The diffraction of light by quasiperiodic ultrasound

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The diffraction of light by quasiperiodic ultrasound

Rémy Mosseri, Francis Bailly

To cite this version:

Rémy Mosseri, Francis Bailly. The diffraction of light by quasiperiodic ultrasound. Journal de

Physique I, EDP Sciences, 1992, 2 (9), pp.1715-1719. �10.1051/jp1:1992239�. �jpa-00246653�

Classification Physics Abstracts

78.35-42.20G

Short Communication

The diffraction of fight by quasiperiodic ultrasound

Rdmy

Mosseri and fhancis

Bailly

Laboratoire de Physique des Solides de Bellevue, CNRS, ^F92195 Meudon Cedex, France

(Received

30 April1992, accepted19 June

1992)

Al~stract We calculate the diffraction of _a light beam by _a liquid filled with _a quasiperiodic ultrasonic _wave, within _a simple phase

grating

approximation. The results, qualitatively different from the periodic _{case, are} obtained

along

two lines, _a direct calculation and _a

straightforward

application of the Cut and Projection method developed in the field of quasicrystals.

1. Introduction.

The first

experiments

about ultrasonic diffraction of

light

_were done 60 _years ago

by

Lucas and

Biquard [I]

and

Debye

^and _{Sears [2]} as a test of _{an even} older

prediction, by

^Brillouin [3],

that, broadly speaking,

the net effect of the ultrasonic _wave is to modulate the refractive index of the

liquid

such that it would act _{as a} diffraction

grating.

Within

simplifying assumptions,

_as shown

by

Raman and Nath [4], ^the

amplitude

of the n~~ order diffracted beam is

proportional

to the n~~ order Bessel

function,

whose argument is itself

proportional

_to the

amplitude

of the ultrasonic _wave and to the width of the

liquid

vessel.

In this letter _we

extend,

within the _same

approximations,

the above calculation to the

case of _a

quasiperiodic

ultrasound. New

phenomena

_are

expected

which could be

probed experimentally, namely

the fact that the diffracted beams form

a dense set indexed

by

two

integers,

their

amplitudes being proportional

to

products

of Bessel functions. We also show that _our result _can be very

easily

derived in the framework of the Cut and

Projection

method [5],

developed

in the

quasicrystal

field.

2. Direct calculation of the diffraction spectrum.

We shall follow the

general

line

given by Berry

[6]. ^A

parallel light

beam

propagating

in the

z direction is

represented,

in the scalar

approximation, by

_{a wave} function

e'~~

_{It is incident}

on the ultrasound filled

liquid

vessel of width D and infinite in the transverse direction _z

(Fig.

I).

The main effect of the ultrasonic _wave is to modulate

along

_z the refractive index of the

liquid

_as follows:

(a) p(z)

_{= po} + pi sin bz

(b) p(z)

_{= po + Pi}

(sin

biz + sin

b2z).

^~~~

1716 JOURNAL DE PHYSIQUE I N°9

+x

~ d.b.

I. +1+z

,- ---~-- ~

,

~

_~

_z

~---

t

~

U-S-

P

Fig.

1. A schematic view of the standard experiment. An inddent monochromatic fight along _z is diffracted by _a liquid filled by _an ultrasonic _wave in the perpendicular direction. The diffracted beams

hit the _screen P. They _are ^indexed by _an

integer

_n and their intensity is proportional to the _square

of the Bessel function of order _n; the argument ^{of the function is the} product of the incident

light

wavevector, ^the ultrasound amplitude and the liquid vessel width.

