂τ2E̲-2ξμ-1∂τ(d⋀E̲)+d⋀μ-1d⋀E̲.
The first two terms of this expression become using the Hodge star operators ε,ξ and its inverse μ-1:(33)(ε+ξ2μ-1)∂τ2E̲=(ε+ξ2μ)∂τ2×[Ex(dy⋀dz)+Ey(dz⋀dx)+Ez(dx⋀dy)],ξμ-1∂τ(d⋀E̲)=ξμ∂τ[(∂yEz-∂zEy)(dy⋀dz)+(∂zEx-∂xEz)(dz⋀dx)+(∂xEy-∂yEx)(dx⋀dy)].
In the last term of (32), we have, still using  the inverse Hodge star operator μ-1,(34)μ-1d⋀E̲=1μ[(∂yEz-∂zEy)dx+(∂zEx-∂xEz)dy+(∂xEy-∂yEx)dz],
so that Δ and ∇. are the Laplacian and the divergence operators, and we get(35)d⋀μ-1d⋀E̲=1μ[(-ΔEx+∂x∇·E)(dy⋀dz)+(-ΔEy+∂y∇·E)(dz⋀dx)+(-ΔEz+∂z∇·E)(dx⋀dy)].
Then, substituting (33) and (35), multiplied by μ into (32) gives with n2=εμ(36)WeE̲=[(n2+ξ2)∂τ2Ex-2ξ∂τ(∂yEz-∂zEy)-ΔEx+∂x∇·EΔEy+∂y∇](dy⋀dz)+[(n2+ξ2)∂τ2Ey-2ξ∂τ(∂zEx-∂xEz)-ΔEy+∂y∇·E](dz⋀dx)+[(n2+ξ2)∂τ2Ez-2ξ∂τ(∂xEy-∂yEx)-ΔEz+∂z∇·EΔEy+∂y∇](dx⋀dy),
and, substituting (29a) into (36), we get finally(37)WeE̲=[(n2+ξ2)∂τ2Ex+2ξ∂τ2Bx-ΔEx+∂x∇·E]×(dy⋀dz)+[(n2+ξ2)∂τ2Ey+2ξ∂τ2By-ΔEy+∂y∇·E]×(dz⋀dx)+[(n2+ξ2)∂τ2Ez+2ξ∂τ2Bz-ΔEz+∂z∇·E]×(dx⋀dy).
Similarly, eliminating E̲ from (31) gives the wave equation WhH̲=0 with(38)WhH̲=[(n2+ξ2)∂τ2Hx+2ξ∂τ2Dx-ΔHx+∂x∇·H]×(dy⋀dz)+[(n2+ξ2)∂τ2Hy+2ξ∂τ2Dy-ΔHy+∂y∇·H]×(dz⋀dx)+[(n2+ξ2)∂τ2Hz+2ξ∂τ2Dz-ΔHz+∂z∇·H]×(dx⋀dy).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.


### 3.2. Pseudo-TE/TM Fields

To get the pseudo-TM fields we suppose that the coefficient of thedx⋀dy term in (37) is null which gives(39)(n2+ξ2)∂τ2Ez+2ξ∂τ2Bz-ΔEz+∂z∇·E=0.
In addition, these fields must satisfy the conditions(40)∂zE=0,∂zH=0,Hz=0,
and then, (39) reduces to the wave equation (14) satisfied by Ez. To get the other components, we further assume Dx=Dy=0 so that(41)Ex,y=-ξHx,yε,Bx,y=(n2+ξ2)Hx,yε.
Then, the first two terms of the Maxwell equations (29b) are null and, we are left with(42a)(∂xHy-∂yHx-ε∂τEz)(dx⋀y)=0,
while we get from (29a), still taking into accont (40),(41)(42b)[ε∂yEz+(n2+ξ2)∂τHx](dy⋀dz)-[ε∂xEz-(n2+ξ2)∂τHy](dz⋀dx)+ξ(∂yHx-∂xHy-ε∂τEz)(dx⋀y)=0.
The set (42a), (42b) supplies two equations to determine the components Hx, Hy in terms of the solutions Ez 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 Ez=0 instead of Hz=0, supply the wave equation (9)  satisfied by Hz. We further assume Bx,y=0 so that(43)Hx,y=ξEx,yμ,Dx,y=(n2+ξ2)Ex,yμ.
Then the first two terms of (29a) are null and one is left with(44)(∂xEy-∂yEx+μ∂τHz)(dxd⋀y)=0,
while we get from (29b)(45)[μ∂yHz-(n2+ξ2)∂τEx](dy⋀dz)-[μ∂xHz-(n2+ξ2)∂τEy](dz⋀dx)+ξ(∂xEy-∂yEx-μ∂τHz)(dx⋀y)=0.
Thus, we obtain a set of two equations (44), (45) to determine Ex,y once known the solution Hz 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 [8–10] where further references can be found.


