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UC-NRLF



B M 533 Dlh



FRESNEL'S THEORY OF
DOUBLE REFRACTION

by
W* Steadman Aldis






1




FRESNEL'S THEORY OF

DOUBLE REFRACTION.



W. STEADMAN ALDIS, M.A.

TRINITY COLLEGE, CAMBRIDGE.

SECOND EDITION.




With the Authors Compliments.



A CHAPTER

mmmn "':
JUL18 1897

OS




FRESNEL'S THEORY OF

DOUBLE REFRACTION.
8102



BY



W. STEADMAN ALOIS, M.A.

TRINITY COLLEGE, CAMBRIDGE.

PRINCIPAL OF THE UNIVERSITY OF DURHAM COLLEGE OF PHYSICAL SCIENCE AT NEWCASTLE-
UPON-TYNE AND PROFESSOR OF MATHEMATICS IN THE SAME,



SECOND EDITION.



CAMBRIDGE :

DEIGHTON, BELL, AND CO.

LONDON : G. BELL AND SONS.

1879



PRINTED BY C. J. CLAY, M.A.
AT THE UNIVERSITY PRESS.



Mr GRIFFIN'S tract on Double Refraction has been
for some time quite out of print. The following pages
are published with a view to supply the deficiency
thus caused. It is hoped that they may serve as a
useful companion to the latter part of the Astronomer
Royal's treatise on the Undulatory Theory of Light.

W. S. A.

CAMBKIDGE,

April, 1870.



Preface to the Second Edition.

THIS Second Edition is an almost verbatim reprint
of the first. It was the author's intention to incorporate
this chapter in a larger work on the Wave Theory of
Light. Unexpected hindrances have delayed the pro-
gress of this work beyond his expectation, and this
reprint is therefore issued to satisfy the demand of
present students.



NEWCASTLE-UPON-TYNE,
April, 1879.



12



FBESNEL'S

THEORY OF DOUBLE REFRACTION.



1. FRESNEL'S Theory of Double Refraction supposes that
the phenomena of light are produced by the vibrations of par-
ticles of ether under the influence of their mutual attractions.

The hypothesis is first made that the particles of ether are
arranged in such a manner that each of them is in stable equi-
librium under the influence of the attractions of the others. Let
R be the potential of all the system of particles with respect
to a point. Then the resolved parts of the force on the particle
at this point parallel to axes arbitrarily assumed will be, if

x ' y, z be the co-ordinates of the point, -^ , -j- , =- re-

ax ay dz

spectively, tending towards the origin. Hence we have
dR dR dR .

-T- = 0, -j-0, -T- = (1).

dx dy dz
Let the single particle at x, y, z be displaced to a point

a + u, y + v, z + w,

while all the other particles remain at rest. Then if we suppose
u, v, w so small that we may neglect their squares and higher
powers, the force on this displaced particle parallel to the axes
will be



dR d*R t _.

~T~ + u T~a + jj- + W jj-

ax dx dxdy dzdx

dy dxdy dy* dy dz
dR



respectively.



j ^ | ^ t ^

dz dzdx dy dz



6 Theory of Double Refraction.

Of these the first term in each vanishes by (1), and putting

-



we get, if X, F, ^denote the forces parallel to the axes on the
displaced particle,



.... ........... (2).

)

Now if we construct the quadric whose equation is

Ax 2 + By* + Cz* + 2 A 'ye + ZB'zx + Wxy = 1 ... (3)
the direction of the resultant force whose components are
X, F, Z is perpendicular to the plane which bisects all chords
of the surface (3) parallel to the direction of displacement of
the particle ; for the equation of this plane is

%(Au+C'v + B'w) + 97 (C'u + Bv + Aw) + f (B'u +A'v + Cw) =0.
The resultant force on the particle will therefore not usually
coincide with the direction of its displacement ; and if we suppose
the particle free to move under the action of this force it will not
usually return to its old position. There will be however three
directions of displacement with which the directions of the force
of restitution will coincide, namely the directions of the three
principal axes of the surface (3).

