A New Construction for Scalar Wave Equations in Inhomogeneous Media

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A New Construction for Scalar Wave Equations in Inhomogeneous Media

S. de Toro Arias, C. Vanneste

To cite this version:

S. de Toro Arias, C. Vanneste. A New Construction for Scalar Wave Equations in Inhomogeneous Media. Journal de Physique I, EDP Sciences, 1997, 7 (9), pp.1071-1096. �10.1051/jp1:1997110�.

�jpa-00247383�

A New Construction for Scalar Wave Equations in

Inhomogeneous Media

S.

de Toro Arias and

C.

Vanneste

(*)

Laboratoire de

Physique

de la Matikre Condensde

(**),

Universitd de

Nice-Sophia Antipolis,

Parc

Valrose,

BP 71, 06108 Nice Cedex

2,

France

(Received

13 March

1997,

received in final form 21

May 1997, accepted

23

May 1997)

PACS.03.40.Kf Waves and _wave propagation:

general

mathematical aspects PACS.42.25.Fx Diffraction and

scattering

PACS.02.70.-c

Computational techniques

Abstract. The paper describes _a formulation of discrete scalar _wave propagation ⁱⁿ an inho- mogeneous medium

by

the _use of

elementary

_processes

obeying

_a discrete

Huygens' principle

and

satisfying

fundamental

symmetries

such _as

time-reversal, reciprocity

and

isotropy.

Its

novelty

is the systematic derivation of _a unified

equation

which,

properly

tuned

by

a

single

parameter, leads to either the Klein-Gordon

equation

_or the

Schr6dinger

equation. The

generality

of this

method enables _one to consider its extension to other types of discrete _wave equations on any kind of discrete lattice.

Rdsumd. Cet article formule _un modkle discret de

propagation

^d'ondes scalaires dans _un milieu

hdtdrogAne

h l'aide de processus dldmentaires qui obdissent h _un principe de

Huygens

discret et satisfont h certaines

symdtries

fondamentales telles que le renversement temporel, la

rdciprocitd

et

l'isotropie.

Son

originalitd

consiste en la ddrivation

systdmatique

d'une dquation unifide

qui,

selon la valeur d'un seul

paramAtre,

conduit soit h

l'dquation

de

Klein-Gordon,

soit h

l'dquation

de

Schr6dinger.

La mdthode est suflisamment

gdndrale

_pour

pouvoir

envisager _son

extension h d'autres types

d'dquations

d'onde _sur toute sorte de rdseau discret.

1. Introduction

The formulation of _wave

propagation by

the _use of

Huygens' principle

has stimulated extensive work in the

past.

Such _an

approach

was first

pioneered by

Johns and coworkers who introduced the Transmission Line Matrix

Modeling

method

(TLM)

_11,2] to solve the Maxwell

equations

in

large electromagnetic

structures [3] which

iTas

_later extended to the

description

of diffusive processes [4]. Linear _wave

propagation

^{is mediated}

by

short

voltage impulses

which

propagate along

the bonds and _are scattered _on each node of _a Cartesian mesh of transmission lines. The

main idea behind such _a formulation is _a discrete

equivalent

of

Huygens' principle, namely

_a

principle

of

action-by-proximity obeyed by

the

pulses

when

they propagate

at each time

step

from one node to the

neighbor

nodes and the emission of

secondary

wavelets at each node _as

(*)

Author for

correspondence (e-mail: vannestetlondine.unice.fr) (**)

^CNRS UMR 6622

©

^Les

iditions

_de

Physique

1997

described

by

the

scattering

_process which radiates the incident _energy in all directions. Besides the

appealing viewpoint

^of

Huygens' principle,

the

advantages

of the method _{as a}

powerful

tool for the numerical

computation

^of wave

propagation

in

complex

^and

large

structures have

already

been

emphasized

in the TLM method [3]. A similar

approach

in which the

pulses

_were

just

scalar

quantities propagating along

the bonds of _a Cartesian lattice _was

recently

advocated for

studying time-dependent

_wave

propagation

in

large inhomogeneous

media

[5-7).

Here,

_we focus _on the nature of the _wave

equations

^which result from such _a

formulation, especially

_on the

possibility

of

retrieving

^standard wave

equations.

This

question

^has

already

been raised in

previous

works where the scalar _wave

equation,

the Klein-Gordon

equation

and Maxwell

equations

have been exhibited. In the TLM

method,

the

equivalence

between the

voltage

and current

impulses

and the electric and

magnetic

fields of Maxwell

equations

_was

initially

demonstrated _{on a} two-dimensional mesh [1]. ^The method _was then extended to three dimensions and different features have been

incorporated by

_many authors in order to describe

losses, inhomogeneous

media and

boundary

conditions [3]. Another recent formulation _[8] lead their authors to the classical _wave

equation,

^the Klein-Gordon

equation

^and to the

Schr6dinger equation. However,

due to

restricting assumptions,

these last results _were limited to scalar

waves in

homogeneous media,

and the

Schr6dinger equation

_was ill-defined at the continuum

limit.

This paper describes what is

essentially

_a formulation from "first

principles"

of such _an

approach.

