Optical properties of anisotropic periodic helical structures

HAL Id: jpa-00209997

https://hal.archives-ouvertes.fr/jpa-00209997

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Optical properties of anisotropic periodic helical structures

C. Oldano, P. Allia, L. Trossi

To cite this version:

C. Oldano, P. Allia, L. Trossi. Optical properties of anisotropic periodic helical structures. Journal de Physique, 1985, 46 (4), pp.573-582. �10.1051/jphys:01985004604057300�. �jpa-00209997�

573

Optical properties ^of anisotropic periodic ^helical

^structures

C. Oldano (*,°), ^{P. Allia}

(+,°)

^{and L. Trossi}

(*)

(*) Dipartimento ^di Fisica, Politecnico di Torino, Italy (+) Istituto Elettrotecnico Nazionale G. Ferraris, Torino, Italy

(°) Gruppo Nazionale Struttura della Materia del C.N.R., ^{U.R. 24} Torino, Italy

(Reçu ^{le 6 août} 1984, accepté ^{le 22 novembre} 1984)

Résumé. ²⁰¹⁴ On considère un modèle optique ^{décrivant une} large classe de milieux optiquement anisotropes ^tels

que les cristaux liquides smectiques ^{chiraux et, à la} limite, les cristaux liquides cholestériques ^et les milieux homo-

gènes anisotropes. Ce modèle décrit une structure présentant ^une périodicité ^le long ^{d’un axe} donné, engendrée

par ^une rotation uniforme du tenseur diélectrique. ^Les équations de Maxwell étudiées jusqu’à présent ^seulement

pour des ^cas particuliers ^sont résolues ici dans le cas général ^{où la} direction de propagation ^{des ondes et} l’orientation de l’axe de rotation du tenseur diélectrique ^{forment un} angle arbitraire. Le processus de résolution ^{met en} jeu ^le

calcul des modes propres de l’onde électromagnétique, c’est-a-dire les ondes de Bloch propres à la ^structure pério- dique, qui ^{se réduit aux ondes} ordinaires et extraordinaires dans le cas limite d’une structure homogène anisotrope.

De la relation de dispersion des modes propres ainsi déduits ^on tire les propriétés optiques ^de cette structure sur une base générale. ^Les bandes de réflexion de Bragg ^sont formées d’une alternance de singulets ^{et de} triplets. ^En général, les bandes d’ordre pair ^{sont des} triplets ^{dont les} pics ^latéraux correspondent ^aux instabilités de Bragg

des modes propres considérés tandis que le pic ^{central est commun aux} deux modes propres ^et conduit à la réflexion totale avec échange de mode. Les bandes d’ordre impair ^sont formées de singulets ^{dont les} caractéristiques ^sont

très voisines de celles du pic central des triplets. ^Les propriétés ^de polarisation ^des modes propres ^sont étudiées dans le cas particulier d’un milieu localement uniaxe pour lequel les ondes de Bloch présentent ^un brusque ^chan-

gement de polarisation ^{à une valeur} particulière ^de l’angle ^{entre l’axe} optique ^{et l’axe de rotation.}

Abstract ²⁰¹⁴ An optical ^{model is} considered describing ^a wide class of optically anisotropic media such as chiral smectic liquid crystals and, ^{in the} limiting cases, cholesteric liquid crystals ^and anisotropic homogeneous ^media.

It describes a structure having ^a periodicity along ^a given ^axis generated by ^a uniform rotation of the dielectric tensor. Maxwell’s equations ^{for this} model, ^{studied so far} only ⁱⁿ particular cases, ^are here solved for the general

case where the direction of the propagating ^{waves and the} orientation of the dielectric tensor make an arbitrary angle ^with respect ^to the rotation axis. The resolving procedure involves the evaluation of the eigenmodes ^{of the} electromagnetic wave, i.e. the Bloch waves intrinsic to the specific periodic structure, which reduce to the ordinary

and extraordinary ^{waves in the} limiting ^{case of} anisotropic homogeneous structures. The dispersion ^{relation for}

the eigenmodes ^is deduced, allowing ^the study ^{of the} optical properties ^{of this structure on a} general ^{basis. The} Bragg reflection bands are found to be constituted alternatively by singlets ^and triplets. ^In general ^{the even order} bands ^are triplets whose lateral peaks correspond ^{to the} Bragg instabilities of each eigenmode, while the central

peak ^{is common to both} eigenmodes ^and gives ^total reflection with a mode exchange. ^{The odd order bands are} singlets whose characteristics are very similar to the central peak ^{of the} triplets. ^The polarization properties ^{of the} eigenmodes ^are studied in the particular ^{case of} locally ^uniaxial media, ^{where the Bloch waves show an} abrupt polarization change ^{for a} particular ^{value of the} angle between the optical ^{axis and the} rotation axis.

