JAMES MAC CULLAGH'S ETHER:
AN OPTICAL ROUTE TO MAXWELL'S EQUATIONS?
INTRODUCTION
There are few modern references to MacCullagh.
Feynman's Lectures: The correct equations for the behavior of light in crystals
were worked out by McCullough in 1843.
Whittaker 1910: He succeeded in placing his own theory [of the refraction of
light by crystals] on a sound dynamical basis; thereby effecting that
reconciliation of the theories of light and dynamics which had been the dream of
every physicist since Descartes.
Was he a mere technician of the intricacies of crystal optics?
Or did he truly break new ground?
DOUBLE REFRACTION
1669: Berthelsen discovers
double refraction of Iceland spar.
E O
1690: Huygens constructs
extraordinary waves and rays
by taking the envelope of
ellipsoidal pulses originating
from the surface of the
crystal.
FRESNEL'S THEORY (1821-1822)
Basic ideas: - Transverse waves,
- Correlation between polarization and velocity of propagation.
Assumption: The propagation velocity (υ ) of transverse plane waves only
depends on the direction of vibration (α , β , γ ) .
Huygens's construction for uniaxal crystals yields υ 2 = a 2 (α 2 + β 2 ) + c 2γ 2 .
Generalization to biaxal crystals: υ 2 = a 2α 2 + b 2 β 2 + c 2γ 2 .
For a given orientation of the wave planes, the two possible directions of the
vibration are those for which υ is a minimum or a maximum.
Hence, the vector n normal to the wave planes and of length 1 / υ must belong to
the fourth-degree, double-sheeted index surface:
n 2 (b 2 c 2 n x + a 2 c 2 n y + a 2b 2 n z ) − [n x (b 2 + c 2 ) + n y (a 2 + c 2 ) + n z (a 2 + b 2 )] + 1 = 0
2
2
2
2
2
2
FRESNEL'S WAVE SURFACE
By inversion of the Huygens construction, the locus at time t = 1 of a pulse
emitted from the origin O at time t = 0 is the envelop at time t = 1 of all the plane
progressive pulses that pass through O at time t = 0 .
For the equation of this surface, Fresnel found
s 2 (a 2 s x + b 2 s y + c 2 s z ) − [ s x a 2 (b 2 + c 2 ) + s y b 2 (a 2 + c 2 ) + s z c 2 (a 2 + b 2 )] + a 2b 2 c 2 = 0
2
2
2
2
2
2
Fresnel deplored the difficulty of deriving this equation.
But he gave the following simple prescription:
Draw the ellipsoid whose axes are equal to the phase velocities in the directions
of the principal axes of the crystal. Cut the ellipsoid by a plane through its center,
and on a perpendicular to that plane mark the two points whose distances from
the center are equal to the axes of the elliptic intersection. When the orientation
of the plane varies, these two points describe the two sheets of the wave surface.
FRESNEL'S WAVE SURFACE
Fresnel's prescription: Draw the ellipsoid whose
axes are equal to the phase velocities in the
directions of the principal axes of the crystal. Cut
the ellipsoid by a plane through its center, and on a
perpendicular to that plane mark the two points
whose distances from the center are equal to the
axes of the elliptic intersection. When the
orientation of the plane varies, these two points
describe the two sheets of the wave surface.
Q•
B
•
A•
P•
•
O
Verdet: The most elegant and rapid procedure for arriving at the equation of the
wave surface is undoubtedly the one used by MacCullagh.
MAC CULLAGH'S FIRST MEMOIR (1830)
MacCullagh's gives a purely geometric justification of Fresnel's prescription of
the wave surface.
Two surfaces are said to be reciprocal if they meet the condition:
At point Q of the first surface, draw the tangent
plane and the perpendicular to this plane that
passes through the origin O. Call P the
intersection of the perpendicular with this
plane, and R its intersection with the second
surface. For any choice of Q, the distances OP
and OR are inversely proportional.