The aim is to

generalize

the

previous

results to various kinds of _waves in inhomo- geneous media

by considering

the most basic

properties obeyed by

the

scattering

nodes. Such

a formulation not

only

leads to the _wave

equations already

mentioned but also results in _a

time-dependent Schr0dinger equation

which is well defined. Another

advantage

is the

possible generalization

of the method from the scalar _waves considered in this paper ^to

spinor

_{[9] or}

vector _waves.

At each time

step,

the

time-dependent

discretized _wave is defined _{as a} linear combination of incident currents at each node of the Cartesian lattice. The nature of the _wave

depends

on the number of distinct

propagating

currents _on each bond of _a

given

^node. For

example,

one or several kinds of currents

impinging

_{on one} node

by

the _same bond will describe _a

scalar,

_a

spinor

or a vector field. Since

they

need

only

_one

type

of

current,

scalar _waves

correspond

to the

simplest

situation. For the sake of

illustration, they

will be the focus of this paper. In addition to the

propagating currents,

a current is attached to each node in order to describe _an

inhomogeneous

medium. This current does not

propagate

^but

participates

in the

scattering

_process which involves the

propagating

currents. We shall demonstrate that

a few

elementary properties

and

symmetries

of the scatterers like time reversal

symmetry, reciprocity

and

isotropy

lead _to the classical _wave, Klein-Gordon _or

Schr6dinger equations.

At that

stage,

_we find that these discrete

equations

_are not

yet completely general

because the mass, the

velocity

or the

potential

remain correlated. We show that it is

possible

to _remove this limitation

by attaching

_a second current to each node.

The paper is

organized

_as follows. In Section

2,

we introduce the currents which

propagate

on the mesh of _a Cartesian lattice. Scatterers _are described

by scattering

matrices located _on each node of the lattice. The field is then defined _{as a} function of incident currents. In order for the field to

obey

_a discretized _wave

equation,

the currents must be eliminated from its time evolution. We show in Section 3 how this condition

strongly compels

the form of the

scattering

matrices. Time-reversal

symmetry, reciprocity

and

isotropy

of the scatterers _are introduced in Sections 4. 5 and 6

respectively.

The

resulting

discrete _wave

equations

are discussed in

Section 7. This leads _us to attach _a second current to each node _as described in Section 8. We

conclude

in Section 9

by discussing possible

extensions of this work.

Fig.

1. Currents propagating

along

the bonds of

a d-dimensional Euclidean lattice.

E4 54

scattering _Si )

~

E2

~ ~ ~

~

_{~ ~}

53 E~

Fig.

2.

Scattering

_process.

2. Currents and Fields

2. I. DEFINITION _{OF THE} CURRENTS. A _current is _a real _or

complex number,

which prop-

agates along

a bond of _a d-dimensional Cartesian lattice. For scalar _waves, it is sufficient to consider that each lattice bond carries _{two currents}

propagating

in

opposite

directions

(Fig. 1) [10].

^At any time

t,

the

system

is

completely

^defined

by

the values of all _currents in the lattice. We _assume that time and Space variables _are discrete. In _one time

Step

_T, a current

propagates

^between two nodes of _a

given

^{bond. All} currents _are

Synchronized.

In other

words,

the currents are

simultaneously outgoing

^from the nodes at Some time t and become incident

currents on

neighbor

nodes _at time t + T.

Each node of the lattice iS _a Scatterer. The _{Scatterers are} identical in _a

periodic System

or are

distinct,

and

randomly

chosen in _a disordered

quenched System.

Each Scatterer iS described

by

a 2d _X 2d matrix S which transforms the 2d incident currents

Et,

at any time

t,

in 2d

outgoing

currents

Sk

at the _Same time

(Fig. 2) according

to

2d

Sk

#

~ _skiEi

_k

=

1,.

2d

1=1

where the _{ski are} the matrix elements of the matrix S.

It turns out that the model iS not Suflicient to describe wave

propagation

in _an

inhomoge-

neouS medium like _a material in which the index of refraction iS _a function of the

position.

AS

shown

later,

this

problem

iS Solved

by attaching

to each node _an additional _current that will be called

throughout

this _paper the node current. For notational convenience in _{sums over} the

currents,

we Shall write the node currents

E2d+1

and

S2d+I However,

we

emphasize

that the

subscript

2d+1 iS not ascribed to a

spatial

dimension in contrast with the other indices

ranging

from 1 to 2d. The node current

participates

in the _Same

Scattering

_process aS the other _cur-

rents but does not propagate to

neighbor

nodes.

Instead,

the

outgoing

current

S2d+1

^becomes

1)

E3

#

S2d+1

E4 _E2 Scatterlng s~~ ₅₂ ~~~~~~~~~~~ ~~

____

j j)

_(~

~~ S2d+1

~~

~~

l)

time t time t time _t + T

Fig.

3. Time evolution of the currents.

the incident current

E2d+1

_on the _same node at the next time step. ^{In other}

words,

the prop-

agation

process for the node _current reads

E2d+1(t

₊

_T)

=

S2d+1(t).

We can

imagine

the node

current as

propagating

in _one time

step along

a

loop

attached to the node. This

loop

is

equiv-

alent to the

"permittivity

stub" introduced in the TLM method _to describe

inhomogeneous electromagnetic

structures

[11].

In

conclusion,

the

general

scheme of the time evolution of the currents is

suitably represented

in

Figure

3. We write

2d+1

Sk

#

~ _skiEi

_k

=

1,.

