Pedestrian Analysis of Harmonic Plane Wave Propagation in 1D-Periodic Media

doi:10.4236/jmp.2011.24027

Journal of Modern Physics, 2011, 2, 188-199

doi:10.4236/jmp.2011.24027 Published Online April 2011 (http://www.SciRP.org/journal/jmp)

Pedestrian Analysis of Harmonic Plane Wave Propagation

in 1D-Periodic Media

Pierre Hillion

Institut Henri Poincaré, Le Vésinet, France

E-mail: pierre.hillion@wanadoo.fr

Received June 2, 2010; revised July 8, 2010; accepted July 10, 2010

Abstract

The propagation of TE, TM harmonic plane waves impinging on a periodic multilayer film made of a stack

of slabs with the same thickness but with alternate constant permittivity is analyzed. To tackle this problem,

the same analysis is first performed on only one slab for harmonic plane waves, solutions of the wave equa-

tion. The results obtained in this case are generalized to the stack, taking into account the boundary condi-

tions generated at both ends of each slab by the jumps of permittivity. Differential electromagnetic forms are

used to get the solutions of Maxwell’s equations.

Keywords: Periodic Slabs, Multilayer Film, TE, TM Waves, Propagation

1. Introduction

The modern approach to harmonic plane wave propaga-

tion in periodic materials such as photonic crystals [1,2]

relies on the Floquet-Bloch modes [1,2,3] and on a

quantum mecha-ics-like technique. We present here for

1D-periodic media, made of a stack of slabs with

al-ternate but constant permittivity, in brief a multilayer

film, a less powerfull pedestrian tech-nique but providing

the explicit expressions of the electromagnetic TM and

TE fields. We start with the analysis of a TM plane wave

propagation inside an horizontal x,y-slab of thick-ness a,

permittivity ε(z) and afterwards, the results obtained in

this case are transposed to the stack of slabs.

Harmonic plane wave propagation in a multilayer film

has been known for a long time [4], the traditional ap-

proach being to consider the multiple reflections that

take place at the inter-faces [4], using for instance the

S-matrix propagation technique [5], but because of the

per-mittivity periodicity, we proceed differently dwelling

on boundary conditions at both ends of each slab where

exists a jump of permittivity. A particular attention is

given to evanescent waves because of their interest in

meta-materials with negative permittivity and permeabil-

ity.

In adddition, we start this paper with a succinct intro-

duction of electromagnetic differential forms [6,7] more

efficient than the conventional formalism to tackle the

kind of problems to be discussed here. We only use the

strong solutions of Maxwell’s equations supplied by the

differential-form formulation so that we have no need of

a computational tool as required by the weak solutions

[8].

2. Differential-form Formulation of

Maxwell’s Equations

We work with the subscript j,k,l, taking the values 1,2,3

associated respectively to the coordinates x,y,z. The

summation convention is used and jkl is the antisym-

metric Levi-Civita tensor.

The 3D differential-form formulation of Maxwell’s

equations is [6,7] in absence of charge and current with

the exterior derivative operator d and



= ct

d0a)d



 

EB B

HD D

0b)

(1)

In these relations dd

ii

x



, E, H are the 1-forms









,,

ii

EH xEH d

i

(2a)

and (,

B

D

) the 2-forms





1

,,d

2jkli ik

BD xx



BD



d

(2b)

We consider these equations in a medium with permit-

tivity



(r) (r is written for x, y, z) and constant perme-

ability µ.

Then, let *h be the Hodge star operator [6,7]

P. HILLION

189





1

*dd d

2



j

jkl kl

hxx x



 (3)

supplying the permittivity and permeability operators *



,

*µ



**,*rh

 

*h (3a)

from which the 2-forms D, B become

*, *







D

EB H (4)

so that the coefficients of the differential terms in (2b)

are



,,

iii i

DBrE H



 



(4a)

Taking into account (4), the Maxwell Equations (1a)

become for harmonic fields exp(it)



d*=0,d *icic r

 

 EHH E=0 (5)

that is according to (2a,b) and (4a)





i2 dd

jkk jjklljk

EE cHxx

 

 



0

(6a)







i2 dd

jkk jjklljk

HH rcExx

 

 



0

0

(6b)

Finally, a simple calculation gives for the second set

(1b) of Maxwell’s equations



123

ddd0,

ddd

jj

Hx xx

rEx xx











(7)

Electromagnetic 2-forms supply weak solutions of

Maxwell’s equations by integration on 2D-manifolds,

understood as limit over 2D-small simplexes made of

triangular elements as used in numerical electromagnet-

ics when physical regions are approximated by finite

elements [8]. Strong solutions, the only ones considered

here, are obtained by making null the cofficients of the

differential terms, they are solutions of conventional

Maxwell’s equations but easier to get as shown here be-

low.

