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The logical grounds of Fresnel’s theory of light can be presented by starting from the fundamental idea that the motion representing light is a local periodic motion. This was indeed the conclusion intuitively extracted from the reflection and refraction experiments with light. One can come up with it approximately along the following line of reasoning. The time equation of a vibratory motion is given by

(1)

where q is the relevant coordinate of the motion, ω its frequency and a, b two constants giving the maximum q (the amplitude). This equation can be easily cast in a simpler form, making the amplitude explicit and involving only a single periodic function

(2)

Now, what we observe in the case of light which passes through a thin circular hole for instance is just a succession of bright and dark circular zones – fringes. A natural idea, borrowed from mechanics, is that these fringes represent the energy of light in the space points where they are observed. And, since this energy – also known as intensity of light – is proportional with the square of the elongation of the periodic motion,

(3)

we certainly have to deal here with a periodic process in space, showing brightness alternating with darkness over a constant intensity background, in any point in space at any time. In reality though, the things are a little more complicated, and this from quite a few points of view. The genius of Fresnel addressed all of these points in turn.

The first of these points of view is that the periodic process is actually a space vector described not by a single equation, but by three:

(4)

for k = 1, 2, 3. The second form of these equations is written here for convenience, in order to show that these components of the motion are not linearly independent. Indeed, one can eliminate the time between these three equations. The elimination is even possible in two different ways. First of all, in order that the system of linear equations (4), considered in the unknowns sin(ωt) and cos(ωt), be compatible, one must have satisfied the following algebraic condition:

(5)

This is a linear relation between the components of vector . As long as we don’t have a geometrical or physical interpretation of this vector the equation does not mean more than that. Fresnel chose however to interpret the vector as a displacement, according to the rules of mechanics, and this led him to the conclusion that equation (5) is actually the equation of a plane in space. Whence, even the deeper physical conclusion that the light is given by a plane motion, exactly like the Kepler motion which describes the motion of the planets.

The similarity between the motion defining the light and the Kepler motion is actually richer than the mere flatness of the motion: in a way, even the shape of the light motion is a conic, which is the natural trajectory of the Kepler motion. Indeed, there is a second possible way to eliminate the time in equation (4), based upon the fact that the unknowns, being trigonometric functions, satisfy the basic trigonometric identity. This identity has here three possible explicit forms, namely

(6)

and two more equations for the other two pairings of the components of vector . According to known theorems of geometry, these are three ellipses representing the projections on the coordinate planes of an ellipse situated in the plane (5). Therefore, not only the light motion is a plane motion but, more than that, the end of the light vector describes a conic in the plane of motion. Only, here we cannot call it a trajectory, for there is not a material point which follows it. As a matter of fact, from this moment the things become weird by comparison with the dynamics of material points.

The first thing to be noticed in experiments is that the light does not show fringes quite in any instance. In order to make the light do fringes we need to force it to run through thin holes or slits, as we said before. The process requires certain dimension of the holes or slits. The explanation of this phenomenon started with the explanation of the propagation of light. And that was offered by the old Huygens’ Principle, more or less in the following terms. The core assumption of the principle is that when a physical source of light acts, every point from the space adjacent to this source becomes itself a source of light. The set of points in space reached by light at a certain time is a surface in space called the wave surface. Thus, every point of a wave surface becomes itself a secondary source of light creating a subsequent or secondary wave surface exactly like the primary source. The envelope of the secondary wave surfaces created by the point of a primary wave surface is a new primary wave surface, and so on. Now, leaving aside the tough question about why the light from the newly created sources goes always forward and never comes back to its source, this image of the propagation had first to be reconciled with the idea of light ray. The most natural light ray is actually a straight line passing through the source of light. This is the way we perceive the propagation of light between two points in space, and this is the Newtonian way of explanation of light, according to the principles of dynamics. Therefore the wave surface could not be imagined but only as the surface cutting the light rays in all directions in space at a certain moment of time. If the light propagates with the same speed in all directions, then the wave surface is a sphere and the light rays are perpendicular to it.

It is this property that allows us to describe the directional propagation as a pencil of parallel rays representing a stream of light particles according to the idea of the classical corpuscular theory of light. Indeed, for the rays of such a pencil, we can say a priori that the speed of light is about the same. Therefore, the equivalent of the wave surface here is a portion of surface cutting each ray from the bundle at a right angle. This property is maintained all along the bundle of rays, as long as its thinness is preserved. If the bundle is thin enough, ideally approximating a ray, it makes sense to say that the wave surface is very much plane, in the sense that it has a small curvature. This is the way in which the concept of plane wave entered the realm of natural philosophy, and it is certainly the way Fresnel considered it.