2d + 1

(1)

1=1

where the _{ski are} the matrix elements of the

(2d +1)

X

(2d +1) scattering

matrix S. This matrix is _a function of the

position

in _an

inhomogeneous

medium.

2.2. DEFINITION OF THE FIELD. The

previous

definition of the currents includes the three essential features of the model: the

principle

^of

action-by-proximity,

which is described

by

the

current

propagation

between

neighbor nodes,

the

scattering

_process in which the nodes act like

secondary

_sources ^of

"spherical"

wavelets and the

linearity

of this process. These features _can be viewed _{as a} discretized realization of

Huygens' principle. However,

additional

ingredients

and choices _are needed _to achieve the

description

of the model. For

example,

the form of the field

propagation equation

will

strongly depend

_on the choice of the

scattering

matrices.

Among

all the

questions

to be answered in _our

approach,

the first _one iS how to define the field.

The field iS

Supposed

to

obey

_a discretized linear _wave

equation

which

depends

on the

physical problem

_{we are} interested in: the scalar _wave

equation

for _an acoustic _wave, the Klein-Gordon

equation

^for a

zero-spin

massive

particle,

etc. At time

t,

the field is defined _{as a} real _or

complex quantity ((ii,12 id, t)

_on each node of the

lattice,

where

ii

12

id

_are the discrete indices which locate the node

position

_xk

#

ikl,

k

=

1,. d,

_{as a} function of the mesh

parameter

I.

For

briefness,

_we note r the set of indices

ii,

₁₂

id

of _an

arbitrary

node. There is _{no reason} for the field to be identified with _one of the currents

previously

defined.

Actually,

_we

just

postulate

that the value of the field

#(r, t)

is _a function of the incident and

outgoing

currents at node _r and at time t.

Therefore, by taking

account of

linearity,

the most

general

definition

reads

2d+1

#(r,t)

=

£ [I[(r)Ek(r,t)

+

I[(r)Sk(r,t)]

k=1

where

I[

and

I[

_are

complex

coefficients to be determined.

bond k

I

bond k

k

, ,

',

,

, i , i

' '

propagation

' _rlO~_{e Tk}

- - - M

node _r

,

t) Sp(rk,

^node rk node _r

,

t +

T)

^{t +}

T)

time t time t + T

Fig.

4. Notation for the bond

linking

the central node _r to _one of its

neighbor

nodes _rk and the

propagating

currents defined _on this bond.

The field is _a linear

superposition

of the

currents,

which vanishes when all currents vanish.

By noticing that,

due to the

scattering

_process

(Eq. (1)),

the

outgoing

currents

Sk(r,t)

_can

be

expressed

_as linear combinations of the incident currents

Ek(r, t)

at the _same time

t,

^it is sufficient to define the field

solely

_{as a} function of the incident currents

2d+1

#(r,t)

=

£ lk(r)Ek(r,t). (2)

k=I

It is necessary ^to

point

out that the

lk

_are functions of the node

position.

This

property

is

required

to describe _an

inhomogeneous system.

3. Closure of the Wave

Equation

The

general

form of the discretized _wave

equations

_{we are}

looking

for _can be written:

4(r,

t +

T)

₌

f(4(r', t'), 4(r", t"), 4(r"', t"'), (3)

where the field _at node _r and time t + T is _a function of the field _on the _same node

and/or

on nodes at

previous

times

t', t",

etc. Such _an

equation

^is

straightforwardly

derived from the discretization of the

corresponding

continuous

equation. Equation (3)

^is a closed

equation,

which _means that

f

is _a function where _{no currents} appear

explicitly (so

that

(3)

involves fields

exclusively),

and that the number of terms in the r.h.s. of

(3)

must be finite.

Usually,

and at the lowest order of the discretization

procedure,

the involved fields _are fields _on the

same node _r

=

(ii,12

_id

)

and _on the

neighbor

nodes

ii ~1,

12

~1, id

~1 at times t and t _T. In the remainder of this section _we

derive, by

_use of _our

model,

_a closed

equation

of the form of

equation (3). Also,

_we show that

imposing

the closure conditions

strongly

constrains the form of the

scattering

matrix.

For this purpose, we start to derive _an

equation

of the

type

of

equation (3)

from the evolution rules summarized in

Figure

3 of the model. First of

all,

let _us introduce _some notations. In _a

d-dimensional Euclidean _space, node _r is linked to the

neighbor

nodes

by

2d bonds. We _note rk the set of indices of the

neighbor

node which is linked to node _r

by

the bond

k, along

which the currents

Ek(r)

_or

Sk (r) propagate (Fig. 4). Moreover,

bond k of node _r is named bond k for node _rk, which

simply

_means, for

example,

that bond 2 of the central node in

Figure

3

corresponds

to bond 1 of the

right

side node

(k

=

2, k

=

1).

This notation

immediately

leads to

Ej(rk,t

+

T)

₌

Sk(r, t)

and

Ek(r,

t +

T)

₌

Sj(rk, t).

Note that for the node current

(k

₌ 2d +

1),

_we have r2d+1 # r and

I

= k.

The first

step

in

attempting

to derive _an

equation

like

(3),

is to write down the definition of the field

#(r,

t +

T)

on a node _r at time t + T

2d+1

i~(r,t

+

r)

₌

~ lk(r)Ek(r,t

+

r)

k=1 2d+1

=

~ lk(r)Sj(rk,t)

k=1

2d+1

"

2d+1

~ _~k(r) ~

_Ski

(~k)Ei(rk> t) (~)

k=I I=1

where the

propagation

and the

scattering

rules have been used.

