Caustics and symmetries in optical imaging. The example of convective flow visualization

HAL Id: jpa-00246961

https://hal.archives-ouvertes.fr/jpa-00246961

Submitted on 1 Jan 1994

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.

Caustics and symmetries in optical imaging. The example of convective flow visualization

A. Joets, R. Ribotta

To cite this version:

A. Joets, R. Ribotta. Caustics and symmetries in optical imaging. The example of convective flow visualization. Journal de Physique I, EDP Sciences, 1994, 4 (7), pp.1013-1026. �10.1051/jp1:1994181�.

�jpa-00246961�

J. Phys. I Fiaiic.e 4 (1994) 1013-1026 JULY 1994, PAGE 1013

Classification Fhysics AbsU.acts

42.20C 42.30V 47.80

Caustics and symmetries in optical imaging. The example of

convective flow visualization

A. Joets and R. Ribotta

Laboratoire de Physique ^des Solides, Bât. 510, Université de Paris Sud, 91405 Orsay Cedex, France

(Received19 Januaiy J994,

accepted

J8 Mait.h 1994)

Abstract. lt is shown that

qualitative

and

quantitative

information _can be callected from the study ^{of the} fuit caustic formed

by light

_rays deflected

by

some medium. The

study

necessitates the calculation of the ray trajectones along with _a detailed balance of the

symmetries

of the whole _set- up. As _an example _we consider the _case of the caustic

produced

by parallel rays interacting with _a

periodic

convective flow _{in a}

liquiô

crystal. The

amplitude

^{of the} flow and ail its essential features

can be deduced, indicating that _a caustic carries _more information than _a simple image.

l. Introduction.

Light

_rays deflected either

by

a transparent

inhomogeneous

medium _or

by

_a deformed surface

con

provide

information _on the

inhomogeneities

_or the deformations. This is the basis of imaging

techniques

that _are

widely

^used in macroscopic

experiments. Usually,

transmitted rays are deviated because of index

change

ⁱⁿ _space, ^{while reflection} _images are used to

analyse

surface deformations.

Among

the _most familiar

examples

is the visualization of flows

(open flows,

confined

penodic

flows,

turbulence...).

Another _case is

provided by

the

optical techniques developed

to

study

the elastic deformations _{in a}

liquid crystal subjected

to some

externat

field,

with

applications

_to

optical

devices. In

generaL

these

techniques

can give

qualitative

information about the structures, but _{it is} difficult _{to extract} quantitative

measurements, except may be concemmg

geometncal

characteristics

(distances, angles).

And, actually,

there _is a need _{to increase} the information contained in the images and for instance, it is important to trace back to the

amplitude

of the deformation.

What _is called _an

image

_is the section

by

_some

plane

of the

family

of emerging rays.

Generally,

this

family

admits _a surface

envelope,

the caustic, named _so because _at every of its points the

light

intensity ^is

sharply

increased. It _is this surface which

instead,

contains the information conceming the structure of the deformation under

study.

This surface may itself

contain

smgulanties

of lower dimension that appear

brighter.

Now, a focussed

image

of _a

given

object

is obtamed when the

plane

_is

positioned

in such _a way that it contains these

brightest singularities.

An

example

is

given by figure1.

One may presume that _a better

knowledge

of the fuit

caustic, including

ail the

possible singularities,

is necessary to have _a

more correct

interpretation

of the images. As _an illustration of this

approach

we shall consider

a rather

simple

case of 2-D

periodic

flow _structure

developed

in _a

liquid crystal.

This

study

was

first motivated

by

the interpretation ^{of the different}

light

_patterns caused

by

the _various flow

structures

developed

on the way ^to the chaos.

B A

focal

plane

virtual caustic

a) b)

Fig. a) Photograph of _an image of the conveciive layer. The bright thick line~ _are the sections of the virtual _cones

by

the focal plane of the micro~cope, ^located

slightly

below the cusps. The twmning is

indicated

by

the different separation ^distances A and B, Some fines _are destabilized into _some points clo~e to a defect of

penodicity

(di;location), where the flow is

complex

and 3-D (see text). b) Sketch of the

c;tu;tic _in 3-D.

2. Basic elements _on caustics in

periodic

flows.

An

optical

caustic _is defined _as the

envelope

of _a given

family

of

nonparallel

_rays

[1] (Fig. 2).

If the rays are

parallel

the caustic is

formally

_a point

rejected

to

infinity. Any

system ^{that has} the property ^of

deflecting

incident

parallel

_rays for _instance, will

produce

_a caustic in the _near field, 1-e- this caustic _is

pulled

back from

infinity.

The first

example

^that cames

naturally

to

mina is the focal point ^of a converging lens. The process of ideal image formation

through

_a

perfect

lens associates _a point caustic with every source

point. However,

this caustic _is unstable again~t infimtesimal deviations from

perfect

focahsation,

producing

then the well- known effects of

spherical aberration,

_coma, astigmatism.

rays

causti

Î'

Fig. ?, A caustic _is the envelope ^of a

famfly

of

non-parallel

_rays.