Now the obvious observation needs explanation, that the motion of light along the ray is anything but periodical, as it should be in order to provide fringes. Rather, the motion along the ray seems to be uniform, which according to classical dynamical principles means that between light particles and the source of light there is no force. On the other hand a periodic process is actually a consequence of an elastic force. The only rational idea is that the vibratory motion of light, no matter of its direction in space, is somehow propagated along the ray. Due to the fact that the bundle of rays associated with the plane wave is very thin, the motions attached to adjacent rays are interacting by interference: they just add as vectors, so that the resultant displacement is the vector sum of the two displacements. Let us follow this process. For two adjacent rays, we have two motions given by two vectors like that from equation (4) above

(7)

The resultant displacement is then the sum of the two vectors

(8)

which can be written in the form

(9)

which is a vector just like that from equation (4), having the amplitude and phase components given by

(10)

Therefore the intensity of a two-ray bundle is given by

(11)

From the point of view of the motion, this two-ray bundle can be combined with another adjacent ray, giving a three-ray bundle characterized by a new vector like (10), with a new intensity (11). The process can be continued until the rays of the bundle are exhausted. The final conclusion is that a thin pencil of rays can be characterized by an equation of motion given here in equation (1), whose intensity is given by (11). The thinner the pencil the closer to reality is this representation.

Therefore, what we may tell about a stream of light particles can very well be read on the equations (10) and (11). First of all, if the light rays of the stream are too far away, the sum of the displacement vectors representing their vibrations does not even make sense: in order to be possible to be added, the origins of the two displacement vectors have to be close, ideally in the same point. On the other hand, now it becomes more obvious why a thin bundle of rays shows constant brightness when falling upon a certain surface: in modern terms it is completely coherent, i.e. all the vibratory motions associated with its rays have the same phase, and therefore by interference they offer the same intensity in any cross plane, because the periodical term in equation (11) vanishes. On the contrary, when a bundle of rays is passed through a slit or a hole, at least the marginal rays are forced to interact with matter, which introduces differences in the phases of their associated motions with respect to the other rays of the bundle. As a consequence the alternating brightness and shadow appear when observing the stream of light, due to the periodical character of the very same term in equation (11). Therefore we have even an image of the interaction of light with matter: this last one affects the phases of the motions associated with different rays of the light.

A really strange thing though is brought about with the observation that the plane of vibratory motion of light is the plane of the wave, i.e. the vibration is perpendicular to the light ray. Inasmuch as the light vector is a displacement, this displacement is therefore transversal to the direction of propagation of light. How can one come up with the idea of transversal motion? Well, just by asking: what determines the volume of interference of the light in the general case, i.e. when there are no slits and holes, and how come that some of the bodies are transparent while others not. It was, again, the merit of Fresnel to notice that an ellipse in space, representing the light motion as above, can be obtained by the intersection of the plane of motion with a surface of the second degree – a quadric. All things would fall then in place if we could offer the reason of existence of such a quadric. With a stroke of his inimitable genius Fresnel saw the reason at once: the light propagates through a continuum! How can this change the things? Well, it is just as simple as this: if a periodic motion propagates through a continuum, this continuum has to have the capability of exerting elastic forces in every direction in space, in order to sustain that periodic motion by a sort of rebounding. Otherwise the periodic motion will go extinct. The action in a continuum is characterized by what came later to be known as stresses, known as elasticities of the continuum in the time of Fresnel.

A continuum has, in general, different elasticities in different directions in space. Every point of that continuum has then a scale of the things which can happen in the immediate neighborhood of that point, and that scale is established by the elasticities of the continuum in that point. If, for instance, we denote by a, b, c the displacements allowed by the elasticities of the continuum in three orthogonal directions in space which, by the way, give also a local orthogonal reference frame, then the ratios

(12)

or a function of them, will reflect the existence of that scale, therefore the existence of a gauge of space, at the point of coordinates x, y, z in the given reference frame. The equation

(13)

is just such function, implicitly expressed as an algebraic combination. Spatially it is a second order surface (quadric), more precisely an ellipsoid, whose intersection with a plane in space is always an ellipse. As the only plane in the propagation of light is that of the wave, and as the motion giving light is an ellipse, voilá! The ellipse representing the motion of light is given by the intersection of the plane of wave with the ellipsoid of elasticities of the continuum that makes possible the propagation of light. We only have to accept that the motion of light takes place in the plane of wave, i.e. perpendicular to the ray, and that the light is somehow represented by periodic variations of position.