2.1. TM Field

The wave equation for the magnetic field is obtained by

eliminating E from (5) which gives

122

d* d*0c





HH (8)

Now, according to (2a) and (3a), the second term on

the right hand side of (8) is



22 22

1

*d

2



d

j

kl jkl

ccHx

 



Hx



d

(9)

while in the first term



dd

j

kkjj k

H

HxxH (10)

and, using the inverse Hodge star operator 1

*







11

*d d

j

kl klj

rH





 Hx (10a)

Then, we get in Appendix A



11

d* ddd

2ikl jkl



x





H (11)

in which with the Laplacian operator

j

 













12 2

j

jkkj jj

rHr HrH



 

 k



(11a)

Subsituting (9) and (11) into (8) gives the differential

form of the wave equation







22 dd

jkl jjkl

cH x x



0





(12)

Let us now consider the TM field in which



(z) is an

arbitrary function



0, expi

yz xy

H

HH ky



 z (13)

in a medium with permittivity (z) depending only on z.

A simple look to (12) shows that this equation reduces

to







22 dd

xxyz

cH x x



0





x

(14)

with the strong solution 22 0

x

cH







, that is ac-

cording to (11a) and (13)

 

12221

dd0

dy

zz ckzz

z

 



 





 

(15)

Once obtained the solutions of (15) and consequently

Hx according to (13), the electric field is provided by the

2-form (6b)





 

1

dd 0,dd

dd 0

xzxy

yx z

EyzHiczE zx

HiczE xy







 





 





(16)

with the strong solution



0,

,

x

y

zx

z

yx

E

Eic zH















(16a)

which achieves to determine the TM harmonic plane

wave Ey, Ez, Hx. From now on, µ = 1.

2.2. TE Field

The wave equation for the electric field is obtained by

eliminating H from (5)



122

d* d*0cr







 EE (17)

According to (2a) and (3a), the second term in (17) is

190 P. HILLION

 



22 22

1

*d

2d

j

kl jkl

cr crExy

 



E (18)

while in the first term



d



dd

j

kkjj k

EExx E (19)

and using the inverse Hodge star operator 1

*





11

*d d

j

kl klj

Ex





 E (19a)

Then



11

1

d* ddd

2

1dd

2

mjjklkl j

jklmlkmj

x

Ex

Ex x







 

 

E

(20)

so that with A

2, A3 deduced from A1 by a circular per-

mutation of x,y,z, a simple calculation gives



1

123

d* d,,,,,,



A

xyzAyzxAzxy





 E(21)

with



1,,,,d d

z



A

xyzxyz x y (22)

in which



12

,, 2

z

zxx yyzz yz

x

yzE EE E





 



(22a)

and, taking into account the divergence Equation (7)

0

jjj j

EE



  this expression be-comes





1

,,

zzz

xyzE E



jj











(23)

supplying x y by a circular permutation of x, y, z so

that according to (21), (22):



11

d* ddd

2ikl jkl



x





E (24)

Substituting (18) and (24) into (17) gives the different-

tial form of the wave equation for the electric field

 



22 dd

ikl jjkl

crrExx







 



0

(25)

For the TE wave



0,exp i

yz xy

EE Ekyz



  (26)

in a medium with permittivity



(z) depending only on z,

0

jj

E



 in (23): Equation (25) reduces to







22 dd0

xx

EczEyz







 



(27)

and, taking into account (26), this differential form has

the strong solution

 

2222

0

zy

zczkz

 





 



(28)

The components Hy, Hz of this TE field are in terms of

Ex:

0, 0

zxy yxz

Ei cHEi cH











z

a

(29)

3. Harmonic Plane Waves in a 1D-Periodic

Medium

3.1. TM Plane Wave in a Slab (0 < z < a)

Suppose that the plane wave (13), (16a) with



(z) = exp(ikiz)

impinges of the z = 0 face of an horizontal slab of

thickness a and constant permittivity



1 endowed in a

medium with permit-tivity



0 for z < 0 and



2 for z > a,

that is



0

1

2

:0;

:0

:

z

za























(30)

So, according to (13) and (16a), the components of the

incident and reflected fieds in the half space z < 0 are (i =

1



)









0

exp i,

exp i

i

xi i

i

yii

i

z

yi i

HA

EckA















(31)

22 22

0

,

,0

iy z

yi

ky kz

kkc z













(31a)

and







0

exp i,

exp i

r

xr r

r

yir

r

z

yr r

HA

EckA















(32)

22 22

0

,

,0

ry i

yi

ky kz

kkc z













(32a)