N° 7 CAUSTICS, SYMMETRIES AND FLOW VISUALIZATION 1015

Other familiar

examples

of caustics _are the

dancing bright

[mes at the bottom of _a

pool

due to the focalisation of _Sun rays

by

the

irregular

surface of the water, and also the

sparkling

of the reflected Sun rays

by

the _same surface. Less obvious

examples

in the

atmosphere,

_are the formation of mirages and of rainbows.

From _a mathematical

standpoint

caustics _are

singularities

that may be either stable _or unstable

[2-4].

It is shown that the stable

singularities belong

to a small number of classes

folds,

_cusps,

swallowtails,

umbilics...

However,

the focal

point

of _an ideal

converging

lens does _not appear in this classification because it is unstable

(degenerate)

and the effect of _a

perturbation

is _to

decompose

it into _{a set} of those stable

singularities.

An index modulation in space is

commonly

assimilated to a lens. Periodic convective flows realise _a

periodical

set of such lenses. Indeed, a prototype ^{is the}

Rayleigh-Bénard

convection of _a

loyer

of

isotropic

fluid where the thermal

gradients

modulate the index of refraction.

Another

example

is given

by liq,Jid crystals

^{in which} the index of refraction _is

spatially

modulated

by

convective rolls. These rolls _are

produced

under the action of _{a tran~verse} electric field _{across a}

loyer,

and _are

periodic

ⁱⁿ _one direction of space and invariant

by

translation

along

their axis

[5, 6].

The

problem

of caustic formation reduces _{to a} 2-D _one

corresponding

to the basic

experimental

situation. We shall

give

a

complete description

of the caustics formed

by

the emergent rays and also

by

their continuation the real and virtual caustics. Then _we shall calculate the ray

trajectories

inside the nematic

loyer

and _we deduce the

envelope

of the rays.

By comparing

the simulation with _our

expenmental

values _{we are} able to estimate the

amplitude

of the convection. We Shall aise

study

the

relationship

between the symmetry of the caustic and the symmetry ^{of the index} modulation. As _an

example,

the _so- called _«

squint

effect _» shall be

remterpreted.

As the _structure of the flow becomes _more

complicated (at higher

values of the electric field)

one observes that the _caustic becomes unstable and _{now contains} other types ^of

singularities

that necessitate _a 3-D

analysis

which will be

presented

in another

publication.

3. Observation of caustics in the electroconvection of _a nematic

layer.

Nematics _are

composed

of

elongated

molecules, whose local orientation defines _{a new}

macroscopic

variable,

the director

usually

denoted

by

a unit vector _m. It _is understood that

directors of opposite sense are

equivalent

_mm m.

Optically,

the nematics _are uniaxial

materials,

the

optical

_axis

lying along

the director. The nematic

(Merck

Phase

V)

has _an

ordinary

refractive index n~ =

1.65 and an

extraordinary

refractive index _n~ 94. The

liquid crystal

is sandwiched between two

conducting semi-transparent plates (plane (.r,y))

separated by

a

typical

distance of d

=

100 _~Lm.

By

an appropriate treatment of the

plates

_one

imposes a uniform molecular

alignment

in the

plane

of the

plates (planar anchoring) along,

say, the >.-axis. In the _{rest state} the nematic

loyer

constitutes a

homogeneous

uniaxial medium of

optical

_{axis m}

= x.

When _an AC

voltage

v, of

typical frequency f

=

100 Hz (« conduction regime »

[5]),

is

applied

across the

loyer

and fixed _{at some} value

just

above the threshold

v,

₌ 6 V, the rest

state is destabilized and the convection

develops

_in the form of _a

stationary

structure of

periodic rails,

of

period

A

= 2 d. The _axis of the rails is normal to the initial direction,; of the molecules and this is the _« normal roll structure _»

[6].

Because of the

coupling

between the

hydrodynamic velocity

and the director, the molecular

alignment

_is

periodically

distorted

along

_x, the _maximum of the distortion

being

at the roll centers

(Fig. 4).

A

light

beam,

polanzed along

_x, is sent

normally

to the

plates.

Inside the

loyer,

the effective local index for each

incoming (extraordinary)

_ray is

periodically

modulated

along,;.

In a rather primary

description,

the

loyer

con be reduced to a set of

alternately diverging

and converging lenses

[7].

After traversing the nematic the rays refract at the

liquid crystal-glass

interface

Fig. 3.-Microphotograph

of the _caustic. The sampie is tilted around.< by ^40°. The image is

topologicaliy equivalent

to a vertical section of the _caustics. Note that the virtual cusps are destabilized into unfolded butterflies.

= =

d

(contrary

to the incident rays which _are normal bath _to the

glass-liquid crystal

interface

= =

0 and to the

optical

axis _m

=

x).

The

outgoing

_rays are focalized

along

a real caustic and their

prolongation

below the upper interface appear focalized

along

a vii.tuai caustic. The

light

pattern is shown usmg a

polarizing

microscope, which grues the section of the _{caustic in} its focal

plane (Fig. l).

In the literature these caustics _are

usually

reduced to a set of _« foci _{», or}

« focal

points

_»

(in

the

(x, z) plane).