## 3.1. Wave Equations for Fields

We get according to (23) and (27)(31)∂τ-1(d⋀E̲)+μH̲-ξE̲=0,∂τ-1(d⋀H̲)-εE̲+ξH̲=0,
and, eliminating H̲ from (31) supplies the differential-form formulation of the wave equation WeE̲=0 for the E-field. A simple calculation gives after multiplication by ∂τ2(32)WeE̲=(ε+ξ2μ-1)∂τ2E̲-2ξμ-1∂τ(d⋀E̲)+d⋀μ-1d⋀E̲.
The first two terms of this expression become using the Hodge star operators ε,ξ and its inverse μ-1:(33)(ε+ξ2μ-1)∂τ2E̲=(ε+ξ2μ)∂τ2×[Ex(dy⋀dz)+Ey(dz⋀dx)+Ez(dx⋀dy)],ξμ-1∂τ(d⋀E̲)=ξμ∂τ[(∂yEz-∂zEy)(dy⋀dz)+(∂zEx-∂xEz)(dz⋀dx)+(∂xEy-∂yEx)(dx⋀dy)].
In the last term of (32), we have, still using  the inverse Hodge star operator μ-1,(34)μ-1d⋀E̲=1μ[(∂yEz-∂zEy)dx+(∂zEx-∂xEz)dy+(∂xEy-∂yEx)dz],
so that Δ and ∇. are the Laplacian and the divergence operators, and we get(35)d⋀μ-1d⋀E̲=1μ[(-ΔEx+∂x∇·E)(dy⋀dz)+(-ΔEy+∂y∇·E)(dz⋀dx)+(-ΔEz+∂z∇·E)(dx⋀dy)].
Then, substituting (33) and (35), multiplied by μ into (32) gives with n2=εμ(36)WeE̲=[(n2+ξ2)∂τ2Ex-2ξ∂τ(∂yEz-∂zEy)-ΔEx+∂x∇·EΔEy+∂y∇](dy⋀dz)+[(n2+ξ2)∂τ2Ey-2ξ∂τ(∂zEx-∂xEz)-ΔEy+∂y∇·E](dz⋀dx)+[(n2+ξ2)∂τ2Ez-2ξ∂τ(∂xEy-∂yEx)-ΔEz+∂z∇·EΔEy+∂y∇](dx⋀dy),
and, substituting (29a) into (36), we get finally(37)WeE̲=[(n2+ξ2)∂τ2Ex+2ξ∂τ2Bx-ΔEx+∂x∇·E]×(dy⋀dz)+[(n2+ξ2)∂τ2Ey+2ξ∂τ2By-ΔEy+∂y∇·E]×(dz⋀dx)+[(n2+ξ2)∂τ2Ez+2ξ∂τ2Bz-ΔEz+∂z∇·E]×(dx⋀dy).
Similarly, eliminating E̲ from (31) gives the wave equation WhH̲=0 with(38)WhH̲=[(n2+ξ2)∂τ2Hx+2ξ∂τ2Dx-ΔHx+∂x∇·H]×(dy⋀dz)+[(n2+ξ2)∂τ2Hy+2ξ∂τ2Dy-ΔHy+∂y∇·H]×(dz⋀dx)+[(n2+ξ2)∂τ2Hz+2ξ∂τ2Dz-ΔHz+∂z∇·H]×(dx⋀dy).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.


## 3.2. Pseudo-TE/TM Fields

To get the pseudo-TM fields we suppose that the coefficient of thedx⋀dy term in (37) is null which gives(39)(n2+ξ2)∂τ2Ez+2ξ∂τ2Bz-ΔEz+∂z∇·E=0.
In addition, these fields must satisfy the conditions(40)∂zE=0,∂zH=0,Hz=0,
and then, (39) reduces to the wave equation (14) satisfied by Ez. To get the other components, we further assume Dx=Dy=0 so that(41)Ex,y=-ξHx,yε,Bx,y=(n2+ξ2)Hx,yε.
Then, the first two terms of the Maxwell equations (29b) are null and, we are left with(42a)(∂xHy-∂yHx-ε∂τEz)(dx⋀y)=0,
while we get from (29a), still taking into accont (40),(41)(42b)[ε∂yEz+(n2+ξ2)∂τHx](dy⋀dz)-[ε∂xEz-(n2+ξ2)∂τHy](dz⋀dx)+ξ(∂yHx-∂xHy-ε∂τEz)(dx⋀y)=0.
The set (42a), (42b) supplies two equations to determine the components Hx, Hy in terms of the solutions Ez 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 Ez=0 instead of Hz=0, supply the wave equation (9)  satisfied by Hz. We further assume Bx,y=0 so that(43)Hx,y=ξEx,yμ,Dx,y=(n2+ξ2)Ex,yμ.
Then the first two terms of (29a) are null and one is left with(44)(∂xEy-∂yEx+μ∂τHz)(dxd⋀y)=0,
while we get from (29b)(45)[μ∂yHz-(n2+ξ2)∂τEx](dy⋀dz)-[μ∂xHz-(n2+ξ2)∂τEy](dz⋀dx)+ξ(∂xEy-∂yEx-μ∂τHz)(dx⋀y)=0.
Thus, we obtain a set of two equations (44), (45) to determine Ex,y once known the solution Hz 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 [8–10] where further references can be found.


## 4. 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 componentHz satisfying in some domain of R3 the wave equation (9) with proper boundary conditions. Then, since, according to (8), Hx,y are obtained at once from Ex,y, we are left to get these last two components in terms of Hz.In the conventional formalism, this requirement leads, according to (21), to perform the numerical integration of the expressions:(46)∫-∞∞dτ(n2+ξ2)-1∂yHz,∫-∞∞dτ(n2+ξ2)-1∂xHz,
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].



---

*Source: 832487-2009-11-17.xml*