Treating it in a quantum, "sum over all histories" way:
The light acts as if it follows all paths from point A to point B, but most of those paths cancel out: the wave arriving by almost every path interferes destructively with the wave arriving via some other route. The minimum time path, however, can't cancel out: the wave arriving that way arrives first, before any other candidate for interference. So the light wave traveling along the minimal time path is the one that actually occurs, the one that is actually observed.
Perhaps someone with a stronger background in quantum electrodynamics could vet that paragraph, but I believe it to be basically correct, and it obviates any need to assume light rays that somehow "know" before they select a path which one will end up being the least-time path.
KC
-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@phys-l.org] On Behalf Of Ludwik Kowalski
Sent: Sunday, 13 April, 2014 12:59 PM
To: phys-L new
Subject: [Phys-L] Each ray travels "as if it knew" all values of t=t1+t2
1) On 4/12/2014 John Denker wrote: " ... Snell's law was invented and re-invented several times, long before there was any connection to waves. In particular, one can postulate Fermat's principle of least time as a first principle. One can apply it on a ray-by-ray basis, as a way of deriving and/or explaining Snell's law ... without mentioning waves."
2) Suppose the air-water boundary is a horizontal plane. Suppose a ray (representing a collimated beam of light), travels from a point A (in air above water) toward a point B, at the water surface. This defines the plane of incidence. The refracted ray, passing through an underwater point C, defines the plane of refraction. It turns out that the plane of refraction coincides with the plane of incidence, and that the angle of incidence and the angle of refraction satisfy Snell's law. These are experimental facts.
According to Fermat's principle, "light travels between two points along the path which requires the least time, as compared to other conceivable nearby paths." In this particular situation(see the figure below) the near-by paths are all imaginable "AB plus BC."
3) What does the phrase to "postulate Fermat's principle" stands for? Does it mean to assume that each individual "ray," emerging from the point A, is intuitively choosing a path (choosing the point B) for which the time of travel from A to C will be the shortest?
4) Browsing the Internet I found an interesting answer to this question. The author wrotes: "I think it's a little misleading to say that light 'knows' the end point in advance since 'knowing' also implies some kind of consciousness." In other word's, the Fermat's principle is not an explanation; it is only a description. Each ray travels "as if it knew" all conceivable values of t=t1+t2.