Subwavelength Hexahedral Plasmonic Scatterers: History, Symmetries, and Resonant Characteristics †
<p>The transformations of a superquadric sphere for increasing values of the rounding factor <span class="html-italic">p</span>. The bottom left and bottom right cases depict the studies cases used in this work: a “smooth” <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math> and a “sharp” <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> cube. The figure indicates a constant <span class="html-italic">z</span>-directed electrostatic excitation <math display="inline"><semantics> <mrow> <mi mathvariant="bold">E</mi> <mo>=</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <msub> <mi mathvariant="bold">u</mi> <mi>z</mi> </msub> </mrow> </semantics></math>.</p> "> Figure 2
<p>The absolute value of the polarizability at the <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>0</mn> <mo>]</mo> </mrow> </semantics></math> domain of a smooth cube with <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math>. The color figures at the top color depict the surface potential of the six most pronounced resonances <math display="inline"><semantics> <msub> <mi>C</mi> <mn>1</mn> </msub> </semantics></math>–<math display="inline"><semantics> <msub> <mi>C</mi> <mn>6</mn> </msub> </semantics></math>, for the regular mesh, while at the bottom the same surface potential are shown for the refined mesh. Blue and red colors indicate negative and positive values of the potential. The inset figure illustrates the used meshes, i.e., regular and refined ones.</p> "> Figure 3
<p>As in <a href="#photonics-06-00018-f002" class="html-fig">Figure 2</a>, the absolute value of the polarizability at the <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>0</mn> <mo>]</mo> </mrow> </semantics></math> domain for the sharp cube (<math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>). The color figures at the top depict the surface potential of the six most pronounced resonances <math display="inline"><semantics> <msub> <mi>C</mi> <mn>1</mn> </msub> </semantics></math>–<math display="inline"><semantics> <msub> <mi>C</mi> <mn>6</mn> </msub> </semantics></math>, extracted with a regular mesh, while the bottom figures are extracted with a refined mesh. The <math display="inline"><semantics> <msub> <mi>C</mi> <mn>4</mn> </msub> </semantics></math> resonance is actually different between the two mesh setups (inset sketches top and bottom). Note also the disagreement of the <math display="inline"><semantics> <msub> <mi>C</mi> <mn>3</mn> </msub> </semantics></math> resonance between the smooth case (<a href="#photonics-06-00018-f002" class="html-fig">Figure 2</a>) and the sharp case (here) [<a href="#B15-photonics-06-00018" class="html-bibr">15</a>]. The observed inconsistencies provide evidence that the higher-order modes residing at <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>∈</mo> <mo>(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> are sensitive to both the mesh and the sharpness of the cube.</p> "> Figure 4
<p>A sketch describing the proposed categorization scheme: vertex-edge vectors <math display="inline"><semantics> <msub> <mi mathvariant="bold">V</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi mathvariant="bold">V</mi> <mn>2</mn> </msub> </semantics></math> are denoted with red, and face (hedral) vectors <math display="inline"><semantics> <msub> <mi mathvariant="bold">F</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi mathvariant="bold">F</mi> <mn>2</mn> </msub> </semantics></math> with green. The blue vector indicates the axis of rotational symmetry due to the used excitation (<math display="inline"><semantics> <msub> <mi mathvariant="bold">u</mi> <mi>z</mi> </msub> </semantics></math>-polarized electrostatic field). A mode is characterized by four numbers, denoting the number of nodes (sign changes of the surface potential distribution) along the four vectors <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>. The combination of the four numbers and the particular symmetries of cube, allow the recreation of every mode. Please note that in the case of a sphere all <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">V</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="bold">F</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi mathvariant="bold">F</mi> <mn>2</mn> </msub> </semantics></math> are parallel (or antiparallel) representing the elevation vector <math display="inline"><semantics> <msub> <mi mathvariant="bold">u</mi> <mi>θ</mi> </msub> </semantics></math>, while <math display="inline"><semantics> <msub> <mi mathvariant="bold">V</mi> <mn>2</mn> </msub> </semantics></math> becomes the azimuthal vector <math display="inline"><semantics> <msub> <mi mathvariant="bold">u</mi> <mi>ϕ</mi> </msub> </semantics></math>.</p> "> Figure 5
<p>The six main resonances and the three particular clusters for the case of a smooth cube with <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math> and a regular mesh. A total of 22 resonances and their surface potential distribution is presented. Above each resonance the categorization number is given, described in <a href="#photonics-06-00018-f004" class="html-fig">Figure 4</a>. Please note that all resonances in clusters are more sensitive to modeling mesh density, than the main six resonances <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>C</mi> <mn>6</mn> </msub> </mrow> </semantics></math>.</p> "> Figure 6
<p>The six main resonances and the three particular clusters for the case of a sharp cube with <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> and a refined mesh. A total of 17 resonances and their surface potential distributions are presented. Above each resonance the categorization number is given, described in <a href="#photonics-06-00018-f004" class="html-fig">Figure 4</a>. Please note that all resonances in clusters are more sensitive to modeling mesh density, than the main six resonances <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>C</mi> <mn>6</mn> </msub> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Superquadric Surfaces and the Cube
3. Plasmonic Spectra of Smooth (p = 15) and Sharp (p = 50) Sube
3.1. Case p = 15
3.2. Case p = 50
4. Historical Remarks: Plasmonic Resonances of a Cube (Years 1970–2004)
5. Categorization of the Resonances
6. Discussion and Conclusions
Author Contributions
Conflicts of Interest
Appendix A
Modes | Regular Mesh | Refined Mesh | Modes | Regular Mesh | Refined Mesh |
---|---|---|---|---|---|
Cluster 1 | Cluster 1 | ||||
… | |||||
… | |||||
… | – | – | |||
… | – | – | |||
… | – | – | |||
Cluster 2 | Cluster 2 | ||||
… | … | … | … | ||
… | … | … | |||
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Tzarouchis, D.; Ylä-Oijala, P.; Sihvola, A. Subwavelength Hexahedral Plasmonic Scatterers: History, Symmetries, and Resonant Characteristics. Photonics 2019, 6, 18. https://doi.org/10.3390/photonics6010018
Tzarouchis D, Ylä-Oijala P, Sihvola A. Subwavelength Hexahedral Plasmonic Scatterers: History, Symmetries, and Resonant Characteristics. Photonics. 2019; 6(1):18. https://doi.org/10.3390/photonics6010018
Chicago/Turabian StyleTzarouchis, Dimitrios, Pasi Ylä-Oijala, and Ari Sihvola. 2019. "Subwavelength Hexahedral Plasmonic Scatterers: History, Symmetries, and Resonant Characteristics" Photonics 6, no. 1: 18. https://doi.org/10.3390/photonics6010018