# What is incorrect about the original statement of Fermat's principle?

Here are some statements about Fermat's Principle taken from Eugene Hecht's Optics book.

1. The original statement of Fermat's Principle : "The actual path between two points taken by a ray of light is the one that it transverses in the least time."

2. Reformulated by the book author: "Light, in going from point A to B, traverses the route having the smallest optical path length (OPL)."

3. Fermat's Principle in its modem form reads: "A light ray in going from point A to point B must traverse an optical path length that is stationary with respect to variations of that path."

The books states that "The original statement of Fermat's Principle of Least Time has some serious failings and is in need of alteration" but does not explain those failings clearly enough.

Questions:

• The OPL divided by $$c$$ equals the total time, so, to state that the time must be a minimum is the same of saying that the OPL is a minimum. Doesn't it? If so, statement "1" must be the same of "2".

• What is wrong with Fermat's principle of least time and why do we need to consider the statement "3" and not "2" or "1"?

• When will the OPL will be a maximum instead of a minimum?

## 1 Answer

The classic example of when the correct path should maximize the time is inside of a mirrored ellipse. There are four possible paths for a light ray which begins and ends at the center (shown below). Two of those paths are maxima and two are minima. The fact that the original statement of Fermat's principle does not account for this is probably what Hecht is referring to when he says that it has serious failings. • You could equally use a rectangle or any other arrangement with multiple plane mirrors. It would be interesting if you could construct a refractive example. Is it possible? – Floris Apr 16 '15 at 23:42
• @Floris A refractive example would be interesting. The only one I can think of just exhibits a local minimum. Consider holding a focusing lens outside of your line of vision with a point. You can see the point both directly (true minimum) and through the lens (local minimum). – Chris Mueller Apr 17 '15 at 1:55
• I as wondering the same, as I'm working with refraction, I think the OPL will always be a minimum. But I'm not sure I can affirm that on my text. – Pedro77 Apr 20 '15 at 14:34