Fermat's Principle, Snell's Law, and the Prism

[Roger Dodger]

I have a conundrum of sorts that has made me feel like an idiot and I am hoping someone can point out my mistake.

Suppose a light source is placed to the left of a prism and a detector is placed on the opposite side. I have seen plenty of pictures of this sort, and they all appear to show the light passing through the prism obeying Snell's Law. But in doing so, they have the blue light sampling a more substantial portion of the prism than the red light. To me, this makes no sense. Blue light is slower than red light when passing through glass so if anything it would want to sample those paths that reduce its time in the prism.

I fail to see how Snell's Law applies here. Every derivation I have seen of Snell's Law has the detector placed inside the glass. This will give a local path of least action for light refracting from the air into the glass. However, this path would not appear to be related to the path of least action for the entire trip from source to the detector placed to the right of the prism.

Using Feynman's QED description of light, the probability amplitude for reaching the detector would be the product of (1) the probability amplitude for reaching a detector inside the prism and (2) the probability amplitude for light emanating from that detector into the air and reaching the other detector.

I realize we can see the light as it passes through the detector, but that's a completely different problem, since there is no guarantee that the light that reaches the detector sampled the same paths that the light that reaches our eyes sampled.

Or do I have it all wrong?

 

[ehild]

I have a conundrum of sorts that has made me feel like an idiot and I am hoping someone can point out my mistake.

Suppose a light source is placed to the left of a prism and a detector is placed on the opposite side. I have seen plenty of pictures of this sort, and they all appear to show the light passing through the prism obeying Snell's Law. But in doing so, they have the blue light sampling a more substantial portion of the prism than the red light. To me, this makes no sense. Blue light is slower than red light when passing through glass so if anything it would want to sample those paths that reduce its time in the prism.

Fermat's Principle says that the time the light travels between points P1 and P2 along a path that the time shortest. The blue light travels in the prism along different path than the red light, If you want to make that time shortest, you need to choose a different angle of incidence.

I fail to see how Snell's Law applies here. Every derivation I have seen of Snell's Law has the detector placed inside the glass. This will give a local path of least action for light refracting from the air into the glass. However, this path would not appear to be related to the path of least action for the entire trip from source to the detector placed to the right of the prism.

You need to apply Fermat's principle for the entire path from the source and detector, without placing a detector inside the prism. No light emanates from a detector. It is absorbed there.
Snell's Law can be derived from Fermat's principle at an interface between two media, but there is a simpler approach, based on the wave vector k of the light wave. kh serves as momentum of the corresponding photon. When the photon strikes the interface at angle of incidence θ the interface exerts only normal force, so the component k parallel with the interface remains unchanged. This component is ksin(θ) which is the same at both sides of the interface. And k=n/(λ₀) where λ₀ is the vacuum wavelength and n is the refractive index of the medium. So n₁sin(θ₁)=n₂sin(θ₂)
Snell's Law applies at both interfaces the light strikes when enters and leaves the prism. The blue light has refractive index different from the red light, it follows different path inside the prism.

 

[Roger Dodger]

Fermat's Principle says that the time the light travels between points P1 and P2 along a path that the time shortest. The blue light travels in the prism along different path than the red light, If you want to make that time shortest, you need to choose a different angle of incidence.You need to apply Fermat's principle for the entire path from the source and detector, without placing a detector inside the prism. No light emanates from a detector. It is absorbed there.

That much I realize. The problem is that Snell's law is derived by placing the detector inside the glass. As I pointed out, that is not where the detector is actually located. So why does Snell's Law work at the interface between the air and glass?

Snell's Law can be derived from Fermat's principle at an interface between two media, but there is a simpler approach, based on the wave vector k of the light wave. kh serves as momentum of the corresponding photon. When the photon strikes the interface at angle of incidence θ the interface exerts only normal force, so the component k parallel with the interface remains unchanged. This component is ksin(θ) which is the same at both sides of the interface. And k=n/(λ₀) where λ₀ is the vacuum wavelength and n is the refractive index of the medium. So n₁sin(θ₁)=n₂sin(θ₂)
Snell's Law applies at both interfaces the light strikes when enters and leaves the prism. The blue light has refractive index different from the red light, it follows different path inside the prism.