A _more careful

study

^of these points

[8]

has shown

by

a reconstruction

technique

that

they

_are in fact the

tips

^of a small portion ^{of the} caustic, in the forrn of _{a cane.} In order _to obtain at _once an

image

of the _{caustic we}

simply

rotate the

sample

^around the x-axis

by

an

angle

of about 40°. Due _to the _mvariance of the normal roll

along

_y, the

resulting image

is

topologically equivalent

to the section of the caustics in the vertical

ix, z) plane (Fig. 3).

We

see

clearly

that the caustic is

composed

of _a sequence of _canes

alternately

oriented

upwards

and downwards. In the

language

of

singularities,

ail points ^{of the caustic} are « fold

points

_», except ^the tips ^{of the} canes which _{are «} cusp points ». In practice, ^the term cusp means rather

some finite portion ^{of the} caustics around _a cusp point.

According

to the

theory

of

singulanties [9],

the two fold-curves of _{a cane connect}

tangentially

at the cusp-point, as con be

seen in

figure

3. The cusps are the so-called

« focal points » where the

light

intensity is much

higher

thon

anywhere

else _on the fold-curve. We check that the real cusps are located above the separation fines between _two

adjacent

rails

(Fig. 4).

We _name them up-cusps or

down-cusps

N° 7 CAUSTICS, SYMMETRIES AND FLOW VISUALIZATION loi?

real caustics

outgoing down~cusp

rays

~

up"cusp g<ass

~

,"'~"', ,"'~"i

liquid

' '

ctysta<

~

,

/ ,, /

x

', ,,' ,,_,,,'

glass

inccming

rays

virtual caustics

Fig. 4, Simulation of the ray

trajectories

(thin hnes) and of the _caustic (thick fines). The model for the molecular distortion _{is q~}

= q~o sin Lt sin q= (with _q~o 21,2°, A

=

2 «IL

=

393 _~Lm, d ar/q

=

308 ~m). The _arrows represent the director _in the midplane of the layer. The convection is indicated by

the dashed fines. The small circles indicate the expenmental points ^{from the observed caustics. The} fit is made by adjusting at once the

majonty

of the

experimental

_{points on} the theoretical caustic _curve. The typical error bar _is mdicated _on the lowest point on the left side of the

figure,

The _mirrors M passmg through each real cusp are intercalated with the virtual cusps.

depending

on whether the flow _moves

along

z or z. The canes are net limited

along

z.

They

con

easily

be observed up to distances of the order of 20 d. The caustic presents ^remarkable

geometncal

charactenstics. For instance, the down-canes are net identical to the up-canes their cusp-points ^have a distinct

height.

This _is net

expected

at first from _a

rapid

consideration of the model for the flow, We aise observe that the virtual cusps are net

equidistant along

x.

This effect _was named _« squint _»

il 0]

and attributed to some

nonlineanty

in the electroconvec-

tive mechanism,

although

it is observed _{even at or near} the threshold. In the

followmg

model,

we shall show that this effect _con be

explained differently.

It is observed that, for values of the _constraint

F =

(v v, )/v~

m

0.15,

each virtual cusp is

replaced by

a ensemble of three cusps

(unfolded butterfly singularity) (Fig. 3).

This observation will be discussed later.

4.

Equations

for the rays.

We first calculate the ray

trajectories

inside the nematic

layer.

This

problem

can be soJved

by

several

general

methods

il 1-13]. Here,

we

apply

Fermat~s

pnnciple

which _can be shown _to be valid in _an

anisotropic

medium

[12].

It _states that the rays are the extremals of the

optical path

[

= n ds. where _{n ii} the local ray index and ds _is the curvilinear abscissa

along

the

path.

Here, _ii is given

by iii.

n ₌

,/nj _sin~

_p

+

nj cos~ _p

_Il

where p is the

angle

between the local

optical

^axis m and the local tangent vector

r =

(ài/ds, dj'/ds)

to the ray.

Assuming

that ail the rays pass

through

the

loyer,

which is checked in the expenment, we can choose _{= as} the

independent

variable and _{>. as} the unknown function _r =,r(=). The

optical path

now takes the form

[

=

~~

L d=

= ii

(=,,;(=),

^~

jl

+

~°~ ^~

dz.

(2)

=,, =~

d= d=

It _is convenient to mtroduce the

angle

_q~ between the,;-axis and _{m, so} that

p

(=, _>.,

il

=

q~(,;, _z) + arctan Il,i. (the dot indicates the derivation with respect to =). As

usual,

we assume

that the director _is

harmonically

distorted _{: q~}

= ~ao sin kx sin _q=

(k

= 2 «IA, _q

=

«Id).

The equation ^{for the} rays ^is obtained

by

writting the

Euler-Lagrange equations

for the functionnal L, i.e. that the first variation of the

optical path

is _zero for the actual rays. The equation ^{for the}

rays is then _:

~2,,.

(i +.i.2) (n~ ^(.i~= v~,)

_{npp (.i.~a~ + v~=))}

~ ~~~

I' + nôô

where the

partial

derivatives are noted

by subscripts

_~a, éq~/é>., n~ = én/ép, etc. An

equivalent

form of equation

(3)

_was

given by Kosmopoulos

and

Zengmoglou,

who

analyzed

the grating action of such _a nematic

loyer [14].

