# The electric field in the focal length of a strongly focused lens

I have a Laguerre Gaussian beam and I want to focus it by using a strongly focusing lens. But my question is that I don't know what will happen to the electric field structure in the focal length after passing through a lens. Can anyone help me with that? I know the lens will add a phase to it but I want to know the exact electric field.

• Just curious if this relates to your previous fiber question? My understanding is pretty basic but if you use another lens on the output of the fiber you would restore the E field, but I guess inside the fiber ( because the lens adds angle to the Z direction) you will get different amounts of attenuation etc. Also would you need to worry about diffraction at each aperture. Interesting problem, good luck, hopefully the experts will answer. – PhysicsDave Oct 3 '18 at 13:41
• @ PhysicsDave Actually it is not related to my previous question. I faced with it when I was studying about lenses and their effects on the electric field. Anyway thank you very much for your answer – sara Oct 3 '18 at 13:55

## 1 Answer

So long as you don't focus the beam too strongly, the Laguerre-Gauss solution is a perfectly valid solution for the focusing of the beam, i.e. if you put a lens in front of a Laguerre-Gauss beam (or at least, one without any radial nodes) then the only effect will be a re-shuffling of the focal spot and waist size, exactly as with Gaussian beams, and including a (modified version of the) Gouy phase.

This holds, of course, so long as the wavelength $$\lambda$$ of the light is considerably smaller than the beam waist $$w_0$$. If you focus down to focal spots where $$\lambda/w_0$$ is no longer $$\ll 1$$, then the light will start stepping out of the paraxial regime, and you will need to amend your solution, including vector effects such as a longitudinal component of the polarization.

You don't really say how tight you need your focus to be, though, which makes me suspect that you're not actually looking to get into that regime. If you do, then you need to have a clear idea of which non-paraxial modifications you're concerned with (and why), and then go directly to the primary literature to look for solutions (or, most likely, approximate solutions) which suit your needs.