If these directions be taken as axes of co-ordinates the equa- "
tion (3) reduces to .

Ax* + By 2 + C^ = 1,

and the equations (2) reduce to

X=Au, Y=Bv, Z=Cw.

Now it is evident that if u, v, w are all positive, X, F, ^must
all tend towards the origin, since the equilibrium is stable, and
A, B, C must be all positive. They are usually denoted by the
letters a 2 , ft 2 , c 2 . The equation (3) thus becomes

V + &y + cV = l .................. (4).

This surface is usually called the ellipsoid of elasticity, and
its axes the axes of elasticity. It is assumed that the directions



Theory of Double Refraction. 7

of these axes and the values of a, b, c are constant throughout
the medium. A medium in which a, 6, c are all or any of them
different is called a crystal. If all are unequal it is called a
biaxal crystal. If two of them are equal and the third different
it is called a uniaxal crystal.

If the particle be displaced parallel to the axis of x and the

other particles be undisturbed it will oscillate in a time , for
its motion is given by the equation -^ = a*u.

2. It is then assumed that under these circumstances a
particle so displaced will draw an adjacent particle into a pre-
cisely similar state of displacement, and that this again will
draw the next, and so on ; that thus a series of vibrations will
be propagated through the medium, the velocity of propagation
being connected with the constant a and the wave length by the
simple relation

X 2?r \

- = or v = 7T~ . a.
v a 2-Tr

For it is supposed that the wave travels over a wave length
while one particle performs a complete oscillation.

If the particle be displaced through a space p in a direction
inclined at angles (a, 0, 7) to the axes of elasticity, the forces on
it parallel to the axes are

a*p cos a, I 2 p cos /3, c*p cos 7,

respectively, and the force on it in the direction of displacement
will be p (a 2 cos 2 a + 6 2 cos 2 fi + c 2 cos 8 7),

and for its motion in that direction we have therefore



~P (a* cos2 a + & 2 cos2 ft + c 2 cos 2 7).

If therefore the motion in that direction alone be considered the
time of the particle's oscillation will be

_ 27T

Va 2 cos 2 a + 6 2 cos 2 + c 2 cos 2 7 '

and if a wave of such vibrations can be propagated through the
medium its velocity of propagation will as above be propor-
tional to Va 2 cos 2 a. + 6 2 cos 2 ft + c 2 cos 2 7.



8 Theory of Doulle Refraction.

But if r be the central radius vector of the ellipsoid of elas-
ticity drawn in the direction of this displacement, we have

- = a 2 cos 2 a + 6 2 cos 2 & + c 2 cos 2 7.

Hence the velocity of propagation of the wave corresponding
to any given direction of displacement, if such a wave exist, is
inversely proportional to the central radius vector of the ellipsoid'
of elasticity drawn in that direction.

3. At this point it will be well to notice the important
assumption made. The force on any particle is made to depend
on its absolute displacement, and is supposed to be the same as if
the other particles were undisplaced. It is evident that the real
force will depend on the displacement of the particle relative to
the surrounding particles, and quite a different equation of
motion from that given above will arise. A particular case of
the investigation is given in Airy's Undulatory Theory of Optics,
Art. 103, and the general problem has been discussed by Cauchy.
The simplicity and beauty of the Mathematical results of
Fresnel's hypothesis probably more than counterbalance, from
the point of view of a mathematical student, the possible superior
accuracy of the more complicated hypothesis.

4. In considering the propagation of light through media of
any kind, it is necessary to examine not the motion of one par-
ticle alone, but to imagine a series of particles simultaneously
vibrating similarly. The most simple hypothesis that can be
made is that all the similarly displaced particles at any instant
lie in a plane, the case ordinarily called a plane wave. It is
evident that by the combination of a number of such plane
Waves we can represent any other form of wave.