J. Physique ⁴⁶ (1985) ^{573-582 AVRIL} 1985,

Classification

Physics ^Abstracts

41.10H 2013 61.30 ^- 42.10Q

1. Introduction.

Considerable attention has

recently developed

^{for a}

wide class of

optically anisotropic

media, characterized

by

^a

periodicity along

^one direction,

generated by

a uniform rotation of the dielectric tensor around this axis.

Many

structures are known, ^{which are}

suitably

described

by

^{such an}

optical

model. The

simplest

ones are the twisted nematics and the cholesteric

liquid crystals

(N*), ^{where one of the}

principal optical

axes of the dielectric tensor is

parallel

^{to the rotation}

axis.

The general ^case

corresponds

^{however to more}

complicated

structures, i.e. ^{to some} smectic

phases.

Among

^{them, the most}

widely

^studied systems ^{are the} chiral C-smectics

(S*)

^{which are} constituted

by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004604057300

monolayers

^of

prolate-shaped

molecules whose

long

axes are tilted

by

^{a constant} angle ^with respect ^to the normal to the

layers.

^The

typical

helicoidal struc- ture of this

phase originates

^{from the presence of a}

progressive,

uniform rotation of the

long

^{axes of the}

molecules of different

planes

around the normal to the set of

layers,

which is the helical axis of the struc- ture (and ^the z-axis of a suitable reference

frame).

The rotation of molecules is

completed

^{in one full}

pitch length p,

which is here coincident with the

spatial periodicity

^{of the} structure. In cholesteric

liquid crystals,

^this

periodicity

is instead

p j2.

The optical

properties

^{of these}

anisotropic

^media

are very

peculiar

^and

interesting

from different view-

points.

Most of the _papers

published

^{so far are concerned}

with theoretical and

experimental

studies of the

propagation

^of

light along

the z-axis.

Although

^this

is

only

^a

special

case, it is in fact of great interest both in

theory

^and

application.

^A very extensive literature

exists,

particularly

^{in the case} of cholesteric

liquid crystals.

Detailed references can be found in recent

comprehensive

^reviews [1, 2].

As far as the

light propagation

^is along ^the z-axis,

smectic

liquid

crystals neither show new

optical properties

^nor pose substantial

analytical problems

with respect ^to cholesterics [3-5]. Difficulties arise instead in the case of

light obliquely propagating

^with

respect ^to the helical axis of the structure.

Berreman

numerically

solved the Maxwell _equa- tions for

S*

^{in the}

general

^{case of}

oblique

incidence,

and calculated in a direct _way the reflection spectra of model

S*

^{and N*}

samples having

surfaces normal to the helical axis

[3].

^In

oblique

incidence, the reflec- tion and transmission spectra ^{of these} structures are

rather different. In particular, ^{a second set of} Bragg

reflection

peaks

^adds up, in

S*,

^{to the set present}

in cholesteric

liquid crystals.

^This

prediction

^{has been}

experimentally

^verified [6, 7].

While

simple inspection

^{of the}

symmetries

^{of the}

starting

^set of Maxwell’s

equations

used in numerical

computation

is sufficient to

explain

^the presence and location of the reflection bands in both _cases, the number and the

polarization properties

^{of the}

peaks

of each band cannot be

adequately interpreted

^without

explicitly considering

the details of wave

propagation

in these

crystals.

This is done here

by considering

^the

electromagnetic

wave as a

superposition

^of

eigen-

modes, ^{which are the Bloch waves}

corresponding

^to

the

periodic

^{structure under} consideration. This method has the obvious advantage, ^{over numerical}

approaches,

^of

giving extremely

detailed information about

light propagation

within the

crystal.

^Further-

more, it

provides

^a

deeper understanding

^{of results}

referring

^to

specific optical properties,

^{such as}

typically

the behaviour of the reflectance as a function of the

wavelength

of incident

light.

As a matter of fact, ^this

general

approach ^{has been}

applied

^{so far to the case} of cholesteric

liquid crystals,

allowing

various features of

light propagation

ⁱⁿ

these systems

[8-12]

^to be studied in detail. The close

analogy existing

^between

S*

^{and N*} structures sug-

gests ^{that a similar}

approach

should be suitable also in the case of chiral smectic

crystals.