The ellipsoids x 2 a 2 + y 2b 2 + z 2 c 2 = 1 and x 2 a −2 + y 2b −2 + z 2c −2 = 1 are reciprocal.
Consequently, the surfaces derived from these ellipsoids through Fresnel's
prescription are reciprocal.
The first of these surfaces is the index surface. Hence, the second is the
reciprocal of the index surface, which is the wave surface by definition.
CONICAL REFRACTION
Hamilton 1832: When the elliptic section in Fresnel's prescription becomes a
circle, there are an infinite number of refracted rays forming a cone of light. Soon
verified by Lloyd.
Easy to analyze in MacCullagh's geometrical approach.
MacCullagh frustrated.
(James Lunney and Denis Weaire, from Preston 1890)
FRESNEL'S BOUNDARY CONDITIONS (1823)
Fresnel determines the relative amplitude of incident and reflected waves in the
isotropic case through the boundary conditions:
- Equality of the elastic constants on both sides (different densities).
- Equality of the parallel components of the vibration.
- Equality of the energy fluxes.
When the vibration is perpendicular to the plane of incidence:
a1 '
sin(i − r )
.
=−
a1
sin(i + r )
When the vibration is in the plane of incidence:
a1 ' tan(i − r )
.
=
a1 tan(i + r )
These formulae are empirically valid if the vibration is perpendicular to the plane
of polarization.
MAC CULLAGH'S BOUNDARY CONDITIONS (1835-1837)
In the isotropic case, he retrieves Fresnel's formulas under the assumptions:
- Equalities of the densities on both sides (different elasticities),
- Continuity of the vibration,
- Equality of the energy fluxes.
The vibrations must then be in the plane of polarization.
In 1837 MacCullagh works out the consequences of these conditions in the
anisotropic case and find laws agreeing with Brewster's and Seebeck's
experiments (after a first unsuccessful attempt in which he had replaced energy
conservation with the continuity of "lateral pressure").
Polar-plane theorem: "Beautiful" (FitzGerald), "remarkably elegant" (Poincaré)
This is the first published, complete theory of refraction for anisotropic media.
Neumann obtains similar results earlier but publishes them later.
MAC CULLAGH'S BOUNDARY CONDITIONS (1835-1837)
For this work, MacCullagh receives a medal of the Royal Irish Academy and
Hamilton's praise:
It may well be judged a matter of congratulation when minds endowed with
talents so high as those which Mr. Mac Cullagh possesses, are willing to apply
them to the preparatory but important task of discovering, from the phenomena
themselves, the mathematical laws which connect and represent those
phenomena, and are in a manner intermediate between facts and principles,
between appearances and causes.
MacCullagh's Keplerian approach
MAC CULLAGH'S BOUNDARY CONDITIONS (1835-1837)
This echoes MacCullagh's own perception of his work:
If we are asked what reasons can be assigned for the hypotheses on which the
preceding theory is founded, we are far from being able to give a satisfactory
answer. We are obliged to confess that, with the exception of the law of vis viva,
the hypotheses are nothing more than fortunate conjectures. These conjectures
are very probably right, since they have led to elegant laws which are fully borne
out by experiments; but this is all that we can assert respecting them. We cannot
attempt to deduce them from first principles; because, in the theory of light, such
principles are still to be sought for. It is certain, indeed, that light is produced by
undulations, propagated, with transversal vibrations, through a highly elastic
ether; but the constitution of this ether, and the laws of its connexion (if it has
any connexion) with the particles of bodies, are utterly unknown. The peculiar
mechanism of light is a secret that we have not yet been able to penetrate…. In
short, the whole amount of our knowledge, with regard to the propagation of
light, is confined to the laws of phenomena: scarcely any approach has been
made to a mechanical theory of those laws…. But perhaps something might be
done by pursuing a contrary course; by taking those laws for granted, and
endeavouring to proceed upwards from them to higher principles.