A plane wave of light consists of vibrations of the particles
of ether in the plane of the wave front, the displacements and
velocities of all the particles in that plane being parallel and
equal. This wave is propagated with a velocity which in a
crystalline medium depends, as above explained, on the direction
of the displacement of the particles.



Theory of Double Refraction. 9

5. The fact that the vibrations which produce light are
transversal to the direction of propagation, is deduced from the
experimental result that two rays of light polarised in planes at
right angles do not interfere. The methods of practically pro-
ducing polarised light are explained in Airy's Undulatory
Theory. We assume that polarised light consists of vibrations
of the particles of ether in a fixed direction in the plane of the
wave front, and that this direction is perpendicular to the plane of
polarisation. The former assumption is sufficient to explain the
experimental fact, the latter is usually accepted as true.

6. If a series of particles all lying in a plane within a
crystalline medium be equally displaced in parallel directions,
the force on each of these particles according to Fresnel's hypo-
thesis will not usually be in the direction of displacement, or
even in the plane. It may happen however that the resolved
part of this force in the plane may coincide with the direction of
displacement ; and we will prove presently that there are two
directions of displacement for which this is the case. If the
particles be displaced in either of these directions the force per-
pendicular to the plane will produce vibrations perpendicular to
that plane, which therefore do not produce light ; the other parts
of the force will cause all the particles to oscillate equally in the
plane front, and will thus produce a wave of light, if we assume
that the particles oscillating in this plane immediately put in
motion those in a contiguous parallel plane. The velocity of
propagation of the wave will also, by what has preceded, be
inversely proportional to the radius vector of the ellipsoid of
elasticity drawn in the direction of the displacement.

7. Suppose that DPD' represents the central section of the
ellipsoid of elasticity by a plane parallel to the wave front, and
let C be its centre, CP the direction of displacement, CD the
diameter of the section conjugate to (7P,and(7Q the diameter of the
ellipsoid conjugate to the plane PCD. Then the" force of restitution
is perpendicular to the plane QCD, since this is the plane to
which CP is conjugate, and if the resolved part of this force in
the plane of the wave front coincide with CP, we must have CP
and CD at right angles, or CP must be an axis of the section

lo



10



Theory of Double Refraction.




DPD'. Hence the two directions of vibration with which the
resolved part of the corresponding force in the plane coincides
are the axes of the section of the ellipsoid of elasticity by the
plane front, and the velocities of propagation of the corresponding
waves are inversely proportional to the lengths of those axes.

8. If the equation of the plane front at first be

Ix + my + nz = (1),

and \, fjL } v, the direction cosines of either axis of the section, we
have (Aldis, Solid Geom. Art. 56) the equations

0'










(2)



to determine X, //<, v, the direction cosines of the lines of displace-
ment ; and if v be the velocity of propagation of either wave we
have to determine v the equation



-c 2

(Aldis, Solid Geom. Art. (56), Formula (10), altering a 2 , b'\ c\ r*

1111 .-IN

into -, 77,, ,, -o respectively.)

or 1> c 11 r * '



Theory of Double Refraction. 11

If all the particles in the plane (1) be displaced in any other
direction than either of those given by (2), these displacements
can be resolved into two, one in each of those directions, and
there will then result two sets of vibrations travelling with the
velocities given by (3). Hence if at any instant there be a
series of particles in the plane (1) vibrating equally in parallel
directions, after a unit of time vibrations will be excited in the
two planes

Ix + my -f- nz v v

Ix -f- my -\-nz- # 2 ,

v x v 2 being the values of v obtained from (3). Also each of these
sets of vibrations will compose a wave of polarised light, the
planes of polarisation being perpendicular to the two lines whose
direction cosines are given by (2).

9. If the envelope of the plane

Ix + my \-nz = v..... (1)

be investigated, where Z, m, n, v are connected by the equations

= 1 (2),

= (3),

we shall obtain the equation of a surface which all the wave
fronts touch after a unit of time, in whatever direction the
original wave front may have been situated.