However, ^the

striking

differences observed ⁱⁿ

light propagation

between

S*

^{and N*} structures

imply

^{that the}

study

of the

S* crystals

^{cannot be}

merely

considered as a

straightforward

extension of the formalism

developed

to

investigate

cholesteric

liquid crystals,

but deserves

a

fully independent

^treatment.

To our

knowledge,

^the

only existing theory

^for

S*

structures,

explicitly following

this line of

approach,

was

proposed by Taupin et

^al. [13], who established

a set of

equations

for the coefficients of Bloch waves.

However, ^{this set was solved}

only numerically by

means of an iterative

procedure.

^{In the} present paper, the

dispersion

relations of the

independent eigen-

modes of the

problem

^{are instead}

given

in the form of

an

analytical

characteristic

equation,

which is derived

by taking

^{into account all} the essential

symmetries

of the

problem.

^The

optical properties

^{of the structure}

are then discussed on a much firmer basis. In

particular

the features of the reflection bands are

fully analysed

and their most

important polarization properties

evidenced.

2. Propagation equations.

Let us consider a

sample

^{of a} chiral smectic

liquid crystal having

planar surfaces normal to the helical axis. Let (x, y, z) ^{be a} reference frame such that the

sample

^{surfaces are}

parallel

^{to the} (x,

y)

plane, ^the

helical axis of the

liquid

crystal

being along

the z-axis.

The

optical properties

ⁱⁿ

S*’s

^are determined, ^at

any

point, by

the local dielectric tensor, i.e.

by

^its

principal

^values E1, E2, 83 and

by

^the

corresponding principal

directions, which define a local frame (1, 2, 3).

Transformation from (x, z, z) ^to (1, 2, 3) ^is

performed by properly rotating

(x, y, z)

by

^{the set of Euler}

angles

(0,

4>, 03C8)

defined in reference [3].

Specifically,

three successive rotations are involved,

as shown in

figure

^{1 :} a) By 0 around the x-axis.

b)

By ql

around the 3-axis. c)

By 0

around the z-axis.

In

S*

^{both 0}

and 03C8

^{have constant} values,

while

is

proportional

^{to z,}

going through

^{2 nc} radians in one

full

pitch length,

p ; i.e.

~>

⁼ qz

with q

⁼

2 nIp.

^The

components ^{in the} (x, y, z) ^frame of the dielectric tensor have been evaluated

by

^Berreman [3]. ^{In the} present paper it is

assumed ql

^{= 0. For} these condi-

tions, the dielectric tensor

E(~)

^{can be} written, ^when

4> = 0 as :

575

As a consequence, ^{E can} be

expressed,

^for

arbitrary 0

values as :

where

The structure of this tensor differs from that for cholesteric

liquid crystals

because of the terms Exz, Byz, ^which are nonzero in the present ^{case. Let us notice} that

precisely

^{these terms are} characterized

by

^a

different

periodicity

^with respect ^to the other z-

dependent

elements of the tensor. This feature has relevant consequences ^on the

degree

^of

complexity

of the formalism needed to

study light propagation

in

S*. According

^{to Berreman}

[3],

^Maxwell’s

equations

for

plane

^{waves in}

S*

^{can be written as}

where c and m are the

velocity

^and

frequency

^{of the}

incident

light,

^T is the column 4 vector and d is

the 4 x 4 matrix :

By

making

^{use of}

equations

(1) ^and (2), the elements of A can be written :

where

ni is the refraction index of the external medium

and 0;

the

angle

of incidence of

light

^{on the}

sample’s

^surface.

In addition :

Equation (4)

^{can be solved}

numerically [3].

However,

a

deeper knowledge

of the details of wave

propagation

in

S*,

ⁱⁿ

spite

^{of the}

complexity

^{of their structure and}

of the

corresponding

mathematical formalism, may be obtained

by explicitly taking

^{into account the} existing

symmetries

^{of the}

problem.

^In

particular,

the transla- tional symmetry ^{of the structure}

along

any direction

perpendicular

^{to the helical axis} suggests ^{that the} solutions of

equation

(4), ^for

plane

^wave

propagation

ⁱⁿ

the (x, z) plane, will take the form :

where both E(z) ^and

H(z)

^are

only

functions of z.

The

wavevector kx

is related to m ⁼ ni sin

0; by

^the

relation m ⁼

kx c/m.

