Stable GSTC formulation for Maxwell’s equations

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https://hal.inria.fr/hal-03203013

Preprint submitted on 20 Apr 2021

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Nicolas Lebbe, Kim Pham, Agnès Maurel

To cite this version:

Nicolas Lebbe, Kim Pham, Agnès Maurel. Stable GSTC formulation for Maxwell’s equations. 2021.

�hal-03203013�

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Stable GSTC formulation for Maxwell’s equations

Nicolas Lebbe

⇤

_{, Kim Pham}

†

_{, and Agn`es Maurel}

‡

_,

⇤

_{Universit´e Cˆote d’Azur, Inria, CNRS, LJAD, 06902 Sophia}

Antipolis Cedex

†

_{IMSIA, CNRS, EDF, CEA, ENSTA Paris, Institut Polytechnique de Paris, 828 Bd des}

Mar´echaux, 91732 Palaiseau, France,

‡

_{Institut Langevin, ESPCI Paris, PSL University, CNRS, 1 rue Jussieu,}

75005 Paris, France,

Abstract—We revisit the classical zero-thickness Generalized

Sheet Transition Conditions (GSTCs) which are a key tool for

efficiently designing metafilms able to control the flow of light in

a desired way. It is shown that it is more convenient to use

an enlarged formulation of the GSTC in which the original

metafilm is replaced by GSTCs that exclude the layer from the

physical or computational domain. These new ”layer” transition

conditions have the same form as their ”sheet” analogues hence

they do not necessitate additional complications in their use;

their advantage is that they provide a well-posed problem

hence guaranty the stability of numerical schemes in the

time-domain. These assessments are demonstrated for an all-dielectric

structure; the effective susceptibility tensors are derived thanks

to asymptotic analysis combined with homogenization technique

and bounds for the susceptibilities entering the balance of energy

are provided. While negative constant susceptibilities appear in

the classical zero-thickness GSTCs, their values in the enlarged

formulation are always positive which ensure the stability of the

effective problem. Validation of the effective model is provided

by means of comparison with direct numerics in two and three

dimensions.

Index Terms—Metasurface, metafilms, generalized sheet

tran-sition conditions (GSTC), two-scale homogenization technique,

asymptotic analysis, numerical implementation of the enlarged

GSTCs.

I. I

NTRODUCTION

Metasurfaces are smartly engineered two-dimensional

struc-tures that offer an attractive alternative to their bulk

three-dimensional analogue. Because of their subwavelength

thick-ness, they are less lossy, more compact and of relatively ease

of fabrication. Besides these thin composite layers exhibit

great abilities for controlling the flow of light (see review

papers [1], [2], [3], [4], [5], [6]) and new perspectives are

thought using metasurfaces with spatial gradients [7], [8],

temporal variations [9], [10] and non-linearities [11]. To

describe metasurfaces, effective models have to encapsulate

the microscopic distributions of currents and fields resulting

in transition conditions relating the field differences at the

two opposite sides of the film to their mean values. Through

this homogenization process, the effects of the microscopic

distributions are encoded in macroscopic parameters which can

be interpreted in terms of effective surface currents and surface

polarisations. In their most general form, these conditions are

called General Sheet Transmission Conditions (GSTCs) [12].

The interest for effective transition conditions predates the

emergence of metafilms and the derivation of reliable models

for thin homogenous layers has given rise to a vast

litera-ture, see e.g. [13]. In a general perspective, that is leaving

Corresponding author: N. Lebbe (email: [email protected]).

aside the actual composition of the film, Idemen and

co-workers postulate the validity of the Maxwell’s equations in

the sense of distributions; doing so, they derive universal

transition conditions by introducing Dirac delta-distribution of

fields concentrated on a zero-thickness surface and provide

the first theoretical justification of the GSTCs [14], [15],

[16]. Since these pioneering works, different methods have

been developed to account for actual geometries and

arrange-ments of scattering particles in metafilms, including two-scale

asymptotic homogenization [17], [18], [19], homogenization

of local fields within dipole approximation [20], [21], and a

plethora of retrieval methods see e.g. [22], [23], [24], [25]

(a historical overview is provided in [20]). In most cases,

for both homogeneous films and metafilms, the transition

conditions are expressed across a zero-thickness interface, or

sheet. The underlying idea is that the actual thickness can

be neglected, besides it may be not well defined. Hence the

determination of an effective thickness would be submitted to

some arbitrary choice. In addition, models with zero-thickness

transition conditions clearly depart from models assimilating

a metafilm to a layer with effective bulk properties

(permittiv-ity, permeability) and effective thickness. Discussions on the

drawbacks of such attempts are provided in [26], [27] and in

[28] with the perspective of classical bulk- versus

interface-homogenizations.

