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Refraction and the travel time of light

Let's say you have two points, and you want light to travel from one point to the other. In this simulation, by the way, the first point is colored red and the other point is colored purple. If you want the light to travel between the points in the smallest possible time, then the light should travel along the straight line connecting the two points. This is true, at least, if the light travels through just one medium (air, for instance) between the two points.

Things get a little more interesting if the light changes medium along the way, however. In this simulation, there are two different media. Point 1, where the light originates, is in medium 1 (air, for instance), while point 2, where the light ends up, is in medium 2 (glass, for instance). You can adjust the index of refraction of each medium, and you can also adjust the position of points 1 and 2. You can also adjust the position of a third point (colored green in the simulation) that is on the interface (the boundary) between the two media, and through which the light passes on its way from point 1 to point 2.

Your goal is to adjust the position of the point on the boundary so that the light takes the smallest possible time to travel from point 1 to point 2, and then you have to think about how this relates to reality. In other words, which of the many possible paths the light could take does the light actually take?

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Activities

The positions of the points on the screen are specified in meters - if you right-click on any point on the screen, the coordinates of that point will show up in yellow in the bottom left of medium 2 (the lower medium). The travel times for light over the various paths are given in nanoseconds - light is very fast!

  1. When you first start the simulation, the three points are all along a vertical straight line. Without moving the points, adjust the index of refraction of each medium. What effect does this have on the travel time for the light? What does this tell you about the connection between the index of refraction of a medium and the speed at which light travels in that medium?
  2. Now, look at the graph (the graph of light travel time as a function of the position of the point the light passes through on the interface between the two media) as you click and drag the green point left and right. Determine where the green point has to be to minimize the time it takes for light to travel from point 1 (red) to point 2 (purple). Explain why this location for the green point makes sense.
  3. Now, click and drag point 1 (red) to a location in the upper left of medium 1, and move point 2 (purple) to a location in the lower right of medium 2. Once again, move the green point left and right to determine the path the light follows from point 1 to point 2 that minimizes the travel time. Try this in three different situations: (a) when the two media have the same index of refraction, (b) when medium 1 has a higher index of refraction than medium 2, and (c) when medium 1 has a lower index of refraction than medium 2. Is the minimum-time path also the minimum-distance path? Explain.
  4. It turns out, when light travels from a point in one medium to a point in another medium, that it actually follows the path that minimizes the travel time. Based on your investigations with this simulation, what are some implications of that?
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