Wave function of a laser beam?

[DEvens]

TL;DR Summary

Wave function of a laser beam before it hits the diffraction grating

Summary:: Wave function of a laser beam before it hits the diffraction grating

So I'm reading "Foundations of Quantum Mechanics" by Travis Norsen. And I've just read Section 2.4 on diffraction and interference. And he derives a lovely formula for the wave function of a particle after it leaves a Gaussian diffraction slit. Then he puts two slits side-by-side and gets the well-known double slit formula. There's even a lovely color picture of the wave function after the double-slit and it gives a very clear picture of how the interference pattern develops. All very pretty.

I'm wondering about the wave function before the double slit. What kind of wave function is it that allows a laser beam to move along without spreading out, or not very much. I fondly remember from high school, we had a Helium-Neon laser that, with no special optics, could send a beam 100 meters with no noticeable spread. Yet a few cm after the diffraction grating it had spread many cm wide.

Norsen has plane waves before the slits. So that's kind of not what I'm looking for. My laser in high school had a very narrow beam, maybe half a cm across.

What's the wave function of a non-spreading beam? Or even a nearly non-spreading beam? How can I get a narrow circular beam that does not spread out?

 

[mfb]

Half a centimeter is very wide compared to the space between the slits in your double slit. It is also very, very wide compared to the wavelength. To a good approximation your laser beam is a plane wave.
Check how the diffraction pattern of a single slit with 5 mm diameter looks like at 100 m distance for comparison.

 

[DEvens]

Half a centimeter is very wide compared to the space between the slits in your double slit. It is also very, very wide compared to the wavelength. To a good approximation your laser beam is a plane wave.
Check how the diffraction pattern of a single slit with 5 mm diameter looks like at 100 m distance for comparison.

But plane waves are *not* a good approximation. That's the point. The beam does not spread.

The reason that plane waves retain their integrity is because they produce constructive interference with each other. It's the old marching-in-step picture. And the edge goes away because the edge is defined out of existence.

But the beam is not like that at the edges. If that were all that was going on, the beam would need to be diffracting at the edge of the beam. Which would produce exactly the effect that a narrow hole does. That is, if a laser beam hits a pin-hole, it does not continue on as a narrower beam.

There is something more going on here beyond just plane waves.

 

[mfb]

But plane waves are *not* a good approximation. That's the point. The beam does not spread.

An ideal plane wave doesn't spread at all. Sounds like it is a very good approximation here.

Did you do the calculation I suggested?

 

[PeroK]

But plane waves are *not* a good approximation. That's the point. The beam does not spread.

The reason that plane waves retain their integrity is because they produce constructive interference with each other. It's the old marching-in-step picture. And the edge goes away because the edge is defined out of existence.

But the beam is not like that at the edges. If that were all that was going on, the beam would need to be diffracting at the edge of the beam. Which would produce exactly the effect that a narrow hole does. That is, if a laser beam hits a pin-hole, it does not continue on as a narrower beam.

There is something more going on here beyond just plane waves.

I'm not sure about lasers, as that's technically out of the scope of non-relativistic QM, but in terms of a quantum particle:

The wave function could be a 3D gaussian wave-packet, with a mean zero momentum in two dimensions and a non-zero momentum in the direction of motion, $x$ say.

The slit or slits act like a time-dependent 1D infinite square well potential, which lasts only a short time. The wave function picks up a non-zero momentum component in that direction - $y$, say. After the slit or slits, therefore, the wave function still has its momentum in the x-direction and its slowly dispersing zero-momentum gaussian shape in the z-direction; but now a significant y-momentum from its time in the infinite square well. In terms of a high variance of momentum. The mean momentum may still be zero. Once out of the square well it spreads out like a free particle in the y-direction.

That's as close to a QM description as I can get.

I'm not sure whether this helps.

 

[kith]

Typically, lasers produce Gaussian beams.

As @mfb has indicated, it is worthwhile to look at what happens in the case where $r$ is much smaller than the relevant length parameters.

 

[vanhees71]

An ideal plane wave doesn't spread at all. Sounds like it is a very good approximation here.

Did you do the calculation I suggested?

An ideal plane wave is already spread over all space and thus doesn't exist. A laser wave is well described as socalled Gaussian beams of various kinds. For a good introduction, see

J. Garrison, R. Chiao, Quantum optics, Oxford University Press, New York (2008).
https://dx.doi.org/10.1093/acprof:oso/9780198508861.001.0001