# 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?

• Related/possible duplicates: physics.stackexchange.com/q/234996/50583, physics.stackexchange.com/q/421584/50583, physics.stackexchange.com/q/140949/50583 Mar 14 '19 at 20:07
• I am sorry, but non of these answer my question. Mar 14 '19 at 20:08
• The answer is diffraction. 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. Mar 14 '19 at 20:42

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. 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. 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 $\Theta=\pi$. 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. Mar 14 '19 at 22:02
• @ThePhoton correct, I've updated my answer 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 $$w_0$$.

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! 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. 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. Mar 15 '19 at 1:54
• This answer is wrong. You can't focus light into a single point, even if you try to describe it as a classical wave. There is no classical solution to the wave equation with an abitrary small focus-size. Mar 30 at 18:53
• @Quantumwhisp please tell me then what is the classical solution for ω_0 in my example. Mar 31 at 0:04