In the theory as presented above, the spirit had to create ideal things in order to fit the whole concept of light, as given by our experience. Not the least among these ideal things is the concept of ray which, we might say, is the root of the modern theory of light. This concept, geometrically just as simple as a straight line, asks the natural philosopher to imagine a limiting process for a bundle of rays, in order to come up with the physical idea of physical light ray. However, during the very process of inference of the geometrical properties of light, one casually forgets the fact that we are not able to observe a pencil of rays without constraining the light in a certain way. Specifically, we have to pass the light through a hole. There are here two points of abstraction that have been overlooked by the classical theory of Fresnel. First of all, the only idea we can come up with just by observing the light of a certain source in space, is that of wave surface. This idea is made possible by a direct analogy between the waves on the surface of water and the light in space. That the light is characterized by a motion is a further idea made possible first by this analogy. And so is, in fact, the idea that the motion characterizing the light is perpendicular to the direction of propagation. It is only when we identify the direction of propagation with the light ray that something new, characteristic only to light, comes into play. Indeed, unlike the case of waves on water, where we have to imagine the direction of propagation, in the case of light we can visualize a ray.

We should not forget, however, that in this visualization the experiment plays the essential part. Thus, if from Huygens’ point of view the wave surface is globally defined, by a process of abstraction eliminating the experiments from which we gain the information about the light, Fresnel actually showed not only how to explicitly consider these experiments, but that this consideration is in fact quite necessary. This last idea is indicated by the whole reasoning above, pointing almost explicitly to the usefulness of the classical mechanics in the problem of light, in a manner not suspected even by Newton himself. As to the first idea, it was explicitly illustrated by Fresnel himself, who constructed the wave surface of Huygens, thus giving a physical explanation to its geometrical structure. The construction gives an explicit relation between the elasticities in a point and the light rays passing through that point, suggesting also the weak points of the Fresnel’s philosophy. It goes more or less along the following lines.

Assume that a stream of light passes through a point in continuum, locally characterized by a reference frame with the origin in that point, and by an ellipsoid of elasticities related to this reference frame, as given in equation (13). The plane of one wave touches the ellipsoid in a certain point, then proceeds through the ellipsoid until it reaches the opposite tangent point, X, Y, Z say. Because this exit point is also a point of the ellipsoid, we have by (13)

(14)

The exit wave plane itself is a tangent plane to the ellipsoid in the point X, Y, Z, so the equation of this plane is

(15)

The normal direction of this plane, which, at least in particular conditions, is also the direction of propagation of the wave, is given, up to a factor, by the normal vector of this plane, having the components

(16)

Thus, the equation of the ray, which is perpendicular to the plane of the wave, is given by

(17)

where p is a factor of proportionality. This factor can be found by taking X, Y and Z from equation (17) and inserting into (15). The result is

(18)

and has an outstanding interpretation. If we calculate the projection of the vector X, Y, Z, with components given by equation (17), along the normal (16) to the wave plane, it is 1. This makes out of p the distance between origin, which is the center of the ellipsoid, and the exit plane wave, which allowed Fresnel to assume that this quantity is proportional to the speed of light. This fact was indeed a breakthrough for the wave theory of light, and is the basis of the future elasticity theory, that started being developed, even from the time when Fresnel was still alive, by Cauchy, Navier, and later by Saint-Venant (Saint-Venant, 1872).

However, the real achievement to be noticed here is the fact that the theory contains Huygens Principle in a very harmonious way. Indeed, in the coordinates x, y, z of the current point of a ray, taking p as the distance along the ray, allows one to write the locus of the points where the rays in all directions in space intersect the wave planes. The great achievement here is that the plane of a wave could be conceived as a tangent plane to the ellipsoid of elasticities of our continuum. Such a surface is the so-called pedal surface of that ellipsoid. It can be found by eliminating X, Y, Z from the equation of the tangent plane (15) by using equations (17) and (18). The result is

(19)

This came to be known as the Fresnel’s wave surface. It contains two sheets: a sphere and an ellipsoid, which were indeed used by Huygens, in particular instances, in order to explain the phenomenon of refraction of light. Nothing should be added here in order to underline the success of the wave theory of light over the particle theory.

The philosophers took care to call this surface with Fresnel’s name, and with good reasons. The Huygens’ wave surface is a little more complicated, but the general principle of obtaining it is exactly the same. What is really different is the fact that the wave plane should actually be more general than the plane tangent to the ellipsoid of elasticities. In other words the plane wave is something existing physically by itself, its plane should not be necessarily tangent to an ellipsoid. Fresnel guided his reason as follows.

The equation (19) represents only the local situation. Fresnel showed that the intersection of a wave plane passing through origin parallel to a given tangent plane is a curve having two extreme positions. It is therefore to be expected that these two positions would represent the amplitudes of the light vibrations assisted, and respectively impeached by the continuum supporting the light. If (l, m, n) is the unit vector of the direction of vibration in the wave plane, then the distances of these positions from origin in the wave plane are the roots of the algebraic fourth-degree equation

(20)

The easiest way to prove this is by the Lagrange multipliers. Indeed the equation (19) can be rewritten as

(21)

where α, β, γ are the direction cosines of the position vector of magnitude r, so that we have the natural relations:

(22)

The problem becomes now an extremum one: find the extremum of the function r (α, β, γ) from (21) under the constraints from equation (22). Using the Lagrange multipliers, the result is the equation (20) above.

