I’d like to see Paul Colborne’s excellent Skew Gaussian work being implemented in the code. It would made a much more robust reference frame for POP than the existing pilot beam calculation. 

It is a bit off topic, sorry. But connected.

You can search this site for Paul’s work. There are KB articles and dlls that are included in the OS distribution. And although it isn’t strictly necessary for GBP, as soon as you go off axis you’ll have to address the very issues that Paul’s work makes so much easier.

Hi all,

I’ve been working on GBD in my spare time for quite some time. I have tried to use Paul Colbourne’s approach at first to trace off-axis Gaussian beams, but struggled to compute the phase of said beams with his method. I managed to reproduce the results of the paper: https://arxiv.org/pdf/2106.09162.pdf that @Brian.Catanzaro refered to. However, as far as I can test (and it is most-likely my fault), it only works at specific axial distances. For example, in Fig. 3, I can get the nice Airy disc if the distance is 5.52, which is exactly the EFL of the lens. However, if I move away form this distance, I don’t get the effect depicted in Fig. 5. I did speak with the author @Jaren.Ashcraft (who happens to be on this very forum too :), and there seem to be an issue with the underlying equation that is being used, namely Eq. 12. This equation was described by Cai and Lin in Eq. 9 of their paper https://opg.optica.org/ao/fulltext.cfm?uri=ao-41-21-4336&id=69426. One thing for sure is that the first determinant of Eq. 9 should be raised to the power -½ and not +½. But for the rest, it is hard for me to double check the derivation of that equation, but I had an insight from a mathematician (who is still working on it) that it might be correct assuming Eq. 3 and 6 are correct (so I’m looking to confirm Eq. 3 and 6 next)! One important thing I noticed is the following. Assuming the following system matrices (according to Cai and Lin Eq. 9):

Which describes a thin lens of focal length f = 100.0 mm followed by an air space of length z = 100.0 mm. If we use a beam like so:

Which describes a non-decentred symmetric Gaussian beam at its waist position with a Rayleigh range of 1 / 0.0031831 = 100 * pi (the wavelength is 0.01 mm, and the waist is 1 mm, leading to the 100 factor). Simply looking at position:

We see that we are left with the first determinant and exponential of Eq. 9. If one calculates the determinant of E(r_2), assuming I did not make a mistake, one gets a value of 1.5707723114645764 instead of the expected Gouy phase shift of 0.3081690711931703 that POP and this equation return:

I’m not sure at all whether this is the problem, but just something I noticed. And that problem doesn’t prevent the correct calculation of the Airy disc at Z = f.

I’m a bit desperate at this point so if @Brian.Catanzaro or @MichaelH or anyone else are interested, I’m happy to share whatever I have done so far (which is Python based).

Take care,

David

Can you help me understand the POPPY and POKE references please? Are these trying to reproduce the built-in POP or Huygens results, or are they trying to add something different to the existing capability?

• Mark

Good luck with that!