Transfer matrix theory of leaky guided waves

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Transfer matrix theory of leaky guided waves

Didier Sornette, Louis Macon, Jean Coste

To cite this version:

Didier Sornette, Louis Macon, Jean Coste. Transfer matrix theory of leaky guided waves. Journal de Physique, 1988, 49 (10), pp.1683-1689. �10.1051/jphys:0198800490100168300�. �jpa-00210849�

Transfer matrix theory ^of leaky guided

^waves

Didier Sornette

(1,2),

Louis Macon

(1)

and Jean Coste

(1)

(1)

Laboratoire de Physique ^{de la Matière} condensée, ^{CNRS UA} 190, Faculté des Sciences, ^Parc Valrose, 06034 Nice Cedex, ^France

(2)

^{also at} Centre de Physique Théorique, ^{CNRS LP 014, Ecole} Polytechnique, F-91128 Palaiseau Cedex, France

(Reçu le 14 avril 1988, accepté ^{le 22} juin 1988)

Résumé. ²⁰¹⁴ On présente ^une approche générale ^{en terme} de matrice de transfert pour décrire la propagation

d’ondes guidées ^en présence d’inhomogénéités ^et de diffuseurs. Nous nous focalisons particulièrement ^{sur le} problème ^du couplage ^avec les modes de rayonnement conduisant à des fuites de l’onde guidée dans le milieu environnant. Des exemples ^sont fournis par les ondes acoustiques ^{de surface,} les excitations electromagnéti-

ques ^et acoustiques ^{ou les} ondes évanescentes proches ^d’une frontière, les ondes guidées dans les fibres

optiques... ^A partir de considérations de symétrie ^et de lois de conservation, ^nous obtenons, ^{dans le cas de} diffuseurs symétriques, ^{la forme} générale de la matrice de transfert comme fonction de quatre paramètres indépendants. Pour des réseaux périodiques unidimensionnels, les fuites disparaissent ^aux bords de bandes : cet effet cohérent provient des interférences destructives entre les rayonnements émis dans le volume par

chaque diffuseur. Ce résultat montre clairement que le problème ^{de fuite est} profondément différent d’un effet de dissipation ^usuel. Finalement, ^nous discutons la compétition ^entre la localisation d’Anderson et les effets de fuites cohérents, ^en présence de désordre. Proche des bords de bandes, la longueur d’atténuation due aux

fuites est beaucoup plus grande que la longueur ^de localisation. Nous suggérons ^{une situation} expérimentale

où ces effets pourraient être observés.

Abstract. ²⁰¹⁴ We present ^a general transfer matrix approach ^{for the} propagation ^of guided ^waves in presence of

inhomogeneities ^{or «} scatterers ». We particularly address the problem ^{of the} coupling with radiation modes

leading ^{to a} leakage ^{of the} guided ^{wave to the} surrounding bulk medium at each scattering. Examples ^are

surface acoustic waves, electromagnetic ^or acoustic excitations or evanescent electromagnetic waves near a

boundary, guided ^{waves in} optical ^{fibers... From} symmetry and conservation laws, ^we obtain, ^{in the case of} symmetric scatterers, the general ^form of the transfer matrix in terms of four independent ^real parameters. ^For 1D-periodic lattices of identical scatterers, ^we show that leakage ^{vanishes at the band} edge : this coherent effect

stems from the complete destructive interference between the converted radiations at each scatterer. This result demonstrates that coherent leakage ^is deeply different from a usual « dissipation ^{» effect.} Finally, ^we

discuss the competition between Anderson localization and coherent leakage in the presence of disorder. Near the band edges and in the presence of disorder, ^{the «} attenuation length ^{» due to the} leakage ^{is much} larger

than the localization length. ^We suggest ^an experimental situation where these effects could be observed.

