# Laser beam focus

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?

• I am sorry, but non of these answer my question. – Timo_BLN Mar 14 '19 at 20:08
• The answer is diffraction. – Gilbert Mar 14 '19 at 20:16
• 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. – TheEnvironmentalist Mar 14 '19 at 20:42

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.

• Thank you, that was the answer I was looking for! – Timo_BLN Mar 14 '19 at 22:17
• 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. – S. McGrew Mar 15 '19 at 1:35
• 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. – Paul Childs Mar 15 '19 at 1:54

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.

• This does not explain why you cannot focus to a point though. – Paul Childs Mar 14 '19 at 22:05
• @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. – ACuriousMind Mar 14 '19 at 22:15 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)
• Or $\Theta=\pi$. – The Photon Mar 14 '19 at 21:46
• 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. – Paul Childs Mar 14 '19 at 22:02
• @ThePhoton correct, I've updated my answer – Thomas Fritsch Mar 14 '19 at 22:14

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 $$w_0$$.