Absorption computed from internal field
05 December, 2023
Source:vignettes/112_absorption_field/112_absorption_field.Rmd
112_absorption_field.RmdObjective
This example compares the calculation of far-field absorption against that of internal electric field intensity, for a core-shell dimer. The calculation is done for a fixed incidence direction.
The standard far-field simulation uses the following
input file
ModeAndScheme 2 3
MultipoleCutoff 12
Wavelength 300 800 100
Incidence 0 0 0 2 # z axis, CP
Medium 1.7689 # for water
Verbosity 1
OutputFormat HDF5 cross_sections
#ScattererCentredCrossSections
Scatterers 2
Au@Au 0 0 0 21 18
Au@Pd 55 0 0 32 30 Note that the two particles are different: the first scatterer is a
homogeneous Au sphere, artificially separated into two regions. The
second scatterer is a Au@Pd core-shell.
Mackowski’s partial cross-sections
Formulas derived by Mackowski provide partial absorptions for individual spheres, which we average with a numerical cubature. We verify that the total coincides with the previous plot.
Internal field intensity
Absorption can also be calculated from the internal electric field intensity as
\[ \sigma_\mathrm{abs}(\lambda) = \frac{2\pi}{n \lambda}\iiint_V \varepsilon''(\lambda) |\mathbf{E}(\mathbf{r})|^2 \,\mathrm{dV} \] with \(n\) the refractive index of the incident medium, \(\varepsilon''\) the imaginary part of the material’s dielectric function, and \(\mathbf{E}(\mathbf{r}\) the internal electric field.
We apply this formula below to the four distinct regions (2 cores, and 2 shells). The first step is to generate a grid of points for TERMS to compute the electric field intensity in the regions of interest. One could use a regular grid extending over a larger area, and select the different regions in post-processing, but this is not efficient as many points would have to be sampled, including in non-absorbing regions (ambient medium). Instead, we create a numerical cubature over the ball (or shell) region, by taking a cartesian product of a spherical cubature in \((\varphi, \theta)\), and a Gauss-Legendre quadrature over the radial range.
We define 4 such cubatures, one for each region of interest, and
convert the points to cartesian coordinates before exporting them in a
position file for use in TERMS
Show positions file (click to open)
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47.580697518 24.330396469 1.939633e+01
28.602756920 7.869840374 1.626293e+01
69.367073469 -9.005159154 2.712427e+01
54.452046646 31.596438881 4.959482e+00
41.244645633 9.732141419 2.719021e+01
47.634645458 -27.792875459 1.401925e+01
78.488769694 0.771216900 -2.170057e+01
61.451647173 -24.201957892 -1.989656e+01
82.999476801 -14.866035999 -4.273399e+00
45.713105558 -26.915943331 -1.457797e+01
38.189119390 18.838100322 -1.964057e+01
86.406825983 1.554613678 -5.867378e+00
62.996944122 -2.281729641 -3.088809e+01
52.474661717 -17.349264259 2.675551e+01Note that TERMS will automatically detect the material region corresponding to each point, and return this information in the near-field map. This is helpful to combine multiple regions as in this example, without needing to run separate simulations, or keep track of each point ourselves.
The input file for this near-field simulation is shown below.
ModeAndScheme 1 0
MultipoleCutoff 15
Wavelength 350 750 20
Incidence 0 0 0 2 # z axis, CP
Medium 1.7689 # for water
Verbosity 1
OutputFormat HDF5 map
#ScattererCentredCrossSections
SpacePoints positions
MapQuantity 2 E
Scatterers 2
Au@Au 0 0 0 21 18
Au@Pd 55 0 0 32 30 Processing
The output near-field maps can be split by volID and
scatID fields, corresponding to our four regions of
interest. We visualise below the electric field intensity at the sampled
points for the outer shell of scatterer 1,
Rows: 13,230
Columns: 10
$ phi <dbl> 0.0000000, 0.0000000, -3.0572515, 1.1557061, -1.5105198, -2…
$ r <dbl> 30.01201, 30.01201, 30.01201, 30.01201, 30.01201, 30.01201,…
$ theta <dbl> 0.0000000, 1.2012247, 1.3864755, 1.7418184, 1.5963487, 1.57…
$ wavelength <dbl> 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350, 350,…
$ x <dbl> 0.000000, 16.801474, -17.649841, 7.160165, 1.085051, -10.21…
$ y <dbl> 0.000000, 0.000000, -1.492148, 16.247384, -17.979418, -14.8…
$ z <dbl> 18.01801000, 6.50839400, 3.30232100, -3.06647700, -0.460352…
$ ScatID <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
$ volID <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ I <dbl> 0.4750168, 1.2593607, 0.5449848, 0.7380284, 0.5376374, 0.54…
From the volumetric intensity maps, we apply the integration formula by multiplying with the cubature weights, and compare the results to the partial absorption cross-sections obtained above.
Note that we only sampled a few wavelengths for clarity, but the
near-field calculation is relatively fast, especially with the
reasonable number of points afforded by the use of an efficient
cubature, and the use of Scheme=0 for solving at fixed
incidence.
Last run: 05 December, 2023