Thus,

the field

#(r,t

₊

_T)

at time t + _T is

expressed

in terms of the incident currents at time t _on the _same node _r and _on its

neighbor

_rk. One realizes

immediate(y

that

using

the

same rules to express the incident currents at time t _{as a} function of the incident currents at time t _T and

iterating

the

procedure,

will

give

incident currents at times t T, t

2T,

on nodes _as distant from node _{r as} desired. Such _{a process} is not bounded and _{seems to}

prevent

us from

writing

a closed

equation

for the field

#.

This statement is true

except

for

some

particular scattering

matrices of the

type

we describe below. For this _{purpose, we} shall rewrite the

scattering equation (1)

in order to exhibit the field

#:

2d+1 2d+1

Sk

"

~ _sklEl

"

(skm/~m)l§

+

~ _(ski ~lskm/~m)El ₍₅₎

1=1 1=1

where _1n is any of the 2d +1 current indices.

Suppose

that the

following

condition is satisfied:

Vl _ski

Ill

" constant _{m pk}

(6)

then, equation (5)

^becomes:

Sk

"

Pkl§

and substitution in

equation (4)

leads

directly

to:

2d+1

l§(~>t

~

T)

_~

~ ~k(r)Pk(~k)l§(~k>t) (~)

k=1

which is _a closed

equation

of the

type

we are

looking

for.

It turns out that condition

(6)

is too restrictive and _can be

replaced by

Vi

#

k _ski

/~i

_# constant m pk,

(8)

Then, equation (5)

becomes

8k

~

Pkl§ #kEk (9)

where pk #

Pklk

_skk.

According

_to

(9),

the

outgoing

current _{on one} bond is not

only

_a function of the field

#

but also

depends

_on the incident _{current on} the _same bond _as sketched

Figure

5. It _seems at first

sight

that _we went backward in

writing

^condition

(8)

instead of

(6),

since a current appears

" -/1k +

pk4

Fig.

5. The

scattering

_process

depicted according

to

equation (9).

2d

_{~ l}

2d+1

~k(r )Ek(Tk,t)

§§(T,I

~

T)

_#

~ ~k(r)pp(Tk)§~(Tk,t)

k=1 ~ ~~

Fig.

6. Sketch of equation

(10).

in

equation (9)

in addition to the field

#. Actually,

substitution of

equation (9)

in

equation (4) gives

2d+1 2d+1

l§(~,t~ _T)

"

~ ~k(r)Pk(~k)l§(~k,t) ~ ~k(r)#k(~k)Ek(~k>t) (~°)

k=I k=I

which is

represented schematically

in

Figure

6. It is obvious from

Figure

6 that the 2d +1 incident currents

Ej(rk,t)

at time t

(bold arrows)

_are

deduced, according

to the

propagation rules,

from the currents

Sk(r,

t

T) leaving

node _r at time t T, aS

depicted

in

Figure

7.

Thus,

the additional currents due to the looser condition

(8) eventually

involve

only

the initial

r

Fig.

7. The

outgoing

currents

leaving

node _{r at} time t T, which

give

after

propagation

the

input

currents at time t in equation

(lo).

node _r instead of

propagating

_new terms to remote nodes.

Using again equation (9), equation (10)

becomes

2d+1

2d+1

l§(~,t

~

T) ~ ~k(r)Pk(~k)l§(~k>t) ~ ~k(~)#k(~k)Pk(r) l§(Gt _T)

k=I k=1

2d+1

~

~ ~k(~)#k(~k)#k(r)~k(r>I T). (~~)

k=1

The last term iS _a linear

Superposition

of 2d +1 incident currents upon node _r at time t _T.

In order for

equation (11)

to be _a closed field

equation

without current

terms,

it iS Sufficient

2d+1

for this

superposition

of _currents to be

proportional

to

#(r,

t

T)

₌

~ lk(r)Ek(r,

t

T).

Therefore,

_we must set k=I

Vk

=

1,.

,

2d + 1

pk(rk)Pk(r)

#

P~(r) _(~~~

where _we denote

p~(r)

_{a constant} which is _a

priori

a function of _r. In

fact,

it is

simple

to prove that

p~ (r)

does not

depend

_{on r.} To _prove this _we

begin by recognizing

that

equation (12)

_can

be written for _one of the

neighbor

nodes _rk Vi

=

1,.

,

2d +1

pi(rki)pi(rk)

#

p~(rk).

Now _rk

Plays

the role of the central node _r of

equation (12)

^and rki denotes its

neighbor

nodes.

Among

the

neighbor

nodes _rki, the initial node _r

(I

₌

k)

^{is of}

particular

interest and the last

equation

reads

#k(r)#k(~k)

~

#~(~k).

Comparison

with

equation (12)

leads _to the

required identity

Since the

node r

iS

bitrary, quation

where _now the constant

p~

iS

independent

of the node's

position.

One notices that for k

= 2d +

1,

_rk

= r,

I

=

k,

and

pj(rk)

=

pk(r).

Therefore

#2d+1(r)

=

£2d+1(r)p (14)

where

e2d+1(r)

= ~1.