By

using

^the

following

relations for

Hx

^and

Hy (easily

obtained from

equations

(4) ^and

(5)) :

the set of linear

equations

(4) ^can be reduced to a

pair

^of

coupled,

second-order differential

equations

^{for the}

field components

Ex, Ey :

The

periodicity

^{of the structure}

along

the z-axis allows the solutions of

equation

(11) ^to be written in the form :

i.e. as

superpositions

of Fourier components,

according

^{to the}

Bloch-Floquet

^{theorem. In}

equation

(12)

kz

^{is the}

component,

parallel

^to the helical axis, ^of

the

wavevector of incident

light.

Substitution of

equation (12)

^{in the}

coupled

difterential

equations (11) gives

^{rise to an infinite set of}

algebraic equations

^for the unknowns

(x,,, y,,) :

where

The form of

equation (12)

is such that in

equations

(13)

all the coefficients of the unknown

(xn, y")

^{are real}

functions

of kz

if the medium is non

dissipative,

^{i.e. if the}

dielectric tensor is real. The infinite set (13) ^{of homo-}

geneous

equations

for the unknown

quantities (x", y,,)

admits nontrivial solutions if and

only

^{if the deter-}

minant of coefficients, D,

regarded

^{as a function of the}

complex

^variable

kZ,

^is

equal

^{to zero.} This condition

gives

the secular

equation,

from which both inde-

pendent eigenvalues

^{can be}

simultaneously

^obtained

for _any value of the

frequency

of the incident

light.

The set

(13)

^is

remarkably

^more involved than the

three-diagonal

^set

corresponding

^{to the case of}

cholesteric

liquid crystals.

^{For this reason the ad hoc}

methods used in references

[9,

10] ^{to find the}

eigen-

values of the

three-diagonal

^set

corresponding

^{to the}

case of a N* cannot be

directly applied

^{to the more}

general

^{case of a}

Sg.

^{In reference}

[13)

^a

purely

^numeric

iterative

procedure

^is

suggested.

This method is

clearly

^not suitable in the cases where the

eigenvalues

of the

equation

system ^are

degenerate

^or

nearly degenerate,

^{as for instance near} the boundaries of the reflection bands.

In the next section a

simple analytical

form for the secular

equation

^is

given.

^{In this}

equation,

which is the characteristic

equation

^{of the}

problem,

all essential symmetry

properties

^{of the}

problem

^are

clearly

^shown.

This is the main

advantage

^{of the} present ^{method over} all

previous

attempts. The coefficients which _appear in the characteristic

equation

^are

straightforwardly

deduced from the coefficients of the recurrent relations

(13),

^thus

avoiding

any iteration

procedure.

3. Characteristic

equation

^{for the} eigenmodes.

The

problem

^of

obtaining

^the

dispersion

^relations

kz(w)

^{for the}

eigenmodes (xn, yn)

^of

electromagnetic

waves

propagating

within the

S*

^{may be}

^suitably

reduced to the

analysis

of the solutions of a

simple biquadratic equation

for the variable sin nk, ^where

k ⁼

k2/q.

^{This is} the central

point

^{of the} present

approach,

^and represents ^a remarkable

improvement

with respect ^to conventional methods of numerical

computation.

The characteristic

equations

^reads

where the functions U and V can be obtained, ^{in terms}

of

D(k), through

^standard

application

^{of an ad hoc} formalism,

expressly developed

^{to solve the}

problem

of

light propagation

ⁱⁿ

periodic

structures, whose main features are summarized in the

following.

^{For further}

mathematical details, the reader is referred to paper

[11]

]

and to references therein. First, ^the convergency of the

577

coefficients’ determinant

D (k)

^{is ensured}

by

^alterna-

tively dividing

^{the recurrent} relations,

(13a)

^and

(13b) by

^the

quantities An

^and

B., respectively.

^{One obtains a}

new coefficients’ determinant

D * (k)

in which all the main diagonal ^elements

Dn

^are unity.

Two

auxiliary

^functions

da and db

may ^now be defined as

where both limits (k -->

ka, k -+ k6)

^{can be shown to}

exist and to be

generally

^{nonzero, since}

D *(k)

^turns

out to be an

analytical

function of the

complex

variable k, with four infinite sets of

simple poles

^{on the}

real k-axis, ^{for k =}

± kp

^{+ n and k =}

1: kb

^{+ n} (n =

integer).

