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Why isn't it possible to focus a laser beam to an infinitely small point in space? I am familiar with the shape of a gaussian beam, but why can't my $w_0$ be equal to zero?

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  • $\begingroup$ I am sorry, but non of these answer my question. $\endgroup$ – Timo_BLN Mar 14 at 20:08
  • $\begingroup$ The answer is diffraction. $\endgroup$ – Gilbert Mar 14 at 20:16
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    $\begingroup$ It's amazing how the two current answers to this question, by Thomas Fritsch and @ACuriousMind, answer the question with the same underlying principle approached from two entirely different sides. Thomas Fritsch explains it from the mathematical model, with the result that you'd need a wavelength of $0$ to have a beam with a width of $0$, and ACuriousMind states that at the most extreme case, you'd have only a single wave, which logically would be bounded in width by wavelength and amplitude, thus requiring the same wavelength of $0$. This is why I love physics. $\endgroup$ – TheEnvironmentalist Mar 14 at 20:42
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Ok, I'm not happy with the answers so far so here's my 2 cents.

Yes $\omega_0$ can be zero. If you have a perfectly spherical mirror with a single particle lasing source inside, then by classical optics it will focus light to a infinitely narrow point at the centre.

The real reason then why it is not truly infinitely narrow (manufacturing defects aside) is that even if we are to ignore entanglement re: the photon and assume it to not ne governed by the exclusion principle, the emitter will be, and cannot be isolated to a infinitely narrow non-moving position.

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  • $\begingroup$ Thank you, that was the answer I was looking for! $\endgroup$ – Timo_BLN Mar 14 at 22:17
  • $\begingroup$ That answer is not correct. It is possible to cause a single atom in a crystal to emit light, and to know that the atom is localized to within a fraction of a nanometer, but it is not possible to focus the emitted light back to a region the same size. $\endgroup$ – S. McGrew Mar 15 at 1:35
  • $\begingroup$ How does that explain why it is incorrect? Uncertainty in location and momentum cause respective spatial and temporal decoherences. These will govern how tightly you can focus light. $\endgroup$ – Paul Childs Mar 15 at 1:54
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An ideal Gaußian beam is diffraction-limited - its wavefront inevitably spreads out due to Huygens' principle. An infinitesimally small beam would diffract infinitely strongly, i.e. not resemble a beam at all: By Huygens' principle a single point (i.e. a "beam source" with zero radius) as a source simply emits a single, spherical wave, not a beam.

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  • $\begingroup$ This does not explain why you cannot focus to a point though. $\endgroup$ – Paul Childs Mar 14 at 22:05
  • $\begingroup$ @PaulChilds I linked three other questions about focusing to a point vs. conservation of etendue in the comments to the question, and the OP said they didn't answer their question, so I didn't see the point of explaining that yet again. $\endgroup$ – ACuriousMind Mar 14 at 22:15
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According to Wikipedia:Gaussian beam the beam waist ($w_0$) and the total angular spread of the beam ($\Theta$) are related by

$$ \pi w_0 \tan \frac \Theta 2 = \lambda $$

From this formula you see: The wave-nature of light (via its wavelength $\lambda$) is responsible for the non-zero waist.

That means you would have a beam waist $w_0 = 0$ only for these cases:

  • wavelength $\lambda = 0$ (so that there is no diffraction)
  • total angular spread $\Theta = 180°$ (meaning we have a spherical wave from a point source)
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    $\begingroup$ Or $\Theta=\pi$. $\endgroup$ – The Photon Mar 14 at 21:46
  • $\begingroup$ I think we can assume that the wavelength is not zero in which case the only non-trivial solution is $\Theta = 0$ as The Photon has said. $\endgroup$ – Paul Childs Mar 14 at 22:02
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    $\begingroup$ @ThePhoton correct, I've updated my answer $\endgroup$ – Thomas Fritsch Mar 14 at 22:14
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The Gaussian beam is not a precise solution of the Maxwell equation, it can only derived in the paraxial approximation, and this approximation is not applicable for small $w_0$.

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