The differential equation ^{of second order}

(3)

is

integrated

with the classical

Runge-Kutta

method of order 4. The step ^of

integration

d= is taken

equal

to d/200. The incident rays are

Parametrized by

the,r-coordinate A of their entry point. ^The _rays

being

^normal to the

loyer,

the initial data

are.;(0)

= A,

,k(0)

=

0

(the loyer

is defined

by

0 _{w = w} dj. The abscissa of the entry point is taken between 0 and A.

Figure

4 shows an

example

of such _a simulation of the ray

trajectories.

Before

Jeaving

the nematic

layer~

the rays are refracted _on the

liquid crystal-glass

interface

= =

d. In order to test the model

quantitatively

this refraction _must be included in the

calculation. The relation between the

angle

of incidence _i

=

arctan1.(d)

and the

angle

of

N° 7 CAUSTICS, SYMMETRIES AND FLOW VISUALIZATION 1019

refraction

a con be extracted from the

general

^{laws of}

crystals optics Ii

n(

tan i

sin _a

=

~

(4)

n~

(ni

₊

n( ^{tan~ 1)'~}

where n~ ^{is the refractive index of the}

glass.

We _now consider a set of incident rays

parametrized by

A. After

passing through

the

layer

and

refracting

_at the

liquid crystal-glass interface,

the rays follow again a rectilinear

trajectory

characterized

by

two parameters : the

abscissa,ro =.;(d)

of the exit

point

and the

angle

of refraction

a. Of _course,,~o and

a

depend

_on A.

5. Calculation of the caustic _curve.

The _set of the

emerging

_rays constitutes _a one-parameter

family

of

straight

fines, Its equation ^is

>.

=,~o(Àj+ (d-z>tan a(À), (5>

The caustic is

by

definition the

envelope

of the

emerging

rays, that _is, the locus where

« two

neighbouring

_rays intersect

». This is

expressed by

_zeroeing the

partial

derivative of,r with respect to

(see [15]

and

appendix)

é>./éA

= 0

(6)

Equations (6)

and

(5)

define the caustic _{as a curve}

parametrized by

A.

Explicitly,

we obtain

>~j(À)cos ce(À)sin

a(À)