Finally, utilizing equation (13), equation (11)

reads

2d+1

2d+1

l§(~,t

_~

_T)

~

~ ~k(r)Pk(~k)l§(~k,t)

_~

#~ ~ ~k(r)Pk(r)/#k(r) l~(r,t T). (15)

k=I k=1

Equation (15)

is _a closed field

equation

in which _currents do _{not appear.} As the field

#

is exhibited at times _{t + T, t} and _{t T,} this _new

equation

is _an

improvement

_over

equation (7)

in the _sense that _a discrete second time derivative

#(t

+

T) 2#(t)

+

#(t _T)

_can be obtained

as

required in,

for

example,

the discrete

time-dependent

_wave

equation.

Making

_use of condition

(8),

the

general

matrix element reads

Ski "

Pk~l #kbkl (16)

where pk #

Pklk

_skk and Ski ^is the Kronecker

symbol.

Instead of

(2d +1)2 elements,

the

scattering

matrix

depends

_{now on} the 3 _X

(2d+1) complex

coefficients _pk,

lk

and pk. These coefficients _vary

independently

from _one node to another in _an

inhomogeneous system except

^{for the} coefficients _pk which

obey

condition

(13) relating

neighbor

nodes.

Furthermore,

_p2d+1

#

~p, according

to

equation (14).

In summary,

equation (15)

is the

general _equation

which describes the time evolution of the field

#.

This closed

equation

has been derived from the evolution rules of the

currents,

the definition of i~

(Eq. (2))

and from conditions

(8, 13).

Let _us discuss the status of these conditions. Condition

(8)

cannot be avoided if _we look for _a closed

equation

in

i~,

^since a looser condition would make terms appear on all nodes of the

system. Meanwhile,

condition

(13),

which allows _one to

replace

the currents in the last _term of

equation (11 by

the field

i~(r,

_t

T),

is sufficient but _{not necessary.}

Indeed,

_{suppose we} consider that these incident currents defined

on node _{r at} time t _T are the result of

propagation

and

scattering

^of currents at

previous

times t 2T and _t 3T _on the

neighbor

nodes and _on node _r.

Reproducing

the _same calculation

steps

as those which lead to

equation (11),

the last term may be

replaced

_on the _one hand

by

_new terms

involving

the field i~ on node _r at time t 3T and _on its

neighbors

_rk at time

t 2T

respectively,

and _on the other hand

by

_{a new} linear

superposition

of incident currents _on node _{r at} time t 3T.

Therefore,

it is

possible

to recover a new closure condition

(analogous

to

Eq. (13)) by taking

this _new linear

superposition

of incident currents to be

proportional

to

i~(r,t 3T). Additionally,

the _same

procedure

_can be iterated and similar

superpositions

appear ^at times t

5T,

t

7T,

etc. _so that the

resulting

field

equation

can be closed at any

step

of the iteration

procedure.

The net result is to open the

possibility

of

describing high

order time derivatives of the field i~. ^We shall not consider these

possibilities

further and restrict

ourselves with

equation (15)

^{which is sufficient} to obtain the second order time derivative of

#.

The

description

of the model is far from

being

achieved. To make further _progress, ad- ditional

properties

of the scatterers are needed. The most basic

properties

to consider _are suitable

symmetries

of the

scattering

_process which will restrict

again

^the number of

indepen-

dent parameters and make

precise

^the

physical

content of

equation (15).

We shall

successively

introduce _some of the most natural

symmetries

_{we can} think

of, namely

time-reversal _symme-

try, reciprocity

and

isotropy.

It is worthwhile to

point

out that such

symmetries,

like the results obtained _so

far,

will

only

involve the discrete

geometry

^of the Cartesian lattice and _assume neither _any

topological

_{space nor any} Euclidean _or Riemannian structure.

Nevertheless,

_we shall need the definition of

underlying

^manifolds

provided

with such structures when

taking

later the continuous limit of the discrete _wave

equation obeyed by

the field

#.

4. Time-Reversal

Symmetry

In this

section,

we introduce the time-reversal

symmetry by formulating

its natural action _on both currents and fields.

4.1. CURRENTS. Time-reversal is

naturally completed by reversing

^the direction of the current _arrows in such _{a way} that the currents

propagate

^and are scattered backward in time.

They

will

obey

time-reversal

symmetry only

if

they

describe the

past

states of the

system

in

reverse order. It is then obvious that the

scattering

_process must be reversible.

Therbfore,

we state that the forward

scattering

process

depicted

in

Figure

8a

implies

the existence of

1)

54

E2 ~~~[~~~~

^~~

^~~ (a)

(i, j) ₊

lj)

E3 _S2d+1 83

ii,J I)

~* j~*

4

2d+1

~* /~* _scattering

~

M

(~)

~*

_~

3

j~*

^E

2d+1 ³

Fig. 8. Processes involved in the time-reversal symmetry:

a)

forward process;

b)

time-reversed

process.

the _reverse process

depicted

in

Figure

8b. Since the currents are, in

general, complex quantities,

complex conjugate

currents _are involved in the time-reversed _process. If the currents _are

real,

this will be

equivalent

to

reversing

the

original

currents.

Using

matrix

notation,

time-reversal invariance reads

(s~j

~

s(E~j

_~

(Ejj

~

s(sjj

where

(Sk)

and

(Ek)

_are the column _vectors of

outgoing

and incident currents

respectively.

