On birefringence phenomena, associated with total reflection

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On birefringence phenomena, associated with total reflection

B. Julia, A. Neveu

To cite this version:

B. Julia, A. Neveu. On birefringence phenomena, associated with total reflection. Journal de Physique,

1973, 34 (5-6), pp.335-340. �10.1051/jphys:01973003405-6033500�. �jpa-00207388�

LE JOURNAL DE PHYSIQUE

ON BIREFRINGENCE PHENOMENA, ASSOCIATED WITH TOTAL REFLECTION

B. JULIA and A. NEVEU

Laboratoire de

Physique Théorique

^{et Hautes}

Energies, Orsay (*) (Reçu

le 10 novembre

1972)

Résumé. ²⁰¹⁴ Nous étudions les

propriétés

^de

biréfringence

de la réflexion totale d’un faisceau lumineux

monochromatique,

^{sur le}

plan

^de

séparation

de deux milieux

homogènes isotropes,

par des méthodes de

déphasage.

^Ceci

explique

à la fois l’effet Goos-Hänchen

longitudinal

^dans

lequel

une source de lumière

monochromatique

^non

polarisée

donne deux images ^de

polarisations rectilignes orthogonales,

^{et l’effet} transverse étudié récemment par

Imbert,

pour

lequel

^{les deux}

images

^sont

polarisées

circulairement. Nous donnons une méthode

simple

pour déterminer les

polarisations

^{et les}

positions

^des

images

^{d’une source}

cylindrique.

Abstract. ²⁰¹⁴ We

study by phase-shift

methods the

birefringence properties

of total reflection of a

light

^{beam at the}

separation plane

^{between two}

isotropic homogeneous

media. This includes the Goos-Hänchen

longitudinal

effect in wich a source

of unpolarized light

^is

split

^{into two}

images,

each of them

linearly polarized,

^{as well as the} transverse shift

recently investigated experimentally by Imbert,

where the effect is between left and

right

^circular

polarizations.

^We give ^a

general simple

method for

determining

^the

polarizations

^and

positions

^{of the}

images.

Classification Physics ^Abstracts

08.10 ^- 08.20 ^- 18.10

Introduction. ^- The existence of

birefringence

pro-

perties

of total reflection of

light

^{on the}

separation plane

^{of two}

isotropic homogeneous

^{media has been}

known for some time

[1 ],

^and

experimental

^evidence

was first demonstrated

by

^{Goos and Hanchez}

[2].

These authors considered a

large

^number of successive total reflections of a

pencil

^of

light

^{between two}

parallel planes,

^{and found that a source of}

unpola-

rized

light

^is

split

^{into two}

images,

^one

image being polarized parallel

^to the incidence

plane,

^{the other}

perpendicular. Using

^{a different}

apparatus,

^Imbert

[3]

has

recently

found evidence for a

splitting

^between

right

^and left circular

polarizations (see

^also

[10]) :

ⁱⁿ

his

experiment,

^the successive total reflections of the

pencil

^of

unpolarized light

^take

place

^on the sides of a

regular

dielectric

prism.

These

effects,

^{which are} of the order of one wave-

length

^{at each}

reflection,

have been

computed by

various methods

[4], [6], [7]

among which

[4]

^{the use}

of

Poynting

^{vectors is} very

popular.

^{In our}

opinion however,

^{if the}

Poynting

vector may be useful to

obtain an order of

magnitude

^{of the}

effect,

^{it cannot}

provide

^a

completely

consistent treatment of the

problem :

ⁱⁿ

particular

because it does not tell

clearly

which

polarization

^{states are}

relevant ;

^another

(*) Laboratoire associé au Centre National de la Recherche

Scientifique. Postal address : Laboratoire de Physique Théorique

et Hautes Energies, ^Bâtiment 211, Université de Paris-Sud, 91405 Orsay, ^France.

unphysical point

^{in the}

computation

of the various effects with the

Poynting

^{vector is the use of}

plane

waves of a finite

width, neglecting

what may ^occur

near the

edges.