x(À

=

xo(À

~, ~

>j(À cos~

a (À

~~~

Z(À)

_# d+

~,~~~

where the

prime

denotes the derivation with respect to A. For each value of A, the values

,ro(A )

^and _a (A are calculated

by

the method described in the

preceding

section. Then, the

caustic _curve is deduced

by equation (7) (the

derivatives with respect to are

numerically

obtained

by

the central-difference

method).

The result _is shown in

figure

4.

Experimentally,

the position ^of points ^{of the} caustic is determined

by varying

the

height

of the local

plane

of the

microscope.

These _are

reported

in the _same

figure.

A fit is obtained

by adjusting only

_one parameter ^the

amplitude

_q~o of the molecular distorsion _so that the

majority

of the

points

lie _on

the _curves. Since the caustic _is the ensemble of the _cones the fit _is not made for each _cone

separately,

but rather for ail the _{cones at once.} A rather

good

fit is obtained for q~o =

21.2°. The asymptotic ^{branches of the caustic}

correspond

to

a'(A

=

0. The real and virtual

cusp-points correspond respectively

to the local _minima and _to the local _maxima of the

height

_z.

They

are then given

by

~~

=

0

(8)

(see

the

appendix

for a

rigourous presentation).

In previous articles

[7,

16,

17],

_in which

approximations

were made, _some

analytical

formula _was given for the

height

of the cusps.

Here,

_on the contrary, ^the

height

_can be deduced

directly

from the calculation of the whole

caustic and without any approximation. ^{However another} more

powerful

method _can be

denved if _{one is} interested

only

in

calculatmg

the

height

of the cusps. This method will be described in _a

forthcommg

_paper.

6. Caustics and

symmetries.

The presence of

symmetries

is, _among the essential

geometrical

characteristics of the caustics, the _most noticeable feature.

Here,

the caustic is

periodical along

_>., with the _same

penodicity

A as that of the flow pattern

(Fig. 4),

measured

by

another method

[6].

We notice from the

experiment

that the real cusps are

equidistant

with

penod

^A/2

along,;,

while the virtual cusps

are twinned

(twins separated by A)

and each pair

separated by

B such that A ₊ B

=

A. The whole set of _cones

(real

and virtual

ones)

has the mirror symmetry ^{the mirror is vertical and} passes

through

_a real cusp

(a

vertical fine _m the

plane

of the

figure). Obviously

there _{is a mirror} for each real cusp, and _no mirror for the virtual _ones. This may indicate _a symmetry

breaking,

as

proposed by

Ben-Abraham

[101.

One also notices that the

down-cusp

and the up-cusp are not

equivalent by superposition

and _{are at} different

heights.

In order to understand the

light

pattern, one must examine ail the

possible symmetries

of the fuit system.

Owing

to Curie's

principle il 8]

_one must indude in this

description,

the

liquid crystal,

the

light

beam, the interaction

light-matter

and the

boundary

conditions. This

principle

states that

« the symmetry elements of the _causes must be present ⁱⁿ the effects

produced

» In _our system, ^{it is clear that} we may take the

light

_{pattem as} the _« effect _», and the director

field, conjointly

with

boundary

conditions and the incident beam, _as the _{« cause ».} Now the

problem

to solve is the

relationship

between the effect and its cause. In

particular,

_is the symmetry ^of the tlow

directly

reflected in the _caustic ?

The nematic

liquid crystal

_is characterized

by

its director field which bears the local

optical

axis. A

simple

_way to define _a symmetry of the director field is to consider how it is

transformed

by

_a

displacement f (punctual

affine transformation of the

i-i, z) -plane

_preserving

the

distance). By

the transformation

f,

a

point

P is transformed into the point

f(P)

and any vector attached _to P is transformed

by

the first derivative

f'(P) (1.e.

the Jacobian matrix)

off

into _a vector attached _to

f(P) (Fig. 5).

We _note that the derivative

f'(P)

transforms also the directions attached _to P into directions attached _to

f(P) (this

results from the

linearity

of the operator

f'(P)).

One _can say that the director field

m(P)

is _invariant

by

the transformation

f,

if

111~f(P)

#

f'jlll(P))

₁₁

9j.

Let _{us now assume} that the director field _m has _a symmetry

f,

and let us compare the

optical

path along

some curve C,

parametrized by

_s, and the

optical path along

^the transformed _curve

f(C), parametrized by

s'.

Observing

that the local index _n

depends only

on the

angle p

between the director _m and the local direction ds

=

(dà., dz)

of the _curve, and also that

f'(P)

_preserves this

angle (or

at least its

magnitude),

we can wnte

1'

,

ÎIÎ

^~

^~~

^{~~' "}

~

n

lP

ds

~

(9)

where

p'

=

(m(s'), ds') (Fig.

5). We have also used the fact that ds is

preserved by f.

The

optical paths along

C and

f(C

are

equal.

As _a consequence, the rays, which _are the extremals of the

optical path,

_are

exchanged

into _one another

by

the transformation

f.

If _now the

boundary

conditions and the incident beam both have the symmetry ^{of the director} field, then the ray

trajectones

form _a set

globally

mvariant under the _same transformation. In this _case, the

envelope

of the rays, i e. the caustic, _is aise invariant under the transformation. More

generally,

^the symmetnes ^{of the} caustic reveal the symmetry ^elements common to the director field, the incident

light

beam and the

boundary

conditions.

Let _{us now} interpret ^the