Differentiating (1), (2) and (3) we have
xdl + ydm + zdn dv = 0,
Idl + mdm + ndn 0,
Idl . mdm . ndn , f Z 2
a 2



n* l_/\

-c 2 ) 2 ] ~



Whence, using indeterminate multipliers, we obtain

Q (4),



.(5),



12



Theory of Double Refraction.



(6),



l+Bv



m



Multiplying tlie first three of these equations by I, m, n
respectively and adding, we get,

v + A = 0.

Transposing the third terms of these same equations, squaring
and adding, we get, if r 2 = a? + y* + /,



= B>{^



m



i ;

by (4) x = w { 1 + -5 si- = w

J v '



= mv



(6)*



Again from (4), (5), and (6) multiplying them by x, y, z
respectively and adding



or



Whence, putting for B its value, the equation required becomes

*** *\ / 2 i <f = *

This can be reduced into a different form, for multiplying by
r 2 = x z + y z + 2 it becomes

-ce 2 + -2-f;T2-# 2 +
J !



or



r a



2 + -f^T 2 + -f^- 2 = ( >-
r b r 2 c



Theory of Double Refraction. 13

The equation of the wave surface can also be deduced in the
following manner.

The perpendicular on any tangent plane to the surface being
inversely proportional to a principal axis of the parallel central
section of the ellipsoid of elasticity, it follows that the polar
reciprocal of the wave surface with reference to the origin is an
apsidal surface of this ellipsoid. Whence by Salmon, Solid
Geometry, Art. 463, the wave surface is also an apsidal surface of
the reciprocal surface of the ellipsoid of elasticity, that is of the
ellipsoid whose equation is



From this property its equation can be easily deduced by
eliminating I, m, n between the equation



6 2 m 2 cV



which gives the lengths of the axes- of the section of (9) by the
plane whose equation is Ix + my -f nz = 0, and the equations

x lr t y = mr, z = nr,
whence we get

aV IV cV _
^- a 3-lV-& 2 + r 2 -c 2 ~ U '

where r 2 = # 2 + y z + z*.

10. If with the different points of the original wave front
as centres we describe a series of equal wave surfaces it is evi-
dent that the plane

Ix + my + nz = v

will touch them all. That is, the new wave front may be re-
garded as the envelope of these wave surfaces. This is analogous
to the case of propagation of light through a homogeneous
medium, in which case the wave surfaces are spheres. We may
also fairly suppose that the point in which the wave surface
having any given point of the original wave front as centre touches
the second wave front, is the point at which the disturbance in
the second wave front is produced by the disturbance at the

1-7



14 Theory of Double Refraction.

given point of the first front, and the line joining these points is
the direction of the ray proceeding from the first point. A ray
must be considered as a small portion of a wave separated from
the rest. The existence of such rays must be accepted as a fact ;
the theoretical explanation of the separation of a portion of a
wave from the rest need not be considered here, belonging rather
to the question of diffraction.

It is not difficult to see that the reciprocal ellipsoid



has important properties relating to the ray velocities analogous
to those which the ellipsoid of elasticity possesses with relation
to wave velocities. These the student can develope for himself.

11. If a wave of light be incident from vacuum into a
double refracting medium, we may suppose the vibration of each
point of the incident wave to produce after a time, a vibration at
some point of the wave surface described with the point of
incidence as centre.

Let the plane of the paper be the plane of incidence, and let
J.jBTbe the trace of the front of the wave on the plane of the paper,
AB the trace of the face of the crystal. Also let PQ be the





wave surface to some point of which the disturbance produced by
A has arrived when the disturbance at H has reached B. The
vibrations at intermediate points will have reached points of
wave surfaces similar and similarly situated to PQ, but sue-



Theory of Double Refraction. 15

cessively diminishing in size. Any plane drawn through B
perpendicular to the plane of the paper touching the surface
PQ will touch all these other surfaces and will be a front of the
refracted wave. There can be two such planes drawn, and thus
one incident wave will produce two refracted waves. The
corresponding refracted rays will be obtained by joining A with
the points of contact of these planes with PQ.