Classification

Physics ^Abstracts

03.40K ^- 71.55J ^- 84.40T

1. Introduction.

Wave propagation ⁱⁿ arbitrary 1D-systems ^{has been}

the focus of intense researches

[1]

and is still the

subject ^of interesting developments especially ^when

one considers this problem ^{in novel contexts}

[2]

which can raise ^new experimental and theoretical

questions.

In this respect, ^we address here the general

problem ^{of the} competition between 1 D-wave propa-

gation and coherent leakage ^{to the bulk.} Experimen- tally, ^{no real 1D} system ^exists rigorously : ^1D- scattering is often bound to lead to a coupling ^with

bulk radiation modes in addition ^to the pure 1D- reflection. The interest in this problem ^{is also} suggested by ^the practical ^{fact that} guided ^{waves are}

often used to develop devices with many industrial

applications. ^For example, real time signal proces-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490100168300

1684

sing in radars or in TV sets use Rayleigh ^acoustic

waves propagating ^at the surface of a piezoelectric

solid

[3].

^In particular, surface acoustic waves

(SAW),

propagating ^on reflector arrays constituted of parallel grooves, have ^numerous applications

(reflective

array compressors, bandpass filters, ^filter banks, high performance ^SAW resonators, ^oscil-

lators...) [3].

^The behaviour of electromagnetic

waves near surfaces and interfaces is important ^to

the operation ^of integrated optics ^devices

[4].

^Guided electromagnetic ^{waves in} optical ^{fibers are} developed

for efficient communication, sensing ^and signal pro-

cessing purposes.

This problem ^{of the} competition between 1D-

wave propagation and coherent leakage in the bulk has been partly addressed for example ^{in the context}

of SAW in

[5]

^{within an} approach ^which integrates ^a

field theory ^{with the} theory of two-mode coupling

and also recently for dielectric waveguides ^{within a} coupled ^mode theory

[6].

^{In the} present paper, ^we extend these studies to full generality ^{within a} macroscopic model and show how to incorporate ^the specific ^{coherent nature of the} guided ^{wave to bulk}

radiation leakage in the usual transfer matrix formal- ism of 1D-wave propagation. From space reversal symmetry ^{and a} local energy flux conservation, ^we

obtain the general ^{form of the} single symmetric

scatterer transfer matrix in terms of four independent

real parameters

(compared

^{to two} parameters ^for symplectic ^matrices describing space-symmetrical

scatterers without leakage corresponding ^to the pure

one dimensional

(ID) case).

^This formalism is relevant for the optimal design ^of high quality

resonators and oscillators, ^to non-destructive testing

methods

(where

^the leakage ^{of the} Rayleigh ^wave

due to scattering by ^{a defect acts as a} signature ^{of its}

presence),

^{to the} achievement of high ^{contrasts in}

ultrasonic microscopes

(where

^{the contrast is} largely

increased by ^the leakage of excited Rayleigh ^waves by

discontinuities),

^{to surface} optical ^{devices as well}

as to modulated optical ^fibers...

Two among the four parameters of the transfer matrix have a simple meaning

(similar

^{to that for} symplectic

matrices),

^one being ^{related to the local}

value of the 1D-wave velocity and the other to the

typical

longitudinal

^scale of the 1D-scatterer. The existence and meaning ^{of two} additional parameters

can be understood physically ^as follows : the coupl- ing of the 1D-wave with the bulk implies ^{that the} propagation ^{is no} longer purely ^{1D. This means that}

one must therefore specify both the value of a scale transverse to the 1D-direction

(for

example ^{the size}

of the core in an

optical fiber)

^{which measures a}

« distance » from 1D-behaviour and a physical prop- erty of the bulk medium

(the

sheath of the core in an

optical

fiber)

^{such as its wave} velocity ^{or index of}

refraction.