geometrical

features of the _caustics observed in _our experiment. The

penodicity

of the _caustics

along

the ~;-axis _is that of the director field. The presence of the vertical _mirrors passing

through

the real cusps indicates that the director field has this _mirror symmetry. Examination of

figure

3 does _not reveal any other symmetry. ^{Now [et} us mtroduce

N° 7 CAUSTICS, SYMMETRIES AND FLOW VISUALIZATION 1021

L

i i i i i

_ -

d~

fi fi'

~ f ~

~

Ù~~~~~~~

i

Fig. 5.

Example

of _a tran~formation (symmetry ^with respect to L) facting on the point ^{P and} on the director _m,

the symmetry ^{elements of the}

boundary

conditions and of the incident

light

beam. The

boundary

^conditions _concern the lower and upper

separation planes

between the

hquid crystal

and the

holding plates.

There, in addition _to the

previous

vertical _{mirrors, one} has the horizontal _mirror located in the middle of the

layer

and

consequently

_an inversion

point.

The

boundary

conditions _are also invariant

by

_continuous translations

along

_x.

Indeed, they

have the maximal

possible

symmetry. ^For a source located

only

_on one side of the

layer,

as it _is in the expenment, ^{the incident beam} is

symmetrical by

^ail translations

along.~

and in every

vertical _mirror,

But,

_in order _to be

complete,

_one must aise consider _an identical _source located

on the other side. There _is indeed _{no a}

priori

_reason for the structure in the

layer

to have the reversai

symmetry,

for

example

the _mirror symmetry ^{about the median fine. This} is achieved

experimentally, by simply

_tuming the

layer upside

down. In this

operation,

the _{new caustic is}

clearly

deduced from the previous one

by

an inversion relative _{to a} point located in the centre of the rolls. At _some abscissa _x, a up-cusp of the former _caustic is

replaced

_on the other side of the

layer by

a

down-cusp (Fig. 6).

The up-cusp is recovered

by translating

the caustic

by

,

~. _" ..

_"

L,1: ~,

: j[ ;(

Î

," ~. ;~ ~.

; source is below

~, 'l ^~ :

i

._ / i

"_ i source <sabove

O

,

.- -'

> J fl '

]r l'

.lÎ ,[

." '.

' Î

.'. f

l Il ÎÎ _Î

1 Z ~ Î

'fl

Fig. 6. The fui] _caustic produced by ^{the incident beam} coming from below (resp. above) ^{is indicated}

by the thick

(resp.

dotted) fines, The director field

corresponds

to the standard model. lt shows the symmetry 5, generated by a mirror (the dashed vertical fine), and _{an inversion} around the _center of the point ^{located in the}

mid-plane

of the layer. The _nematic

layer

_is ^confined between the two horizontal fines, The fui] caustic does have the _same symmetnes.

,4/2. Moreover the

twinning

effect is still present ^and

undergoes

the _same transformation. In conclusion, the

knowledge

of the fuit caustic

(the effect),

and that of the two components ^{of the}

cause, enables _us to deduce the symmetry ^{elements of the} last component ^of the _{cause, i-e-} the

structure.

This

study

_can be

supplemented,

in this _case, where _we know _some of the features of the flow structure,

by using

the classification of the symmetry groups for _a 2-D

layer.

Here, there

is a

periodicity along

an axis,~, and then, the classification

gives

_seven symmetry groups

[20].

These _are the groups of the patterns ^obtained

by repeating

_some motifs _{on a} fine. Here _we choose _to represent ^{the motifs}

by appropriate alphabet

characters. Thus the _seven patterns are

1)

...LLLL..

2)

...rLrL...

3)

VVVV...

4)

...NNNN...

5)

AVAV..

6)...DDDD...

7)

HHHH... We _notice that the symmetry groups 6 and 7 _{are not}

compatible

with _{a non-zero} molecular distortion _at the roll _centers and that the symmetry groups 1, 2 and 4 do _net contain the mirror symmetry. ^{The choice is between the} symmetry _group 3,

generated by

two vertical

parallel

mirrors, and the symmetry group

5, generated by

one vertical _mirror and _one inversion

(Fig. 6).

In order to reveal _a

possible

inversion

point,

one must insure that the two components

of the cause under control have this _inversion

point.

This is what has been made

by reversing

the

sample.

One _is left with

groups.

This is the maximal symmetry

corresponding

to a

convective flow in form of rolls

filling

the whole thickness of the

sample.

Let _{us now} examine the

particular

distribution of the virtual cusps.

First,

_{one notices} that the real cusps possess the symmetry ^with respect to the vertical mirror passing

through

them. This

is associated with their

equidistance along.i (periodicity A/2).

On the contrary, there is _no mirror

passing through

the virtual cusps, but rather there is _{a mirror}

passing

inbetween with the

same

periodicity

A/2. This does _not

impose

_an

equidistance

between the virtual cusps, in

which _case the

periodicity

^{of the} cusps is A. This

particular

distribution of the virtual cusps is

due to the convective flow cell characterized

by

an inversion

point,

the centre, and the absence

of any ^vertical mirror. It

might

appear in

experiments

that the virtual cusps are very close to

equidistant,

but this would be rather due _{to some}

particular

values of the parameters. ^The

generality

is that those cusps are net

equidistant

but _are rather twinned.

Then,

there _{is no} need to invoke the effect of the nonlinearities, contrary to what _is assumed

m reference

[2 Ii

in order to

interpret

^this result. In the paper cited

here,

the

non-equidistance

was ascribed to some

particular

flow _structure

(squint)

that would

imply