Now, in the other two intervalls (30) where j, j = 1, 2

is constant, Equation (15) reduces to

222

222 2

dd 0,

1, 2

j

jjy

z

ckj







 

 (33)

and, we shall consider the three situations

222222

121212

0, 0;0, 0;0, 0



 (33)

1) taking into account (30), the solutions of Equation

(33) in the first situation , j = 1, 2, are with

amplitudes A, B, At

20

j











22 22

1

exp iexpi,

,0

y

zA kzBkz

kk cza









 



(34a)

P. HILLION

191



22 22

2

exp i,

,

ttt

yt

zA kz

kkc za









 

(34b)

Remark 1. If the region with permittivity



2 above the

slab is limited at z = 2a,



t(z) is chan-ged into







exp iexpi,2

tttt t

zA kzBkzaz



a (34c)

a remark of interest in the next section. Now, taking into

account (34a), the components of the TM wave for 0 < z <

a are, acccording to (13) and (16a)









 

1

exp iexpi,

exp iexpi

x

i

yi

zy

HA B

Eck AB



 









 











(35)

22 22

1

,

,0

y

ky kz

kk cza











 

(35a)

while for z > a, taking into account (34b)







2

exp i,

t

xt t

t

yt t

t

zy t

HA

Eck A



t













(36)

22 22

2

,

ty t

yt

ky kz

kkc za









 

(36a)

We now have to take into account the boundary condi-

tions at z = 0 and z = a, imposed by the continuity of the

Hx, Ey components of the electromagnetic field at per-

mittivity jumps.

Then, according to (31), (32), (35), we get at z = 0 the

two relations





11

00

,

iriir



A

AABkAA kAB





  (37a)

while at z = a, taking into account (35), (36) it comes





 



11 1

11 2

exp iexpiexp i

tt

tt t

AkaBkaAka

A

kkaBkkaAk k

 

 



 a

(37b)

The four relations (24(a,b)) supply in Appendix B the

four amplitudes Ar, A, B, At in terms of the incident am-

plitude Ai which achieves to determine the fields (32),

(35), (36). These boundary conditions impose no con-

straint on frequency when all the possible values of ky

are considered.

2) In the second situation, 2

10



, , the com-

ponent kt of the wave vector is pure imaginary and, ac-

cording to (36a)

2

20

 

exp ,

tt t

zAkzaza







Remark 2: similarly to the previous remark, for a upper

region bounded at z = 2a













expexp 2,

2

tt tt t

zAkzaBkaz

az a







 









(38a)

So, according to (38), the field above the slab is eva-

nescent, does not propagate and the components (36)

t

x

H

, t

y

E become







2

exp iexp

exp iexp,

t

xt yt

t

yttyt

HA kykza

EickA kykzaz









a









(39)

and, using (35) in 0 < z < a, the boundary conditions at z =

a imply





 

11

21

exp iexpi,

exp iexpi

t

tt

AA kaBka

ikAkAka Bka







 









(40)

from which we get







21

21i expi

1iexp i

tt

t

AkkAka

kk Bka





 (41)







21

01i expi

1iexpi

t

kk Aka

kk Bka





 (42)

These relations have to be made complete with the

boundary conditons (37a) at z = 0 from which A, B are

provided in terms of Ai and Ar so that according to (41),

A, B, Ar, At are ob-tained in terms of Ai. Explicitly, sub-

stituting into (41) the relations (B.1) of Apendix B, we

get with the



functions



10

11

2i

kk





















21

1i expi

1iexp i0

tir

kkAAka

kkA Aka

 











(43)

supplying Ar from which the amplitudes A, B, A are ob-

tained.

In this case also, the boundary conditions impose no

constraint on  when ky takes all the possible values but,

the situation is different when ky = 0 for propagation in

the z-direction. Then ky = k and

j

ck





, j = 1, 2,

with j = 1 in 0 < z < a and j = 2 for z > a so that if



2>



1

there is a frequency band gap in the intervall (2

ck



,

1

ck



).

3) Finally in the third situation: , , the

TM plane wave (13), (16a) ge-nerates in the slab an

evanescent plane wave with the components Hx, Ey de-

duced from (35), (35a) by changing k into ik and it

comes (for simplification, the coefficient exp(ikyy) pre-

2

10 2

20



(38)

192 P. HILLION

sent in each component is discarded)







 



exp exp

exp exp,

0

x

y

HA kzBkaz

EikA kzBkaz

za









 





(44)

Then, according to (31), (32), (44), the boundary condi-

tions at z = 0 supply the two relations



11

00

exp ,

iexp

ir

iir

AA ABka

kAA kABka





 

 









(45)

from which we get



01 01

21i1i

21i 1i

ii i

ri i

,

A

kkAkk B

A

kkAkk B

 

 

  (46)

Now, at z = a, the boundary conditions imply









11

12

expexp i,

iexp expi

tt

tt t

AkaBAka

kAkaBkA ka













(47)

which gives





21 21

expii 0

tt

Akak kBk k

 





(48)

together with (46), this last relation supplies A, B in

terms of Ai which achieves to determine Ar and At ac-

cording to (46), (47).

