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Picture this: a localized, collimated laser-beam, say one light-second long (there is a range of different frequencies or TE/TM modes), is sent into an infinite ideal vacuum. By an ideal vacuum, I mean a vacuum containing no real particles. The spacetime is flat.
Will the beam of photons diverge in the course of time?

Some thoughts:
*It will because the total wave function of the photons will spread out in time if we consider the fact that the photons have a spread in frequencies.
*It will because the photons have a chance in promoting virtual particle pairs to real pairs. The chance is small though but we have an infinite amount of time at our disposal. This doesn't make the beam diverge but it makes its intensity smaller.
*Nothing will happen to the beam.
*Something other than what I've mentioned happens.

Deschele Schilder
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3 Answers3

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Yes, the beam will diverge. Not through imperfections in the laser or interactions with virtual particles, but simply due to geometry and the wave nature of light. Look into Gaussian beams for the behavior of typical laser beams.

A Bessel beam would not diffract, but a true Bessel beam is infinite in extent and requires infinite power. In reality (or in a finite thought experiment), Bessel beams can only be approximated over a finite distance.

Deschele Schilder
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Christopher James Huff
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  • I'm sorry, I wanted to upvote you! In 50 minutes. – Deschele Schilder Sep 09 '20 at 11:42
  • So my fourth thought was correct? – Deschele Schilder Sep 09 '20 at 13:45
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    @DescheleSchilder please keep in mind that the classical beam can also be described matematically in the quantum framework with photons. The classical description is simpler, the quantum requires quantum field theory. see https://motls.blogspot.com/2011/11/how-classical-fields-particles-emerge.html – anna v Sep 09 '20 at 15:57
  • @annav I didn't read your link. How can photons be reconciled with classical electromagnetics? Can one describe a photon in a classical way? – Deschele Schilder Sep 09 '20 at 17:01
  • @DescheleSchilder if you learn quantum field theory and manage to read the linked discussion you will see that the opposit happens, the classical emergers from the quantum, a synergy of photons creates the classical em wave.see the lastpart of an answer of mine, after the "Edit"https://physics.stackexchange.com/questions/577312/feynman-couldnt-explain-how-the-photon-becomes-instantaneously-aware-of-the-gla/577327#577327 – anna v Sep 09 '20 at 18:04
  • @annav This is true poetry (aside of the wordy)! I'm not kidding. – Deschele Schilder Sep 09 '20 at 18:11
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Let's assume that your beam is 1.) a beam (finite cross section) and 2.) single frequency. Yes it will diverge.

The key point is (1.). A beam with finite cross section is not a plane wave, so it must be comprised of many plane waves that interfere in such a way as to produce a finite cross section. We specified single frequency. That means that the wave vectors that combine to form the finite cross section all have the same magnitude. Since they all have the same magnitude but we don't have a plane wave, we must conclude that wave vectors pointing in different directions are combining. Those vectors represent plane waves propagating in different directions. That is, the beam spreads.

garyp
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Here are some more points to consider:

  • Laser beam is a vague term: are we talking here about plane waves (which exist everywhere in the vacuum) or about wave packet released at a certain instant? Can we really have plane waves, given that every emission event has finite time, and we cannot wait for an infinitely long time for the plane wave to form?
  • Photons emitted via stimulated emission are identical
  • There are maybe multiple modes in the laser, and some randomness in the energy shifts of the atoms, so photons will not be identical.
  • In vacuum phase and group velocities of light are equal, i.e. pulses do not spread (within classical electrodynamics framework)
  • And then, as mentioned, there is scattering from vacuum fluctuations (i.e. from virtual electron-hole pairs, etc.)

In other words, the real world is imperfect, and the beam will certainly diverge.

Deschele Schilder
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Roger V.
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  • I edited the question and replaced laser by localized laser. This doesn't answer my question. It's pretty obvious that in the real world this happens. But I'm explicitly asking for the ideal case. – Deschele Schilder Sep 09 '20 at 10:51
  • I only tried to look at your question from the point of view of basic laser physics and EM: what is a beam, what it means for a beam to spread (phase vs. group velocity), how it is generated in a laser. These are all well-defined physical concepts. – Roger V. Sep 09 '20 at 11:03
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    @DescheleSchilder The point is that there's lots of allowable "ideal cases" and since you require a finite time/length of your collection of photons, it's not ideal. – Carl Witthoft Sep 09 '20 at 12:23