the absence of any inversion point. ^{On the} contrary, the

experiment clearly

shows the presence of such symmetry.

Moreover, it _can

hardly

be the result of nonlinear effects, _smce the twinnmg is observed _even within fl above threshold.

The _{same reason, i-e-} the absence of any vertical mirror

passing through

the _centre of the

roll, implies

that the up and

down-cusps

_are not

necessarily

identical. Indeed the experiment indicates that

they

_appear at different

height (Fig. 3).

7. Discussion.

Let _us

point

out that these results _can be

generalised.

^{We first} consider the effects of _an

anharmonicity

of the molecular distortion

along,;.

As

already

mentioned, far above the _convection threshold nonlinearities _come into

play

and the virtual cusps evolve towards _more

complex shapes (Fig. 3).

This effect _can be

expected

from the presence of

higher

harmonies _in the molecular distortion. The numerical simulation of

the _{caustic is now} made after

mtroducing

_two harmonies _in the previous model for the

molecular distortion _~

= (q~o sm k,; _{+ a sin} ? k,; ₊ b sin 3 k>. _{sm qz.} The result shown _in

figure

7 agrees quite ^{well with the}

experimental

observation. Then the

anharmonicity

can be reflected

by changes (instabilities)

in the

topology

of the caustic. The symmetry is conserved,

N° 7 CAUSTICS, SYMMETRIES AND FLOW VISUALIZATION 1023

Fig.

7, -Simulation of the caustics for _a molecular distortsion _q~

containing

3 harmonics _{: q~}

=

(q~o sin k~r + a sin 2 kV ₊ b sin 3 k> sin q= with q~o 0.377, _a

=

0.005, h 0.0285. The virtual cusps

are replaced

by

_« butterflies

» (3 connected cusps). Compare ^with

figure

3.

because of the symmetry ^{of the chosen harmonies. In the}

general

_case, it _is

likely

^{that another} type ^of harmonic with a different

symmetry

^{would induce} a symmetry

breaking

of the caustic,

We have studied the case where ail the rays are

parallel

to the

(,;,

₌

-plane

and the

problem

is

essentially

2-D. The basic

equations

for the caustic

(6)

and

(8)

have there _a

simple interpretation

and do _not

justify

_a mathematical demonstration.

However,

for other convective

structures that break the continuous translational invanance

along

the

j,-axis,

for _instance the

vancose structure

[6],

the

problem

becomes 3-D, It needs a mathematical formulation based _on

the

theory

^of the

singularities.

Such _an

analysis

is sketched in the

appendix

for the 2-D

light

patterns, ^{and will be}

developed

in detail for 3-D

light

_{patterns in a}

forthcoming

_paper.

However,

_our statement that the fuit caustic has the

symmetries

_common to the director field, the

boundary

conditions and the incident

light

beam, _is

mdependent

of the dimension of the

physical

_space.

Consequently,

it _remains valid in the much _more

general

_case of the

3-D-light

patterns. ^{It is also vahd for other} types ^of materais,

provided

that the refractive index

depends only

_on

angles,

_so that equation

(9)

holds. In

particular,

our results

apply

to the _case of biaxial

liquid crystals.

The symmetnes ^{considered in the}

previous

section are

generated by punctual

transforma- tions,

namely displacements. They

act on the

points,

the _vectors and the directors. We may also define local transformations which

keep

the

points

invanant and which act

only

on the

vectors _or on the directors. For

example

[et _us consider the standard model for the director field

q~ = q~o ^sm k.r _sin qz and the local _mirror symmetry t around the _>.-axis

(Fig. 8).

This director field is _not invariant

by

t.

Composing

t with the mirror f _{in the}

mid-plane

of the

layer,

we

define the transformation

f

*

= t _o

f.

_{The director field is invariant}

by

^{Î * while the}

optical path

is net conserved

by ^f* Then,

the reasoning that _we have used

throughout

this article is _no

longer

valid, and the symmetry f * cannot be evidenced in the caustic.

Generally,

the caustic will _not

necessarily

contain these local symmetries. This

implies

that the _caustic

might

not

reveal the fine structure, for instance of the flow _or of the index

distribution,

thus

restrictmg

^the

amount of information that _{one can} extract from _{a caustic.}

A wider class of

symmetries

_is

provided by diffeomorphisms,

_{i e.}

bijective

and differentiable point transformations. It _is obvious that here too, our conclusions _on the

symmetries

of the

caustic _are identical,

provided

that equation

(9)

holds. This

problem

would deserve _a

deeper

G ~Î----x

~ ai

@--u--x

~

~~p~=p t(m>

i(P)

nô)

~~

--/-~

x

~

Fig.

8. -a) Definition of the local mirror symmetry t around the _{~< axis,} and of the global mirror

symmetry Î _around the,t axis. b)The director field in the convection flow does not possess both symmetnes t and

Î,

but the symmetry Î*

= ta Î.

study.

We _note

only

that it _can be formalized in terms of isometries of the metric space, whose fine element is defined

by

the

optical path

element nds

(cf. [13]).

For the state of comparison with _our

example,

[et _us

finally

consider the much

simpler

case

of thermal convection in

isotropic

fluids

(Rayleigh-Bénard convection)

close _to the threshold.

There,

the index is _a function of the

temperature

distribution. Its

period along

_x is the

penod

A of the flow. The real caustics now

correspond

_to

only

_one direction of the vertical flow

(downwards)

while the virtual _one

corresponds

to the

upwards

^motion. There _are no other cusps. With each _cusp is associated _a vertical mirror. Thus ail cusps are

