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

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 equation. The results obtained in this case are generalized to the stack, taking into account the boundary conditions 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.


Introduction
The modern approach to harmonic plane wave propagation 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 approach 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 permeability.
In adddition, we start this paper with a succinct introduction 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].

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 antisymmetric 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 In these relations d d i i x   , E, H are the 1-forms and ( , B D ) the 2-forms We consider these equations in a medium with permittivity (r) (r is written for x, y, z) and constant permeability µ.
Then, let *h be the Hodge star operator [6,7]   supplying the permittivity and permeability operators *, *µ   so that the coefficients of the differential terms in (2b) are Taking into account (4), the Maxwell Equations (1a) become for harmonic fields exp(it) (5) that is according to (2a,b) and (4a) Finally, a simple calculation gives for the second set (1b) of Maxwell's equations 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 electromagnetics 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 below.

TM Field
The wave equation for the magnetic field is obtained by eliminating E from (5) which gives Now, according to (2a) and (3a), the second term on the right hand side of (8) is while in the first term (10) and, using the inverse Hodge star operator (11) in which with the Laplacian operator Subsituting ( 9) and ( 11) into (8) gives the differential form of the wave equation Let us now consider the TM field in which (z) is an in a medium with permittivity (z) depending only on z.
A simple look to (12) shows that this equation reduces to with the strong solution Once obtained the solutions of (15) and consequently H x according to (13), the electric field is provided by the 2-form (6b) with the strong solution which achieves to determine the TM harmonic plane wave E y , E z , H x .From now on, µ = 1.

TE Field
The wave equation for the electric field is obtained by eliminating H from (5) According to (2a) and (3a), the second term in (17) is while in the first term and using the inverse Hodge star operator so that with A 2 , A 3 deduced from A 1 by a circular permutation of x,y,z, a simple calculation gives and, taking into account the divergence Equation ( 7) 0 supplying  x   y by a circular permutation of x, y, z so that according to (21), (22): Substituting (18) and ( 24) into (17) gives the differenttial form of the wave equation for the electric field For the TE wave in a medium with permittivity (z) depending only on z, and, taking into account (26), this differential form has the strong solution The components H y , H z of this TE field are in terms of (29)

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(ik i z) 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 So, according to (13) and (16a), the components of the incident and reflected fieds in the half space z < 0 are and Now, in the other two intervalls (30) where  j , j = 1, 2 is constant, Equation (15) reduces to and, we shall consider the three situations 1) taking into account (30), the solutions of Equation (33) in the first situation , j = 1, 2, are with amplitudes A, B, A t 2 Remark 1.If the region with permittivity  2 above the slab is limited at z = 2a,  t (z) is chan-ged into 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) while for z > a, taking into account (34b) We now have to take into account the boundary conditions at z = 0 and z = a, imposed by the continuity of the Hx, Ey components of the electromagnetic field at permittivity jumps.
Then, according to (31), (32), (35), we get at z = 0 the two relations while at z = a, taking into account (35), (36) it comes The four relations (24(a,b)) supply in Appendix B the four amplitudes A r , A, B, A t in terms of the incident amplitude A i which achieves to determine the fields (32), (35), (36).These boundary conditions impose no constraint on frequency when all the possible values of k y are considered.
2) In the second situation, 2 1 0   , , the component k t of the wave vector is pure imaginary and, according to (36a) Remark 2: similarly to the previous remark, for a upper region bounded at z = 2a So, according to (38), the field above the slab is evanescent, does not propagate and the components (36) and, using (35) in 0 < z < a, the boundary conditions at z = a imply from which we get These relations have to be made complete with the boundary conditons (37a) at z = 0 from which A, B are provided in terms of A i and A r so that according to (41), A, B, A r , A t are ob-tained in terms of A i .Explicitly, substituting into (41) the relations (B.1) of Apendix B, we get with the  functions supplying Ar from which the amplitudes A, B, A are obtained.
In this case also, the boundary conditions impose no constraint on  when k y takes all the possible values but, the situation is different when k y = 0 for propagation in the z-direction.Then k y = k and there is a frequency band gap in the intervall ( 3) Finally in the third situation: , , the TM plane wave (13), (16a) ge-nerates in the slab an evanescent plane wave with the components H x , E y deduced from (35), (35a) by changing k into ik and it comes (for simplification, the coefficient exp(ik y y) pre- Then, according to (31), (32), (44), the boundary conditions at z = 0 supply the two relations from which we get Now, at z = a, the boundary conditions imply together with (46), this last relation supplies A, B in terms of A i which achieves to determine A r and A t according to (46), (47).
Of course, if the region with permittivity  2 is bounded In this situation also, there is a frequency band gap if  1 >  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 inside the slabs 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).
The equation (15) becomes inside the slabs since We consider the three situations (33a), 1) In the first situation , so that according to (13) and (16a) we get for the x y H E components intervening in the boun-dary conditions Now, the intervalls     and these relations made explicit in Appendix C supply . and changing m into m  gives a similar result at To achieve the determination of these amplitudes, we need the boundary conditions at where the incident field impinges on the multilayer film, and at 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 and consequently   A A are obtained in terms of 2) In the second situation, is pure imaginary and, in the intervalls and the boundary conditions (54) at z = 2ma are changed into from which we get in terms of which takes the place of (C.3) in Appendix C while in (C.6) k 2 has to be changed into ik 2 to get in terms of .,  , 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 and the boundary conditions at a are from which we get in terms of These relations take the place of (C.6) while to get in terms of , one has to change into i in (C.3).
It is implicitly assumed that the first and last slabs of the stack have the permittivity 1 Finally, at 2 z M  , 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 0   There is also a frequency band gap if  > 2  for vertical propagation.Remark 3: According to (36) the permittivity is periodic in the multilayer film so that, is a plane wave solution of the differential equation and is periodic if

