Calculation of the T-matrix using the DDA

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It is possibly to calculate T-matrix using the DDA, which can be used for 
repeated simulation of different particle orientations. Originally it was 
proposed by Dan Mackowski in 2002 - http://dx.doi.org/10.1364/JOSAA.19.000881 . 
However, the procedure was quiet cumbersome. 

Recently, Dan showed that it can be significantly simplified by using 
plane-wave spectrum instead of spherical harmonics. Details can be found at ELS 
12 proceedings http://www.helsinki.fi/els/articles/38/ . It probably will also 
be published in special JQSRT issue after this conference.

So that is an interesting idea to implement. However, it is still not trivial. 
To perform such calculation efficiently, significant changes of the code 
internals are required. Moreover, in many cases T-matrix is not really needed. 
For example, if one wants to perform orientation averaging it is probably 
faster to do it directly by a quadrature than to calculate T-matrix and use 
analytical formulae. The most viable application for T-matrix calculation seems 
to be databases. Complete information about the particle can be stored in a 
relatively small T-matrix, which can be further used to obtain any light 
scattering patterns for any particle orientation.

Original issue reported on code.google.com by yurkin on 12 Jul 2010 at 5:41

• Blocked on: Expanding scattered field into series of spherical harmonics #138

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This may be especially useful for metallic nanoparticles, when order of the 
T-matrix is relatively small, while Niter can be up to a few thousands. Then 
calculating T-matrix and performing analytical orientation averaging may be 
faster than doing straightforward orientation averaging by multiple DDA 
simulations.

Original comment by yurkin on 6 Jan 2011 at 7:23

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138

Original comment by yurkin on 4 Jul 2013 at 9:46

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Spherical-harmonics expansion of the scattered field can help.

Original comment by yurkin on 4 Jul 2013 at 9:58

• Now blocked on: Expanding scattered field into series of spherical harmonics #138

[myurkin]

Another related idea is compressive sensing, see e.g.
Carin L., Liu D., Lin W., and Guo B. Compressive sensing for multi-static scattering analysis, J. Comput. Phys. 228, 3464–3477 (2009).

[myurkin]

Such calculation with ADDA (using plane-wave excitation) has been performed in
Stilgoe A.B., Nieminen T.A., and Rubinsztein-Dunlop H. Controlled transfer of transverse orbital angular momentum to optically trapped birefringent microparticles, Nat. Photon., 1–6 (2022).

[myurkin]

Another application of ADDA for such problem is described in
Asenchik O.D. Using the method of discrete dipoles to approximate solutions of the problems of light scattering and absorption by particles, Opt. Spectrosc. 122, 294–302 (2017).
However, it focuses not on the (far-field) T-matrix, but on the internal-field calculation for any incident field (corresponds to the inverse of the interaction matrix or to spherical-harmonics expansion of the transition operator).

[myurkin]

Another related issue is, whether such constructed T-matrix is always meaningful. For instance, Farafonov et al. showed that for some particle shapes (even much smaller than the wavelength) the T-matrix cannot be constructed (with any of the standard methods) due to failure of the analytic continuation of the fields expressed in series of vector spherical wave functions. While the DDA is not explicitly mentioned there, it may nevertheless be affected. For instance, I would not be surprised if the T-matrix obtained through the DDA would not uniformly converge with increasing its order (or the scattering quantities won't converge).

Farafonov V., Il׳in V., Ustimov V., and Prokopjeva M. On the analysis of Waterman׳s approach in the electrostatic case, J. Quant. Spectrosc. Radiat. Transfer 178, 176–191 (2016).

[myurkin]

Python script for such calculation is available from https://gitlab.com/k.czajkowski/addatmatrix/ . Currently, it uses plane-wave incident fields (with some optimal sampling) and direct calculation of spherical-harmonics expansion of the scattered fields from dipole polarizations.