^In particular, D * (k)

diverges

^as

1 l(k - k.,b) when k -> ku,b.

Finally,

^{the U and V functions are}

given by

In

principle,

^{the four}

independent

^roots of equa- tion

(15) give

^{rise to} four infinite sets of k values,

which are the

eigenvalues

^{of the}

problem :

However a change ^{of n}

merely corresponds

^{to a re-}

indexing

^{of the unknowns} x,,,

y,, and kt, Ki

^corres-

pond

^{to waves}

propagating

forwards and backwards

respectively.

^{As a} consequence, for each

point

^{of the} (w,

0;) plane,

^{i.e. for each}

frequency

of incident

light

and each

angle

of incidence

Oi, only

^two

really

^inde-

pendent

^{modes are} excited,

corresponding

^{to the}

eigenvalues k1 and k2 respectively.

It can be

easily

shown that three different types ^of solutions of the characteristic

equation (15)

may ^occur,

depending

^on the values of the coefficients U(co,

Oi)

and F(co,

Oi). Actually,

^the

resulting eigenvalues k,,2

may

independently

^be either real or

complex.

^A

real k

corresponds

^{to a}

propagating

^{wave i.e. to a stable}

solution. A

complex ki corresponds

^{to an attenuated}

wave, i.e. to an unstable solution. Two different types ^of instabilities can occur

depending

^on the value taken

by

^{the real} part ^{of k} which,

according

^{to reference} [11]

are referred as B- and

C-type

instabilities

respectively.

In both _cases, the

instability

^{occurs where the waves}

reflected from

layers

^{which are one}

pitch

apart ^add

coherently, giving

^{rise to a} strong reflection. In case B, only ^{one mode is} involved, the incident and reflected

waves

having opposite k

values. The real

part k’

^{of k}

therefore satisfies the Bragg condition : 2 k’ = 0,

± 1, ± 2, ^{... In case C} instead, the coherent reflection is the one that gives ^{rise to a mode}

exchange,

^{so that both}

eigenmodes

^are

necessarily

involved, and total reflec-

Fig. ^{1. - Euler} angles ^{of the} dielectric tensor principal ^axes (1, 2, 3) ^with respect ^{to the} sample (x, y, z) ^frame (after Berreman, ^Ref. 3).

tion of incident

light

^{occurs. In this case, the}

polariza-

tion states of the incident and reflected

light

^are

almost

opposite (either

^{from mode 7r to mode} Q, ^or vice _versa, as

typically

ⁱⁿ

S*

^{and in N* at}

sufficiently high

^incidence

angles,

^{or from left to}

right

^circular

polarization,

^{or vice versa, as in N* at lower incidence}

angles).

^{This case is}

mathematically

characterized

by

^a

negative

value of the discriminant of the characteristic

equation (15),

^{so that the}

Bragg

condition is no

longer

satisfied. A

physically quite

similar situation occurs

if both

eigenmodes simultaneously undergo

^a

B-type instability

with total reflection of the incident

light.

In this _case, too, ^the

polarization

^{state of} the incident and reflected beams ^can be _very different.

A more detailed discussion of

equation (15)

^{and a}

complete

classification of its solution ^can be found in reference

[12].

4. Structure of the Bragg reflection bands.

The formalism

developed

^{in the}

previous

^sections

allows one to

perform

^{a detailed}

analysis

^of

light propagation

^{in a} wide class of

liquid crystals. Actually, setting

^{0 =}

n/2

^{in the}

defining equations

^{for the}

dielectric tensor is

equivalent

^to

considering

^a

perfect

cholesteric

liquid crystal.

^On the other hand,

by setting

⁰⁼⁰

and 81

⁼ 82 ^one obtains a homogeneous

medium. Intermediate values of the tilt

angle

⁰

obviously

^describe different chiral smectic

liquid crystals.

Within the present framework, ^{it is then}

possible

^{to follow in} great detail the behaviour of the

dispersion

relations of a set of

S*, by continuously varying

^a

single

parameter, 0, between the limits above described.

Typical examples

of the results which ^can be extracted from the present

approach

^are reported

in the

following.

First the solution of the characteristic

equation

^are

analysed

and both the real and

imaginary

part ^{of the}

independent eigenvalues k,,2

^are

^plotted

^as

functions of

plA

⁼ wlqc. ^A reduced Brillouin zone

scheme is used throughout. ^The

principal

values of the dielectric tensor

always

are 81 ⁼ B2 ⁼ 2, E3 ⁼ 3.

