Consider an elliptic reflecting surface. Consider the propagation of light from one focus (S) to another (S') by reflection from the surface. Choose any point P on the reflecting surface,

SP + S'P = constant

So Choice of point P is arbitrary, light can take any path, i.e. if light starts from S, after reflection, it will always end up at S'.

I think this is also true for focus to focus propagation of a thin convex lens.

I have been thinking about this. It is not really the sort of example I was expecting but I should at least respond.

Firstly the lens.

The path from object to image in a lens is a path of minimum time, irrespective of the angle the ray makes to the optical axis.

Indeed this becomes the definition of the image.

The refraction that occurs at the lens surfaces obey Snell's Law which is exactly the requirement that the path takes a local minimum time.

Whilst you are right that there are many such paths (rays) from object to image, each one is a local minimum.

If one considers a ray perturbation from one ray to another then one can argue that the geometry of the path space is a saddle point, albeit one with no second derivative in the direction that is not a minimum.

So when we come to the ellipsoid the same is true. For all straight lines from one focus to the surface and then to the second focus the path is constant.

However, light never takes a path that involves two reflections. One can apply the principle of minimum time to the reflection in a mirror also and so obtain the laws of reflection.

So the collection of all single reflection paths is a local minimum in all directions except the angle of emission from the first focus. In this dimension the geometry of the paths is again flat.

So you again have a saddle point with no second derivative in the dimension that is not a minimum.

What I was really looking for was an example of a true non minimum path but one that is stationary.

Thanks for your contribution. It certainly made me think about things.

The notion itself of "ray" is just an approximation for light, because light is a WAVE. In gneral we may introduce the notion of ray when the properties of the medium vary very slowly with respect to the wavelength. Then inteference is ruled out in the ray approximation (usually called the Geometric Optics Approximation, GOA). The Huygens principle (minimum optical path) is a zero order approxiamation for light propagation and is good for refracrion, reflection, mirages, ecc. but not, for example, to explain diffraction, i.e. the tendence of light to turn around opaque obstacles (you can see the shadow of your finger on the wall, but try to see the shadow of one hair!). A better approximation for light propagation is to consider the rays as lines perpendicular to a set of constant.phase surfaces obeing the eikonal equation. In this case, the rays are still paths which minimize some line integral and the formalism is very similar to the minimum action principle of Hamiltonian dynamics. In the eikonal approximation, the rays may be curved even in vacuum, but still they minimize some line integral (not the optical path, of course). An example is the focusing of a laser beam: near the focus the rays do not cross, but they pass from convergent to divergent being locally parallel at focus. The eikonal is a better approximation than the minimum optical path, because the eikonal function is to be identified with the phase of the optical wave. Rays are perpendicular to the planes of constant optical phase. This is the best approximation we can do still retaining the ray picture. Beyond we must use wave optics.

Another point is that even when the minimum action principle applies, this applies only locally, i.e. only in a limited region of space, where rays do not intersect. If we go to farr along a given ray anything can happen and the minimum principle no longer holds. For example, if we go too far along a ray starting from the focus of an elliptical cavity so to reach the other focus, we loose the unicity, because more rays intersect in the same point: in this case the minimum principle does not apply, as noticed in a previous comment. Similarly, if we follows too much the rays focused by a lans, we arrive, in general, to a caustic, where all rays are tangelnt to one curve (the caustic). In this case too the minimum principle fails The study of caustics is beyond the GOA: .

Dear Professor Anthony Dymoke-Bradshaw

Really, very nice opinion, I am fully agreed with you nice veiws.

If you consider light as a WAVE, then a part of the light can go to local non minimum. A previous poster metioned diffrqaction, at an edge diffraction really scatters light (as a wave) in many directions. For example look at http://www.dtic.mil/dtic/tr/fulltext/u2/a230189.pdf or http://www-sop.inria.fr/reves/Nicolas.Tsingos/research/GTD.html

An interesting response from Enrico Santamato.

I agree that the concept of a ray is only suitable for diffraction free situations and that the principal of least action cannot, as far as I can see explain, for example, how a diffraction grating works. Yet the principal behind the two concepts is very similar.

For a grating one observes light travelling to in such a way that there is only additive interference at the “orders” and destructive interference elsewhere. This is exactly how the principal of least action explains the path of a ray through a medium of constant or slowly changing properties, (and some sharp discontinuities such as mirrors and lenses).

For paths well away from the minimum time path, the phase changes rapidy with small changes in path and so the waves sum to zero. Near a path of minimum time, however, this is not the case and the paths around the minimum sum to a finite amplitude; but it doesn’t have to be a minimum, any stationary point will give the same effect, I just asked for an example of one that is not a minimum.

