Analytical vs numerical angular averaging

05 December, 2023

Objective

Here we compare analytical orientation averaging of far-field cross-sections with a numerical cubature.

This simulation uses the following input file

ModeAndScheme 2 3 
MultipoleCutoff 5
Wavelength 300 700 200
Medium 1.7689 # epsilon of water
Incidence file incidence
OutputFormat HDF5 cross_sections

Scatterers 4
Au 100.0000  0.00000 -30 20.0
Au  80.9017 58.77853 -10 20.0
Au  30.9017 95.10565  10 20.0
Au -30.9017 95.10565  30 20.0

Incidence angles and the corresponding weights are specified in an external file incidence,

Show incidence file (click to open)

38 
 0.00000000  1.57079633  0.00000000  0.00952381
 3.14159265  1.57079633  0.00000000  0.00952381
 1.57079633  1.57079633  0.00000000  0.00952381
-1.57079633  1.57079633  0.00000000  0.00952381
 1.57079633  0.00000000  0.00000000  0.00952381
 1.57079633  3.14159265  0.00000000  0.00952381
 0.78539816  0.95531662  0.00000000  0.03214286
 0.78539816  2.18627604  0.00000000  0.03214286
-0.78539816  0.95531662  0.00000000  0.03214286
-0.78539816  2.18627604  0.00000000  0.03214286
 2.35619449  0.95531662  0.00000000  0.03214286
 2.35619449  2.18627604  0.00000000  0.03214286
-2.35619449  0.95531662  0.00000000  0.03214286
-2.35619449  2.18627604  0.00000000  0.03214286
 1.09313802  1.57079633  0.00000000  0.02857143
-1.09313802  1.57079633  0.00000000  0.02857143
 2.04845464  1.57079633  0.00000000  0.02857143
-2.04845464  1.57079633  0.00000000  0.02857143
 0.47765831  1.57079633  0.00000000  0.02857143
-0.47765831  1.57079633  0.00000000  0.02857143
 2.66393434  1.57079633  0.00000000  0.02857143
-2.66393434  1.57079633  0.00000000  0.02857143
 0.00000000  0.47765831  0.00000000  0.02857143
 0.00000000  2.66393434  0.00000000  0.02857143
 3.14159265  0.47765831  0.00000000  0.02857143
 3.14159265  2.66393434  0.00000000  0.02857143
 0.00000000  1.09313802  0.00000000  0.02857143
 0.00000000  2.04845464  0.00000000  0.02857143
 3.14159265  1.09313802  0.00000000  0.02857143
 3.14159265  2.04845464  0.00000000  0.02857143
 1.57079633  0.47765831  0.00000000  0.02857143
 1.57079633  2.66393434  0.00000000  0.02857143
-1.57079633  0.47765831  0.00000000  0.02857143
-1.57079633  2.66393434  0.00000000  0.02857143
 1.57079633  1.09313802  0.00000000  0.02857143
 1.57079633  2.04845464  0.00000000  0.02857143
-1.57079633  1.09313802  0.00000000  0.02857143
-1.57079633  2.04845464  0.00000000  0.02857143

This file was obtained using the cubs package, with the following command:

terms::export_cubature(cubs::cubs(N = 38, 'lebedev'), out = 'incidence')

The command to run the example is simply

../../build/terms input > log

The full log contains basic details of the calculations, and finishes with the timing.

Show log file (click to open)

readInputFile> Parsing file input

readInputFile> Detected keyword ModeAndScheme
               mode=2 => spectrum_FF for far-field quantities
               scheme=3 => Seek T^(j) using Mackowski's approach
readInputFile> Detected keyword MultipoleCutoff
               Supplied ncut(1)=  5
               Setting ncut(2)= ncut(1)
               Setting ncut(3)=  -8
readInputFile> Detected keyword Wavelength
               Wavelength LB (nm):   300.000000
               Wavelength UB (nm):   700.000000
               nsteps, step:       200    2.0000
readInputFile> Detected keyword Medium
               Constant host epsilon=  1.7689E+0
readInputFile> Detected keyword Incidence
               Incidence filename= incidence
               Expected incidence count=     38
               Incident Euler angles and weights:
                  alpha        beta        gamma        weight
               0.00000000   1.57079633   0.00000000   0.00952381
               3.14159265   1.57079633   0.00000000   0.00952381
...
termsProgram> Program run time (CPU & real in s):  1.056E+01  9.246E+00

The output consists as usual of a number of files storing the far-field cross-sections, for each angle of incidence.

Orientation-averaged results

We compare the analytical orientation-averaged spectra (solid lines), with the numerical cubature (dashed lines):

apart from the blue end of the spectrum, the accuracy is good.

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Last run: 05 December, 2023