These values are coincident with those used in Berre- man’s _paper

[3]. Figures

^{2 to 5} describe the

dispersion

relations of a set of different structures with

decreasing

tilt angles (0 ⁼ 90°, 75°, 45°, 150), ^for

obliquely

^inci-

dent

light.

For the sake of

simplicity,

and in order to evidence the

dependence

^on 0 of the

dispersion

relations, ^the

angle

of incidence

Oi

^was

kept

^constant

and

equal

^to 60° for all curves.

Figure

2 shows the

typical

behaviour of the

disper-

sion curves of a

perfect

cholesteric

liquid crystal

(0 ⁼ 90°) ^at

high

^incidence

angle

^{and for a}

frequency

range

corresponding

^{to the first two} reflection bands.

Each band is a

triplet,

whose external

peaks

^{are due}

to the

Bragg

reflection of each one of the two inde-

pendent

eigenmodes

^(and

correspond

^{to the} B-type

instabilities of the two

dispersion

curves), ^{while the}

central

peak

^{is a} total reflection

peak (corresponding

to the

C-type instability,

^{which is common to both}

dispersion

^curves; notice that the real part ^{of k} does not

satisfy

^the Bragg condition, ^{i.e. 2} k’ :A

integer).

Fig. ^{2. - Real and} imaginary part, k’ ^{and k",} of the z-compo- nent of the reduced wavevector versus p/ À. ^{for the} following

set of parameters: 81 ^{= E2 =} 2, 83 ⁼ 3, ^tilt angle ^{8 =} n/2,

incidence angle Oi ⁼ 60°. B and C indicate the type ^{of the} instability regions, n ^{and a the dominant} polarization

states of the eigenmodes.

Fig. ^{3. - Same as} figure 2, ^{with 0 = 75°. The indices of B} and C refer to the order of the Bragg reflection bands.

In

figures

3-5, ^{where 0 :0 900 a new set} of reflection bands is observed to add _up to the pattern

analysed

in

figure

^{2. This new set of}

instability regions

^derives,

as is indeed

expected,

^{from the terms of the}

starting equations,

which contain the

periodicity 0

^{= qz in}

addition to the

periodicity 0

⁼ 2 qz. As a consequence, the

dispersion

relations of an

S* crystal

^at

oblique

incidence look rather more

complicated

^{than those}

of a N* in the same conditions. As a matter of fact,

the set of

peaks

^of k",

intrinsically

associated with the

periodicity

^qz, becomes of

increasing

relevance with respect ^{the other set} when 0 decreases. For 0 ⁼ 15°,

corresponding

^{to a}

nearly homogeneous

medium,

only

the first

peak

of k" is

significantly

present, the

dispersion

^{curves for k’}

being

^rather

simple,

^as

expected.

The

instability regions

^{intrisic to}

S*

possess ^remar- kable features. These are in fact isolated _pure

C-type

instabilities, while those associated with the 2 _qz

periodicity

^are

composed

^of B-C-B, ^or B-(B ⁺ B)-B

triplets

[11]. ^{In other} words, ^no additional B-type

reflection

peaks

turn out to be related to the _qz

periodicity

^{of the} system. ^{It seems}

interesting

^to

analyse

^{the reason} for such behaviour.

Let us consider for instance the first

peak of figure

^3.

At sufficiently ^low

frequencies,

i.e. well before the first

instability region,

^{the two}

independent

branches of the

dispersion

^relations

correspond

^{to almost}

linearly polarized eigenmodes.

^In

particular,

^the upper branch is associated with mode Q, the lower one to mode n.

This

polarization

^state is well defined and

preserved

until the first

instability region

^is

approached.

^{At a}

given

^value

of plA,

^the

upper k

^branch

actually

^crosses

the line k’ ⁼

1/2.

^{However no} B-type ^band appears, the

corresponding

^k"

remaining identically

^{zero there.}

Finally,

in the reduced Brillouin zone scheme both

independent

^modes

collapse

^{and stick}

together,

^defin-

ing

^a

typical C-type

reflection

peak.

Within the

peak,

a mode

mixing

occurs, ^as

previously

discussed. On the

opposite

side of the

peak,

^{the modes}

again

^become

distinguishable

^and stable, ^their

polarizations being

now reversed. ^The

upper k’

branch,

corresponding

now to mode _n, will cross the line k’ ⁼

1/2, again

without

generating

^a B-type

instability.

Fig. ^{4. - Same as} figure ^{2, with 8 = 45°.}