12. The preceding Article gives the refracted rays when a
ray passes from any homogeneous medium into a double re-
fracting crystal. The following construction applies when a ray
passes from any medium into any other.

With the point of incidence of the ray on the common sur-
face of the media as centre, describe in the second medium the
half of the wave surface belonging to each medium. Produce
the incident ray to cut the surface belonging to the first medium,
and at the point of intersection draw a tangent plane. This
tangent plane will cut the bounding plane of the media in a
straight line. Through this line draw tangent planes to the
wave surface of the second medium. The lines joining the point
of contact of these tangent planes to the point of incidence of the
ray will be the refracted rays.

It would appear at first from this construction that a single
ray passing from one double refracting medium into another
would give rise to four rays, since the incident ray would meet
the wave surface of the first medium in two points. We shall
see however presently (Art. 15) that if a ray proceeding in any
direction within a crystal have originally been refracted from
air, it must be polarised in one or other of two definite planes
according as it is considered to be proceeding to one or other of
the points in which its direction cuts the wave surface; and thus
if the given ray be polarised in either of these planes we must
only take one of the points as the point to which the incident ray
corresponds. If the given ray be either unpolarised or polarised
in any other plane it must have arisen from two rays of common
light, and must be considered to consist of two rays polarised in
the required planes travelling with different velocities. We



16 Theory of Double Refraction.

should in this case expect four rays, which the construction would
give. The construction includes the last article as a particular
case.

13. Returning to Art. 8, we see that for all ordinary
positions of the wave front, there are two velocities of propaga-
tion of the wave. These two will be equal if the wave front
coincide with one of the circular sections of the ellipsoid of
elasticity, and in that case, whatever be the direction of the
vibrations in the plane of the front, only one wave will be
propagated. The two lines perpendicular to these positions of
the wave front are called the optic axes of the crystal, or the lines
of equal wave velocity.

The planes of polarisation of the two rays corresponding to
any given wave front are connected with the optic axes by a
very simple relation, which we will now investigate.

Let CN})Q the normal to the wave front, CO, CO' the optic
axes of the crystal. Then the planes of polarisation of the two
rays are planes which contain CN and the axes of the sec-




tion of the ellipsoid of elasticity by a plane perpendicular to
CN. This section will evidently cut the circular section per-
pendicular to CO in a line perpendicular to the plane OCN.
Similarly it will cut the other circular section in a line perpen-
dicular to the plane 0' CN. Hence the radii of the section by



Theory of Double Refraction.



17



the wave front perpendicular to the planes OCN, O'CN are
equal and therefore they are equally inclined to the axes of the
section. The planes of polarisation of the two rays are therefore
planes through ON bisecting the angles between the planes
OCN and O'CN.



14. Again let v v v 2 be the velocities of the two waves
corresponding to the same wave front. We can express these
velocities in terms of the angles CN } and 0' CN, as follows.

The equation of the ellipsoid of elasticity being
the equations of the planes of circular section are
and that of the wave front is

Hence if we denote the angles OCN, O'CN by 0, & respectively,
we have

COS0 = -

V a 2 - c 2

cos & = .

Va 2 -c 2

/. (cos 9' + cos 6) Vet? c 2
(cos & cos 6} Va 2 c 2

Again v v v 2 are the roots of the equation
I 2 m 2 n 2 ^ Q

Hence

v 2 + v 2 = I 2 (b 2 + c 2 ) + m 2 (c 2 + a 2 ) + n 2 (a 2 + b 2 )

= a 2 + c 2 -(a 2 -c 2 )cos0cos<9'by (4) (6).

v 2 v 2 = I 2 b 2 o 2 + m 2 c 2 a 2 + nV6 2 ,

= aV - c 2 ^ 2 (a 2 - b 2 ) + nV (b 2 - c 2 ) ;



