Is optical length maximized for any ray reflected off a concave mirror?

Question

Statement: "In optics, you can take the example of a concave mirror: the optical path chosen by the light to join two fixed points A and B is a maximum."

The statement gives the impression that the light ray travels the path with maximum optical length (maximum time) between any two fixed points when reflected off the surface of a concave mirror. (Usually, a light ray takes the path of least optical length, such as the case when reflected off a flat mirror or a refraction between mediums with positive refractive index). However, if you use a ray simulator you can configure an infinite number of point pairs so that the optical path is not maximised, as I have found out. So the statement can't be correct. right?

A generative AI answer for the query above (namely Bing AI):

The statement is false. The optical path chosen by the light to join two fixed points A and B is not always a maximum when reflected off a concave mirror. It depends on the position and shape of the mirror, as well as the position of the points A and B. There may be other ray paths that have longer or shorter optical paths than the one chosen by the light. The correct statement is that the optical path chosen by the light to join two fixed points A and B is stationary with respect to variations of the path when reflected off a concave mirror. This means that a small change in the path does not change the optical path length significantly. This is a consequence of Fermat’s principle of stationary time, which states that light travels between two points in such a way that the time taken is stationary with respect to variations of the path.

Original link to the statement: https://physics.stackexchange.com/a/144362/366787

Someone (earlier I also erroneously thought so) can say "The statement is made regarding symmetrical points off the axis and for such points, the ray should contact the pole of the mirror in order to traverse both points and such that, it is the maximum distance that the ray can traverse assuming only one reflection off the mirror." But such an assumption is wrong because we can always introduce kinks to the path of the ray like zigzag or bending to increase path length hence it is still an inflexion point. Legendary Richard P. Feynman had this say on a similar scenario in one of his lectures at Caltech :

"Actually, we must make the statement of the principle of least time a little more accurately. It was not stated correctly above. It is incorrectly called the principle of least time and we have gone along with the incorrect description for convenience, but we must now see what the correct statement is. Suppose we had a (flat) mirror. What makes the light think it has to go to the mirror? The path of least time is clearly AB. So some people might say, “Sometimes it is a maximum time.” It is not a maximum time, because certainly a curved path would take a still longer time! The correct statement is the following: a ray going in a certain particular path has the property that if we make a small change (say a one percent shift) in the ray in any manner whatever, say in the location at which it comes to the mirror, or the shape of the curve, or anything, there will be no first-order change in the time; there will be only a second-order change in the time. In other words, the principle is that light takes a path such that there are many other paths nearby which take almost exactly the same time."

Further explanation: The question is not about whether or not such rays have the ability to form usable images for practical applications, but about the stationery action of a light ray reflecting off a concave mirror. It is about whether the stationary action of such a ray is minimum, maximum or in-between (saddle point). For a plane mirror, for example, any ray travelling from point A to B through point C (Point C must be on the surface of the mirror) takes always the geometrically shortest path no matter where you geometrically configure those two points. (The point C automatically selected by the ray will be on the shortest path or rather only the ray that reaches B will be the one that reflected off point C on the shortest geometrical path). So what about a concave mirror? My guess is the stationary action of a ray being minimised, maximised or saddle point depends on the configuration of the mirror and the three points. So the statement "In optics, you can take the example of a concave mirror: the optical path chosen by the light to join two fixed points A and B is a maximum." is poorly conceived.

Answer 1

If I understand properly your question, there is a very simple example: the elliptical mirror. It is obvious that the foci $F_1$ and $F_2$ are conjugate since the optical path is constant: neither minimum nor maximum. This is a well-known property of the ellipse: the distance $F_1MF_2 = 2a$ is constant if $M$ is on the ellipsoid.

If $M$ is inside the ellipsoid, $F_1MF_2 < 2a$ and If $M$ is outside the ellipsoid, $F_1MF_2 > 2a$

Now, consider a curved mirror tangent to the ellipsoid (any shape, but tangent) at a point $H$: the ray $F_1HF_2$ verifies Descartes' law for this new mirror: it is a path actually followed by light.

If this tangent mirror is entirely inside the ellipsoid, for a point $M$ close to $H$ on the mirror, the distance $F_1MF_2$ is always smaller than $F_1HF_2$: we are therefore dealing with a maximum.

If this mirror is entirely outside the ellipsoid, for a point $M$ close to $H$ on the mirror, the distance $F_1MF_2$ is always greater than $F_1HF_2$: we are therefore dealing with a minimum.

If this mirror is on one side outside the ellipsoid and on the other inside, we are dealing with a saddle point.

EDIT : I specified that the tangent mirror in $H$ is not necessarily a plane mirror.

Hope it can help and sorry for my poor english.

Answer 2

The answer for a concave, rotational symmetric mirror:

The geometric path is the shortest path between A and B and a mirror point C on the surface, where the length is measured by multiples of the wavelength. Point C can be determined by the reflection principle: The path triangle has to be perpendicular to the tangent plane in C and both rays against the tangent plane in C have to be equal. Variation of C in a small circle around the point of reflection gives a quadratic variation of the distance of C.

This proof follows the same principles as for plane mirrors. The shortest path is the shortest time of course by the constancy of the speed of light c in vacuum for all wavelengths.

I have to admit, that the reflection pinciple for the shortest straight path between two points and a point of any concave manifold is not quite transparent to me by elementary geometry. Needs an afternoon of minimum detection in differential geometry.

Answer 3

It is possibly helpful to stress that [within the set of (virtual) paths between 2 points A and B] there is never$^1$ a path of maximal time/optical length in the Fermat's principle. It is always possible to construct an even longer (virtual) path for the variational principle.

--

$^1$ If space is 1-dimensional $d=1$, one can consider back-and-forth paths. If one rules out back-and-forth paths, then assume that $d>1$, so that paths are not spacefilling.