The notion of an arbitrary choice for the thickness of

the film boundaries involved in the field jumps and mean

values is perfectly legitimate. Indeed, approximate models

result from an asymptotic process with some parameter ⌧ 1

measuring the subwavelength dimensions. Being conducted

up to order O( ) (in general), an infinite family of models

exists and these models are all equivalent up to terms in

O(

2 ). We shall see that shifting the film thickness from

zero to any O( ) provides such family of equivalent models.

This could be an incidental remark but it turns out that these

approximate problems can be discriminated in terms of their

well-posedness or stability. Up to now, this problem has been

addressed in the context of metafilms in elastodynamics [29],

[30] and in acoustics [31], [32] and it has been shown that

unstable formulations may foster numerical instabilities in the

time domain. Such instabilities have been reported in [33]

for homogeneous thin films, and in [34], [35] for structured

thin films. In all cases, instabilities appear when a negative

constant coefficient entering the interfacial energy is involved.

In [34], the authors conclude that negative and constant surface

susceptibilities represent a non-physical system, and thus are

not allowed (where constant means without dispersion). This

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conclusion is in close relation with the present study since we

shall see that devices as simple as dielectric homogeneous and

structured films may lead to negative constant susceptibilities

when expressed in zero-thickness GSTCs; in contrast we shall

prove that constant negative susceptibilities are not possible

when using enlarged GSTC formulation in which at least the

film thickness has been restored.

The paper is organized as follow. The main results of the

analysis are presented in §II for an all-dielectric metafilm

composed of scattering particles embedded at the interface

between two different substrates (figure 1). The form of the

effective susceptibility tensors entering in the GSTCs and

the resulting balance of energy are given when considering

enlarged transition conditions across an excluded region that

contains the particles. Explicit positive bounds for the effective

susceptibilities entering the energy balance are provided. The

derivation of the model is detailed in §III. To derive the

effective susceptibilities we use a matched asymptotic analysis

combined with two-scale homogenization. As a warm-up and

to set the main ingredients of the asymptotic procedure, we

first envision a homogeneous thin film sandwiched between

the two substrates. Next we incorporate the two-scale

homog-enization tools to deal with a film composed by scattering

particles. This procedure allows us to recover the expected

symmetry of the susceptibility tensors and to establish the

bounds thanks to minimization principles. We provide in §IV

validations of the effective problem by means of comparisons

with direct numerical calculations of the Maxwell equations in

two- and three-dimensions (the details of the numerical

imple-mentations of the direct and effective problems are freely

avail-able https://www.comsol.fr/community/exchange/841/). Some

technical calculations are collected in the appendices.

II. S

UMMARY OF THE MAIN RESULTS

A. The actual problem

We consider the Maxwell’s equations in three-dimensions

x = (x, y, z)

for dielectric materials with permittivities

de-noted "

0 "

±

r

in the two substrates, "

0 "

sc

r

in the scattering particles

and a uniform permeability µ = µ

0 ("

0 and µ

0 are the free

space parameters). The particles are evenly distributed at the

interface x = 0 between the two substrates, with spacing

d

y

along y and d

z

along z. We define S = d

y

d

z

the area

of the periodic cell and d =

p

S. In the time domain, the

electric E, displacement D and magnetizing H fields satisfy

the Maxwell’s equations in the actual problem (figure 1),

namely

rotH =

@D

@t

,

D = "

0 "

r

(x)E,

rotE = µ

0 @H

@t

,

divH = 0,

H

and E ⇥ n, continuous at the interfaces.

