Back in 1887, the Scottish physicist William Thomson proposed his geometric model of the structure of the ether, which supposedly was an all-pervading medium, the vibrations of which appear to us as electromagnetic waves, including light. Despite the complete failure of the ether theory, the geometric model continued to exist, and in 1993 Denis Ware and Robert Phelan proposed a more advanced model of a structure capable of filling space as much as possible. Since then, this model has mostly interested mathematicians or artists, but a recent study suggests that it could be the basis for future technologies that use light instead of electricity. What is Weir-Phelan Foam, why is it unusual, and how can it be used to capture light? We will find answers to these and other questions in the report of the research group. Go.
Research basis
Literally a hundred years ago, in the scientific community there was a very curious theory about some matter of everything around. This theory was aimed at explaining the nature of electromagnetic waves. It was believed that the ether surrounds everything and is the source of these waves. The scientific discoveries that followed the ether theory completely destroyed it.
William Thomson
However, in 1887, when the ether theory was full of strength and popularity, many scientists expressed their ideas about how exactly the ether could fill all space. William Thomson, also known as Lord Kelvin, was no exception. He was looking for a structure that would ideally fill the space so that there were no empty areas. These searches were later called the "Kelvin problem".
A primitive example: imagine a box containing cans of cola. Between them, due to the cylindrical shape, there are voids, i.e. unused space.
Thomson, in addition to believing that the Earth is no more than 40 million years old, proposed a new geometric structure, which was improved by Denis Ware and Robert Phelan, as a result of which it was named after them.
At the heart of the Weir-Phelan structure are honeycombs that fill space with non-intersecting polyhedra, leaving no empty space. The honeycombs that we usually think of as hexagons thanks to the honeycomb actually come in a variety of shapes. There are cubic, octahedral, tetrahedral, rhombododecahedral, etc.
Weir-Phelan structure
Weir-Phelan honeycombs are unusual in that they consist of different geometric shapes-elements. At its core, it is an ideal foam of bubbles of the same size.
The ancestor of this foam was the one proposed by Lord Kelvin, already familiar to us. However, his version consisted of shortened cubic honeycombs. The Kelvin structure was a convex uniform honeycomb formed by a truncated octahedron, which is a space-filling four-faced polyhedron (tetradecahedron), with 6 square faces and 8 hexagonal faces.
This option for maximum space filling was considered ideal for almost a hundred years, until in 1993 Ware and Phelan opened their structure.
Pentagondodecahedron and tetrahedron
The main difference between the Weir-Phelan honeycomb and its predecessor is the use of two types of constituent elements, which, nevertheless, have the same volume: a pentagondodecahedron (a dodecahedron with tetrahedral symmetry) and a fourteen-hedron with rotational symmetry.
In the work we are considering today, scientists from Princeton University decided to use Weir-Phelan foam in photonics. First of all, it was necessary to find out whether such a foam has photonic band gaps (PBG), which block the propagation of light in all directions and for all polarizations in a wide frequency range.
In their study, the scientists demonstrated that the Weir-Phelan foam-based 16,9D photonic network resulted in significant PBG (XNUMX%) with a high degree of isotropy*, which is an important property for photonic circuits.
Isotropy* - the same physical properties in all directions.
Kelvin foam and C15 foam also performed well in PBG, but were inferior to the Weir-Phelan structure in this measure.
Previously, similar studies have already been carried out, but in them attention was paid to two-dimensional dry foam. Then it was found that two-dimensional amorphous dry foam exhibits PBG only for transverse electric polarization. The problem is that there are two polarizations in XNUMXD foam.
Despite the possible difficulties, three-dimensional foam can be safely considered a promising material in the field of photonics, according to the researchers. There is a reason for this: Plateau's laws ensure that edges form exclusively tetrahedral vertices. And this is a big plus for photonic networks. A striking example of this is a diamond with a PBG of 30%.