^{For these} reasons, ^an

approach using phase-shifts

^{seems safer} to us, in

particular

^because

it is more

closely

^{related both to the exact solution of}

Maxwell’s

equations,

^{and to} the mechanism of for- mation of

images

ⁱⁿ

geometrical optics.

In this paper, ^we show that one can

compute

^all effects which have been

experimentally

^observed

by using

^{the exact solutions} of Maxwell’s

equations

^for

the reflection of

polarized plane

^{waves on} the sepa- ration

plane

^{of two}

homogeneous isotropic media, together

^{with some}

simple geometrical optics approxi- mations,

^{which turn out to be}

always

valid in expe- rimental situations. We deal with the usual

longitu-

dinal Goos-Hânchen effect in section

1,

^{and with} Imbert’s transverse shift in sections 2 and 3. It turns out that the

geometrical arrangement

^{of the}

experimental apparatus

is of crucial

importance

ⁱⁿ the determination of the

polarizations

^{of the two}

images

^{of an un}

pola-

rized source.

1.

Longitudinal

^shift

(Goos-Hânchen).

^{- The lon-}

gitudinal

shift is the easiest to

compute, owing

^{to the}

simpler geometrical

^{structure of the}

system :

^we consider

(Fig. 1)

^a rectilinear ^source

of light

^S

(narrow

slit for

instance), emitting

^a

pencil

^of

light

^{of small}

aperture

Brn. This

pencil

^is

totally

^{reflected on} the

plane

surface between the ^vacuum and an

isotropic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003405-6033500

336

medium with index n. This

separation plane

^is

parallel

to the source ;

Oy

is its intersection with the

plane

ⁱⁿ

which

figure

^{1 is drawn.}

Let i be the ^mean

angle

of incidence of the narrow

pencil

^emitted

^by

^{S. We can}

decompose

^this

pencil

into ^a linear

superposition

^of

plane

^{waves, which have}

slightly

^{different incidence}

angles

^on

Oy.

^Each

plane

wave can then be reflected on the surface

Oy,

^accord-

ing

^{to Maxwell’s}

theory.

^We then consider the set of the reflected

plane

^{waves. In the}

approximation

^of

geometrical optics (to

^be

justified later),

^{these reflected}

waves recombine

by superposition

^{to a}

pencil which,

ⁱⁿ

lowest order in _Brn, comes from the virtual

cylindrical

source 1 : 1 is

just

^{the center of curvature of the}

envelope

of the reflected

plane

^{waves. 1 will in}

general

be different from

S’,

^the

symmetric

^{of S with}

respect

to

Oy,

^{and its}

position

^will

depend

^{on the}

polarization,

thus

giving

^{rise to} the effect.

It is

straight

^{forward to}

compute

^the location of I : its coordinates in the

(S’ X,

^S’

Y) system (see Fig. 1)

are :

where À is the

wavelength

^{of the}

light

^{in vacuo, and}

5(i)

the

phase

^{shift of a}

plane

^{wave at total}

reflection,

^as

computed

from Maxwell’s

theory. Going

^to the expres- sion of 5 in terms of i and n for

polarizations

perpen- dicular and

parallel

^{to the incidence}

plane [5],

^one

finds the

explicit expressions :

B"1’)

As i

varies, I1

^and

III

^{define two caustic surfaces}

which are the

images

^{of S}

through

^our

anisotropic optical system.

Several remarks ^{are in} order : formulae

(2)

^and

(4)

do not seem to exist in the literature ^on the

subject.

Our results for the

experimentally

^{observed quan-}

tities

L1 LL

^and

All1

^agree with those of Hora

[6]

^and

Boulware

[7]

^{who also} derives them via ^a

phase-shift analysis. ^However, they

^differ

by

^{a factor}

^cos2 i/cos2 ij,

with

sin il

⁼

1/n

^{from the formula of Renard}

[8]

^and

Imbert

[3] who, however,

^{derive them}

using

^{a heuristic}

argument

^with

Poynting

^vectors.