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e

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0

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y

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z

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Fig. 1: Actual problem of a metafilm composed by dielectric

scattering particles evenly distributed at the interface between

two dielectric substrates.

We recall that the above equations imply an

energy-conservation law expressed in a bounded domain ⌦ as

dE

dt

+

Z

⌃

S

_{· n dS = 0,}

with E =

1 ₂

Z

⌦

"

0 "

r

|E|

2 + µ

0 |H|

2 dx, S = E ⇥ H,

(2)

where E is the electromagnetic energy and S is the Poynting

vector (n is the unit vector normal to ⌃ = @⌦). In the absence

of energy flux through ⌃ that is to say if ⌃ is associated

with boundary conditions E ⇥ n = 0 or H ⇥ n = 0, the

electromagnetic energy is conserved in ⌦.

B. The effective problem

The asymptotic analysis combined with homogenization

tools provides an effective model in which the array is replaced

by transition conditions of the GSTC type (this will be detailed

in the forthcoming §III). The effective problem applies outside

the thin layer of thickness e bounded by

-

_and

+

_{, which}

has been excluded from the physical space (figure 2). As

previously said, the excluded region contains entirely the

array of particles which means that e is equal or larger than

the extend of the particles along x. Outside this region, for

x <

e

-

and x > e

+

_{, the Maxwell equations apply}

rotH = "

0 "

±

r

@E

@t

,

rotE = µ

0 @H

@t

,

divH = 0.

(3)

"

-r

"

+

r

y

z

0 x

e

n = ˆ

x

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Fig. 2: Effective problem in which the GSTCs (5) apply across

(4)

Next, across the excluded region, transition conditions have

to be specified, hence we define for any field A = D, E, H,

JAK = A|

e

+

A

|

_-e

-

,

A

av

=

1

2 ( A

|

e

+

+ A

|

_-e

-

) ,

(4)

and with the above definitions we obtain conditions of the

GSTC type, namely

JH ⇥ ˆ

x

_{K = "}

0 @P

k

@t

r

k

M

x

⇥ ˆ

x,

JE ⇥ ˆ

x

_{K = µ}

0 @M

k

@t

r

k

P

x

⇥ ˆ

x,

(5)

where ˆ

x

is the unit vector along Ox, P = P

x

x + P

ˆ

k

, M =

M

x

x + M

ˆ

k

(with P

k

· ˆ

x = M

k

· ˆ

x = 0), and the operator

r

k

M

x

⇥ ˆ

x = @

z

M

x

y

ˆ

@

y

M

x

z

ˆ

1 . The vectors P and M refer

to the electric and magnetic polarization densities, respectively.

Expectedly, we obtain zero cross-susceptibility tensors hence

we have

P =

ee

E

˜

av

,

M =

mm

H

av

,

(6)

where the subscript “av” refers to the average values of the

fields at both sides of the interface, and where

ee

, and

mm

are the electric-to-electric and magnetic-to-magnetic surface

susceptibility tensors whose forms are

ee

=

0 B

B

@

xx

ee

xy

ee

xz

ee

xy

ee

yy

ee

yz

ee

xz

ee

yz

ee

zz

ee

1 C

C

A

,

mm

= e

0 B

B

@

1 0 0

0 1 0

0 0 1

1 C

C

A

,

(7)

and where the 6 effective susceptibilities are defined in terms

of the solutions of so-called elementary problems; they are

specified in the forthcoming Eqs. (47)-(48). Next, if the

mi-crostructure is invariant when z ! z we have

xz

ee

=

yz

ee

=

0 , if it is invariant when y ! y then

xy

ee

=

yz

ee

= 0

(this is

shown in appendix C). We recover that for both symmetries

are realized, the susceptibility tensor is diagonal. Eventually,

the average electric field has to be understood as

˜

E

av

=

1 "

0 D

x,

av

x + E

ˆ

k,

av

,

(8)

as in [19], since it involves the fields being continuous at the

dominant order corresponding to vanishing effect of the thin

metafilm.

