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Huygens’ Principle at Work

The Huygens’ Principle is all about the evolution of the wave surface, as referred to the source of light. In its classical formulation, that evolution is locally described in the global manner, i.e. each point of the wave surface becomes a new source of light. Then Fresnel theory of light can be regarded as a better, experimentally sound, replacement for this image of local evolution of wave surface. However, when it comes to local properties of a surface, these are described in terms of the two fundamental forms of that surface, which are quadratic forms defined in the tangent plane. There are therefore reasons to believe that the Fresnelian description of light is somehow contained in the algebra of the two fundamental differential forms of the wave surface. The results of the present essay illustrate the idea.

Let’s assume therefore that we have an arbitrary quadratic form

(1)

which is defined in every point of the surface. This “definition” means that we know the coefficients X, Y, Z as functions of the global coordinates and we know also the local “differentials” s¹ and s² representing the differentials of the components of position vector. These differentials are the basic instruments we work with in the local theory of surfaces. They allow for instance the definition of curvature matrix (Guggenheimer, 1963): this is the matrix of transition between s¹ and s² and the components of the variation of normal unit vector of surface. The normal to wave surface varies, for instance in refraction and diffraction processes, and thus the curvature matrix can be taken as an instrument of describing these processes. It also can vary if the wave surface deforms accidentally by the presence of matter. In the first case we have a variation of the second fundamental form while in the second case we have a variation of the first fundamental form. Therefore the quadratic form in equation (1) can represent either one of the two fundamental forms or some other quadratic form just as well.

If there is a variation of this quadratic form, it is described both by the variation of the physical parameters: X, Y, Z → X + dX, Y + dY, Z + dZ and by the variations of the components of the position vector. The total variation of this form will then be obtained by differentiation

(2)

Thus, the variation of our quadratic form is exclusively due to the variation of the coefficients, along the locus of points satisfying the following differential equation

(3)

so that equation (2) becomes:

(4)

Thus the quadratic form (1) is constant in case where the coefficients are constant and the coordinates satisfy equation (3), which gives a Hamiltonian motion in the tangent plane. If the coefficients X, Y, Z are not constants, then the equation (3) must have some conditions of integrability showing the fact that the coefficients can be expressed as functions of coordinates or vice versa. More precisely, by this condition of integrability it is understood that the fundamental differential forms are ds¹ and ds², i.e. the second degree differentials of the position vector, and that the first degree differentials, s¹ and s² have to be expressed in terms of the second degree differentials, or vice versa. The natural condition of integrability is given by the vanishing of the exterior derivative of left hand side of (3), which shows that that differential form is an exact differential:

(5)

According to one of Cartan’s theorems, we then have

(6)

where λ, μ, ν are three external parameters making a symmetric matrix which “assists” in the integrability. Therefore s¹ and s² must satisfy the system of differential equations

(7)

Performing the matrix multiplication gives

(8)

The symplectic form (elementary area variation) corresponding to this differential system is

(9)

This quadratic form is algebraically conjugated to both the variation

(10)

and to the quadratic form generated by the newly introduced parameters:

(11)

This last quadratic form may represent some other physical conditions enforcing the integrability. It can be, for instance, a quadratic form characterizing the surface of separation between two transparent media.

If we are to solve the system (8), then the matrix of the system is essential. That matrix can be written as

(12)

where by ω_1,2,3 we denoted the differential forms

(13)

In cases where the two quadratic forms (9) and (10) are conjugated, the essential matrix of evolution (11) is given by

(14)

The parameters λ, μ, ν may represent new conditions of a physical or geometrical nature, entering the process of deformation of the surface.

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