Then, taking

the

complex conjugate (Ek)

#

S*(Sk),

_one obtains SS*

= S*S

=

I2d+1

where

I2d+1

is the

(2d +1)

X

(2d

₊

₁₎

unit matrix.

This condition and the

general

form

(16)

^for the

scattering

matrix elements leads to the

following

conditions

Vk

(k

=

1,.

2d +

1)

pkl(/lk "

=

~~~

2d+1

~~ ~ _~kP~

" e~' ₊ e-1~2

~"~

(~~j

Again,

let _us

point

out that all the

quantities appearing

in these relations _are functions of the node _r. This statement holds in

particular

for the constants

Cl

_# e"i and

C2

_" e"2

4.2. TIME-REVERSAL SYMMETRY _{oF THE} FIELD.

Equation (15)

^{which describes the} time evolution of the field _can be viewed _{as some}

general

relation which expresses the field

#

at time _{t + T as a} function of

#

at times t and t _T

ifi(r>^{t +}

T)

=

fiifi(rk, t), ifi(r,

t

T)I (21)

In

general, #

is

complex

valued function _so that

equation (21) obeys

time-reversal

symmetry

if the reverse-time

equation

is verified

by

the

complex conjugate

field

#*, I.e.,

ifi*(r,t T)

=

fiifi*(rk,t)>

_ifi*

(r,t

₊

_T)I (22) Therefore,

_our task is to check that the time-reversal

symmetry

^translated on

currents,

which

was considered in the

previous section,

enables _us to obtain

equation (22)

from

equation (21).

For this _{purpose, we} first substitute the definition

(2)

of the field

#

in

(21)

2d+1 2d+1 2d+1

~ _lk(r)Ek(r,

_{t +}

_T)

=

f ~ ~i(rk)Ei(rk, t), ~ lk(r)Ek(r,t T) (23)

k=1 l=I k=1

Then, Figure

8 tells _us that

reversing

the direction of current

propagation

and

substituting

the

complex conjugate

currents

S(

^for

Ek gives

^the current values of the reverse-time

system.

In other

words,

the time-reversal

symmetry applied

to the currents,

appearing

in

equation (23),

leads to

2d+1 2d+1 2d+1

~ lk(r)S((r,t T)

₌

f ~ ~i(rk)S/(rk,t), ~ lk(r)S((r,t

+

T) (24)

k=1 l=I k=1

2d+1

This

equation

is similar to

equation (22). However,

it involves the

quantities ~ _lks(

_instead

k=1 2d+1

of _~l*

=

~ _{l(E(. Therefore,}

_to demonstrate the time-reversal invariance of the field

#,

it

k=I

is necessary ^to show that these two

quantities

_are identical.

Actually, they just

need to be

proportional

to each other since the function

f

is linear.

Using

the

scattering

formula

(9),

_we

can write

2d+1 2d+1

~ _lks(

=

Aiifi* ~ _lkp(E(.

k=I k=I

Since

lkp(

~

Gil( according

to

equation (18),

one obtains

2d+1

~ _lks(

=

(Al C()#*

=

Ciifi*

k=1

where _we have used

equation (20).

However,

_we have

previously pointed

out that

Cl

is _a function of the node _r.

Therefore,

2d+1

substituting Ci#*

for

~ _lks(

in

equation (24)

leads to k=1

Ci(r)ifi*(r>t _T)

=

fiat(rk)ifi*(rk,t), Ci(r)ifi*(r,t

+

T)I

This

equation

will be

equivalent

to

equation (22)

if _we

impose

the condition

Vk

l7i(rk)

=

Ci(r).

Thus,

the constant

Cl

must be the _{same on} the node _r and _on its

neighbors

_rk, for _any node

r in the

system.

One _must conclude that

Cl

is _a constant _over the whole lattice.

Therefore,

the relation

(17)

becomes

Vr,

Vk

»klr)Pllr)/Pklr)

= e"~

(25)

4.3. FORM oF THE TIME-DEPENDENT

EQUATION.

We _now

proceed

to show that the

results obtained _so far have

already _strong

consequences on the form of the evolution

equation

of

#. First,

the coefficients of

#(r,t _T)

and

#(rk, t)

in

equation (15)

_can be recast in the

following form,

with the

help

of identities

(18, 20, 25)

2d+1

»~

i

L >k(r)Pk(r)/»k(r)

=

-»~/lcic~(r)I

k=1

and

>k(r)Pk(rk)

=

£k(r)I>k(r)llPk(rk)li»~/(CiC2(r))i~/~

where the modulus and

phases

of I and p have been

separated

in the above

expression.

The

sign ek(r)

= ~1

depends

_on the choice made in

taking

the _square root.

Plugging

these two results into

(15)

leads to the time-reversal invariant

equation

of the field

2d+1

ifi(r>t

+

T)

+

1»~/(CiC2(r))tiff(r>t _T)

=

L >k(r)Pk(rk)ifi(rk> t)

2d+1

~

iP~/(CiC2(r))i~~~ ~ £k(rjilk(rjiiPk(rkj14(rk,tj (26)

where

(p~/[CiC2(r)](

=1since (p( =

(Cl

#

(C2(r)(

= 1.

It is _now easy ^to convince ourselves that the form of

(26)

is either

fi~#(r)/fit~

=

L#(r)

_or

ifi#(r) /fit

=

L#(r)

where L is _a real

operator.

^Let us notice that the left hand side of

equation (26)

^{contains the} terms which _are needed to build the time derivatives of the field. There _are

only

two

possibilities: p2/[CiC2(r)]

= +1 _or

p~/[CiC2(r)]

= -1. If

p~/[CiC2(r)]

=

+1,

the

left hand side of

equation (26)

becomes

#(r,

t +

r)

₊

#(r,

t

r)

which allows _us to construct _a second order time derivative:

[#(r,

_{t +}

r)

₊

ifi(r,

t

r) 2#(r, t)] /r~.

If

p2 / [Cl C2 (r)]

=

-1,

the difference

#(r,t

+

r) #(r,t r)

is

directly

related _to the first order time derivative:

[((r,

t +

r) #(r,

t