Of course, if the region with permittivity



2 is bounded

at z = 2a,



exp i

ttt

A

kz





 

p iexpi

is changed into



ex

ttttt

A

kz Bkz





m



a

. In this situation also,

there is a frequency band gap if



1 >



2.

3.2. TM Wave Propagation in a Periodic

Multilayer Film

We now consider a stack of slabs with each the same

thickness a but with an alternate value



z) constant in-

side the slabs

 

 

1

2

:2210,1, 2,

:2 122

ma zma

z

mazma





 





 





(49)

so that .



2zma z





The TM plane wave (13) (16a) is assumed to impinge

on the z = 0 face of this stack (m = 0) and the following

notations are used for the field



(z).

  

 

2

21

:221

:2 122

m

zmazm a

zzmazm







 





 





(50)

The equation (15) becomes inside the slabs since

dd 0z



 and

222 2

1,1,

jy

ckj





 



 

2222222

2121

222 2222

21221 22

dd 0,,

221

dd 0,

2122

mmy

zkkc

ma zma

zkk

maz m a

 

1

,c













 

 

 

(51)

We consider the three situations (33a),

1) In the first situation

22

12

0, 0 2

10,





2

20



, the equations (51) have the solutions with ampli-

tudes





22

,

mm

AB and





21 21

,

mm

AB













 



22121

21212 212

exp iexpi,

221

exp iexpi,

(2 1)22

mmm

zA kzBkz

ma zma

zA kzBkz

mazm a



 



 



 



(52)

so that according to (13) and (16a) we get for the

x

y

H

E

components intervening in the boun-dary conditions







2

exp i,

m

xym

H

ky z











21

exp i

m

xym

H

ky z













2

11

exp i

m

yy

ky



†

2mz



Eck



 , (53)



21 †

22 21

exp i

m

yy

Eck ky

 



 m

z

in which









 

†

22121

†

21212 212

exp iexpi

mm m

zA kzBkz



 



(53a)

Now, the intervalls









21,2mama and









2,2 1ma ma have a joint boundary at 2zma



and the continuity of Hx, Ey at these boundaries implies











21 2

††

2221112

22,

22

mm

ma ma

kmak



 







ma

(54)

and these relations made explicit in Appendix C supply





22

,

mm

AB in terms of





21 21

,1,2

mm

A

Bm M





1

. and

changing m into m



gives a similar result at





2zm1a

for





21 21

,

mm

AB



 in terms of





22mm

,

A

B.

To achieve the determination of these amplitudes, we

need the boundary conditions at where the inci-

dent field impinges on the multilayer film, and at

0z

2zM

a



output of the stack.

The first and last slab in this stack being assumed to

have the permittiviy



1, we get at according to

(37a) with an evident change in notations

0z





11

00 01000

,

iri ir



A

AABkAAkAB





  (55)

from which





00

,

A

B are obtained in terms of





,

ir

A

B

and consequently





22

,

MM

AB reas-cending the series

2

P. HILLION

193

t

(C,3), (C,6) of Appendix C for . 0,1, 2,,mM



Then,



exp ii

ty

A

ky kz being the field outside the

stack for , the boundary conditions are similarly

to (37b),

2zM

0

a



being the permittivity outside the stack











 



22

11

21 2110

exp2 iexp2 i

exp2iexp2i

MM tt

MMtt

aB MkaAMka



1

exp 2i

AMk

t

A

kMkaBkMkaAkM







 ka



(56)

from which



,

rt

A

A are obtained in terms of 2) In the second situation, is pure

imaginary and, in the intervalls

22

12

0,0, k 



21ma

2

2zma

i

A

which

achieves to determine the amplitudes ,

M1, 2m



, the function





21mz





becomes according

to (38a)

 

222121

,, ,

mmm m

ABA Bm

 1,2 1M

















212212

2 1exp2

m

kzm aBkmaz









21

exp

mm

zA













 (57)

and the boundary conditions (54) at z = 2ma are changed into











  

2 121221

11212 221221

exp2 iexp

expxp2 iiexp

mm

Am akAkaB

kAmak k AkaB





 

 





21

1 2

exp 2i

2 ie

mm

m akB

makB



(58)

from which we get in terms of



22

,

mm

AB



21 21

,

mm

AB



















2112221211221

2eexp 1i

2ei exp1i

mm

mmm

kkkaAkkB

kkkaA kkB

 

21

21

xp2i 1i

xp2i1

m

Amak

Bmak



 

 









21



22

,

mm

AB



21

,

m

i

(59)

which takes the place of (C.3) in Appendix C while in

(C.6) k2 has to be changed into ik2 to get

in terms of .