Now the the argument goes in another direction: the very wave plane can be arbitrary, both as orientation and distance from the origin. This expresses the fact that the plane wave propagates indefinitely in every direction in space. Nevertheless, the Huygens’ Principle tells us that in every point the situation is the same, therefore we need to find the envelope of the planes given by equation

(23)

this time under the condition (20) itself. The result is again the Fresnel’s wave surface (Smith, 1835; Hamilton, 1841; Salmon, 1882), but this time a little more general:

(24)

The equation (20) is a quartic equation, while (24) is plainly a sextic. The two are certainly different, as they should be indeed. To point out the particular character of the equation (19), it is sometimes designated as Fresnel’s elasticity surface. The surface from equation (24), like that given by equation (20), has indeed two different sheets among others, one of which is a sphere, the other an ellipsoid. It therefore contains Huygens’ construction representing the double refraction as a particular case.

There is nothing more to say in order to underline the undeniable success of the wave theory of light, except the somewhat hidden message of the Fresnel theory. Indeed, Fresnel, like Newton a century and a half before for the case of motion of bodies, showed that the human experience, insofar as technological, has a specific way to enter the realm of light. It is not the assumption that in the propagation of light the space points become sources of light, not even the geometry of the problem itself that counts here. The truly important fact is that the construction of the wave surface demands the simultaneous consideration of disparate results of observation, a characteristic of the future quantum and wave mechanics. The geometry comes with the method, one might say it is only accidental. Indeed, in order to observe physical details of light it is necessary to perform experiments, and the experiments a fortiori entail thin pencils of light, closer to the light ray as defined by Fresnel than to the global wave surface of Huygens. It is this synthesis of the wave surface the one that places Fresnel alongside Newton when it comes to natural philosophy. The one-dimensional infinitesimal of Newton just found through Fresnel, the natural extension for three dimensions. The modern differential geometry and gauge theories draw their entire driving force from here. One might say that the Fresnel theory liberated the science from the shackles of phenomenology of passive observations, thus enlarging it by introducing the class of directed experiments which, after the initial success of the theory, acquired an amazing expansion.

From the point of view of the geometrical theory of surfaces per se, the theory of light starts from observations just as simple as the following:

1. When passing through holes and slits, the light shows diffraction patterns

2. When passing from a transparent medium into another the light shows refraction phenomenon, i.e. it changes the direction of propagation.

In the language of wave surface, these properties can be translated into following:

1′. If the wave surface is limited to a small portion of it, then we observe diffraction patterns

2′. When passing from a transparent medium into another, the normal to surface wave changes its direction.

These observations have consequences which can be developed exclusively within the differential geometry of surfaces, without recurring to any extraneous hypothesis. Indeed, by interaction with matter, the wave surface is locally disturbed. This means that its local metrical characteristics, given by the first and the second fundamental forms, are changed. Everything must be then described by the variation of these quadratic forms.

The most important Fresnelian lesson should be finally mentioned with overwhelming consequences widely noticeable in todays theoretical physics. From the equation of the ellipsoid of elasticities (13) to the equation (19) of the Fresnel’s surface of elasticity, one can pass without too much algebra. It is only necessary to perform an inversion of the current coordinates with respect to an arbitrary sphere, followed by a special homographic transformation of the semiaxes of the ellipsoid:

(25)

Here R is the radius of our arbitrary sphere. These transformations reveal two different groups of transformations, playing essential parts in physics. Both of them refer to forces, but in two different instances, springing from the same theory of light: forces as vectors and forces as tensions.

References

Hamilton, W. R. (1841): On a Mode of Deducing the Equation of Fresnel’s Wave, Philosophical Magazine, Vol. 19, pp. 381–383; Reprinted in «The Mathematical Papers of Sir William Rowan Hamilton», A. W. Conway & J. L. Synge Editors, Cambridge University Press 1931, Vol I pp. 341 – 343

Saint-Venant, B. de (1872): Sur les Diverses Manières de Présenter la Théorie des Ondes Lumineuses, Gauthier-Villars, Paris

Salmon, G. (1882): A Treatise on the Analytical Geometry of Three Dimensions, Hodge, Figgis & Co., Dublin

Smith, A. (1835): Investigations of the Equation to Fresnel’s Wave Surface, Transactions of the Cambridge Philosophical Society, Vol. 6 (1838), pp. 85 – 89