The foam has the tetrahedral property of the diamond lattice coordinates, but differs in having curved edges and somewhat unequal bond lengths. It remains only to find out how and to what extent such differences affect the photonic properties.
If the ribs of the 17D dry foam are made thicker, then photonic networks (images below) can be created that exhibit pronounced photonic PBGs of up to XNUMX%, comparable or superior to those in typical examples of self-assembling photonic crystals.
Image #1: Photonic foam networks obtained by thickening the edges of the Weir-Phelan structure (left), Kelvin structure (center) and C15 foam (right).
In order to implement such a model in practice, dry foam must first be crystallized and then coated with a dielectric material. Naturally, the PBG performance of the foam will be less than that of the photonic crystal, but this drawback can be covered by a number of advantages. First, the self-assembly of the foam can allow for the rapid production of large samples. Secondly, photonic foam heterostructures, given previous studies, may have a wider range of applications.
Results of the study
First of all, it was necessary to study dry foam, which is defined as local minima of the interfacial region tessellations* subject to volume constraints, so that the final geometry obeys Plateau's laws.
Tessellation* - splitting the plane into component parts that completely cover the entire plane, leaving no gaps.
To construct the Weir-Phelan, Kelvin, and C15 foams, the scientists started with weighted Voronoi tessellations for BCC, A15, or C15 crystals, respectively.
Voronoi diagram
The parameters were chosen in such a way that all separation cells had the same volume.
Networks formed from curved foam ribs and from straight tessellation ribs of their predecessors have been studied. To evaluate the topology of all types of foam was used ring statistics*.
Ring Statistics (Ring Statistics)*Analysis of the topological characteristics of network materials (liquids, crystalline or amorphous systems) is often based on graph theory, using knots for atoms and bonds for interatomic bonds. The absence or existence of a connection between two nodes is determined by the analysis of the functions of the total and partial radial distribution of the system. In network material, a sequence of nodes and links connected in series without overlap is called a path. Following this definition, a ring is simply a closed path. If you carefully study a particular node in the network, you can see that this node can participate in multiple rings. Each of these rings is characterized by its size and can be classified based on the relationships between the nodes and the links that make it up.
The first way to define a ring was given by Shirley W. King. To study the connectivity of glassy SiO2, she defines a ring as the shortest path between two nearest neighbors of a given site.
In the case of the study under consideration, calculations were made of the number of shortest rings per vertex in an elementary cell.
One cell in the Kelvin model has 2 squares and 4 hexagons per vertex, while TCP (tetrahedrally close-packed) foam has only pentagonal and hexagonal faces (mean values: 5.2 and 0.78 in Weir-Phelan foam; 5.3 and 0.71 in C15 foam). Voronoi tessellations A15 and C15 are TCP structures with the largest and smallest number of faces (f) per 1 cell. Thus, the Weir-Phelan structure has the largest number of faces (f = 13 + 1/2), and C15 is the smallest number of faces (f = 13 + 1/3).
Having finished with the theoretical preparation, the scientists started modeling the photonic network based on dry foam ribs, i.e. foam-photon network. It was found that at a PBG value of 20%, system performance is maximized, and at 15%, the Weir-Phelan foam becomes unstable. For this reason, scientists did not consider wet foam, where the Plateau boundaries have tricuspid sections. Instead, all attention was focused on dry foam structures, where scientists could gradually increase the thickness of the ribs.
In addition, each edge is the medial axis of the spherical cylinder (capsule), where the radius is the setting.
The researchers remind that such foam networks are not literally foam, however, in their report, for the sake of simplicity, they will be referred to as “foam” or “foam network”.
During the simulation, the parameter was taken into account ɛ (dielectric contrast) is the proportion of dielectric constants of materials with high and low insulation value. It is assumed that the dielectric contrast is between 13 and 1, which is commonly used in the literature as a standard when comparing the performance of various designs of photonic materials.
