### Math Blues I

Gaaah.

So I'm trying to figure out a bit about wave dynamics - namely how a wave can move in a collimated fashion, such as in a light ray or laser beam.

I was playing around with the wave equation a bit. I set up field sims for a 3d wave and advanced time, reproducing all sorts of interesting effects. Diffraction, reflection, interference, refraction ect.

Then something started bugging me - if you have a travelling wave, how do you get it so that it retains a collimated shape? It seems to me that the divergence of the gradient on a point outside the beam is going to be nonzero due to the difference between the zero and nonzero amplitudes inside and outside the beam, and so the region outside the beam should be sucking up energy and spreading the wave as it travels.

At first I thought it was just that I was looking at a scalar wave, and light is a more complicated vector wave operating off of different rules. But in my optics book, they eventually transform maxwells equations into a set of 6 scalar equations for the electric and magnetic field, and the same laplacian(field) = acceleration(field) behavior results.

Okay, so then I did some reading and discovered that beams usually have gaussian distribution of amplitude along the beam radius.

I want to be able to figure out how the amplitude's radial profile changes as you move along the wave. So I wanted to transform the wave equation into a different coordinate system moving with the beam.

laplacian(field(x,y,z)) = acceleration(field(x,y,z)) -> acceleration(field(x-vt,y,z)) = ???

and here's where I have to quit for today, I need to make dinner.

This better not be something that some old math god fart like Newton or Euler solved in 5 minutes while waiting for the coffee to brew.

So I'm trying to figure out a bit about wave dynamics - namely how a wave can move in a collimated fashion, such as in a light ray or laser beam.

I was playing around with the wave equation a bit. I set up field sims for a 3d wave and advanced time, reproducing all sorts of interesting effects. Diffraction, reflection, interference, refraction ect.

Then something started bugging me - if you have a travelling wave, how do you get it so that it retains a collimated shape? It seems to me that the divergence of the gradient on a point outside the beam is going to be nonzero due to the difference between the zero and nonzero amplitudes inside and outside the beam, and so the region outside the beam should be sucking up energy and spreading the wave as it travels.

At first I thought it was just that I was looking at a scalar wave, and light is a more complicated vector wave operating off of different rules. But in my optics book, they eventually transform maxwells equations into a set of 6 scalar equations for the electric and magnetic field, and the same laplacian(field) = acceleration(field) behavior results.

Okay, so then I did some reading and discovered that beams usually have gaussian distribution of amplitude along the beam radius.

I want to be able to figure out how the amplitude's radial profile changes as you move along the wave. So I wanted to transform the wave equation into a different coordinate system moving with the beam.

laplacian(field(x,y,z)) = acceleration(field(x,y,z)) -> acceleration(field(x-vt,y,z)) = ???

and here's where I have to quit for today, I need to make dinner.

This better not be something that some old math god fart like Newton or Euler solved in 5 minutes while waiting for the coffee to brew.