As previously commented, the excluded domain must

satis-fy two conditions. On the one hand it has to be thin which

means of the same order of magnitude than e. On the other

hand it has to be large enough in order to contain entirely the

particules. Doing so, x 2 ( 1, e

-

₎

_{and x 2 (e}

+

_{, +}

_{1) are}

composed of the substrates only both in the actual and in the

effective problems.

Two explicit parameters will be used in the following which

are the volume averages of the permittivity and of the inverse

of the permittivity within the excluded region. Denoting the

1 _{It is worth noticing that (5) along with (3) and D}

_x

_{= "}

₀

_"

_r

_E

_x

_provide

JD

x

K = "

0 divP

k

,

JH

x

K = divM

k

.

"

sc

r

e

+

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e

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"

-r

"

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r

0

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x

y

z

d

z

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d

y

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+

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-Fig. 3: Volume x 2 ( e

-

_{, e}

+

₎

_{⇥ (0, d}

y

)

⇥ (0, d

z

)

comprising

a single scattering particle, in which '

-

_{and '}

+

_{are the filling}

fraction of the particle for x < 0 and x > 0.

volume fractions '

-

_{and '}

+

_{of the particles for x < 0 and}

x > 0

respectively (figure 3), these averages read

h"

r

i

±

= 1

'

±

"

±

r

+ '

±

"

sc

r

,

h"

r

1 i

±

=

(1

'

±

₎

"

±

r

+

'

±

"

sc

r

,

(9)

and the averages over the whole excluded layer are given by

h"

r

i =

e

-

h"

r

i

-

+ e

+

h"

r

i

+

e

,

h"

1 r

i =

e

-

h"

1 r

i

-

+ e

+

h"

r

1 i

+

e

.

(10)

We now move on to the balance of energy in the effective

problem, that must be the counterpart of (2). Denoting ⌦

e

the

domain ⌦ punctured by the excluded domain x 2 ( e

-

_{, e}

+

_),

the boundary of ⌦

e

is @⌦

e

= ⌃

e

[

+

[

-

. Next, the fluxes

of the Poynting vector through

+

_and

-

_{make the GSTCs to}

appear. Specifically, we have

d

dt

(

E + E ) +

Z

⌃

e

S

_{· n dS = 0,}

E =

1 ₂

Z

⌦

e

"

0 "

r

|E|

2 + µ

0 |H|

2 dx,

E =

1 ₂

Z ⇣

µ

0 e H

av

2 xx

ee

D

2 x,

av

"

0 +"

0 yy

ee

E

y,

2 av

+

zz

ee

E

z,

2 av

+ 2

yz

ee

E

y,

av

E

z,

av

⌘

dx

k

,

(11)

where x

k

= (y, z). In the absence of fluxes through ⌃

e

, the

total energy (E + E ) is conserved. This is why, when the

effective problem is implemented numerically, the interfacial

energy E has to be positive. Indeed, a negative E fosters

nu-merical instability as E diverging to 1 can be compensated

by E diverging to +1 (an exemple of such behaviour is given

in [35]). It is worth noticing that the above balance of energy

holds for constant effective susceptibilities. When resonances

are involved resulting in frequency dependent susceptibilities

in the harmonic regime, E (being the opposite of the energy

delivered by the metasurface) can be negative as reported

in [10] (figure 6 in this reference). It is worth noticing

that for resonant metasurfaces the derivation of the

energy-conservation law has to be conducted differently (this has been

done in [36], [37] for acoustic resonances).

(5)

Coming back to (11), we shall prove that with

M

k

=

e

-h"

r

i

-+

e

+

h"

r

i

+

,

m

k

= e

h"

1 r

i,

m

?

=

e

-h"

1 r

i

-+

e

+

h"

1 r

i

+

,

M

?

= e

h"

r

i

(12)

(and h"

r

i and h"

r

1 i defined in (10)), the diagonal term of the

susceptibility tensor has the following bounds

(

0 <

M

k



xx

ee



m

k

,

0 <

m

?



↵↵

ee



?

M

,

↵ = y, z

(13)

and for non-zero off-diagonal term

yz

ee

, we also have

yz

ee

<

min

⇣p

(

yy

ee

?

m

)(

zz

ee

?

m

),

p

(

yy

ee

?