From the general transfer matrix expression, ^we

obtain the main following ^{new results.} i) ^{For 1 D-}

periodic

lattices of identical symmetric scatterers, ^we show that the leakage ^{vanishes at the band} edges :

this coherent effect ^{stems from the} complete ^destruc-

tive interference between the converted radiations at each scatterer. This result demonstrates that cohe- rent leakage ^is deeply different from a usual ^« dissi-

pation ^» effect. It is interesting ^{to note that this}

cancellation of the leakage in the bulk rationalizes the observed quality ^factors

(as

high ^as

105)

^of

surface acoustic wave

(SAW)

resonators

[3, 5, 7, 8].

ii)

We discuss the competition between Anderson localization and coherent leakage in presence of disorder. Near the band edges and in the presence of

disorder, ^{the «} attenuation length ^{» due to coherent} leakage ^{is much} larger ^{than the} localization length Çlac. ^We suggest ^an experimental ^situation

[9]

^where

these effects could be observed. In section 2 we

expose the general formalism, apply ^{it to 1D-} periodic arrays in section 3 and discuss the competi-

tion between Anderson localization and the coherent

leakage in section 4.

2. Transfer matrix formalism.

Since 1D-systems ^are always topologically ^ordered

[1],

^the equations ^{for the wave} propagation in any

1D-system ^at any energy ^or frequency ^{can be cast}

into essentially ^{the same} transfer matrix formalism

[1].

To be concrete, let ^us consider a wave

amplitude

at point x ^{of the} general ^form

Yl (x )

⁼

Yl

^{eikx +} Yi e- ikx before the scatterer and

Y2 (x ) = Y2+

^{ei kx +} Yi e- ikx ^{after the} scatterer. We look for the general expression of the matrix T defined by

In absence of leakage, symmetry under time reversal

imposes ^{d = a * and c = b *}

[10].

^{In our case, this}

symmetry ^{does not} hold anymore. In the following,

we assume that the scatterers have ^a center of symmetry. ^This spatial symmetry is weaker than the usual time reversal symmetry. ^{In this case, one must} verify

which shows that T verifies 03C31

Tal

^{= T-1 where}

This

implies

^{c = - b and ad - bc = 1.}

(Note

^{that for}

a simplectic matrix, the space symmetry implies

c ⁼ b ^{* = -} b which shows that b is a pure imaginary

number b=ig

and la 12 _I 1 b 12

=1. This last re-

lation is also obtained by the conservation of the

energy flux).

The transfer matrix therefore takes the

following ^form

Det T =1 leads to

We have yet ^to write the conservation of energy flux in the leakage ^{case. In this} goal, ^{we must} specify ^the

nature of the radiation waves. Let us note

Z ( 03B8 )

^the amplitude of the bulk wave radiated in the direction characterized by ^the polar angle 03B8

[11].

^{Due to the} linearity ^{of the} equation ^of propagation, ^{we can} write, without loss of generality, ^the

following

^linear

relation between

Z(0 ), Yi

^and Yi :

Equation (3)

^{is the new} ingredient characteristic of the leakage problem where the total bulk radiation is written as the sum of the scattering ^{of the two 1D-}

waves,

Yi

^and

Y2 ,

^{which are incident on the}

scatterer. Due to the linearity ^{of the} underlying propagation equations, ^the theory ^is independent ^of

the choice

(Yl , Y2 ) :

another choice such as

(Yi , Y1)

^{can be made.} Equation

(3)

completely encaptures the role of interference effects on the coherent radiation of 1D-waves since it involves

waves amplitudes. The coherent radiation of a single

scatterer has been thoroughly ^studied theoretically

[12]

^and experimentally

[13]

^for example ^{for SAW} impinging upon ^a single groove and is characterized

by ^a complicated ^radiation emissivity diagram ^for

the

energy le(O )12 (incident

^{wave from the}

left)

^or

I e’(0 )12 (incident

^{wave from the}

right)

dependent

on the geometry of the grooves. Note that equation

(3)

^{is very} general ^{and can be} applied ^to many other situations. For symmetrical scatterers, ^one has

e’(0) = e(v - 0 ).

