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No matter what lens is put in the beam path of a Gaussian beam, it will always go through a waist of non-zero width.

Why not just a point? I know the maths, I'm wondering whether there is any physics that prevents it.

SuperCiocia
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  • There are no point like objects in physics, that concept only exists in mathematics. Everything naturally occurring always has a finite size, as far as we can tell experimentally. – CuriousOne Dec 21 '15 at 20:58
  • Although the electron seems to be awfully small... – Jon Custer Dec 21 '15 at 22:29
  • @JonCuster: The classical electron radius is about three times larger than the proton radius... what electrons don't have, within that radius, is the kind of parton-structure that one can find in protons, at least not up to the energy range, to which it has been measured with precision, so far. If you do allow for a bit of intellectual nonsense, then you can actually approximate "the real electron" with some order of the QED perturbation series of a "naked" point electron, and then you do get quite a bit of structure from the induced distribution of virtual photons and electrons. – CuriousOne Dec 21 '15 at 23:07
  • @CuriousOne - perhaps I was misremembering scattering cross section measurements of electrons which only result in a maximum upper bound that is still pretty darn small. Always open to corrections... – Jon Custer Dec 22 '15 at 00:17
  • @JonCuster: If we do electron-electron scattering at energies above 1MeV we get plenty of "structure", after all, one can make pretty much everything in the particle zoo with that. So, in some sense, there is even a Higgs in there etc., but that's because "in some sense" is intellectual nonsense, just like calculating the first or higher order perturbation series of the virtual particle cloud would be. Yes, it's all in there, but only because we are simply exciting higher mass particle states of the quantum field that happens to have "electron" as its stable, low energy, charged state. – CuriousOne Dec 22 '15 at 00:30
  • @CuriousOne: And below 1 MeV we still have plenty of particles - photons - in scattering processes. – Vladimir Kalitvianski Jan 03 '16 at 22:45
  • @VladimirKalitvianski: Photons are quanta, not particles. – CuriousOne Jan 04 '16 at 05:02
  • @CuriousOne: "Quanta" means "particles". Whatever. – Vladimir Kalitvianski Jan 04 '16 at 08:38
  • @VladimirKalitvianski: The word "quantum" isn't even a distant relative of "particle". It's in a completely different category. If you mean to say that you don't care about the difference, that's fine, but it doesn't make the difference disappear. – CuriousOne Jan 04 '16 at 08:43
  • @CuriousOne: Quantum (photon) caries away some energy-momentum and angular momentum, so what the difference are you talking about? In QM all particles (excitations) are treated on equal footing. – Vladimir Kalitvianski Jan 04 '16 at 10:00
  • @VladimirKalitvianski: Photons don't carry anything. They only "exist" at the spactime point where we create them or where we measure/destroy them. What carries momentum is the quantum field which is described by photons, but that's not the same as photons which are just quanta of that field and neither are particles. Particles, in high energy physics, are free field states that have such a high momentum that we don't care about the momentum uncertainty when we perform weak position measurements. Like I said, these things are well defined, but they are not equivalent. – CuriousOne Jan 04 '16 at 10:03
  • @CuriousOne: I disagree with you about it, but let's stop our discussion. – Vladimir Kalitvianski Jan 04 '16 at 11:05
  • @VladimirKalitvianski: I already said that whether you want to care to understand the differences between these terms doesn't matter to their definitions. You are, by the way, in good company. There are quite a few people who don't want to understand, either. For some reason these things rattle a lot of cages, even with people who could and should know better. I find that, in itself, an interesting scientific phenomenon. I was one of those who were rattled, by the way. I didn't want to "get it", either, but in time it becomes pretty obvious that quanta are actually a pretty cool thing. – CuriousOne Jan 04 '16 at 11:09
  • @CuriousOne: Don't think of my (mis)understanding; better think of the geometric optics, which is relevant to the question. – Vladimir Kalitvianski Jan 04 '16 at 11:20
  • @VladimirKalitvianski: What you were responding to had nothing to do with the original question. In any case, geometric optics doesn't describe the focal point correctly, for that you need wave optics, which would bring you back to quantum mechanics, if you want to go there... and then you will find that still nothing in physics that is a physical object is of infinitely small size. And there lies the actual answer to the question of the OP: what prevents the convergence on a mathematical point? Quantum mechanics. – CuriousOne Jan 04 '16 at 11:26
  • @CuriousOne: Ha! Now, according to you, I am against QM. Funny you are. What I was responding to was your "bit of intellectual nonsense" where I included photons on equal footing with your Higgs. – Vladimir Kalitvianski Jan 04 '16 at 11:36
  • @VladimirKalitvianski: I have no clue what you are for or against, neither do I care. You equated terms that have very little to do with each other, I pointed that out. You were obviously also not following how we got to talk about QM and now you are throwing in random bits about the Higgs in there, which have nothing to do with the matter, to begin with. OK. :-) – CuriousOne Jan 04 '16 at 11:39
  • @CuriousOne: You said that a point-like bare electron exists in QED, but it anyway gets a structure from "the induced distribution of virtual photons and electrons"... and I just seconded it underlying that photons also belong to the particle zoo. – Vladimir Kalitvianski Jan 04 '16 at 12:07
  • @VladimirKalitvianski: You may want to read my statement again, it does not say anything about bare electrons being physical objects, for sure not particles. An electron is just as much a quantum field state as a photon. It just has different charge properties. You can "make" particles from either by going into an observer system in which these states have high momentum, which, by Lorentz symmetry, happens to be the case for most observers. Just don't forget your weak position detector, or this ain't true, either. – CuriousOne Jan 04 '16 at 12:17
  • @CuriousOne: And you may want to read my statements about pointlikeness and smearing in QM here: http://www.aiscience.org/journal/paperInfo/pj?paperId=2156 – Vladimir Kalitvianski Jan 04 '16 at 12:24
  • @VladimirKalitvianski: Not really, I do not care, at all, about personal hypotheses unless they come with a peer reviewed experimental paper that contains evidence for them. As you know, this site doesn't care for them, either. – CuriousOne Jan 04 '16 at 12:31
  • @CuriousOne: You again pretend that I say something contradicting to or not supported with experiments. In reality I am referring to a typical QM calculation with no hypotheses at all. – Vladimir Kalitvianski Jan 04 '16 at 12:37
  • Let us continue this discussion in chat. – CuriousOne Jan 04 '16 at 12:43

2 Answers2

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Why not just a point?

Uncertanty principle I guess. In a point focus, the momentum would be defined with a finite $\Delta p < \infty$ (either I guess you can relate it to the lens numerical aperture in a purely geometrical picture where all rays wave-vectors coincide in the focus, or you take $\Delta p=0$ if you consider that the waist of a gaussian beam has a plane front). In any case, because $\Delta p \Delta x \ge$ constant, problems arise with a finite $\Delta p$ ,as the position would be perfectly defined ($\Delta x=0$).

scrx2
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While I concur that you may use the uncertainty principle to understand this, it isn't necessary. If you have a classical EM field that's governed by the nice wave equation derived from Maxwell's equations, then you can compute a diffraction integral that tells you that you must have a finite waist, even if the far-field divergence is very large.

JQK
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