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 , ,

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 permittivity, reposes on two principal techniques, both requiring 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 discussed in Remark 3, the function is not assumed periodic a-priori, leading to solutions of Maxwell's equations not taken into account in [1] so that, one may ask whether these extra-solutions play some role in propagation, specially for the frequency band gaps.What kind of incident fields is able to generate these aperiodic solutions?The third point concerns the existence of analyticcal expressions for the electromagnetic field amplitudes in each slab of the stack so that even if numerical calculations are needed to get them, they will not take the importance they have in [1] and [5].
It is easy to transpose the analysis of Sec.3 to harmonic plane waves   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 Maxwell's equa-tions have a classical behaviour provided that the refractive index n and the impedance Z are defined as As a conesquence, 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 1 2 1 n  .Taking into account these conditions, the amplitudes of the electromagnetic field inside and outside the metaslab are still supplied in Appendix B suggesting that only minor differences exist for harmonic plane wave propagation in films and metafilms.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 complex distortion factor for slabs and meta-slabs.This analysis could be generalized to a stack of alternate slabs and, with 2 3 , A A deduced from 1 A by a circular permutation of , , x y z a simple calculation gives , , Using the divergence equation the first term of (A.4) becomes Taking into account these results, has a simple expression in terms of the Laplacian and nabla operators ∆,    , , Substituting (A.6) into (A.2) gives finally with the ad hoc circular permutations

Appendix B
We get from (37a) and (37b) Finally, in a meta-slab k is transformed into k and these relations become

Appendix C
Taking into account (53) and (53a) and introducing the functions the relations (54) become a Similarly, according to (52) and (53a), the boundary conditions at are The relation (C.4) supplies in terms of x y H E have the following expressions (j = 0, 1, 2) The boundary conditions at , give the following relations between the 0 The fields (D.3), (D.4) are invariant under the inversions corres-ponding to the exchange slab  meta-slab so that it does not matter whether the slabs are made of dielectric or meta-dielectric, the solutions of the equations (D.6)-(D.8)will be the same in any case.

  
, , , , , , We get at once from (D.8) in terms of an arbitrary amplitude S ubstituting (D.9) into (D.7) gives But we get from (D.6) and, substituting (D.11) into (D.12a) gives ) which achieves to determine the amplitudes of the TM harmonic 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 1 1 , the components H E in this slab become with , > 0   (D.17a) and, substituting (D.17) into (D.18a) gives Once X obtained from (D.19), the relations (D.17 Let us explicit the amplitudes A. Neglecting the second term that depends on exp(k1a) in the coefficient of Ai, the expression (D.19) reduces to so that according to (D.17) in which may be deleted for the same reason,


being the permittivity outside the stack Using (62), (C.3) and (63a), are obtained so that the relations (51) supply r A , t A which achieves to determine the amplitudes  2 1 , the sequence of amplitudes.
from the second relation (B.2)


TM wave propagation in a two layered filmWe consider the propagation of a TM harmonic plane wave in a two layered film made of two slabs of thickness a with permittivity-permeability couples   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 being either positive or pure imaginary.In addition, the coeffficient that appears in each component is deleted.Different situations exist according that  exp i y k y   is real or not and n positive implying , or negative with .