M

)(

zz

ee

?

M

)

⌘

.

(14)

It is worth noticing that (13) provides absolute bounds for

the diagonal terms of the susceptibility tensor. In contrast,

(14) does not provide absolute bounds for

yz

ee

. Rather, the

inequalities tell us that once (

yy

ee

,

zz

ee

)

are known, then

yz

ee

is bounded

2 III. D

ERIVATION OF THE EFFECTIVE PROBLEM

A. A warm-up: asymptotic analysis for a homogeneous film

The case of a homogeneous film with permittivity "

sc

r

and

thickness e

sc

_{(figure 4) is interesting because it allows us to}

introduce the tools of the asymptotic analysis that will be

used for a structured metafilm. Besides, it provides explicit

expressions of the susceptibilities. The asymptotic analysis is

conducted owing to the small parameter = e much smaller

than

the minimum wavelength imposed by the source. Note

that often, non-dimensional form of the equations are used

with x ! x/ and setting

= e/

_{⌧ 1; without loss of}

generality, we set = 1 to avoid multiple coordinate notations.

Far from the film, the fields have macroscopic variations

(at the wavelength-scale), and we define expansions for the

macroscopic fields A = H, D, E as

A =

1 X

i=0

i

_A

i

_{(x, t),}

₍₁₅₎

In contrast, within the film, rapid variations occur which are

accounted for by introducing the microscopic coordinate ⇠

x

=

x

e

. Accordingly, we define the expansions for the microscopic

fields

A =

1 X

i=0

i

_a

i

_(⇠

x

, x

k

, t),

(16)

where x

k

= (y, z)

is kept to account for slow variations along

the film. The equations satisfied by the macroscopic fields

(H

i

_{, E}

i

_{, D}

i

₎

_{are determined by inserting (15) in (1) and by}

2 _{Equivalently, we shall establish that for any (E}

_y

_{, E}

_z

_{), denoting the}

quadratic form Q.F. =

yy

ee

E

y,

2 av

+

zz

ee

E

2 z,

av

+ 2

yz

ee

E

y,

av

E

z,

av

, we have

e

h"

r

i|E

_k,

av

|

2 Q.F.

e

-h"

r

1 i

-+

e

+

h"

r

1 i

+

!

|E

k,

av

|

2 > 0.

with |E

k,

av

|

2 = (E

y,

2 av

+ E

z,

2 av

).

e

"

-

r

"

+

r

0 "

sc

r

x

y

z

e

sc

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Fig. 4: A thin film with thickness e

sc

_{and permittivity "}

sc

r

at the

interface between two different substrate is the simplest case

of layer.

identifying the terms with same powers in . As "

r

(x) = "

±

r

outside the film, (H

i

_{, E}

i

₎

_{simply satisfy}

rot

x

H

i

= "

0 "

±

r

@

t

E

i

_,

_{in X}

±

_,

rot

x

E

i

=

µ

0 @

t

H

i

,

div

x

H

i

= 0,

(17)

and there are missing boundary conditions when x !

0 ±

. Alike, the equations satisfied by the microscopic fields

(h

i

_{, e}

i

_{, d}

i

₎

_{are determined by inserting (16) in (1) with the}

differential operator

r !

1 @

_@⇠

x

ˆ

x +

_r

k

,

(18)

and identifying the terms with same powers in . The resulting

problems have to be complemented with boundary conditions

when ⇠

x

! ±1. The missing conditions on the macroscopic

and microscopic problems are provided simultaneously by

so-called matching conditions which tell us that the two

expansions have to match in some intermediate regions when

x

_{! 0}

±

_{and ⇠}

x

! ±1, specifically up to terms O(

2 )

A

0 (x, t) + A

1 (x, t)

⇠

x

!±1

a

0 (⇠

x

, x

k

, t) + a

1 (⇠

x

, x

k

, t).

To deduce the resulting conditions at each order in , we use

Taylor expansions A

0 _{(x, t) = A}

0 _{(x, t) + ⇠}

x

@

x

A

0 (0

±

, t) +

. . .

