Special mode for axisymmetric particles

[myurkin]

It seems to be possible to employ the axial symmetry of the scatterer to reduce the volume integral equation (VIE) to 2D one and to keep some kind of FFT acceleration. The FFT is retained for z coordinate, while Hankel transform is employed for ρ. The latter can also be made fast by using the FFT. That is described in
Liu Q.-H. and Chew W.C. Applications of the conjugate gradient fast Fourier Hankel transfer method with an improved fast Hankel transform algorithm, Radio Sci. 29, 1009–1022 (1994).
for axisymmetric incident field.

The related issues:

• for arbitrary incident fields, many azimuthal orders should probably be considered (independently).
• singularity of the modified VIE need to be studied rigorously (as in the standard DDA), it may also affect the accuracy of the fast Hankel transform.
• it seems that this fast Hankel transform implies some resampling of the points from the natural axisymmetric grid. Thus, the acceleration will incur some approximation in contrast to the standard DDA. This error should be controlled.
• overall, it is not clear how significant modifications of various parts of ADDA are required. Can it be naturally done with current modularity of ADDA, or will it be easier to make a separate fork.

The latter issue can be considered in combination with possibility to employ infinitely long dipoles for 2D DDA. See, e.g.,
Wiecha P.R., Majorel C., Arbouet A., Patoux A., Brûlé Y., Colas des Francs G., and Girard C. “pyGDM” - new functionalities and major improvements to the python toolkit for nano-optics full-field simulations, Comput. Phys. Commun. 270, 108142 (2022).
However, we can also do this using rectangular dipoles and IGT with no significant changes of the code. However, such direct integration will probably be not efficient - some analytical formulae for this special case need to be used.

Another relevant reference for "axisymmetric" DDA is
Loke V.L.Y., Nieminen T.A., Heckenberg N.R., and Rubinsztein-Dunlop H. T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry, J. Quant. Spectrosc. Radiat. Transfer 110, 1460–1471 (2009).
but no acceleration is employed there.