## Objective

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.

## Total cross-sections

We first retrieve the total cross-sections, as usual,

## 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)
2520
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51.427827471 -12.112351941 -2.910274e+01
52.585282036  11.897934175 -2.930950e+01
27.157009947  14.748139467 -3.701212e+00
61.134779210  15.865245823  2.677867e+01
74.965557335   6.542223351  2.377003e+01
42.533698639  28.315486266 -7.018778e+00
56.689125851  26.129506080 -1.791185e+01
47.641831116  24.129918772  1.923650e+01
28.820265028   7.804994433  1.612892e+01
69.248691601  -8.930958408  2.690077e+01
54.456561674  31.336090419  4.918617e+00
41.357987056   9.651950482  2.696616e+01
47.695334535 -27.563867614  1.390373e+01
78.295226837   0.764862224 -2.152176e+01
61.398486865 -24.002538504 -1.973262e+01
82.768766601 -14.743542777 -4.238187e+00
45.789627748 -26.694161235 -1.445785e+01
38.327637750  18.682878069 -1.947873e+01
86.148039894   1.541803966 -5.819032e+00
62.931050870  -2.262928636 -3.063358e+01
52.495470010 -17.206309723  2.653505e+01
55.000000000   0.000000000  3.184821e+01
84.697884509   0.000000000  1.150408e+01
23.802561793  -2.637485984  5.837104e+00
67.656137185  28.718487716 -5.420232e+00
56.917909971 -31.779989983 -8.137080e-01
36.938887558 -26.231738825 -1.892468e-02
68.508326904  25.102533337  1.420198e+01
65.673625730 -25.299235480  1.613477e+01
34.780766202  23.064721158  8.573767e+00
38.711417469   2.181118229 -2.728064e+01
79.863985730  17.549541684 -9.386376e+00
35.697616188 -14.087324525 -2.105406e+01
69.470850030  15.615538629 -2.368666e+01
31.851146910 -16.690167484  1.413779e+01
80.650025658  17.153364958  7.883306e+00
74.277475836 -15.393932680 -2.014234e+01
26.905401802 -13.496367780 -6.545981e+00
37.504129641  -8.016272430  2.537602e+01
71.616233532 -27.037751701 -2.677504e+00
26.629619185   3.235047601 -1.410547e+01
78.943136101 -16.093777329  1.349166e+01
51.413888834 -12.159614405 -2.921629e+01
52.575859793  11.944359979 -2.942387e+01
27.048366447  14.805686788 -3.715654e+00
61.158717152  15.927152100  2.688316e+01
75.043463050   6.567751143  2.386278e+01
42.485055062  28.425973448 -7.046166e+00
56.695716830  26.231463556 -1.798174e+01
47.613119500  24.224073847  1.931156e+01
28.718111558   7.835449564  1.619186e+01
69.304290073  -8.965807057  2.700574e+01
54.454441175  31.458364016  4.937809e+00
41.304755847   9.689612445  2.707139e+01
47.666831691 -27.671421977  1.395798e+01
78.386124940   0.767846721 -2.160574e+01
61.423453796 -24.096196541 -1.980961e+01
82.877120481 -14.801072162 -4.254724e+00
45.753688825 -26.798321999 -1.451427e+01
38.262582101  18.755778761 -1.955474e+01
86.269579717   1.547820093 -5.841738e+00
62.961997874  -2.271758596 -3.075311e+01
52.485697320 -17.273448838  2.663859e+01
55.000000000   0.000000000  3.193727e+01
84.780937720   0.000000000  1.153626e+01
23.715314925  -2.644861987  5.853428e+00
67.691531384  28.798801942 -5.435390e+00
56.923273605 -31.868866017 -8.159836e-01
36.888377786 -26.305098600 -1.897760e-02
68.546104339  25.172735172  1.424170e+01
65.703475630 -25.369987413  1.617989e+01
34.724221020  23.129224040  8.597745e+00
38.665864761   2.187217952 -2.735693e+01
79.933520442  17.598620795 -9.412626e+00
35.643635072 -14.126721187 -2.111294e+01
69.511319261  15.659209101 -2.375290e+01
31.786408744 -16.736843266  1.417733e+01
80.721758612  17.201336120  7.905353e+00
74.331387294 -15.436983407 -2.019867e+01
26.826832348 -13.534111770 -6.564288e+00
37.455200627  -8.038690766  2.544698e+01
71.662702550 -27.113365574 -2.684992e+00
26.550278476   3.244094747 -1.414492e+01
79.010095562 -16.138785244  1.352939e+01
51.403859902 -12.193620026 -2.929800e+01
52.569080434  11.977763619 -2.950616e+01
26.970196808  14.847092425 -3.726045e+00
61.175940642  15.971694030  2.695834e+01
75.099516670   6.586118539  2.392952e+01
42.450055722  28.505469628 -7.065871e+00
56.700459077  26.304822562 -1.803203e+01
47.592461324  24.291818979  1.936557e+01
28.644611535   7.857362210  1.623714e+01
69.344293502  -8.990880865  2.708126e+01
54.452915463  31.546340590  4.951618e+00
41.266455678   9.716710450  2.714709e+01
47.646323726 -27.748807975  1.399702e+01
78.451526661   0.769994083 -2.166616e+01
61.441417650 -24.163584051 -1.986501e+01
82.955081736 -14.842464894 -4.266623e+00
45.727830558 -26.873266282 -1.455486e+01
38.215774178  18.808231240 -1.960943e+01
86.357028336   1.552148733 -5.858075e+00
62.984264426  -2.278111804 -3.083912e+01
52.478665812 -17.321755828  2.671309e+01
55.000000000   0.000000000  3.198799e+01
84.828232400   0.000000000  1.155458e+01
23.665632166  -2.649062255  5.862724e+00
67.711686623  28.844536906 -5.444022e+00
56.926327928 -31.919476506 -8.172794e-01
36.859614979 -26.346873350 -1.900774e-02
68.567616713  25.212711631  1.426431e+01
65.720473666 -25.410277126  1.620558e+01
34.692021346  23.165955228  8.611399e+00
38.639924755   2.190691442 -2.740037e+01
79.973117008  17.626568913 -9.427574e+00
35.612895507 -14.149155630 -2.114647e+01
69.534364479  15.684077266 -2.379062e+01
31.749543573 -16.763422807  1.419984e+01
80.762606968  17.228653315  7.917907e+00
74.362087193 -15.461498659 -2.023075e+01
26.782090944 -13.555605099 -6.574712e+00
37.427337983  -8.051456896  2.548739e+01
71.689164349 -27.156423921 -2.689256e+00
26.505097881   3.249246647 -1.416738e+01
79.048225650 -16.164415018  1.355087e+01
51.398148924 -12.212984539 -2.934453e+01
52.565219925  11.996785334 -2.955301e+01
26.925683080  14.870670879 -3.731962e+00
61.185748565  15.997058448  2.700115e+01
75.131436424   6.596577859  2.396752e+01
42.430125336  28.550738755 -7.077092e+00
56.703159552  26.346596873 -1.806067e+01
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+01

Note 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