21

,

mm

AB





22

,

mm

AB

The relations (55), (56) achieve to determine the

amplitudes , , 21m

,

in terms of the amplitude



21 21

,

mm

AB



M

AB



1, 2,m of the

incident field. If 21



, there exists a frequency band

gap for propagation in the z-direction as discussed in

Sec.3.1.

3) In the third situation is pure

imaginary and in the intervalls

22

12

0,0, k 1





m a22ma z1

and





2mz



becomes

A

 









12 1

2exp21

m

zmaBkm az



22

exp

mm

zA k













 (60)



and the boundary conditions at a are



21zm













2112212

expexp21 i=exp

m

A

2 2

21 i

mm

mak

 m

BmakA kaB



  

  (61a)

  



 

22 211211 212

exp2exp21 iiexp

mm

kAmakkAkaB



  

 

2 2

1 i

mm

makB









1m

(61b)

from which we get in terms of



21 2

,

m

AB







22

,0,1

mm

A

Bm M.















 

21121212 212

211221121221 2

2e1iexp 1i

2exp1 iexp1 i

mm

mmm

kkkaAkkB

kkkkaAkkB



 



 





2

xp21i

21 i

m

Ama

Bma















21

,

mm

AB



(62)







0110 110

21i1i

21i 1i

iii

These relations take the place of (C.6) while to get

in terms of , one has to change

into i in (C.3).



22

,

mm

AB

1

k k



21

,

r

A

kk Akk A

BkkAkk

 

 

 A

a

(50a)

1

It is implicitly assumed that the first and last slabs of

the stack have the permittivity 1

Finally, at 2zM



, output of the stack in a medium

with permittivity 0



, we have similarly to (34)



. Then, the boundary

conditions (45) at become with evident notations

0z



 

 

100

ie

xpAB k



00 1

01 1

exp ,

r

iir

AAB ka

kAA ka



 













212

11 2120

expexp i,

iexp expi

MMtt

AkaBAka

kAkaBkA ka





 







t



(64)

(63)









Using (62), (C.3) and (63a), are obtained



22

,

MM

AB

from which we get

194



P. HILLION

in terms of



,

ir

A

A, so that the relations (51) supply r

A

,

t

A

which achieves to determine the amplitudes



21

,

m

A



,



21m

B



22

,

mm

A

B and mm

by running down the sequence of amplitudes.



21 21

,1AB m

1

,2,,M1

There is also a frequency band gap if



> 2



for

vertical propagation.

Remark 3: According to (36) the permittivity is peri-

odic in the multilayer film so that,

 

2zz







2

20

y

zk

a

since the component



22

zc





z





z of the wave vector is pe-riodic 2

k

z

za

kk





.

So, if is a plane wave solution of the

differential equation

 

exp iz



z

kz



 

22 2

dd 0zz



z









, then:



za



z

 

22 expi

za z

zakz





2expik

2a

2







(65)

and is periodic if





22,0,1,

z

kan n







3.3. TE Plane Wave Propagation

For the TE plane wave (26), (29) impinging on a slab

with the permittivity set (30), the wa-ve Equation (28)

reduces to Equation (33) so that all the calculations of

Secs, (2.1), (2.2) hold valid for the TE field. One has just

to change ,,

x

yz

H

EE into ,,

x

yz

EHH.

4. Discussion

As noticed in the introduction, the modern approach to

harmonic plane wave propagation in multilayered films,

made of a stack of slabs with different but constant per-

mittivity, reposes on two principal techniques, both re-

quiring important computational tools. The first one,

mainly interested in frequency bands available for

propagation, mixed solid state physics (Floquet-Bloch

modes) and quantum mechanics (eigenstates of hermitian

operators). The second method [5] working with the

S-matrix propagation technique, an improved version of

the T-matrix algorithm to analyse plane wave scattering

from gratings, is mainly interested in the behaviour of

high intensity lasers impinging on gratings made of

1D-photonic crystals. So, any comparison between the

results supplied by the two techniques, both depending

strongly on the performnces of their computational tools,

is difficult.

The analysis performed in Sec.2 of TM and TE waves

both polarized along ox, with fields of the type

, suggests three comments. When these

waves propagate in a slab with the permittivity set (30).