. It follows that the matching conditions read

lim

⇠

x

!±1

a

0 = A

0 ₀

±

,

(19)

at the zero-order (with A|

0 ±

= A(x = 0

±

, x

k

, t)) and at the

first-order

lim

⇠

x

!±1

✓

a

1 ⇠

x

@A

0 @x

₀

±

◆ = A

1 ₀

±

.

(20)

1) Zero-order problem: Let us start the asymptotic

anal-ysis for the homogeneous film. Because of the form of the

differential operator in (18), inserting (16) in (1) provides

equations at the dominant order O(

1 _{). Specifically, we have}

@

⇠

x

h

0 _{= @}

⇠

x

e

0 k

= 0

and @

⇠

x

d

0 x

= 0. In virtue of the matching

conditions (19), we thus have

h

0 = H

0 ₀

,

d

x

0 = D

0 _{x 0}

,

e

0 k

= E

0 k

0 .

(21)

Because of its thinness in O( ), the film is not seen at

the dominant order O(1). Hence we have to go to the next

(6)

order and we shall consider the fields (A

0 _{+ A}

1 ₎

_which

approximate the actual fields up to O(

2 _{). If we do so, it}

makes sense to restore the actual thickness of the film which

means to express the transition conditions between x =-e

-and x = e

+

_{rather than between x = 0}

-

_{and x = 0}

+

_{. As the}

macroscopic fields are continuous in X

±

_{, they can be Taylor}

expanded between x = 0

-

_{and -e}

-

_{and between x = 0}

+

_{and e}

+

_.

It follows that up to O(

2 ₎

_{we have}

A

_|

_-e

-

= A

|

₀

-

e

-

@

x

A

|

0 -

,

A

|

e

+

= A

|

₀

+

+ e

+

@

x

A

|

₀

+

,

(22)

resulting in

q

H

0 ⇥ ˆ

x

y

= e

+

_@

x

H

0 ₀

+

+ e

-

@

x

H

0 ₀

-

⇥ ˆ

x,

q

E

0 ⇥ ˆ

x

y

= e

+

_@

x

E

0 ₀

+

+ e

-

@

x

E

0 ₀

-

⇥ ˆ

x.

(23)

Expressing the above conditions in terms of the fields being

continuous at x = 0 owing to (17) we obtain

8 >

>

<

>

:

q

H

0 ⇥ ˆ

x

y

=

"

0 (e

-

"

-

r

+ e

+

_"

+

r

) @

t

E

0 k

+ e

r

k

H

x

0 ⇥ ˆ

x,

q

E

0 ⇥ ˆ

x

y

= µ

0 e @

t

H

k

0

0 +

✓

_e

-"

-r

+

e

+

"

+

r

◆ _r

k

D

_{x 0}

0 "

0 ⇥ ˆ

x.

(24)

The right hand-side terms are as small as e =

and have

to be complemented by the contributions of

q

A

1 y

_.

2) First-order problem: The useful problems at the order

O(1)

are set on (h

1 _{⇥ ˆ}

_x)

_{and (e}

1 _{⇥ ˆ}

_{x). They read}

@

@⇠

x

(h

1 _{⇥ ˆ}

_{x) =}

_"

0 "

r

@

t

E

_k

0 ₀

+

r

k

H

_{x 0}

0 ⇥ ˆ

x,

@

@⇠

x

(e

1 ⇥ ˆ

x) =

µ

0 @

t

H

0 k

0 +

r

k

D

_{x 0}

0 "

r

⇥ ˆ

x.

(25)

As the right hand side terms have a dependence in ⇠

x

through

"

r

(⇠

x

)

only, the solutions read

h

1 ⇥ ˆ

x =

(h

1 _{⇥ ˆ}

x)

⇠x= 0

"

0 Z

⇠

x

0 "

r

(⇠)d⇠ @

t

E

0 k

0 + ⇠

x

r

k

H

0 x 0

⇥ ˆ

x,

e

1 ⇥ ˆ

x =

(e

1 ⇥ ˆ

x)

⇠x= 0

+⇠

x

µ

0 @

t

H

0 _k

₀

+

Z

⇠

x

0 d⇠

"

r

(⇠)

r

k

D

0 _{x 0}

"

0 ⇥ ˆ

x.