The _case

(a) corresponds

to the standard

periodic

ultrasound while _case

(b)

is _a modulated index which is not

periodic

whenever

bi

and b2 are incommensurate. Note

that,

due to the very low value of the

speed

^{of sound}

compared

to the

speed

of

light,

_we have

dropped

_any time

dependence

in the modulation. _Let

us introduce the dimensionless parameter

t =

kDpi (2)

The _wave function

beyond

the

object

is

j(z, ~)

=

ei(kz+~(x)) (3)

where the

phase retardation, owing

to the geometry ^{of the}

problem,

is

independent

of _z and reads

7(~)

=

kD(P(~) l). (4)

In the standard

periodic

_{case [6] the} function

#

is

expanded

into Fourier series

+CO

j(z, ~)

₌

£

~vne"(kz+nb~)

(5)

n=-m

and the various

amplitudes

read

tin

₌

e~~~(~°~~)Jn(t) (6)

where Jn is the n~~ order Bessel function of the first kind. In the

quasiperiodic

_{case we} write

4(z> ~)

₌ ^e'~~

~~'V(q)e'~~dq

(7)

which leads to

~~~~

~~~~~

~~

/~

~~~ ~~~~~~~~~~~~~~~~~~~~~~ _~~~

Without loss of

generality,

_{we can} take

bi

_" I and b2 _" o.

Then, using

the

development

+m

~it(sin ^{x+sin ax)}

~

_{j (~)}j

(~)~i(I+ma)x (g)

l _m

I,m=-m

we find the

amplitude

of the Fourier components

~V(q) = e"~~~~°~~~

L Ji(t)Jm(t)6(1+

_ma

q) (lo)

When _tY is _an irrational

number,

_one

gets

_a ^dense Fourier spectrum

(in

the module of I and tY) which

simplifies

as follows:

lV(I +

ma)

₌

e~~~(~°~~)Ji(t)Jm(t) (II)

So,

from _an

experimental point

ofview~ ^the diffraction of

light by quasiperiodic ultrasound,

_as

compared

to the

periodic

_case~ should

display

_{a new} behaviour. The diffracted beams _are, in

principle,

dense and _are indexed

by

two

integers (Fig. 2).

The fact that the

experiments

will

probably

show _a discrete spectrum can be

easily

understood. The dense nature of the spectrum

comes from the fact that _{one can}

approach,

_as close _as

wanted,

_any real value

by

_a combination

I+ _ma, at the

price

of

taking high

values of I and _m.

But,

for reasonable values

oft,

the Bessel functions Ji _goes

rapidly

to _{zero as} grows. The

experimental

resolution will therefore

select,

in the dense spectrum, _a discrete set of beams. This is

completely analogous

to what

happens

for the Fourier spectrum ^of

quasicrystals, theoretically

dense [5] and

experimentally

discrete [7]. ^{And indeed} we now show that the above results _{are very}

simply

derived

along

the lines of

the Cut and

Projection

method [5].

3. The Cut and

Projection approach.

Briefly speaking,

this

approach

_proposes to consider _a d-dimensional

quasiperiodic

structure

as a d-dimensional _cut into _a

higher

dimensional

periodic

structure. In the present _case,

we can even consider _an

experimental picture,

which _can be taken either _{as a}

thought

_or

a real

experiment. Suppose

that the

liquid

extends in the

(~,y) plane

and is filled

by

two

perpendicular

ultrasonic _waves of

equal wavelength

and

intensity (Fig. 3).

The standard I- dimensional

periodic

_case recalled above _can be

easily

extended _{to two} dimensions The refractive index reads

P(~>

Y) " PO + pi

(sin

b~ + sin

by) (12)

The Fourier spectrum of the

outgoing

^beam

displays intensity

at the nodes

(I,m)

of the 2 dimensional

reciprocal

_space

~Y(I>

m)

_"

e~~~'~~°~~~Ji(t)Jm(t). (13)

Let _us

intercept

the incident beam

by

_{a screen} S with _a slit of

slope

_a in the

(~, y) plane.

The selected _rays have

experienced

_a

quasiperiodic

refractive index modulation which

corresponds

to

simply

write _y

= tY~ in

equation (12),

and to rescale in order to get ^{the distance measured}

along

the slit.

Taking

b ₌

(I

+ tY~)I ⁱⁿ

equation (12),

_{we recover}

exactly

the _case studied in

paragraph

2.

Now,

to a cut in the real _space

corresponds

_a

projection

in the

reciprocal

_space.

A node

(I, m) projects

onto the

reciprocal

value _q

= + _ma and _{we recover}

equation II).

Note

that,

at least within the

phase grating approximation, placing

the _screen

immediately

after the

liquid

vessel would lead to the _same result.

So,

from _an

geometrical point

of

view,

the 6 function in

(II) corresponds

to the selection of nodes

perpendicular

to the line of

slope

_tY. Whenever _tY is

irrational, only

_one such node

projects

onto q

(the

whole

projection resulting

in _a dense

set), while,

for _{tY a}

rational,

_an infinite number of such nodes _are colinear for the

projection.

1718 JOURNAL DE

PHYSIQUE

I N°9

t ₌ I

,

c~ =

(~5-1)/2

^t = I

,

cx = 0

0.3 0.5

0.4

0.Z o,3

o-z o-I

o-1

00 _q o-o q

-6 -4 -2 0 2 4 6 -6 -4 -2 0 Z 4 6

t = 3

, c~ =

(~5-1)/2

^{t = 3} _, ^{« = 0}

l~~ten#ty

o.05 o-z

0.04

0.03

o.oz

o.oi

