Mathematical formalism for optical system of an electron microscope

[uellue]

https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis is a good start. However, it only captures radial symmetry. Electron optics are generally not symmetric. The following properties are desirable for a formalism:

• Allow to contract groups of optical elements into a single operator
• Handle optics that are not rotationally symmetric
• Handle the phase. Option: https://en.wikipedia.org/wiki/Complex_beam_parameter ?
• Handle non-linearities such aberrations
• Handle image rotation, if possible as a function of radius

Options:

• Extended 4x4 matrices for astigmatic systems http://www.fbmn.h-da.de/~blendowske/uploads/Papers/OVS_ASAM3.pdf
• Aberration term -- I am not sure if this one is correct or useful, though: https://www.osapublishing.org/DirectPDFAccess/41485276-0ADD-1602-9828EA1ADF831E33_276733/ETOP-2013-EWP28.pdf?da=1&id=276733&uri=ETOP-2013-EWP28&seq=0&mobile=no
• 6x6 matrices as they are used for accelerator design: https://indico.cern.ch/event/266133/attachments/474621/656940/Gillespie_Aachen_CP_vs_P_optics.pdf http://lss.fnal.gov/archive/nal/fermilab-nal-091.pdf
  • This should handle the astigmatic nature well
  • Add in-plane rotation term?
  • Errors with additional / extended matrices that expand the whole thing to a polynom?
  • They become rather large very quickly since they contain all cross-terms, but that's why we have computers...
  • Maybe use sparse matrices to only capture aberrations that are actually relevant?
  • Hopefully allow up to 3rd order, which is the relevant one for optics AFAIK: http://www.quadibloc.com/science/opt0505.htm
  • Those will probably not allow direct contraction since they are not linear anymore
  • However, the parameters of a resulting matrix for a group can probably be optimized numerically
  • Relation to machine learning and training neural networks! Numerical optimization in rather large and complicated function spaces. Perhaps move towards an implementation using those frameworks later?

[uellue]

Linear canonical transformation could be the formalism of choice that can handle near-field Fresnel diffraction, far-field Fraunhofer diffraction and any transitions inbetween using one formalism. It turns out that the expression for a lens in linear canonical transformation closely corresponds to a lens in matrix optics: https://en.wikipedia.org/wiki/Linear_canonical_transformation#Spherical_lens

Currently, NumPy doesn't have dedicated support for calculating such transformations. However, there has been recent work on this with the dedicated purpose of calculating optical systems: https://stacks.stanford.edu/file/druid:fq782pt6225/AykutKoc_PhD_Thesis-augmented.pdf

[uellue]

Polynomial expansion of ray transfer matrices should correspond to the "differential algebra method". Having several optical elements after each other corresponds to composition of polynomials, which yields higher-order polynomials. For that reason, rays should be propagated step-by-step rather than trying to combine operators if non-linear aberrations are to be considered.

Proposed four operation modes that use the same parameters:

1) Linear approximation using ray transfer matrices. This can yield operators that are suitable to do very fast transformations and to easily solve equation systems, such as finding focal points etc.

2) Non-linear step-wise ray propagation with all aberrations. Solving will require a numerical solver / optimizer.

3) Fast wave propagation using the linear operator, likely a single-step linear canonical transform.

4) Exact step-wise wave propagation, including all aberrations. This will likely resemble image simulation as it is used for samples.

Question: How to handle apertures? No representation per se in matrix optics since they are not linear. However, matrix optics can be used to project apertures along the optical system -- as pixmaps or by projecting vertices of vector shapes. They are essential for wave optics, that means they should be handled in a fashion that is intuitive, flexible and correct.

Additional functionality: Numerically optimize a single-step polynomial based on the individual elements to do fast ray transfer and/or wave propagation? In case of wave propagation, this could look like a "virtual Schmidt corrector plate" as an aperture, defined by the resulting polynomial?

[uellue]

Relation to other software or methods: A full physics simulation with a suitable method can yield point-to-point mappings of beams going in and going out of an optical element. Those can be used to calculate the coefficients of the matrices using numerical optimization. That should make them easier to use in practical applications and less critical in terms of IP since they only contain the information what the microscope does and not how.

[uellue]

@davidlanders95 that's the one!