The total energy converted into bulk radiation is

Energy flux conservation imposes

This equation ^{must be} valid for all

Yi

^and Y1 and is therefore equivalent ^{to the matrix} equation

with

and

where

and

p, -- 0

and P2 ^are both real

(we

^consider symmetrical

scatterers).

^Note

that p2

^{is a crossed term} describing

the interference of two bulk _waves, one radiated from a

left-propagating

1D-wave and the other from

a right-propagating ^1D-wave.

Equations

(2)

^and

(4)

with the form

(1)

yield ^a non-homogeneous system ^{of two} complex ^{and two}

real relations between a, b, d, pl

and p2.

Then, ^one

could expect ^the system ^to depend only ^{on two real} parameters. ^In fact, it is both

surprising

and interest-

ing ^{to realize}

that P, and p2

^{can be} expressed

(putting (3)

^and

(4)

ⁱⁿ

(2))

^{in terms} of a and b as follows :

This shows that the initial equations,

(2)

^and

(4),

^are

not independent, ^{which was} already ^{the case in the}

absence of leakage leading ^to symplectic ^matrices.

Equations (5)

^and

(6)

permit ^{one to} determine the radiative coefficients pl

and p2

^{from the} coefficients of the transfer matrix

(say a

^and

b).

Therefore, space symmetry and the conservation of energy flux let free four independent ^real parameters

(a, b ).

^The knowledge of the 1D-reflection

(bid)

and 1D-trans- mission

( 1 /d )

^{over the scatterer} completely ^deter-

mines the system and, ⁱⁿ particular, ^{from the meas-}

ure of the 1D-propagation properties ^{one can deduce}

the radiative coefficients pl

and p2 of

^{the coherent} leakage in the bulk.

Additional constraints

(inequalities)

apply ^{on the}

four free parameters. First, p, is positive ^which yields

Second, the bulk radiation energy is also positive ^or

in other words, ^the quadratic form obtained with the matrix tT* u 3 T - 03C33 ^must be negative. ^This implies

that two additional inequalities ^must be verified

1686

A chain relation insures that if the quadratic ^form

obtained with the matrix ’T* ₍₇₃ T - 0-3 is negative

then the quadratic form obtained with the matrix

tTn* _0’3 Tn - _03C33 is negative ^{for all n. If} expression

(8)

is verified as an

equality I a 12 _ I b12 _ 1 = 0,

^one

recovers the usual simplectic ^{case without} leakage

and we are therefore interested in the ^cases where

inequality

(8)

^holds strickly.

If inequality

(9)

^holds strictly, it is easy ^to show that the energy C radiated in the bulk is always positive ^{and does not} vanish. In this _case, the energy loss per ^scatterer is of order p, in all situations : this is similar to a wave propagation in the presence of ^a

dissipation.

A more interesting ^case happens ^when

inequality (9)

^{holds as an}

equality

^{i.e. when}

It is easy ^to verify ^{that this} happens ^{if and} only ^{if the} parameter a ^{has the} following ^structure

with 0 ^real

(it

^is interesting ^{to note that this recovers}

the case studied in

[6]).

^{Let us note}

f3 = - (a* b + b* d)

^{which is} equal ^to /3==

-

(b + b *) e- icP

using

(10).

^Then, if in addition

kil) with n being ^an integer, then C reads

This shows that for a. particular ^value of the fre- quency such that

i) inequality (9)

^{holds as an} equality,

ii)

equation

(11)

is verified and

iii)

I Yi I - (- l)n e- it/> I Yïl ]

^{= 0,} then the total energy radiated in the bulk vanishes. The simultaneous

validity of conditions

i)-iii)

may ^seem drastic but we

will see in section 3 that it occurs at the band edge ^of

a periodic lattice of scatterers leading ^to expression

(10).

