BscPmParaxial.m

[AStilg]

RE: Our previous conversation, can you add the ability to override he fit points. This will be helpful for people calculating simple SLM beams which may not need many fit points.

[EternalRaininNovember]

Can someone explain how BscPmParaxial spreads field to focal plane and calculate expansion coefficients? I don't have any idea.

[AStilg]

It fits the beam shape coefficients "at infinite" distance from the focus. This means that the input is in the Fourier space of the microscope. Any feature there is in the units of wavenumber, k. This works because at sufficiently large distances the vector field is essentially 2D and thus the paraxial problem simply maps into it. The near field (focal plane) is evaluated directly from weighted sums of the vector spherical wavefunctions.

This function is intended model the effect of a field produced from a device such as a spatial light modulator. This would ordinarily be mapped to the back-focal-plane of the microscope and that's what this class processes.

I what you want is a particular paraxial beam mode, BscPmGauss may be better for you. It has Laguerre--Gaussian, Hermite--Gaussian, and Ince--Gausian beams built in with the ability to change their fill factor.

[EternalRaininNovember]

Thanks for the quick response. But I still don't understand how exactly this method works. What is the physical meaning of "tan theta scaling, thin lens appropriate"? For example, in VOLPE it is written that field propagation can be done using a simple Fourier transform. what's the approach here?

[AStilg]

The fit implicitly results in a Fourier 3D vector field transformation. It will be equivalent to performing the Fourier transform on all three vector components, but assuming they are mapped onto some surface. The thin-lens approximation is that the surface is a flat sheet. For a high-NA "real" lens it is, by design, the surface is a solid angle cut out of a sphere (perhaps with some amplitude modulation towards the edges).

[EternalRaininNovember]

Can you please explain in detail why it is "equivalent to performing the Fourier transform on all three vector components, but assuming they are mapped onto some surface"? Or can you share source of literature in which this is described?

Honestly speaking, I want to comprehend how it's work.
What is "wscaling" (BscPmParaxial, 168)?
Why in 242,243 lines do use sign(cos(theta))? not just cos(theta)?
"Et=(sign(cos(theta)).*cos(phi).*Ex_toolbox+sign(cos(theta)).*sin(phi).*Ey_toolbox);
Ep=(-sin(phi).*Ex_toolbox+cos(phi).*Ey_toolbox);"

[AStilg]

The radiation is far enough away at it can be considered as being a 2D vector field projected from the surface of the sphere. If the field were paraxial, it has no associated "z" field. This is the heart of the assumption. We have a collimated paraxial beam with x and y polarisation entering an optical system from "a long distance away".

The idea is not limited to the problem solved by the toolbox. It also applies to microscopes more generally:

Abbe, E., Hon. (1881), VII.—On the Estimation of Aperture in the Microscope.. Journal of the Royal Microscopical Society, 1: 388-423. https://doi.org/10.1111/j.1365-2818.1881.tb05909.x

It's worth a read, the important part can be found, pp. 393. If you don't have access look for "Abbe sine condition". By default the toolbox assumes a thin lens formulation. However, I found that with the microscopes we use in the lab I get the best match using the sine condition. The default is 'tantheta' because that's what the first version used.

[AStilg]

• "wscaling" (BscPmParaxial, 168)? It controls the beam fit to the back aperture. By default it is tuned to achieve 1/exp(2) peak power at the edge of the aperture.
• Why in 242,243 lines do use sign(cos(theta))? not just cos(theta)? See "Abbe sine condition".
  The equations:
  "Et=(sign(cos(theta)).*cos(phi).*Ex_toolbox+sign(cos(theta)).*sin(phi).*Ey_toolbox);
  Ep=(-sin(phi).*Ex_toolbox+cos(phi).*Ey_toolbox);"
  follow from that reasoning.