r)] /2r.

The first _or second order time derivatives of the field must be the

same on the whole

lattice,

which is achieved

by setting C2(r)

=

C2.

+1)

0 o s

j~

lo _scattering

- - --

(a)

,J

ii,

_{J 1)}

E

o

scattering

_s

--

(b)

Fig.

9.

Reciprocity: a)

direct _process;

b)

reciprocal _process.

Lastly,

_we conclude that

according

to the two values _we

assign

to

p~ /(CiC2), equation (26)

becomes

2d+1

»~/(CiC2)

= +1 -

ifi(r>t

+

T)

+

ifi(r,t _T)

=

L £k(r)I>k(r)llPk(rk)lift(rk>t)

P~/(CiC2j

2d+1

= -1 -

i14(r,t

+

T) 4(r>t

_T)1

~

~ £k(r)i~k(rjiiPk(rkj14(rk,tj

as was claimed above. In

particular,

it is worthwhile to

point

out

that,

due to the square ^root of

p2/(CiC2)

in the

right

hand side of

(26),

the factor I _appears when the

equation

is of first

order in time.

In the remainder of the paper, we shall write

p~/(CiC2)

#

p~e~~~~i+~2~

= e~

(27)

With _£e

~ ~l.

5.

Reciprocity

In this

section,

we go further

by imposing

a new

symmetry, reciprocity,

_on the evolution rules of the currents. The definition is sketched in

Figure

9 where E and S _are

respectively

_an

input

and _an

outgoing

current defined _on two

arbitrary

^bonds of the _same node. We _assume that the direct process in

Figure

9a

implies

that the

reciprocal

_process in

Figure

9b also exists.

In the

reciprocal

_process, the currents E and S have the _same values _as in

Figure

9a but the two

arbitrary

^bonds have been

exchanged.

Formulated in this _way, this

symmetry

is the

counterpart

for the

scatterinj

_process, of the

reciprocity

theorem which is well known for

propagation

in

inhomogeneous

media

(see

for instance

[12-14] ).

^This is our main motivation for

introducing

it. Let _us discuss its consequences.

Using

matrix

notation,

the _process described in

Figure

9a _can be written _as

Si

_#

sikEk

where

Ek

is the

input

current and

Si

is any of the

outgoing

currents. We _can also

imagine

_a

second _process for which the

only non-vanishing input

current is incident _on bond while _we

are

looking

at the

outgoing

current on bond

k,

I.e.

Sk

#

skiEi

If both _{processes are} chosen

so that

Ek

#

Et, reciprocity

tells _us that

Sk

#

St, leading

to ski # sik. Since the two bonds k and I _are

arbitrary,

_one concludes that the

scattering

matrix is

symmetrical

s=s~.

Additionally,

the

scattering

matrix is time-reversal invariant:

S~~

= S*.

Hence,

the

scattering

matrix is

unitary.

As _a direct consequence of this

result,

the _sum of current intensities is conserved in any

scattering

_process on the lattice

2d+1 2d+1

~ _SkS(

=

~ _{EkE~ (28)}

k=1 k=1

Since the current intensities _are also. conserved in the

propagation stage,

^their total _{sum over} the whole lattice

keeps

the _same value _at each time

step.

This

property guarantees

the

stability

of the current

dynamics.

To understand its

importance,

let _us consider _a

system

^{of linear size L.}

The linear evolution of the whole set of currents _can be described

by

_a

(2d +1)L~

_X

(2d +1)L~

matrix which transforms the

(2d

+

1)L~ input

currents at time t into the

(2d +1)L~ _input

currents at the _next time

Step

t + T. Such _a linear

System

iS characterized

by

its

eigenvectors

and the

corresponding eigenvalues. Any arbitrary

initial State of the _{currents can} be

expanded

on the basis of these

eigenvectors.

Let _uS choose _one

arbitrary eigenvector,

characterized

by

its

eigenvalue (.

_as the initial state. The current intensities will

diverge

to

infinity

if

(((

> 1 or

decay

to zero if

(((

< 1. The above

result, equation (28), prevents

_us from such _a scenario since the total _sum of current intensities is _constant

regardless

of the initial State. We conclude that all the

eigenvalues

_are bound to be of modulus 1 and that any initial linear

Superposition

of

eigenvectors

iS

preserved by

the time evolution of the currents. This result iS

pleasing

because it iS

equivalent

to what iS

expected

from the

acceptable

Solutions of the usual linear _wave

equations

Such _aS the classical _{wave or}

Schrbdinger equations.

In order to break the

Stability

of the current

dynamics,

loSSeS _or interactions of the

System

with the "outside world" must be introduced in _Some way.

Note that instead of

introducing reciprocity,

_we could have

required

first the conservation law

(28),

and then

combining

this law with time-reversal

Symmetry

would have led to

reciprocity.

From _ski

# sik and

(16),

_we find

Vk, pk~i

_{= pi}

lk

or

pk/lk

# R et

where R and _{i are}

independent

of k.

Combining

this result with

(17, 18)

leads

directly

to

21

= i~ ii. From

(20)

and the

definition

(27)

of _{e~, one} obtains

2d+1

~ ~ (~~~~ ~ ~