18 Theory of Double Refraction.

..4v 1 V=4aV-(a 2 ~c 2 ){c 2 (cos^+cos^) 2 -a 2 (cos^-cos^) 2 }by(5),
= 4aV+ (a 2 -c 2 ) 2 (cos 2 0+cos 2 0') -2 (a 4 -c 4 ) costfcos 0'. . . (7).
Hence squaring (6) and subtracting (7) we get

/ T 2 - 2\2 ^_ / 2 2\2 f "| 2/3 2 ^' 2/1 2 ^O

= (a 2 -c 2 ) 2 sin 2 0sin 2 0';

* 1 2 \ / * * \ /*

From (6) and (8) we easily deduce by adding and subtracting

/) 4- ff A j_ /?' "1

Vj 2 = a 2 sin 2 -= + c 2 cos 2 -=

2 0+(9' 2 + <9' I (9) '

V 2 2 = a 2 sm 2 - +c 2 cos 2

The results of equations (8) and (9) are easily seen to coin-
cide with those deduced in a different manner in Salmon's Solid
Geometry, Art. 245. Analogous results can be obtained for
ray velocities from the reciprocal ellipsoid. (Lloyd, Wave Theory
of Light, Art. 186.)

15. The formulae of the last article enable us to determine
completely the circumstances of the vibrations of the two rays
corresponding to the same wave front in the crystal. They do
not however determine the plane of polarisation if we are only
given the direction in which the ray proceeds within the crystal.
For this purpose we must revert to the wave surface of Art. 9.

Let a ray meet the wave surface at the point x, y, z, let
I, m, n be the direction cosines of the normal to the wave front
to which the ray belongs, and \, //,, v the direction cosines of the
direction of vibration of the particles in the ray. Then we have,
if v be the corresponding wave velocity,



where l - (b* - c 2 ) + - (c 2 - a 2 ) + - (a 2 - 5 2 ) = 0,
\ fj> v

and l\ + ni + nv= 0.



Theory of Double Refraction. 19

Whence eliminating n we get



& f72 ^22 272 2 21 m f 2 22 212 A 2 21 f\

or - [b X a /* 6 v c \ [a vc fj> o X a j = u,

I m n ,

' *,* ^ = ..^2 nt ^ = ,.^2 ^ by symmetry.



These equations determine X, /*-, z/ in terms of v.

Combining these results with the equations (8) of Art. (9),
we easily obtain

X (V 2 - a 2 ) = p (r* -b*)^v (y* - c 2 )
as y z

which give the direction of vibration in the ray proceeding to
any given point (oc, y, z).

A geometrical interpretation can be given to these equations.
The co-ordinates of the foot of the perpendicular on the tangent
plane to the wave surface at x, y, z are with our previous nota-
tion, Iv, mv, nv, and the direction cosines of the line joining this
point with the point of contact are proportional to

x lv, y mv, z nv.
But we have by equations (8) of Art. (9),



, x (r z - v*
:. x Iv ^ 5-



Similarly y mv



Hence X, ft, v are proportional to xlv, y mv, z nv, or the
direction of the vibration constituting any ray is the projection
of the ray on the tangent plane to the wave surface at the point



20 Theory of Double Refraction.

where it meets it. The plane of polarisation is of course per-
pendicular to this. The plane of polarisation of a ray proceed-
ing in a double refracting medium is usually taken to be the
plane containing the normal to the wave front and perpendicular
to the direction of vibration.

This result may be otherwise obtained. If X, /*,, v are the
direction cosines of the direction of displacement of a particle,
those of the resultant force are proportional to a 2 X, & 2 //,, cV
Hence the direction of displacement is perpendicular to the
tangent plane drawn to the reciprocal ellipsoid at the point
where the line of the resultant force meets it. From this,


1

Online LibraryW. Steadman (William Steadman) AldisA chapter on Fresnel's theory of double refraction → online text (page 1 of 2)