Lenses and uncertainty principle

Question

I saw a video explaining the central fringe width of a single slit diffraction pattern with the uncertainty principle. It explained: as the slit size decreases, the uncertainty of the position of photons decreases, so the uncertainty of the momentum of photons increases, and thus the fringe width increases. This made me think about going the other way round: decreasing the uncertainty of the momentum of the photons to make the uncertainty of the position of photons increase. I thought lenses could decrease the uncertainty of the momentum of photons because lenses "force" photons to be transmitted in definite directions. However, it seems hard to figure out how the growing uncertainty of the position of photons would be manifested in this case if my assumption is true.

1. How would the growing uncertainty of the position of photons be manifested if my assumption is true? Is it like photons can suddenly be in a position different from the position it should have been if nothing happened after entering the lens?

2. As a side point, for a given slit size in a single slit experiment, how is the uncertainty of the position of photons calculated (the Δx part in the uncertainty principle)?

Answer 1

Usually you deal with it classically, but the classical explanation includes the uncertainty principle in disguise. See Interesting relationship between diffraction and Heisenberg's Uncertainty Principle?

Ray tracing is often used when designing lenses. The position of rays and the lens surfaces are perfectly known as they are designed. It is possible to design a lens that focuses light to a perfect point.

But when you build a real lens, it doesn't behave quite like the design says. It is good enough for many purposes, but if you want accurate results you need to add in diffraction. You get a focal spot, not a focal point.

Laser beams are about as close to perfectly collimated light as you can get. Again you can design with rays that are perfectly collimated. But that isn't quite how a real laser behaves.

Real light is a wave. Light propagates according to a wave equation derived from Maxwell's equations. To properly understand a laser beam, you need to solve the wave equation in a laser cavity.

A laser cavity is (usually) bounded by spherical or flat mirrors. The wavefront matches the curvature of the mirrors. This constraint leads to a Gaussian beam solution.

Image from https://www.rp-photonics.com/gaussian_beams.html

A fundamental property of Gaussian beams is a divergence angle and a beam waist diameter. In a Gaussian beam, "rays" follow hyperbolic paths. Almost straight far from the waist, but not quite parallel.

You can focus a Gaussian beam with a lens. The outcome is another Gaussian beam with a much larger divergence angle and a much smaller beam waist.

Image from http://laseristblog.blogspot.com

You can see the uncertainty principle at work. Confining a beam to a small waist reduces the uncertainty of position. It therefore increases the uncertainty of momentum, and therefore increases the divergence angle.