In the following, ^{we restrict our} discussion to the class of problems ^{for which}

equation (10)

^{holds. This}

implies pi

= P2 ^{= -}

(b

^{+ b}

*)/11 - b 12. Conversely,

if pl ⁼ P2

(which

^{occurs for} example ^{when the} emissivity diagram

e (0 )

^is symmetric ^around

7r/2:

e (7r - B ) = e ( B )

as occurs for Dirac point-like

scatterers),

then a is of the form

(10).

^{This case}

recovers the expression of the transfer matrix

(Eq.

(5)

^of

[6])

^{derived within a} coupled ^mode

theory

^of

the

propagation

^of light ^{waves in dielectric} waveguides, ^{as a} special ^{case of our} general ap-

proach. Equation

(10)

implies ^{d =}

-

(1 - b ) e-‘

^{and we} have three free parameters b = ) + I> ^and 0.

It is useful to clarify ^the meaning of the different parameters b =- 6 + i g ^and 0. ^We consider the case

of inefficient or « small » scatterers such that the

typical ^{value of the} reflection coefficient for a single

scatterer is small. This is not a restriction since most

situations fall into this class. Since

1/d

⁼

{ ( 1- b ) e- icf>} -1 ^is

^the amplitude transmission coef-

ficient, ^one

expects 11 Id I - 1

^which implies

I b I

^{1 and 0 has the} meaning ^{of a} pure propagation dephasing. ^{In a} periodic lattice of scatterers, 0 ^{is the} dephasing ^{over one} spatial period. Equation

(10)

implies that, = p2 = p ^{= -}

(b+b*)Ill -bl2 - -2 6111 -bl2. Since lbl -c

^1,

this gives ) = - p /2 (03BE is negative)

^to first order. Now,

I bld 1 2 _ 6 2 + 03BC 2

is the energy reflection coefficient. Since p

(and

therefore

6)

^{has the} meaning ^{of an} energy, it is the square of ^an amplitude and therefore is much smaller than the amplitude reflection coefficient which we identify of the order of _JL. In summary, ^one

has 03BE - p-

u 2.

^To leading order, ^{we have}

a 12 - 1 +g2-p

^{and d =}

(1 +p-igp)a*.

^{The case of}

the absence of any leakage for all values of 0 ^is

found with p ⁼ 0, leading ^{to b =} i u which confirms the interpretation of tk ^{as the} leading contribution to the reflection coefficient. The presence of leakage brings ^a negative ^real part ^{to b of}

order JL 2,

ⁱⁿ

agreement ^{with the} specific model studied in

[6].

To be concrete, consider the case of a Rayleigh

SAW of wavelength ^{A = 2}

7r Ik

propagating ^{at the}

surface of a solid and impinging upon ^a groove of width w and depth ^{h. A} perturbative-type theory ^of

the single scattering ^event

[12],

^confirmed by exper- imental measurements

[13],

shows that a portion

J.L 2

of the energy of the ^wave is reflected and a

portion p of it is converted to bulk longitudinal ^and

shear waves. u

21p --

^{1/20 with tL =}

0.6

(h /A )

^sin

(7TW I À) [13]

^where

h/A

plays ^the

role of the small parameter

(03BC = h/A

and p -

(h/A )2).

Typically, h -- ^{1 Rm} and A =3= 10 >m lead-

ing ^to

h/A

and > in the range 10-1-10- 3.

3. Case of a periodic array of scatterers.

Consider a

periodic

lattice of such identical scatterers with

period

^{A. The} general ^{wave structure in the nth}

unit cell is

[14] Yn (x ) _ { yo eIk(x - nn) + Yo

^{e- ik(x -}

nA)}

^{eiK(x - nA) e- iKx with K given by}

s e± iKA and s verifies the secular

equation

s2 -

(a + d ) s

^{+ 1 = 0} leading ^{to KA =}

cos-1 { (a + d )/2} .