I realize that blue light travels a different path. The problem is that the path it appears to take is the wrong one. In photos of prisms I have seen, the blue light has a longer path in the prism than the red light. But this makes no sense, as blue light has a higher frequency and should slow down more than red light when in glass.

 

[ehild]

That much I realize. The problem is that Snell's law is derived by placing the detector inside the glass. As I pointed out, that is not where the detector is actually located. So why does Snell's Law work at the interface between the air and glass?

The detector is placed inside the glass when you apply Fermat's Principle for a single interface. In case of prism, there are two interfaces and both source and detector in the same medium, outside the prism.

I realize that blue light travels a different path. The problem is that the path it appears to take is the wrong one. In photos of prisms I have seen, the blue light has a longer path in the prism than the red light. But this makes no sense, as blue light has a higher frequency and should slow down more than red light when in glass.

I can not follow you. The blue light has a longer path if its enters the prism at the same point with same angle of incidence as the red line. If you want the shortest time between two points for blue light you need to choose a different point where the blue light enters the prism.

 

[Roger Dodger]

The detector is placed inside the glass when you apply Fermat's Principle for a single interface. In case of prism, there are two interfaces and both source and detector in the same medium, outside the prism.
I can not follow you. The blue light has a longer path if its enters the prism at the same point with same angle of incidence as the red line. If you want the shortest time between two points for blue light you need to choose a different point where the blue light enters the prism.

Both the red and blue light enter the prism at the same point. Because of their differences in frequency, they end up at different points once they exit the prism. So far, so good. But the problem is that it appears the blue light takes a path where it spends more time in the prism and less time in the air. (Look at just about any photograph of a prism and you'll see what I mean.) From Fermat's Principle, this makes no sense.

 

[nasu]

When you apply Fermat principle yo compare various paths between two points, for the given monochromatic light.
It is completely irrelevant to compare the path of the blue light with that of the red.
And they have different end points anyway.
Given the entry point of the blue light and the exit point of the same, the path it takes through the prism takes the minimum time. You can put the detector somewhere outside the prism and the path taken by light through the prism between the source and detector will be the one predicted by Snell's law. And will have minimum time for that blue light. If now you want red light to go between the same points, it will have to take a different path. Not the one shown in a dispersion image.What if you had just monochromatic blue light? Would you still think that it should take a different path just because some other beam does something else?

 

[Roger Dodger]

When you apply Fermat principle yo compare various paths between two points, for the given monochromatic light.
It is completely irrelevant to compare the path of the blue light with that of the red.
And they have different end points anyway.
Given the entry point of the blue light and the exit point of the same, the path it takes through the prism takes the minimum time. You can put the detector somewhere outside the prism and the path taken by light through the prism between the source and detector will be the one predicted by Snell's law. And will have minimum time for that blue light. If now you want red light to go between the same points, it will have to take a different path. Not the one shown in a dispersion image.What if you had just monochromatic blue light? Would you still think that it should take a different path just because some other beam does something else?

I'm not sure it's that simple. First, let's consider the location where blue light appears on the screen. Let's call this point A. Then consider a point slightly above it, where "above" is defined as the vertical direction toward the apex of the triangular prism. Let's call this point B. According to QED, the probability of blue light reaching Point B should be higher than point A, because the paths that emanate from the source to Point B are generally shorter than from the source to Point A.

Or I am missing something obvious.

 

[nasu]

Yes, you are missing it again.
The other path is not going between the same two points.
The paths that matter in Fermat's principle or in QED are all BETWEEN THE SAME POINTS A and B (final and initial),
The fact that you can find another final point C which is closer to A than B is irrelevant from this point of view.

You can also find points very close to A for which the time is even shorter. Not even going through the prism. So what?
Nothing to do with path from A to B.

 

[Roger Dodger]

I need to step back and rethink this. Thanks for the input.