First, in this situation, the Maxwell equations have the

same solutions for TM and TE waves. Second, as dis-

cussed in Remark 3, the function is not assumed

periodic a-priori, leading to solutions of Maxwell’s equa-

tions not taken into account in [1] so that, one may ask

whether these extra-solutions play some role in propa-

gation, specially for the frequency band gaps. What kind

of incident fields is able to generate these aperiodic solu-

tions? The third point concerns the existence of analytic-

cal expressions for the electromagnetic field amplitudes

in each slab of the stack so that even if numerical calcu-

lations are needed to get them, they will not take the im-

portance they have in [1] and [5].



ky



z



exp i y



z



It is easy to transpose the analysis of Sec.3 to har-

monic plane waves





exp ii

yz

ky kz

0, 0

propagating in a

photonic meta-film [9] made of alternate slabs and

meta-slabs. It has been proved [10,11] that in a lossless

meta-material with









, the solutions of Max-

well’s equa-tions have a classical behaviour provided

that the refractive index n and the impedance Z are de-

fined as



12

n and







12

Z





. As a cones-

quence, when the TM plane wave (13), (16a) impinges

on a meta-slab, one has just to change in (35), (35a) k

and 1



into k



and 12

1

n. Taking into account these

conditions, the amplitudes of the electromagnetic field

inside and outside the metaslab are still supplied in Ap-

pendix B suggesting that only minor differences exist for

harmonic plane wave propagation in films and meta-

films. Of course, the same properties hold valid for TE

plane waves.

This result is confirmed in Appendix D where TM

wave propagation in a two layered film with slabs made

of dielectric or meta-dielectric is analyzed. The situation

is particularly inte-resting when so

that kz is pure imaginary. Then, the two layered film

behaves as a deforming mirror with a conjuguate com-

plex distortion factor for slabs and meta-slabs. This

analysis could be generalized to a stack of alternate slabs

22 220

y

nc k













221ma zma and meta-slabs













2a21 2mazm when .

2

10

5. References

[1] J. D. Joannopoulos, R. D. Meade and J. N.Winn,

“Photonic Crystals,” University Press, Singapore, 1995.

[2] H. Rignault, J. M.Loutioz, C. Delalande and A Levinson,

“La nanophotonique,” Lavoisier, Paris, 2005.

[3] A. Foroozesh and L. Shafai, “Wave Propagation in 1D

EBGs: Periodic Multilayer Films Consisting of Two Dif-

ferent Materials,” IEEE Antennas and Propagation

Magazine, Vol. 50, No. 2, 2008, pp. 175-182.

doi:10.1109/MAP.2008.4563628

[4] M. Born and E. Wolf, “Principles of Optics,” Pergamon,

Oxford, 1965.

[5] M. Nevière and E Popov, “Light Propagation in Periodic

Media,” Marcel Dekker, New York, 2003.

[6] I. V. Lindell, “Differential Forms in Electromagnetics,”

IEEE Press, Piscataway, 2004.

P. HILLION

195

[7] F. W. Hecht and Yu. N. Obukhov, “Foundations of Clas-

sical Electrodynamics,” Birkausen, Boston, 2003.

[8] Bossavit, “Differential Forms and the Computation of

Fields and Forces in Electromagnetism,” European

Journal Mechanics. B/Fluids, Vol. 10, No. 5, 1991, pp.

474-488.

[9] S. Linden and M. Wegener, “Photonic Metamaterial,”

International Symposium on Signals, ISSSE. 2007, pp.

147-150.

[10] J. B. Pendry, “Negative Refraction Makes a Perfect

Lens,” Physical Review Letters, Vol. 85, No. 18, 2000, pp.

3966-3969. doi:10.1103/PhysRevLett.85.3966

[11] R. W. Ziolkowski and E. Heyman, “Wave Propagation in

Media Having Negative Permittivity and Permeability,”

Physical Review E, Vol. 64, No. 5, 2001, pp. 1-15.

doi:10.1103/PhysRevE.64.056625

196 P. HILLION

Appendix A

Taking into account (10a), the expression

*1

ddA





 

H

becomes

1

1d

2mmjklk j

d

A

x





 Hx (A.1)

and, with 23

,

A

A deduced from 1

A

by a circular per-

mutation of ,,

x

yz a simple calculation gives



123

,,,, ,,



A

AxyzAyzxAzxy (A.2)

with



1,,,,d d

z



A

xyzxyz x y (A.3)

in which





12

22

,,



2

z

zxx yyxz yz

xxyy zxzxyzy

x

yzH HHH

HH



 











  

H





(A.4)

Using the divergence equation the first term

of (A.4) becomes

0

jj

H

j

jz

H

  while adding 2

z

zz

H





to the last two terms, they become

22

j

jzj zj

H





 