The Bloch equations ^{lead to}

Yo

^{= b and}

Yo

^{= s - a. In} general, K ^is complex

(and

^{not a pure}

imaginary)

^{which reflects the}

attenuation of the 1D-wave due to leakage ^{in the}

bulk. However, ^{for the} particular values 0 ^{= n03C0}

with n integer, ^{one finds s =}

(-1 )n.

^{This corres-}

ponds

^{to the « inferior » band edges at which the}

modulus of the eigenvalue s raising ^to unity ^{reflects a} vanishing leakage. ^This striking ^{result can be recov-}

ered explicitly by computing ^{the wave} energy reflec- tion coefficient of a wave incident from the left upon

a semi-infinite periodic ^lattice

[14] :

It is then easy ^to verify ^{that R = 1 for s = ± 1. The} vanishing ^of leakage ^{at the band} edges ^{shows that} leakage ^and dissipation ^are completely ^different phenomena. Dissipation ^{is not sensitive to coherence}

whereas leakage is controlled by the coherent struc- ture of the 1D-wave. We also verify that conditions

i)-iii)

of section 2, ^{for the} leakage ^{C to} vanish, ^are

verified for the Bloch waves with s ⁼ ± 1 provided §

is negative

(which

^{is true}

since 03BE p /2).

Physi- cally, ^this coherent effect results from the Bragg phase condition which insures that the guided ^stand- ing ^{wave has the} period of the reflective array but is out of phase ^{with it.}

Let us now analyse ^more carefully ^the competition

between leakage ^and reflection in the stop

(forbid- den)

bands. Let us consider the first stop ^{band which} extends from 0 ^{= 0} to cp = - 2 03BC. ^{In the} vicinity ^of

the band edge n ^{2 7T} + ^with 0 small, ^{we obtain}

The leakage ^loss

(plg2) (2 g 10 I )1/2

^{exhibits a}

square ^root singularity as 0 --+ 0. This result allows

us to understand a result obtained previously ⁱⁿ

[5]

for SAW propagating ^over periodic arrays of grooves, within the numerical solution of ^a complex approach ^which integrates ^{a field} theory ^{with the} theory of two-mode coupling. ^{Note that our} general

formalism takes into account the complex ^structure

of the evanescent modes which may exist in the

vicinity ^{of each} scatterer. A difficulty remains in the identification of the band edge ^{with the} explicit physical values of the problem under consideration.

Indeed, ^at

ka

^{A = 2 Tr}

(Bragg

reflection

condition),

one can expect ^{to be} exactly ^{at the band} edge ^{with a}

zero loss. In fact, ^{this is not}

quite

^{true since the}

presence of evanescent modes

[15, 3]

^{created at each}

step

(which

^{are also} solutions of the Rayleigh propagation

equation)

^{lead to a local} stocking ^of

energy ^at each grooves which slightly renormalizes the effective SAW velocity according ^to _Cg ⁼

co(l - ^{K g 2) .}

Therefore, ^{kA =}

ko A (1

⁺

K M 2)

^{and if}

ko

^{A = 2 Tr,} this leads to 0 ^{= kA = 2}

Tr K M 2 with

^K

of order unity. Reporting ⁱⁿ equation

(14)

^{leads to}

1-

R = p (2 K / M )1/2.

Taking the numerical values of

[13],

M ⁼

10-2, P = 10- 3,

^{K = 1, one finds the}

total loss due to leakage of the order of 1 % which is indeed very small for scatterers which individually

leak 20 times ^more than they ^{reflect ! This} explains

the high quality ^{factors ,} 105 of SAW resonators made of two distributed reflective arrays forming ^an

effective ^« cavity ^» surrounding ^a transducer.