equidistant.

8. Conclusion.

We have

analysed

^{in detail the}

g~umetry

^{of the}

hght

_pattem

produced by

a

penodic

flow _{in a}

layer.

We have taken _as an

example

the convection _{in a} nematic

layer mainly

because it also

includes _an

anisotropy,

thus

making

the

example

richer and of _a wider interest than the

isotropic case. The

light

pattern is the section of _a surface, the caustic which is

singular

and which _contains

singularities

of lower dimension. The

equations

for the rays have been derived,

and the fuit _caustic has been calculated. It is found _to be in

good

agreement with the

experimental

observations. The

amplitude

of the molecular distortion has been

directly

deduced

by fitting

with the

expenment.

^The inverse

problem

of

interpretation

of the director field

starting

from the

light

_pattern necessitates a detailed account of the symmetries ^which are

present in the whole set-up : the

light

_pattern, the

light

_source and the

boundary

conditions.

This gives a

powerful

method and the symmetry ^{of the} director field has been identified.

Moreover,

_we have given a

satisfactory interpretation

of the pairing of the virtual cusps. This

approach

is rather

general

and may be valid also _{to caustics} formed after reflections _on

deformed surfaces.

However,

_{in most cases it} has _to be

supplemented by

^other methods _in

order _to grue a

fully

detailed

description

of the deformations that

produce

the

light

pattern.

Inversely,

_{since a caustic carnes more} information, it

might

be

misleading

to restrict the

study

to the image which is _{a mere section} of it. Dur purpose is now to extend this

study

ta the _more realistic 3-D _case.

Acknowledgments.

This work _was

supported by

the Direction des Recherches et Etudes

Techniques

under the

contract DRET/ERS/901616.

N° 7 CAUSTICS, SYMMETRIES AND FLOW VISUALIZATION 1025

Appendix.

Here,

the

general

formalism of the

theory

of

singularities [22, 9]

is used _{to write} the

equations

of the fold and

cusp-points.

^We start from the

equation

for the rays :

4~

(x,

_z, À

f

x xo

(À

+

(z

à) tan _a

(À

=

0

(10)

given in _section 5. This

equation

defines _a 2-D surface M in the 3-D space

ix,

_z, A

).

The surface M is

globally parametrized by

_z and A, equation

(10) giving

the coordinate,r _{as a}

function of

z and _:

x(z,

A

=

xo(A

+

(d

_z tan _a

(A

). The _caustic

points correspond

to the

singularities

of the

projection

wfrom M _onto the

physical

2-D space

ix,

₌

),

that _{is to} say to the points ^{where the Jacobian matrix} w ^* of _{gris not} of _maximum rank

(here 2).

More

precisely,

the

fold-points

are defined

by

the condition _: corank

w*(z, )

=1, where the corank (at the

source)

of _w ^* is defined _as the difference between the dimension of M and its rank. The _matrix

w * reads

~

jéx/éz

^é>./ôA

jé,r/éz

^{éx/éA ~~ ~~}

"

éz/éz éz/éA 0

so that the condition for the corank of _w ^* to be

equal

to 1, is written é,;/éA

=

0,

and _{we recover}

formula

(6).

It defines the _curve

lS'

_{of the}

fold-points.

In order _to find the cusp-points, we repeat ^the same reasomng, the

only

difference

being

that M is

replaced

here

by 3'.

_The

curve

3'

is

parametrized by

A, and the restriction of _gr to

3~

lS

WI

_~i1 À

-

(X(Z(À)j, Z(À )).

The condition

giving

the cusp-points is that the corank of the Jacobian matrix

(w [~,)*

should be

equal

to 1, _or in order

words,

that its rank should be

equal

_to 0. On the other hand _we have

~~

jéx/éz.

^{dz/dA + éx/éA dz}

jôx/ôzj

^~~~~

~ ~~ dz/dÀ dÀ

(using

^{the fact that} ôx/ôA

=

0 _on

lS').

Then the equation ^that gives the set lS~. ^' _{of the cusp-} points is dz/dA

=

0,

and _{we recover} formula

(8).

References

Ii Bom M. and Wolf E.,

Principles

of optics

(Pergamon

Press, 1980).

[2] Thom R., Stabilité structurelle _et

morphogénèse (Benjamin

and Ediscience, 1972).

[3] Amol'd V. I., Funct. Anal. and ils Appl, 6 (1972) 222.

[4] Berry ^{M. V. and}

Upstill

C., Progress ⁱⁿ Optics, E. Wolf Ed., vol. 28 (Amsterdam _: North Holland, 1980) p. 257.

[5] de Gennes P. G., The physics of

liquid

crystals (Oxford Clarendon, 1974).

[6] Joets A. and Ribotta R., J. Phys. Fiance 47 (1986) 595.

[7] Penz P. A.,

Phys.

Rei,. Lent. 24 (1970) 1405.

[8] Kondo K., Arakawa M., Kukuda A. and Kuze E., Jpn J. Appt. Phys. 22 (1983) 394.

[9] Amold V. I., Gusein-Zade S. M. and Varchenko A. N., Singulanties ^of Differentiable Maps, ^vol. l (Birkhàuser Boston, Inc., 1985).

loi Ben-Abraham S. I., J. Phys. ^Fiant.e Colloq. 40 (1979) C3-259.

Il Grandjean M. F., Bull. Soc, Fianç.. Miiiéi. 42 (1919) 42.

[12] Kline M, and Kay I., Electromagnetic theory ^and

geometncal optics

(Interscience Publishers, 1965).

il31 Joets A. and Ribotta R., Opt. Commun, 107 (1994) 200.

[14]

Kosmopoulos

J. A. and

Zenginoglou

H. M.,

Appl.

Opt. 26 (1987) 1714.

il 5] Valiron G., Equations fonctionnelles,

applications

(Masson et Cie, 1945).

[16] Penz P. A., Mol.

Ciyst.

Liq. Ciyst, 15 (1971) 141.

[17] Rasenat S.. Hartung G., Winkler B. L. and Rehberg ^I Experiments _in ^Fluids 7 (1989) 412.

[18] Curie P., J. Phys. Théor. Appt, 3 (1894) 393.

[19] Amold V. I., Chapitres supplémentaires de la théorie des équation~ différentielles ordinaires (MIR, Moscou, 1980).

[20] Coxeter H. S. M., Introduction to

Geometry

(Wiiey, New York, 1969).

[21]

Barley

J. S., Gelennter E. and Ben-Abraham S. I., Mol. Ciyst Liq. Cij,st. 87 (1982) 251.

[22] Thom R., Ann, In.ititut Fow.ter 6 (1956) 43.