THE ETHER AS ELASTIC SOLID
Fresnel 1822
Ether made of molecules interacting through central forces.
Longitudinal vibrations excluded by assuming very high resistance to
compression.
Propagation ruled by linear response to transverse displacements: E = [K ]D ,
where [K] is a symmetric operator.
D ⋅ [ K ]D
2 2
2 2
2
The propagation velocity υ is such that υ =
a
b
c
=
α
+
β
+
γ .
2
ρD
2
For a given orientation of the wave planes, υ reaches an extremum when the
projection of E on these planes is parallel to the displacement:
n 2 E − (n ⋅ E)n = [ K ]−1 E .
THE ETHER AS ELASTIC SOLID
Cauchy 1830 on anisotropic propagation
Molecular theory of elasticity with 21 elastic constants.
Ad hoc relations among the constants in order to (approximately) retrieve
Fresnel's surfaces for the (approximately) transverse waves.
The vibration must be in the plane of polarization, unless the ether is regarded as
originally strained (second theory of 1836).
Green 1838 on refraction
Starts with most general potential as a quadratic function of the strain tensor
eij = ∂ i u j + ∂ j ui . In the isotropic case, there are only two elastic constants.
The Lagrangian method then yields the equation of motion and the boundary
condition that the pressure across the separating surface should be the same on
both sides.
Green eliminates the longitudinal waves by assuming, like Fresnel, an infinite
resistance to compression.
Green retrieves Fresnel's sine formula, not the tangent formula.
THE ETHER AS ELASTIC SOLID
MacCullagh's criticism
Elastic-solid theories are artificial and they cannot retrieve Fresnel's laws for
propagation in crystals and laws of reflection, be they molecular or not. The
molecular theories are even worse, because they exclude rotary power.
As to M. Poisson's objection [that the ether should not be compared to an elastic
solid], it was easily removed by a change of terms, for when the elastic solid was
called an "elastic system" there was no longer anything startling in the
announcement that the motions of the ether are those of such a system. The
hypothesis was therefore embraced by a great number of writers in every part of
Europe, who reproduced, each in his own way, the results of M. Cauchy, though
sometimes with considerable modifications…. This state of things was partly
occasioned by the great number of "disposable" constants entering into the
differential equations of M. Cauchy and their integrals; for it was easy to
introduce, among the constants, such relations as would lead to any desired
conclusion; and this method was frequently adopted by M. Cauchy himself.
MacCullagh nonetheless approves Green's appeal to the Lagrangian method.
THE DYNAMICAL THEORY OF 1839
1837: Perhaps the next step in physical optics will lead us to those higher and
more elementary principles by which the laws of reflexion and the laws of
propagation are linked together as parts of the same system.
9 December, 1839: This step has since been made, and these anticipations have
been realised. In the present Paper I propose to supply the link between the two
sets of laws by means of a very simple theory, depending on certain special
assumptions, and employing the usual methods of analytical dynamics.
V = 12 (∇ × u ) ⋅ [ K ](∇ × u)
⇒ ρu
&& = −∇ × [ K ]D, with D = ∇ × u .
⇒ boundary conditions: same density, continuity of u and ([ K ]D)// .
The resulting laws of reflection and refraction are the same as those of 1837.
Heuristics: V function of ∇ × u only, because this vector completely determines a
plane wave in a crystal (for a given frequency).
RECEPTION
Hamilton's and Herschel's interest.
But the lack of a mechanical model was usually deplored.
MacCullagh to Herschel, October 1846:
With respect to the question which you have put regarding my notions of the
constitution of the ether, I must confess that I am not able to give any
satisfaction—I have thought a good deal (as you may suppose) on the subject—
but have not succeeded in acquiring any definite mechanical conception—i.e.
such a conception that would lead directly to the form of my function V, and
would of course include the actual laws of the phenomena. One thing only I am
persuaded of, is that the constitution of the ether, if it ever would be discovered,
will be found to be quite different from any thing that we are in the habit of
conceiving, though at the same time very simple and very beautiful. An elastic
medium composed of points acting on each other in the way supposed by Poisson
and others, will not answer.