At the center of the stop band, 0 = - g and we

find, using equation

(13), R = 1 - p /,u

^{+ 0}

(tk 2)

indicating that the loss due to leakage ^{are of the}

order of 4. This result can be understood from a

simple ^« incoherent » argument valid also in a pure

dissipation ^{case : the} typical ^« penetration ^» length

of the wave in the semi-infinite periodic lattice, ^due

to coherent backscattering, ^is gloc =

dlg

^since

Is I = exp (- dl 61.,,) - e- ".

^{On each} scatterer, ^the power loss is roughly p =

J.L 2.

Therefore,

by adding

incoherently ^the energies ^radiated by ^each scatterers,

one finds that the total powerloss is p

gloc/ d = p/,u .

Note that a

single

^scatterer may leak much

more than it reflects

(p =

²⁰

03BC 2)

but nevertheless the total loss upon reflection of ^{a wave} incident on a

semi-infinite lattice can still be small if p iz. This type of result is also obtained when equation

(10)

does not hold, i.e., ^when expression

(9)

^{holds as a}

strict inequality.

At the other border, cp = - 2 g and R ⁼ 1 -

(2 p /,- )112 +0 (g )

and the loss are of the order of /jL ¹¹² > u. The incoherent argument ^{valid at the}

center of the stop ^{band is not valid here and} contrary

to the other band edge 0 ^{= 0 at which the} leakage vanishes, the coherence enforces a stronger

leakage.

It is interesting ^{to note that the} vanishing ^{value of}

the leakage ^{at the band} edges ^can be checked from a

direct computation of the total energy radiated in the bulk under the form of ^a radiative array.

Interference at « infinity » ^occurs between the N

leaky ^{bulk waves} emitted from the scatterers. The non-conservation of the bulk energy when changing cp ^{must be} interpreted ^{as a} reactance. It is associated with a recirculation of the 1D-wave power ^source and must therefore be understood as resulting ^from

interference effects in the 1D-wave as described

by

the above formalism. The equivalence ^between

these two view points,

(which

^is implicit

here),

^is

well established for electrical ^antennas and elec-

tromagnetic dipole radiation. Let us summarize the argument by computing C with the following toy model:

where Yn denotes the amplitude of the 1D-wave at the nth point ^{of radiation and , =} cp

{ 1- (Cg/ Cb)

^cos

9 }

^{is the} phase delay ^between

two neighbouring scatterers, cg ^{and cb} being ^the phase velocity ^{of the} guided ^{and bulk wave} respect- ively. ^{If the} Yn ^are of unit modulus and of the form

Yn ⁼

einY,

^the

sum E

ein(E + Y) is always ^zero except

n

when ^E + Y crosses 2 7T. To demonstrate that E ⁼ 0, ^{one has to} verify ^{that s + Y does not cross}

2 ^7T for 0 going ^{from 0 to 7T. Take for} example 0 = ^7T corresponding ^{to an odd band} edge ^and

1688

y ⁼ 0. Then, the condition of absence of leakage ^is

obtained as soon as

cg/cb

^{1. But, this is precisely}

the condition for a guided ^{wave to exist : at the}

surface separating ^the « 1D-medium » from the

bulk, the-radiation of a bulk wave in a direction 0 in the absence of scatterers is realized when the projec-

tion on Ox of the bulk wavevector equals ^{the 1D-}

wavevector. This is never realized in

cgICb

^{1 for}

any ^03B8: this is the general fundamental reason why guided ^{waves, surface waves or} evanescent waves

propagate ^more slowly ^{than the} corresponding ^bulk

waves. Note the loose analogy ^{with the « inhibited} spontaneous ^{emission »} discussed in

[16].

4. Competition ^between localization and coherent

leakage.

Let us now briefly discuss the influence of disorder.

The disorder can enter in the position ^{of the}

scatterers or in their reflective or radiative power.

Consider for simplicity ^{the case of a} position ^disorder parametrized by ^{7y = k 8A} and characterized by ^a

zero

average 17)

^{= 0 and a r.m.s.} amplitude

17 2)

⁼

0.- 2/ g ^(the

notation follows from a

generalized central limit theorem

[17]).