~~~)~~~/ ~ _~k~~

~d~l (~~)

pk #

(1+ ee/p~)pkl(/ ~ _lkl(.

k=1

The matrix element becomes

12d+1

ski _#

pk~i pk&ki

# pk

(1+ e~/p~)I(~i/ £ _lkl(

_Ski

k=1

Remembering that, according

to

(18),

_pk

#

e~~21k/1(,

the

scattering

matrix is

parametrized

by

two constant

phases

₇₁ and 72, and

by

2d +1

complex

coefficients

lk

which _are functions

of the location _r.

6.

Isotropy

Isotropy

is not

required

in order for the model to be solvable.

However,

we

impose isotropy

to

simplify

the model since

anisotropic

scatterers could be considered _as well. Let _us consider _a process where

only

_one of the

propagating input

currents E~ ^{is different from} zero

(Fig. 10a).

This _process is

by

definition

isotropic

if all

outgoing

currents have the _same value

except

for

two

outgoing

currents: S~ on the

input

^bond and the node _current

S2d+1

Vj,k j,k#I, j,k#2d+1 Sj=Sk.

In other

words,

when viewed from _one

particular bond,

all

oiher

_bonds

are

equivalent except

the node bond because of its

special

status. The scatterer will be

isotropic

^{if the}

scattering

process is

isotropic

^for _any chosen

input

bond.

Another _process to consider is the _one where the node current is different from _zero

(Fig. 10b).

This _process is

isotropic

if all

outgoing

currents have the _same value

except

the node _current

S2d+1,

_1-e-

Vj,

k

j,

k

#

2d +1

Sj

=

Sk-

Actually,

both kinds of _processes lead to the _same conclusions because of the form of the

scattering

elements discussed in the

previous

section.

Vj,

k

j,

^k

#

2d + 1 pj = pk,

lj

₌

lk,

_pj

= pk

(30)

The

scattering

matrix _now

depends only

_on two

complex

parameters

Ii

and

12d+1,

and _on the two

phases

₇₁ and ₇₂

(since

_pk and pk can be

expressed

as functions of

lk according

to

(18, 29))

Conditions

(30, 13) provide

_an additional constraint for the value of _pi

First,

_suppose the value of _pi is known at node _r.

Then, according

to

(13),

the value of pi ^on the

nearest-neighbors

of node _r reads

~~

~

_{2d ~ PI}

(~k)

_"

/l~lllk(r)

"

/J~llJi (r).

Since the scatterers _{rk are} also

isotropic, jJj(rk)

=

/Ji(rk)

for k

#

2d

+1,

_we obtain Vk

#

2d + 1

jJi(rk)

"

lL~/lLi(r).

+1)

0 o

Sj

lo _scattering

s,

- --

(a)

i,jj 3) +1, j)

S2d+1

(I,J

-1)

~

/E2d+1

~~

o o

scattering s~

- --

(b)

S2d+1

Fig.

10. The two scattering _processes considered to define

isotropy.

Thus, jJi(rk)

has _a constant value _on the nearest

neighbors

_rk of node _r.

Next, starting

from nodes _{rk, we} obtain for the second

nearest-neighbors

_rki of node _r,

/J1(rki)

=

/J~llJi(rk)

=

/Ji(r).

Finally,~ proceeding

in the _{same way} from node to

node,

_we find that the whole

system

is made of two

intermingled

Cartesian lattices characterized

by

their

respective

^values _pi and

p~ lpi

It is

important

to note that this result is

peculiar

to a Cartesian

lattice,

which has been considered from

tie beginning. Actually,

the model _we have discussed _so far does not

depend

on the

type

of the discrete lattice.

Indeed,

if _we had

considered,

for

instance,

_a

triangular lattice,

a

quasiperiodic

lattice _{or even a} random lattice

(where

the node

connectivity changes

from node to

node,

_see

Appendix),

the time

dependent equation

of the field _or the form of the

scattering

matrices would not have

changed. However,

the conclusion

leading

to the two

values of _{pi on a} Cartesian lattice would need to be modified _{on an}

arbitrary

lattice. Consider

a

triangular

lattice in which the

elementary

cell contains three nodes connected

by

three bonds.

Following

_an

elementary

closed

loop

of three bonds

starting

at node _{r, we} would find after _a

complete cycle

Pi

(r)

=

P~/Pi(r)

I.e.

»1(r)

= £1»

(31)

where _El

= ~1.

Moreover,

_El should have the

same value at the three nodes and _{as a} result

over the whole

system.

As this conclusion is true for almost _any

arbitrary lattice,

we are led