Accounting for the matching conditions (20) and passing to

the limit ⇠

x

! ±1, the (diverging) terms linear in ⇠

x

cancel

and we obtain

8 >

>

<

>

:

q

H

1 _{⇥ ˆ}

x

y

=

"

0 (e

sc

"

sc

r

e

sc

-

_"

-r

e

sc+

_"

+

r

) @

t

E

0 k

0 ,

q

E

1 _{⇥ ˆ}

x

y

=

✓

_e

sc

"

sc

r

e

sc

-"

-

r

e

sc+

"

+

r

◆ _r

k

D

0 _{x 0}

"

0 ⇥ ˆ

x.

(26)

The final transition conditions link JAK =

q

A

0 _{+ A}

1 y

_{to the}

average fields as defined in (4). Gathering the contributions

(24) and (26) we eventually obtain

8 >

>

<

>

:

JH ⇥ ˆ

x

K = "

0 e

h"

r

i @

t

E

k,

av

+ e

r

k

H

x,

av

⇥ ˆ

x,

JE ⇥ ˆ

x

K = µ

0 e @

t

H

k,

av

+ e

h"

r

1 i

r

k

D

x,

av

"

0 ⇥ ˆ

x,

(27)

since in the present case

e

_h"

r

i = e

sc

"

sc

r

+ (e

-

e

sc

-

_)"

-r

+ (e

+

_e

sc+

_)"

+

r

.

By identifying the above transition conditions with the GSTC

in (5), we obtain

mm

= e

0 @

0

1

0

1

0

1

1 A ,

_ee

= e

0 @

h"

1 r

i

0

0 h"

r

i

0

0 _h"

r

i

1 A .

(28)

It results that the interfacial energy in (11) reads

E =

1 ₂

Z

e

⇣

µ

0 H

2 av

+ "

0 h"

r

i E

2 av

⌘

dx

k

.

The conditions (27) are those obtained in [38] in which the

authors directly implement the transition conditions in a FDTD

scheme from the Ampere’s and Faraday’s law (Eq. (4) is

our JH

z

K and Eq. (7) is our JE

z

K up to a second gradient

term), see also [13] for a comparison of several schemes

among which the unstable zero-thickness condition of [33].

The identification with (27) is made easy identifying e to the

grid step x.

B. Asymptotic analysis for a metafilms

We now move on to the case of a structured metafilm. Most

of the work has been done in the previous section and we shall

follow basically the same procedure. The additional difficulty

comes from the necessity to keep microscopic coordinates

along the three dimensions, as the evanescent field excited

in the vicinity of each particle have rapid (or microscopic)

variations in the three directions.

According to what has been said, we introduce the

micro-scopic coordinate ⇠ =

x

_{and we now choose}

_{= d}

_where

d =

p

d

y

d

z

; this provides a rescaled unit cell Y with |Y| = 1

(figure 5). Next, far from the metafilm, the macroscopic fields

are expended as in (15). In the vicinity of the film, expansions

being the equivalent of (16) are used, with

A =

X

i

_a

i

_{(⇠, x}

k

, t).

(29)

As in the classical two-scale homogenization of bulk-

meta-materials, the tangential in-plane microscopic variations fields

are handled by ⇠

k

while the macroscopic ones are handled by

x

k

. Hence, we add the classical hypothesis that the terms a

i

are ⇠

k

-periodic. It follows that analogues problems of the cell

problems in the classical homogenization will appear. We call

them elementary problems and they are now set on ⇠ 2 Y a

strip infinite along ⇠

x

with ⇠

k

2 Y (figure 5).

The macroscopic fields (H

i

_{, E}

i

_{, D}

i

₎

_{satisfy (17) as in the}

previous section. The equations satisfied by the microscopic

fields (h

i

_{, e}

i

_{, d}

i

₎

_{are determined by inserting (29) in (1) with}

now the differential operator being

r !

1 r

⇠

+

r

k

.

(30)

Eventually, the matching conditions between the

macro-scopic and the micromacro-scopic fields read as in (19)-(20). Note