RECEPTION
Contemporaries nonetheless stick to the elastic solid.
Green's and Stokes's jelly-like ether.
Aberration and diffraction favor vibration perpendicular to plane of polarization.
Stokes 1862
MacCullagh's V implies the antisymmetric stress system
σ ij = ε ijk Ek (with E = [ K ](∇ × u) ).
[MacCullagh's ether] leads to consequences absolutely at variance with
dynamical principles…. MacCullagh himself disclaimed having given a
mechanical theory of double refraction. His methods have been characterized as
a sort of mathematical induction, and led him to the discovery of the
mathematical laws of certain highly important optical phenomena. The discovery
of such laws can hardly fail to be a great assistance towards the future
establishment of a complete mechanical theory.
RECEPTION: FitzGerald 1879
If H = u& , D = ∇ × u , [ε ] = [ K ]−1 , μ = ρ ,
Then ρu
&& = −∇ × ([K ]∇ × u )
⇒ ∇×E = −
∂D
∂B
, ∇×H =
with B = μH , D = [ε ]E .
∂t
∂t
The correspondence yields a Lagrangian formulation of electromagnetism and
the boundary conditions (continuity of H and E // ).
FitzGerald praised "the rare instinct of genius" that enabled MacCullagh to show
that the complicated phenomena of optics "all flowed from a few elementary
equations." He mentioned that "doubts have been raised as to the validity of the
physical basis MacCullagh advanced for these equations," and countered that
they were still of great value: "Even if not well founded, they were well found."
He also alluded to their electromagnetic interpretation: "The most recent theories
of Maxwell, while attaching new meanings to MacCullagh's symbols entirely
confirm his results."
RECEPTION: Larmor 1893-1895
Theory of ether and matter based on MacCullagh's rotational ether, illustrated by
Kelvin's gyrostatic model (1890). Electrons as centers of radial twist in the
rotational ether.
[MacCullagh's memoir of 1839 contained] an extremely powerful investigation, which
was independent of and nearly contemporary with those of Green, and I think, of at least
equal importance.
The credit of applying with success the pure analytical method of energy to the
elucidation of optical phenomena belongs to MacCullagh…. He arrived at a complete
solution of this problem [determination of the potential V], and one characterized by that
straightforward simplicity which is the mark of all theories that are true to Nature.
To his success two main elements contributed; the bent of his genius led him to apply the
methods of the ancient Pure Geometry, of which he was one of the great masters, to the
question, and this resulted in simple considerations, such as the principle of equivalent
vibrations already explained, which are applicable to the most general aspect of the
problem; while the variety and exactness of the experiments of Brewster and Seebeck on
the polarization of the light reflected from a crystal gave him plenty of material from
which to mould his geometrical views.
DID MAC CULLAGH DISCOVER MAXWELL'S EQUATIONS?
In the crystal's system of axes, MacCullagh wrote
with
[Y = ]
and the continuity of the parallel components of the vector (a 2 X , b 2Y , c 2 Z ) .
In vector notation, ρu
&& = −∇ × [ K ]D, with D = ∇ × u , and continuity of ([ K ]D)// .
If H = u& , D = ∇ × u , [ε ] = [ K ]−1 , and μ = ρ ,
∇×E = −
∂B
∂D
, ∇×H =
with B = μH , D = [ε ]E .
∂t
∂t
CONCLUSIONS
MacCullagh's style: Geometrical, inductive, and dynamical.
Lasting results in optics: Complete laws of reflection and refraction by crystals.
Simple dynamical foundation of these results. Theories of rotary power and
metallic reflexion.
Role as a critique: the first to see the impossibility of the molecular and elasticsolid approaches.
Methodological innovation: Lagrangian approach, conceived inductively.
Anticipation of Maxwell's equations in optics.
MacCullagh far ahead of his times.