°

Since the

image

^{of a beam with a finite}

^aperture

^{Brn is}

FIG. 1. ^- Goos-Hânchen Reflection

a finite

portion

of the caustic

surface,

^the

separation

of

polarizations

^is

complete only

^if the distance between the two

images

^is

larger

than their

spreading

out :

If this condition is not

satisfied,

^{one can still see} the Goos-Hânchen effect

by using light polarized,

perpen- dicular or

parallel,

^{to the incidence}

plane.

Let us now

briefly

discuss the

validity

^{of the}

approxi-

mation of

geometrical optics

^{used in our derivation}

of formulae

(1)-(4). Indeed,

it is in

principle

^unsafe

to use

geometrical optics

^{to obtain a result like}

(1)-(4)

which is of the order of a

wavelength,

^{that is to} say of the same order of

magnitude

^as diffraction.

However,

in the

experimental apparatus,

^{the beam is reflected} N times between two reflection

planes

^like

Oy,

^{in order}

to

amplify

the effect. In this process, it is clear that indeed

phase-shifts

^add up, whereas the

magnitude

^of

diffraction,

^{which is set}

by

^the

aperture

^{of the}

beam,

remains constant. Thanks to this

multiplying effect,

the two

images

^are

split by N(L11.L -

¹¹

111),

^{which can}

be made much

larger

^than diffraction.

Hence,

^whereas

it would be

meaningless

^{for a}

single reflection,

^our

approximation

^of

geometrical optics

^is

justified

^{for the}

full

experiment.

2. Transverse shift and

approximate analysis

^of

Imbert’s

experiment.

^{- In}

[3],

Imbert describes ^a nice

experiment

which shows that total reflection can also

give

^{rise to a}

splitting

^{between the two circular}

pola-

risations. He also

computes

^{this effect}

by using

^the

Poynting

^{vector of the} evanescent wave. What we

want to show in this section is that the effect can be

computed by using phases, just

^{as the}

longitudinal

effect of section 1. The

origin

^{of this new effect on}

circular

polarizations

^{is more subtle than the effect of}

section

1,

^{and rests}

essentially

^on the rather

complex

geometrical

^{structure of Imbert’s}

experiment.

In this

section,

^{we shall}

only give

^an

approximate

treatment of the

effect, leaving

^the

complete analysis

for the next section.

Indeed,

^{we shall now show that}

the

geometry

of Imbert’s

experiment implies

^{that the}

splitting

^{of the two circular}

polarizations

^is

already present,

^{even if the}

phase

^{shifts at} each reflection are

zero,

independent

^{from the incidence}

angle

^{for both}

linear

polarizations, perpendicular

^and

parallel

^{to the}

incidence

plane.

Since in the

experiment

^of

[3]

^all

incidence-angles

^are very close to the

limiting angle,

the

phase

shifts of both transverse and

parallel polariza-

tions are indeed

quite

^small.

However, they

vary

rapidly

^{when the incidence}

angle ^varies,

^{so that the}

treatment of this section is

only approximate

^{in that}

sense : we shall find that it accounts for about two thirds of the

experimental ^{result ;}

^{a more refined}

treatment, taking

^{into account} the variation of

phase shifts,

^{will be}

presented

^{in the next section.}

We now describe

(Fig. 2)

^the

geometry

of the expe-

riment, following

Imbert’s notations whenever pos- sible. The source S is a line in the horizontal

plane

^H.

It emits a

cylindrical

^narrow

pencil

^of

light

^{with the}

angle

^{0’ with}

respect

^{to the} horizontal

plane

^{H. The}

light

^emitted

by

^{S is}

totally

^{reflected on the faces of a}

vertical

prism

^of

glass

^{of index n = 1.8. The cross-}

section of this

prism

^{is an}

equilateral triangle.

