# Light wave characteristics

1. Apr 29, 2006

### yetar

I was thinking about the wave PDE: Utt-c^2*(Uxx+Uyy+Uzz) = 0
And wanted to see in respect to this PDE what is a source of light or a laser.
My question is, does the light waev front always expands like a 3D sphere? Or is a light wave actually a straight line in the 3D space?
If the first is true, then it means that even when you have a laser, the light of the laser should also expand to all the direction from the laser line. Maybe it will very little, but still it would expand in all the direction.
If the lightwave is just a straight line, then a light source like a flash light is just a lot of straight lines coming out in a lot of directions?
Why did I mention the PDE...
Becaue if we look at: Utt-c^2*(Uxx+Uyy)=0
Then a certain point in (x, y, t) is affected by a cone with its axis vertical to the XY plane.
So even if you have a straight line in XY with width larger then 0 (a laser?) then a point which its XY is not on this line projection on the XY plane, will be affected by it, because the cone will "catch" the line.
Maybe you could clear this to me better.

Last edited: Apr 29, 2006
2. Apr 30, 2006

### Astronuc

Staff Emeritus
Photons travel in a straight line. Think of a photon as an intersection of mutually perpendicular electric and magnetic field.

A laser or any other light beam is a 'collection' or group of photons. A group does disperse.

3. Apr 30, 2006

### Claude Bile

There are a couple of approximations we can make when applying the wave-equation to light waves.

Firstly, we can replace the second time derivative with -(omega)^2 since we know the explicit time dependance of a monochromatic wave. The resultant equation is the Helmholtz equation.

http://mathworld.wolfram.com/HelmholtzDifferentialEquation.html

The Helmholtz equation can be further simplified by factoring out the rapid oscillation in E with propagation distance (z), and assuming that the second derivative w.r.t. z is negligibly small (i.e. assuming a beam-type solution). One then obtains the Paraxial Wave Equation.

The Paraxial Wave Equation is the starting point for most forms of beam analysis, with the simplest solution being a Gaussian Beam (sometimes designated the HE00 or LP01 mode). Higher order solutions also exist, based on Hermite or Laguerre polynomials. A basic derivation can be found here;

http://electron9.phys.utk.edu/optics507/modules/m4/gaussian_beams.htm [Broken]

Gaussian optics predicts that a light beam will diffract (spread) as it propagates, in a cone-like manner. The half-angle of this 'cone' is called the far-field diffraction angle and is derived in the link above.

I hope that satisfies some questions.

Claude.

Last edited by a moderator: May 2, 2017