I accept that the wavefronts within the Rayleigh length of a lens are normal to the axis, but one can think of this as coming about by summing many paraxial rays entering the lens at a variety of distances from the axis. For each of these rays the principle of minium time can be applied to show their path from infinity to focus. The normal wavefronts come from summing over the range of incident angles.

I am not sure I understand the comment about caustic curves. Caustics appear where the focussing element is not perfect, e.g. with spherical aberation.

It represents an envelope where many rays pass. If the lens had no aberrations they would cross at a focus or image. Paraxial rays onto a parabolic mirror do not form a caustic, they form a focus. I don’t think this is relevant to the discussion, but please correct me if you think I am wrong.

Generally I am not trying to say what is right or wrong, I have always thought George Box had that figured out ("essentially, all models are wrong, but some are useful”). I am trying to find an example of where the principle of stationary action shows a path of light that is not one of minimum transit time.

A non minimum (in fact a maximum) case appears in the reflexion in a concave mirror.

You can find additional information in "Light in flight or the holodiagram: Columbi egg in optics" by Neils Abramson. And it is a consequence of wave optics, not ray optics.

N. Abramson, Light-in-Flight or The Holo-diagram: The Columbi egg of Optics, SPIE Press, Vol. PM27 (September 1996).

To Anthony.

You are right, but implicitly you are assuming that light is a wave and that dephasing occurs when the ray paths are changed. Huygens minimum-time principle ignores the existence of wave and of interference: it is a least-action purely mechanical principle. Instead,,Your picture where contributes from different paths are added with different phases (like in Feynman's path integrals) belongs to wave optics and not to the Geometrical optics. This infinite path approach accounts for diffraction too, which is a pretty wave phenomenon. The path summing method, indeed, is fully equivalent to solve EXACTLY the paraxial differential equation of optics, which is a WAVE equation, although approximated by paraxiality. In the not paraxial case (i.e. when focusing by a microscope objective is considered), the summing over paths is usually a very good WAVE approximation (mainly used by Fresnel) but still provides not exact solutions to the electromagnetic problem.

About caustics, it is a different issue, which has nothing to do with waves and it is well known also in mechanics. The point is that we can find a unique minimum (or even just an unique extremum) of the action integral only in region of space where rays do not intersect. When rays intersect as for example at foci or along caustics the action extremum is no longer granted or, as it happens in most cases, the extremal path connecting two given points is not unique. For example the action (i.e. optical path) integral along all rays starting from the plane of a perfect thin lens to the common focus has the same value, so that no unique minimum path exists. If you exclude the focus from the considered space region the minimum unicity is recovered for any two fixed end-points. Notice that the Geometric Optics and the minimum-action (opical path) principle can be applied even in the presence of aberrations and beyond the paraxial approximation.

The propagation of rays is governd by Fermat's principle. It reads :the optical path defined by integral nds (n refractive index, s path) is always an extremum. If a ray is reflected by a curved surface, the path between two points can be a minimum or maximum depending on the curvature. Take as an example an elliptical mirror and the two focal points P,Q. All rays starting in P are focused into Q. The path of all these rays is the same, but longer than the direct ray from P to Q.

See: Born/Wolf, Principles of Optics, Chapter 3.3.2.

Horst Weber is right. All ray paths connecting the two foci have same value of the optical path except the one connecting them directly, where the optical path is shorter. In this case, we have two sets of rays connecting the foci: one set contains all infinite rays suffering reflection and the other contains just one ray.. We don't have a unique extremal in this case. To have unique extremal between the foci, we must consider a region of space surrounding them entirely contained in the cavity. Then the extemal paths with reflection are excluded because they travel out form the given region and only the ray directly connecting the foci is left. This unique ray is a local minimum (with respect all varied rays connecting the foci) in the given region. In general, a variational problem must be considered always toghether with the space region where the varied path are confined. To have unique extremal for fixed end-points, this region must be small enough. Variational problems extended to arbitrary large space regions can fail to have unique solution. The elliptical cavity is a trivial example of this. Another trivial exaple is the image by a perfect thin lens of a point light source: alla rays emitted from the source converge in a point in the image plane and all those rays have same optical path from source to image point (the optical waves add constructively in the image point!). The extremum is therefore not unique even in this simple case.

Just thought of a good example. In a single mode fibre the refractive index is a maximum on axis. So this is the slowest path, i.e. one of maximum time. It is a local maximum, adjacent paths are faster. So for a fibre with a small bend the light still travels down the curved axis. The light does not travel down a path along the inside of the axis in spite of this being a shorter and faster path.