^The

...

FIG. 2. ^- Imbert’s Apparatus

(perspective)

arrangement

^is such that the

light

^{follows a helical}

path

^{in the}

prism.

The view of that

path

from under is drawn on

figure

^{3. If the}

phase

^{shifts at} each reflection

are zero, ^a

linearly-polarized plane-wave

^remains

linearly polarized. However,

the direction of the

pola-

rization vector is in

general

^{différent for each} ray of the reflected beam.

FiG. 3. ^- Imbert’s Apparatus (from under) This contrasts with the result of an ideal metallic

reflection,

^{where one of the}

phase

^{shifts is} zero, and the other is _7:, in which case there is no effect.

It is then _easy to

get

convinced that the

polarizations

which should be used are not the linear

polarizations,

but the circular

polarizations

^which

diagonalize

^the

S-matrix,

in Imbert’s

experiment.

^In

fact,

^{after one}

reflection

they

^are conserved for all the _{rays of} the beam.

For all rays issued from the _source, let us take the vector no

orthogonal

^{to the source and to the beam}

as the

origin

^of

phases.

^{For the first}

reflection, phases

are to be counted with

respect

^to the normal ni ^to the incidence

plane.

^{This vector} ni

depends

^on

0’,

^the

angle

^{of the} ray and of the horizontal

plane.

^Let

qJ 1 ((}’)

be the

angle

between no and ni, and n2 the

symmetric

of no with

respect

^to the reflection

plane.

^If

cot(- cvt)

is the

phase

^{of a}

right (left)

^{circular wave with}

respect

to no before the

reflection,

^{it is rot - 2}

qJ 1 ( -

^{rot - 2}

qJ 1)

with

respect

^to n2 after the reflection.

Hence,

^total reflection induces for each circular

polarization

^a

phase

shift which

depends

^on the ray of the beam.

For the second

reflection,

^the

images Si

^of

S,

^and

n2 of no

play

^{the same role as S and} no for the first reflection.

By studying

^the

geometry

^{of the}

system,

one also finds that the

angle

ç between two successive incidence

planes

^is ç ⁼ 2 _{qJ 1.}

Hence,

^{after N reflec-}

tions,

^the

phase

^{shift of a}

right (left)

^{circular ray is}

NqJ((}’) (- Nç(0’)). ^By

^{the same}

^argument

^{as for the}

longitudinal

^shift of section

1,

^{one finds that the source}

is then

split

^{into two}

images, right

^{and left}

polarized,

which are

separated

^{from each other}

by 2 d,

^with

338

The

experimental set-up

^of

[3] gives :

so that one finds

(1)

or,

numerically,

This is to be

compared

^{with Imbert’s}

experimental

value

[3] ]

We thus see that our

approximate

^{treatment which}

neglects

^the variation of l5 with the incidence

angle gives

^{an effect}

^which, though

^too

small,

^{is of the}

correct order of

magnitude.

3.

Complète description

^of the transverse shift. - Let us now come to a more

rigorous

^{treatment of the}

transverse

shift,

^{in which we} shall take into account the variation of

phase

shifts with the incidence

angle.

In the first two

preceding sections,

^{we have used the}

approximation

^of

geometrical optics

to reconstruct the

images

^{from the}

envelope

^{of the}

outgoing plane

^waves.

However,

^{we are}

dealing

^{with a}

system

^{in which}

pola-

rization

plays

^{a crucial}

^role,

and standard

geometrical optics

^{do not tell us which}

polarizations

^{of the out-}

going

^waves should be used to determine the

position

and nature of the

images.