Taking into account these results,



,,

z



x

yz has a

simple expression in terms of the Laplacian and nabla

operators ∆, 



12 2

,,

zzz

xyz HH



 

H

z

 (A.5)

so that



12

1

2

,,

z

Axyz HH

dx dy











 









H

(A.6)

Substituting (A.6) into (A.2) gives finally with the ad

hoc circular permutations





12 2

dd

zzz

xxx

yyy



A

HH xy

H

Hyz

H





 



 





 





 



H

Hzx



(A.7)

Appendix B

We get from (37a) and (37b)



10

,

11

2

ir

i

AAA

BAA

kk





















(B.1)



0,

exp iexpi

tt

AB

A

AkkaBkkta













 (B.2)

in which







21

1exp

t

kk ka



 



i (B.3)

substituting (B.1) into the first equation (B.2) gives r

A

in terms of i

A

and









0

ir

AA

  

  



  (B.4)

Eliminating r

A

between (B.1) and (B.4) gives A, B in

terms of i

A

and substituting this result into the second

equation (B.2) supplies t

A

.

Al these results are valid for 1



, when 1





,



has to be changed into 2

n





 so that according to

(B.1) and (B.3)



22

10

22

21

11,

2

1exp

i

t

nknk

nk nkka









 

i

(B.5)

Finally, in a meta-slab k is transformed into k and

these relations become



22

10

22

21

11,

2

1exp

i

t

nknk

nk nkka





 



i

(B.6)

while we get from the second relation (B.2)

 







exp iexpi

tt t

A

Ak kaBk ka













(B.7)

Appendix C

Taking into account (53) and (53a) and introducing the

functions









112

exp2,exp 2miak miak





 2

(C.1)

the relations (54) become

 

2212211212

22221 22111 1212

mmmm

mm m

ABAB

kABkAB

 

 





 



 

m





 





 

(C.2)

from which we get





22

,

mm

AB in terms of





21 21

,

mm

AB



 





 

11122112 2212112221

+

11122112 2212112221

2

2=

mm

kB kkAkkB

kAkkAkkB

  

 

m







 

 (C.3)

P. HILLION

197



a

Similarly, according to (52) and (53a), the boundary

conditions at are



21zm

 

1212221 221

111211212 22222

mm mm

mmm

AB AB

kABkA

 

 

 



 



 







A

in which

m





(C.4)







1

22

exp21 i,

exp21 i

mak





1



















(C.5)

The relation (C.4) supplies in terms of



21 21

,

mm

AB









22

,

mm

AB











 

222211 112221 11222

222211 1122 21 11222

2

mm

kA kkAkk

kB kkAkk

 

  





 



 

 

m

B

(C.6)

Appendix D

TM wave propagation in a two layered film

We consider the propagation of a TM harmonic plane

wave in a two layered film made of two slabs of thick-

ness a with permittivity-permeability couples





00

,





outside the film,







112 2

,,,





in the first and

second slab respectively. Then, the refractive index



12 0,1, 2j

jj

n



 

j, the plus sign for 0

j



,

0

j



, the minus sign for 0,

jj

0





.

For the TM wave (13), (16a), the components

x

H

,

y

E of the electromagnetic field have the expressions

(31)-(36) in which we use the following notations



12

22 22

,

1,

y

knc

n

kc n

vn



















(D.1)



being either positive or pure imaginary. In addition,

the coeffficient that appears in each com-

ponent is deleted. Different situations exist according

that



exp iy

ky





is real or not and n positive implying ,

or negative with .

0k

0v0, 0v

0n

k

0,

Then, assuming jj



, ,

x

y

H

E have the

following expressions (j = 0, 1, 2)

00

0, i

zkn





 

00

exp i,expi

1exp i,1expi

ir

xixri

ir

yiiyr

HAkz HAkz

EvAkzEvA k



 i

z

(D.2)

111

0,zakn c





 : incident and reflected fields







1

111 1

1

111 11

exp iexpi

1exp iexpi

x

y

HA kzBkz

EvAkzB kz



 









(D.3)

222

2,azaknc





 : field in the second slab







2

2222

2

222 22

exp iexpi

1expi expi

x

y

HA kzBkz

EvAkzBkz



 









(D.4)

00

2,zakt nc



 : field above the two layered

film



0

exp i,

1expi

t

xt t

t

yt

HA kz

EvAk



 t

z

a

(D.5)

The boundary conditions at , give

the following relations between the

0,,2zzaz

the amplitudes A

0z



:



1

11

011

,

11

ir

AA AB

vAAvA B





 





(D.6)

z

a



:

c



 : incident and reflected fields















  

11112222

111112 2222

exp iexpiexpiexpi

1exp iexpi1exp iexpi

AkaBka AkaBka

vAkaBkavAka Bka

 









(D.7)

2za:











 



222 2

222 220

exp 2iexp2iexp 2i

1exp iexpi1exp i

tt

AkaBkaAka

vAka BkavAka









(D.8)

The fields (D.3), (D.4) are invariant under the inver-

sions corres-ponding to the

exchange slab  meta-slab so that it does not matter

whether the slabs are made of dielectric or meta-dielec-

tric, the solutions of the equations (D.6)-(D.8) will be the

same in any case.



