Interference of a fat laser beam: Tilting wave peaks

[Erik Ayer]

TL;DR Summary

To interfere a wide (fat) laser beam, is it possible to alter the wavefunction so it is not perpendicular to the direction of the beam?

I want to split a fat laser beam and interfere it with itself, kind of like this:

The very obvious problem is that the wave peaks shown as black lines would be a whole lot closer together, so the interference fringes would be sub-microscopic. If a couple of glass wedges - oddly-shaped prisms - were inserted into the two sub-beams, could that cause the wave peaks to be almost horizontal such that the interference fringes were visible? Something like this:

I don't know exactly how the waves behave in glass but can take a stab at it. I would think the entire wave would hit it's peak at the same time, and since the light is going through a medium with a refractive index greater than space or air, parts of the thick beam would be delayed more than others. It would vary linearly across the width of the beam. Tuning the orientation of the glass wedges to get exactly the right angle of the wave peaks would be difficult but not impossible.

A related question is about how to make a beam fat in the first place. Two lenses with different focal lengths can make a beam expnader, but what does that do to the wave peaks? When the beam comes out the second lens, would the waves all be perpendicular to the beam direction as they were before expansion, or would they be curved across its width? If the latter is the case, interfering them again would be painful.

I am talking about the probability density wave, not the electromagnetic wave.

If the wedge idea isn't possible, another idea would be to take, pentagonal prisms - not the right-angle pentaprisms used to reverse images - that are kind of a regular pentagon shape and insert the two sub-beams in two of the consecutive faces such that the come out almost parallel and almost on top of each other, then fully interfere after a short distance. That, of course, has its own problems, specifically in that I can't find such a prism and haven't done the math to make sure it would work - see above questions regarding how to do these calculations.

 

[Nugatory]

I am talking about the probability density wave, not the electromagnetic wave.

In that case, it's not clear what you're asking. The mirrors, lenses and prisms that you're discussing are all devices for manipulating electromagnetic waves. If by "probability density wave" you mean the quantum mechanical wave function, that's an abstract mathematical object, not something that to reflect and refract with lenses and mirrors.

 

[Erik Ayer]

Yes, I'm talking about the quantum mechanical wave, and yeah, I kind of don't understand the relationship between the two. Do they have the same frequency and wavelength? I would kind of think so because the double-slit experiment would rely on the quantum mechanical wave to interfere, but the interference pattern can be used to measure the wavelength of the EM wave.

Here's another thought: if a beam is reflected, refracted, or otherwise altered by optics, is the QM wave also not altered? When a beam is reflected off a mirror, does the QM wave keep going in it's own direction? That seems unlikely and would not agree with experiments.

Back to the tilting of the QM waves with respect to the beam's direction of motion, let me assume that the frequency of the QM wave is not affected by the glass wedge. If so, for one photon going through the wedge, all parts of its QM wave will peak simultaneously. The photon will be slowed down by the glass, and different parts of the QM wave will be delayed with respect to how thick the glass is. The QM wave on one side goes through a thin part of the wedge, whereas the QM wave on the other side will go through the thick part of the wedge. That would alter the angle of the wave peaks across the beam.

The alternative is that the QM wave does not peak at the same time across the beam, which would mean the optics (glass) did affect it.

Where am I wrong about this?

 

[tech99]

Summary:: To interfere a wide (fat) laser beam, is it possible to alter the wavefunction so it is not perpendicular to the direction of the beam?

The prism will cause the beam to change direction. The wavefront must be perpendicular to the direction of propagation.

 

[Erik Ayer]

The prism will cause the beam to change direction. The wavefront must be perpendicular to the direction of propagation.

So this would also have to apply to the case of two lenses expanding a beam - the wavefronts would all be straight lines perpendicular to to beam direction after expansion. The wavefronts would not be some type of curve.

Thank you! I should be able to calculate some things...

 

[tech99]

So this would also have to apply to the case of two lenses expanding a beam - the wavefronts would all be straight lines perpendicular to to beam direction after expansion. The wavefronts would not be some type of curve.

Thank you! I should be able to calculate some things...

If the wave is expanding in the manner of a spherical wave, rather than a parallel beam as you have depicted, then of course we have a spherical wavefront. At each point on the surface, the wave front is perpendicular to the radial direction of propagation.

 

[Nugatory]

Yes, I'm talking about the quantum mechanical wave, and yeah, I kind of don't understand the relationship between the two. Do they have the same frequency and wavelength?

An electromagnetic wave isn’t a quantum mechanical wave function; there’s no direct relationship and you‘re posing a question about ordinary classical wave optics. @tech99’s answer is about wave optics as well.

I would kind of think so because the double-slit experiment would rely on the quantum mechanical wave to interfere, but the interference pattern can be used to measure the wavelength of the EM wave.

We need to be careful here because “the double-slit experiment“ refers to two different things. Sometime around 1805 Thomas Young passed light through a double slit to produce a visible interference pattern on a screen; this demonstrated that light was a wave phenomenon. That’s the basis for the theory of light we use to this day to explain collections of mirrors and lenses like the one you’re describing. There’s no quantum mechanics involved; light is classical electromagnetic waves, waves interfere when there are two paths through a barrier, of course we see an interference pattern.

In the quantum mechanical double-slit experiment we shoot particles one at a time towards the barrier with the slits. We put a screen of photographic film or other recording medium behind the barrier so as each particle hits the screen behind the barrier it creates a dot on the screen. Over time more dots appear in some areas than others, and if we give the process enough time we’ll find when we develop the film that the dots have formed a pointillistic interference pattern. This is a purely quantum-mechanical phenomenon: a classical wave would darken the film in proportion to its intensity (as in Young’s experiment) everywhere instead of making dots and classical particles would clump their dots behind the slits instead of building up an interference pattern.

You aren’t doing a particle-at-a-time experiment, you’re using visible light from a laser, so there will be no quantum effects and classical electromagnetic wave theory is what you need. If I’m understanding what you want to do with the phase of your wave, it can’t be done - check the Huygens-Fresnel Principle and look at how phased-array radars steer their beams.

 

[Erik Ayer]

If the wave is expanding in the manner of a spherical wave, rather than a parallel beam as you have depicted, then of course we have a spherical wavefront. At each point on the surface, the wave front is perpendicular to the radial direction of propagation.

In the case where there the beam is expanded, first by passing through a lens with a short focal length, crossing at a point and expanding again until it reaches a lens with a long focal length to turn it back into a beam, those wavefront will be, again, straight lines perpendicular to the direction of the beam. The waves would be spherical between the lenses then flat on either side.

Cool, I understand the rule, so now it should be possible to calculate the paths and delays when going through lenses or wedges using kind of a ray tracing.

So to interfere the two sub-beams afer splitting a thick beam, they can go into two side of a prism and merge at the third side. Getting the position and angle correct so they overlap will be not entirely easy.

 

[Erik Ayer]

An electromagnetic wave isn’t a quantum mechanical wave function; there’s no direct relationship and you‘re posing a question about ordinary classical wave optics.

If a double-slit experiment is done with an intense beam - e.q. not one particle at a time - the interference is in terms of intensity. If it's done a weak, one particle at a time, beam (not really a beam, depending on definition), then the same pattern builds up after a while, correct? The average intensity for the particle version matches the original intensity distribution. That would seem to indicate that the two experiments are not entirely unrelated.

If the two are not unrelated, does the same conclusion that the quantum wave, like the classical wave, are perpendicular to the direction of the beam? Tilting either the EM or QM wave relative to the beam would be impossible, which means I would have to do something else to get two sub-beams to interfere, like use a prism to bring them close together and going in almost the same direction.