^{The T-mat-}

rix formalism is still valid. It leads to a problem ^of products of random matrices which has been well studied in the literature

[1, 2].

It is known that even

in the passing ^{bands, the} waves are completely

reflected

(«

localization ^»

regime)

by ^a semi-infinite system. ^{iK is} replaced by ^{a real} Lyapunov exponent

y

= 03BEloc-1

^inverse of the localization length ^{which is} always positive for any non-vanishing ^{disorder a. It}

remains of the order of pL ^{near the center of the} stop band whereas, ^{in the « old »} passing bands, ^{it is no} longer vanishing ^{and reads}

[1]

Equation (16)

^{recovers the} general behaviour of the localization

length g loc = y - 1 - (À / h )2 _ (. - 2

^at

small frequency ^{to. The} leakage loss appears ^to be of the same order as the localisation effect in the old

passing ^{bands as} discussed in

[6].

^{In concrete cases}

such as the SAW

propagating

^on grooves, it ^can

even dominate since p = 20 J.L 2. _However, a non-

trivial effect occurs when one approaches ^{an « in-}

ferior » band edges ^{at which} leakage ^is

vanishing

ⁱⁿ

the absence of disorder. In this region, it is known

[18, 17]

^{that the Lyapunov exponent has a non-} analytic dependence upon the disorder

Both facts therefore suggest ^that Anderson locali- zation should dominate over leakage. This expec- tation can be made more precise ^{with the} following arguments.

The scaling

(17)

^{can be traced}

[17]

^{to the fact that}

at the band edge, ^the problem becomes invariant under ^a dilatation of the lattice spacing. ^{In our case,} noting

a = (1 + b ) eiTJ = ei (TJ + JL),

^the problem ^is

invariant under a dilatation _{on tk -+ 03BC} /p ^{if the}

variance of the noise is changed according ^to

u -+ U / p 3/2.

^This implies

p = 03C3 2/3

^{and since the}

Lyapunov exponent ^{y scales} as A and therefore as p,

one recovers equation

(17).

Now, radiative losses

are governed by ^{the real} part 6 of ^b. For 6 ^{of order}

03BC 2,

^{, we obtain a} typical ^loss scaling ^as o -4/3

Another

approach

^{relies on the} computation ^of

the total energy leakage analogous ^to equation

(15)

for a localized mode centered at n ⁼ 0 :

with E defined in equation

(15).

We find that the total leakage ^{loss is} La- ⁼

7T ’Y 2/ (1 - cos £ )2 = u 4/3.

^{This can} also be reob- tained by assuming ^{that an} energy leakage ^{of order} 0’2occurs on each scatterer as suggested ^{from the} quadratic expression

(12)

for L. The total loss over a

03C3typical

distance ç loe = y

^{1 is therefore}

(T 2 ç loe =

413. This shows that the effect of the disorder ^on the

leakage ^scales as cr 4/3 whereas it scales with y =

u 2/3 for Anderson localization. The vicinity ^{of the}

band edges therefore offers a promising ^{domain for} observing clearly localization in experimental ^situ-

ations where leakage ^{loss occurs.} This result offers ^a non-trivial signature ^{of the} competition ^{between two}

coherent effects :

i)

Anderson localization and

ii)

guided ^to bulk mode conversion which has been studied in this article.

Experiments ^{on SAW are} presently being ^devel- oped ^{to test these} predictions

[9].

Acknowledgments.

We are

grateful

^{to J.} Desbois, ^J. Peyraud ^{and B.}

Souillard for stimulating discussions and particularly

J. Desbois for useful comments on a preliminary

version of the manuscript. ^We acknowledge ^financial support ^{from DRET under contract n° 86/177 for the} research program «Propagation

acoustique

^en

milieux al6atoires ».