,, ,,,,knABk nBA 

We get at once from (D.8) in terms of an arbitrary am-

plitude X





 



222

202 2

exp i,

exp i

A

vv kaX

BvvkaX



 (D.9)

S

ubstituting (D.9) into (D.7) gives

198 P. HILLION

 

















 



1111 022022

1111120220 22

exp iexpiexpiexp i

1exp iexpi1expiexp i

AkaBkavvkavv kaX

vAkaBkavvvkavvkaX



 







 







(D.10)

from which we get







111221 1122

2,;,,2 ,;,

A

kvk vXBkvkvX



 (D.11)















11 2 212121212

,; ,,expi,expikvkvvvkk avvkk a

 











(D.11a)











1

120221 2

,vvvvvv v





 (D.11b)

But we get from (D.6)











01 101 101 101 1

211 a)211 b

ir

AvvA vvBAvvA vvB  ) (D.12)

and, substituting (D.11) into (D.12a) gives





1

011122011 122

41,;,1,; ,i



X

vvkvkvvvkvkvA







 



(D.13)

Once X ob X tained from (D.13), the relations (D.11)

and (D.12b) supply respectively



11

,



A

B and r

A

. Sub-

stituting (D.13) into (D.9) gives 2



2

,



A

B and finally

t

A

is obtained from the first relation (D.8) which

achieves to determine the amplitudes of the TM har-

monic plane wave in the two layered film, a result that

does not depend, as said earlier, whether one has to deal

with slabs or meta-slabs.

We now assume that  in the first slab , is

pure imaginary giving birth to an evanes-cent wave so

that since , 1 are changed into 11

, the compo-

nents

0za

v

1

k

1

,

viik

1

x

y

H

E in this slab become with , > 0

1

k1

v





 



1

1111

1

11111

exp exp

iexpexp0

x

y

HA kzBkaz

EvAkzBkazz







a









(D.14)

{in a meta-slab where , < 0, (D.14) is defined with

1

k1

v





1

exp kz and





z

1

exp ka







}

The boundary conditions are at

0z







11 1

01111

exp ,

1iexp

ir

AA ABkaX

vAAvABkaX



 









(D.15)

and at za













  

1112222

211122222

expexp iexpi

i1exp1exp iexpi

AkaBAkaBka

vAkaB vAkaBka



 



 



 

(D.16)

Substituting (D.9) valid for a slab or a meta-slab, into

(D.16) gives









†

1112

†

1122

2exp ;,

2;,

2

,

A

kav kv

Bvkv





 (D.17)





†

122 122

12 2

;,,expi

,expi

vk vvvka

vv ka



















1

120221 2

,,,ivvvvvv v





 (D.17b)

But, we get from (D.15)











01 101 11

21i1i exp

i



A

vvAvvB ka (D.18a)













01 101 11

21i1iexp

r

A

vvAvvBka (D.18b)



(D.17a)

and, substituting (D.17) into (D.18a) gives









1

††

01 1221012221

41i;,exp1i; ,expi

X

vvvkvkavvvkvka A







 



(D.19)

Once X obtained from (D.19), the relations (D.17),

(D.18b) supply respectively 11

,

A

B and r

A

. Substi-

tuting (D.19) into (D.9) gives 2

,2

A

B and finally t

A

from the first relation (D.8).

Let us explicit the amplitudes A. Neglecting the sec-

ond term that depends on exp(k1a) in the coefficient of

Ai, the expression (D.19) reduces to





1

†

01 1221

41i;,exp



X

vv vkvka







 



(D.20)

so that according to (D.17) in which may be deleted

for the same reason,

1

B



1

1011

21 i,0

i

AvvAB





 (D.21)

and, substituting (D.21) into (D.18) gives

P. HILLION

199



1

10101

1i 1ii

A

vvvv A



  (D.22)

while according to (D.9) and to the first relation (D.8),

the amplitudes 22

,,

t

A

BA, depend on





1

exp ka.

Then, we get from (D.22) 22

ri

implying that

the two layered film behaves as a mirror giving a distorted

image because of the factor



1

01 01

1i 1ivv vv





changed into its con-juguate complex for a meta-slab.

A

A

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