Let us see, in an

example,

the kind of

problems

^one

runs into when one does not choose the

polarization

to which one

applies geometrical optics appropriately :

in section

1,

^{we could}

imagine

^{to send an incoherent}

light

^{on the}

apparatus,

^{and at the}

exit,

^to

pick

^{up a}

given

^circular

polarization (say, right)

^{with a filter.}

The

envelope

method of section 1

applied blindly

^to

this

outgoing

^{set of}

right circularly-polarized plane

waves would then

give

^a

single image,

^{half way}

between the two

images

^of

linearly polarized light.

This would contradict the result of section

1,

^{which is}

that one should observe

precisely

^the

right

^circular

component

^{of these two}

images.

^This

paradox

^is

easily

^{solved : the two}

images

^found in section 1 are

observable if

they

^{are well}

separated

^with

respect

^to diffraction effects : the beam must have ^a

large enough aperture,

^{and the}

intensity relatively

^{constant over this}

aperture.

^{One then finds that over such an}

aperture

^the filtered

right

^circular

light

^would vary ^a lot in

intensity, passing through

^zero many times.

Hence,

^{such an}

amplitude

^{cannot be}

interpreted

^as

coming

^{from a}

single image,

^but

precisely

^as the inference

pattern

(1) Ly is the shift normal to the slit S ; ^it corresponds ^{to a} shift L’ y

= cos Ly qJ /2

^{normal to} the incidence plane, ^or

that result agrees with Schilling’s [8]. ^{It can be shown to hold} for _any prism whose cross-section is a regular polygon

produced by

^the

right

^circular

component

^{of the two}

linearly polarized images

of section 1.

From this

example,

^{we can} induce the

general

^cri-

terion for the choice of

polarizations

^{to which the}

geometrical optics

^treatment of section 1 and 2 can be

applied ;

^we restrict ourselves to

cylindrical beams,

which is

enough

^{if the source is a slit or a} rectilinear

object

^as in Imbert’s

experiment (2).

^{In this case,}

considering

the incident beam as a

superposition

^of

plane

waves, the action of the

optical system

^{on each}

plane

^{wave can} be described

by

^a

two-by-two

^{matrix B :}

if the

polarization

^{vector of} the incident

plane

^wave

is

a , BP/ ^the polarization

^{vector of the}

outgoing plane

wave is

B a .

^{For a}

cylindrical

^incident

beam,

^the

matrix B

depends only

^{on one}

parameter, 0’,

^which

is the

angle

of the rays of the beam with some reference

plane.

It is clear from the discussion of the

preceding

section that the

polarization

^states

a BP/

^{to which one}

can

apply geometrical optics

^{are those} such that the

outgoing polarization

^states

B(e’) a W

^are

independent

from

0’, except

^{for an overall}

phase,

^{when 0’} spans the beam : for _any other choice of the incident

polariza- tion,

^the

polarization

^would vary

rapidly

^{over the}

outgoing beam,

^{so that}

simple geometrical optics

would be

inapplicable.

^{In other}

words,

^{one should}

look for the

polarizations

^such

that,

^{at the first order}

in e

(3),

(2) ^We hope ^to deal with the more complex ^{case of} spherical

waves in a later publication.

(3) ^{Here we derive} rigorously ^a formulation in terms of geo- metrical optics ^{from Maxwell’s} equations (the ^crude approxi-

mation À ⁼ 0 would give ^{J =} 0).

1) ^{We must} divide the general problem ^{into two} problems

with one parameter (the eikonal) ^each.

2) ^{We can write}

([9]

^p.

119)

^{E and H in the general form}

omitting ^the phase factor e-irot ,(e ^{and h are} complex vectors).

3) ^{In a} homogeneous ^{medium of} index n, ^{Maxwell’s equa-} tions become

-

with

and the analogous equations with h. We use the eikonal 8 for geometrical optics : ^{it means K = 0. We can also} impose

L

^{= 0}

i. e. polarization ^{is constant} along each « ray » [9].