~~~

-6 -4 -2 0 Z 4 6

~ ~6 ~~ ~~ ° ~ ~ ~

t ₌ 5

,

c~ =

(~5-1)/Z

t ₌ 5

,

c~ = 0

Ir~LeI~sity

^Ir~Leru#Ly

o.oz

o-i

o.oi

°"~~

-6 -4 -2 0 2 4 6 ~

~6 ~4 ~~ ° ~ ~ ~

Fig.

2. Comparison of the diffracted spectra ⁱⁿ a periodic (w ₌ 0) and _a

quasiperiodic

(w ₌

(vi

1)/2

for two values oft- In the quasiperiodic _case, I and _m have been given limited values (I, m <

10).

'Y

~'~" ~

~ ~ i.b.

~-

_---~--

~ ⁱ

X ,

,

~ ~

~---

z

~ ~ p

Fig.

3. A schematic view of the experiment which would correspond to the Cut and Projection approach. The liquid is _now filled by ultrasound along the _two perpendicular directions

x and _y. The

incident beam is intersected by _{a screen} S with

a slit of slope _a.

Note

finally

that this

geometrical picture applies

to the well known "addition formula" for Bessel functions. As _an

illustration,

take _one of its

simplest forms,

which reads

+co

Jn(2t)

₌

L Ji(t)Jn-i(t). (14)

Putting

_{tY =} I in _our

problem

results in

doubling

the

amplitude

of the refractive index modula-

tion,

therefore

doubling

the

argument

^{of the Bessel}

function,

solution of the standard

periodic

case. Indeed

expression (lo)

reduces _to

(14)

where the 6 function amounts to

selecting

the

infinite set of nodes in the

(I,I)

reticular

family.

Acknowledgements.

One of _us

(R.M.)

thank A. Zarembowitch for

introducing

to him the Lucas and

Biquard experiment.

References

Ill

Lucas R., Biquard P., J. Phys. Radium 3

(1932)

464.

[2] Debye P., Sears F-W-, Proc. nat. Acad. Sci. U-S- 18

(1932)

410.

[3] ^Brillouin L., Ann. Phys. 17

(1921)

103.

[4] ^Raman C-V-, Nath N-S-N-, Proc. Indian Acad. Sci. 2

(1935)

406.

[5] ^Duneau M., Katz A., Phys. Rev. Left. ₅₄

(1985)

2688;

Elser V. Acta Cryst. A42

(1988)

36;

Kalugin P-A-, Kitaev A-Y-, Levitov L-C-, J. Phys. Lent. France 46

(1985)

L601.

[6] Berry M-V-, The Diffraction of Light by Ultrasound

(Academic

Press,

1966).

[7] Shechtman D., Blech I., Gratias D., Cahn J-W-, Phys.Rev.Lent. 53

(1984)

1951.