For each network, the radius of the ribs (spherocylinders) is optimized for the maximum ratio of the band gap and its middle: ∆ω/ωm, where ∆ω is the frequency band width, and ωm is the frequency within the zone.
Image #2: Photon zonal structure of Weir-Phelan foam (red), Kelvin foam (blue), and C15 foam (green).
The PBG dimensions were further measured and found to be: 7.7% for Kelvin foam, 13.0% for C15 foam, and 16.9% for Weir-Phelan foam. Minimizing the area increases the dimensions of the PBG by 0.7%, 0.3 or 1.3%.
As it became clear from the analysis, TCP networks have much larger PBGs than Kelvin networks. Of the two TCP networks, it is the Weir-Phelan foam that has the largest band gaps, which is presumably due to the smaller change in link length. Scientists believe that differences in bond lengths may be the main reason why in their system, i.e. in the Weir-Phelan foam, PBG is less than in diamond (31.6%) or in the Laves system (28.3%).
An equally important aspect in photonics is the isotropy of PBG, which makes it possible to create waveguides of arbitrary shape. Photonic quasicrystals as well as amorphous photonic networks are more isotropic than classical photonic crystals.
The studied foam-photon structure also has a high degree of isotropy. Below is the formula for determining the anisotropy coefficient (i.e., the degree of difference in the properties of a particular medium) PBG (А):
A: = (√Var[ωHDB] + Var[ωLAB]) / ωm
C15 foam was found to have the lowest anisotropy (1.0%), followed by Weir-Phelan foam (1.2%). Therefore, these structures are highly isotropic.
But the Kelvin structure demonstrates an anisotropy coefficient of 3.5%, which is quite close to the index of the Laves system (3.4%) and diamond (4.2%). However, even these figures are not the worst, because there are also simple cubic systems with an anisotropy coefficient of 8.8% and hexagonal diamond networks with 9.7%.
In practice, when it is necessary to achieve the maximum value of PBG, sometimes it is necessary to change certain physical parameters of the structure. In this case, this parameter is the radius of the spherical cylinders. The scientists carried out mathematical calculations in which they found out the ratio of the photonic band gap and its width as a function ɛ. For each value obtained, optimization of the radius was carried out to maximize ∆ω/ωm.
Image #3: Comparison of ∆ω/ωm of studied foam networks (C15, Kelvin, Weir-Phelan) and other structures (diamond, hexagonal diamond, Laves, SC - regular cubic).
Weir-Phelan foam maintains acceptable PBG dimensions of 8% down to dielectric contrast ɛ≈9, and the radius was increased to reach the maximum PBG value by 15%. PBG disappear when ɛ < 6.5. As expected, the diamond structure has the largest PBG among all structures studied.
For a more detailed acquaintance with the nuances of the study, I recommend looking at и to him.
Finale
The main motivation for conducting this study, scientists say the desire to answer the question - can foam networks demonstrate full-fledged PBG. Converting the edges of dry foam structures into photonic networks has shown that they can.
At the moment, foam is not a particularly studied structure. Of course, there are studies that give good results in terms of amorphous networks, but they were carried out on extremely small objects. How the system will behave with an increase in its dimensions remains unclear.
According to the authors of the study, their work opens up many opportunities for future inventions. Foam is very common in nature and easy to manufacture, which makes this structure very attractive for practical applications.
One of the most ambitious applications of their research scientists call the Internet. As the researchers themselves say, data transmission over fiber optics is not new, but at the destination, light is still converted into electricity. Photonic bandgap materials can guide light much more accurately than conventional fiber optic cables and can serve as optical transistors that perform calculations using light.
As ambitious as the plans are, there is still a lot of work to be done. However, neither the complexity of conducting research nor the complexity of implementing experiments can defeat the enthusiasm of scientists and their desire to improve the world of technology.
Thanks for watching, stay curious and have a great weekend everyone! 🙂
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Source: habr.com