Conclusion : We must then have Ae ⁼ 0 to verify ^Maxwell’s equations

more (

^precisely

Â2 lael « lel 1 . )

That is the case if e(r) ⁼ Cte, that is if the only ^{variation of E is} in the eikonal

or

( BP/ ⁾

^{is thus an}

eigenvector

^of

B-1 B’,

^with

eigenva-

lue ia. B

being

^a

unitary ^{matrix, a}

is real. This

eigen-

value is then related to the

displacement

^{L1 of the}

image

^{of the source}

by

the formula

A

general polarization

^state of the incident beam should then be

decomposed

^{as a}

superposition

^{of the}

two

polarizations al , (Pl

^@

("2) ^l’2

which solve _eq.

(6)

^and

one then observes two

images,

^{for such a}

general pola- rization,

^with

polarizations B (fil a 1 and B a2 ^P2

^which

are

separated by L11

^{1 -}

d 2.

^{It is now}

straightforward

^to

apply

^{those ideas to Imbert’s}

experiment :

^{at each}

reflection inside the

prism,

the relevant matrix A includes not

only

^the

phase

^shifts

given by

^{the solu-}

tions of Maxwell’s

equations,

but also takes care of the rotation ₉ of successive incidence

planes

^{due to}

the

peculiar geometry

^{of the}

experiment.

^{One then}

finds :

where

ôjj ^(b.1)

^{is the}

^phase

^{shift of a}

^plane

^wave

^pola-

rized

parallel (perpendicular)

^to the incidence

plane, b = t (bll

^-

^{b .1). qJ}

^{and ô are} functions of B’

through

the

geometry

^of

figure

2. The total transition matrix of the

apparatus

^is

where N is the number of reflections inside the

prism.

Introducing

^the

angle

^u

by

one finds

where

Using

^Imbert’s

experimental

^{values :}

and the

simplifying

^{fact that}

one

easily

finds that the

eigenstates ^{of B-1 B’}

^are

indeed the left and

right polarizations

when ô L---

0,

and that the distance between the ^two

circularly polarized images

^is

One ^can translate this in terms of the

displacement

per reflection :

From formula

(8),

^and

using

the fact that b N

0,

^one

finds

In this

equation, dç/d0’

^{is the}

geometrical

^term

already given by

section 2. One also has in Imbert’s

experi-

ment :

and,

ⁱ

being

^{close to the}

limiting angle ii,

^{one can use}

the

approximate

^{formulae :}

Hence

(4)

or

numerically,

This value _may be

compared

with Imbert’s result :

The ratio of the two values is 1.03 for n ⁼ 1.8 and i

= il

ⁱⁿ

good agreement

^with

experimental

^results.

Conclusion. ^- We have shown that

simple

argu- ments

using

^classical

optics

^{and a careful}

analysis

^of

the

geometrical

^structure of Imbert’s

experiment

^can

yield

^{a direct}

computation

^and

interpretation

^{of his}

results in terms of the

phase

^{shifts at total reflection}

computed directly

from Maxwell’s

equations :

^these

methods also

apply

^{to the}

longitudinal

Goos-Hânchen

effect,

^and

explain

^{how a}

single phenomenon (total

internal

reflection)

^{from a}

single-unpolarized

^source

(4) ^{That result was} independently ^derived by ^{D. Boulware} [7],

340

can

give

^two

images

^{which are either}

linearly pola-

rized

(Goos-Hânchen)

^or

circularly polarized (Imbert).

Acknowledgements. -

^{We are}

grateful

^to

Pr. C. Bouchiat for a discussion which is at the

origin

of this

work,

^{and to Dr. C. Imbert} and 0. Costa de

Beauregard

^for many informations about their _expe- riment and their calculations.

NOTE : While this work was in progress, ^we received

a

preprint by

^{D. Boulware}

[7]

^{in which a}

phase-shift analysis

^{of Imbert’s}

experiment

^{is also}

proposed.

References

[1]

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Physik

³

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H., ^Ann.

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5

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[6]

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[8]

SCHILLING, H., ^Ann.

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