# How does a non-infinite monochromatic plane ray of light know which way to refract?

I have been making a simulation of light traveling in medium from classical physics perspective and while trying to makes sence of a traveling wave packet i realized i don't understand something very fundamental.

Left side of the image:

Nothing to it, we know which way a light ray will refract/reflect from Snell's law and Fresnel's equations, both of which we can get from Maxwell's equations and border conditions.

Right side of the image:

Googling about ultra short laser pulses I've read that dispersion becomes a big problem. A short pulse of light is by it's definition made of different frequencies, so no problem there, the medium "will do" a Fourier analysis of the light and send them in different directions.

Question:

This means, that a short pulse of light will refract in different directions but a long pulse of light (which are basically a bunch of short pulses one after another) will not. How does the medium know how long the pulse will be?

If it's a short pulse of light it seems to disperse it, but if it's long it just sends it in one direction. How does the medium/light "know" what to do? As long as a physical phenomenon can be approximated using a linear partial differential equation, Fourier analysis works to describe it.

A short pulse of light is by it's definition made of different frequencies, so no problem there, the medium "will do" a Fourier analysis of the light and send them in different directions.

But the medium doesn't actually "do" a Fourier analysis, right? It's just a convenient way to look at it mathematically. So it actually doesn't matter how the incoming wave looks like.

Assuming linearity, it's exactly the same for a soccer/football player hitting the ball. Some momentum goes from the foot into the ball, some gets reflected back into the foot. This could also be described by a Fourier transform of the foot movement and the ball movement. In that case, we prefer looking at it without the fourier transform because it's easy to imagine it. On contrast, we usually imagine light as a wave because we aren't "used" to a single pulse.

So I think this issue is about accepting the following:

• a physical fact: a lot of phenomena can be described with linear approximation.
• a mathematical fact: a linear approximation can be described with Fourier analysis and plane wave is just as valid as a pulse (they are equivalent perspectives, a mathematical fact).
• Thanks for the response, but it doesn't quite clear my dilemma. If I further ask a mathematical question - is there such a thing as a single frequency traveling plane wave? It has to have been traveling from minus infinity in both time and space to be truly monochromatic ergo - it is impossible to simulate an approaching wave of single frequency, since if I define it as approaching, it means it doesn't exist in some space yet - ergo it can not be of single frequency? – KrNeki May 17 at 9:59
• That is correct. A plane wave of single frequency is everywhere and can't "approach" in the sense that you are using. – zonksoft May 17 at 11:15
• You can absolutely simulate a pulse by doing a series expansion (of an approaching pulse) in each half space and taking care of the boundary conditions between e. g. air and glass. – zonksoft May 17 at 11:18
• But in order to make a series expansion, I have to know everything about the pulse before it even reaches the boundary. And how can I do that, if I don't know how long the pulse is. Assuming that in the simulation, one can dynamically turn a laser on and off, the simulation can't know when the user decides to do that so I can't know the length of a pulse. Unless I were to describe my pulse with as consecutive short pulses, in which case each of their bandwidths increases. If you can sense my dilema is, please point me in the correct direction since I am obviously misunderstanding something. – KrNeki May 17 at 11:41
• You absolutely need to know how long the pulse is because you want to solve a so-called "initial value problem". This is a necessary input for the program, the so-called "initial condition" (IC) that adds to the boundary conditions (BC) - together, they make up a "well-posed problem". You solve the wave equation for plane waves and then create a series out of the plane waves that fulfills IC and BC. If you google initial value problem, you'll find lots of examples (it's the math of PDEs, basically), I liked this video. – zonksoft May 17 at 15:10

It is not correct, quite, to say that a long pulse if light is the same as a lot of short pulses.

A short pulse is, indeed, equivalent to the superposition of a continuum of discrete wavelengths, each of which, in isolation, is infinitely long. The individual discrete wavelength components of two pulses overlap and they interfere. They can reinforce one another or cancel each other out, depending on the time spacing between the two pulses. Same goes for a lot of pulses.

In the situation you describe, a long continuous pulse would amount to a lot of short pulses spaced very precisely apart in time: equivalent to one cycle of the central frequency. The wavelength components of the short components would cancel unless they were very close to the central wavelength. That leaves only the central wavelength in the beam, so light exits in one direction and with one wavelength.

• Okay, that is true. But I am still having trouble how to fix my simulation. For an intently long monochromatic ray of light i can not show it as an approaching wave due to the fact that it is infinetely long. I can simulate an approach of a short pulse of light by just not showing freqluencies of amplitude below a certain strenght. But that means that it is by definition impossible to simulate an approaching wave becase accurately because our equations deal with infinitely long signals. – KrNeki May 17 at 8:02
• You have not shown the math of your simulation, so I can't comment on it. However, if you can accurately simulate a short pulse of light, you should be able to simulate longer pulses of light and show a trend toward the longer pulses acting more monochromatic. – S. McGrew May 17 at 13:12