Short intro about COMSOL Multiphysics:

The software COMSOL Multiphysics was used to model the electromagnetic field’s interaction with our structure. COMSOL Multiphysics is a CAE software package that can be used to model any physics based system and the interaction of different physics (COMSOL Multiphysics^®, 2016). For this project the RF Module was used; this module is the best fit for the dimensions of our structure (micrometers) and the intended wavelength (in the visible spectrum). The figure below demonstrates the different options for different structure size and wavelengths (COMSOL, 2013).

3D model – Sphere

For the Electromagnetic field–structure interaction there are two models built. The first and simplest assumes that the shape of our structure is a simple sphere. Using the simple sphere model we have symmetry allowing to model only a quarter of the sphere and we need smaller domain compared to a rod shaped structure resulting in a computational less expensive model and faster conversions. Also the case of spherical lenses is better design due to their independence from their orientation to the incoming radiation compared to the bacterial shaped lenses.

Furthermore parametric method of modelling was used, meaning that the most important parameters for the model were defined and then used for the model design. The parameters used for this model is the radius of the sphere, the wavelength, wavenumber and frequency of the intended light, the thickness of the silicate layer, air layer and Perfectly Matched Layer, the intensity of the intended electromagnetic field, and the material parameters epsilon defined later. The values of those parameters are shown in table 1 below.

Table 1: Values of parameters used in the model

According to the aforementioned parameters a three dimensional model of a layered sphere, representing our structure, was created in COMSOL Multiphysics^®. The 3D model can be seen in Figure 2. In that model the inner part is the r0, representing the cell, the first layer is the silicate layer covering the cell, then the other two layers are the medium with the outermost representing the world further away from the structure. It is important to note that the 3D model shown in Figure 2 is only a quarter of the actual structure but due to symmetry it is possible to use only that part to decrease the computational cost.

Materials

Next the materials need to be determined. The RF module uses three important material parameters for its calculations: relative permeability ( \( \mu_r \) ), electrical conductivity ( \( \sigma \) ) and relative permittivity (\( \epsilon_r \)). The relative permeability is equal or almost equal to unity for most real materials for the optical frequency range that concern us (visible spectrum of the EM field) (Mcintyre and Aspnes, 1971). The values for the electrical conductivity were obtained from the material library of COMSOL Multiphysics^® and from literature and the relative permittivity can be calculated from the refractive index (n) using the following formulas(Griffiths, 1999):

So it is:

Here n is the Real part of the refractive index and k is the Imaginary part. Because we assume that we have non absorbing materials and thus the complex part of the refractive index is zero ( \( k=0 \) ) the relative permittivity can be calculated from the refractive index as:

Concluding the material parameters used for this model are summarized in Table 2 below. The refractive index of water is 1.33 (Daimon & Masumura, 2007) and of the cell 1.401 (Jericho, Kreuzer, Kanka, & Riesenberg, 2012) and the relative permittivity of both is calculated using the aforementioned formula. The same method was used to calculate the relative permittivity of Tin dioxide, with refractive index between 2.33 and 2.8 for 550 nm (Baco, Chik, & Md. Yassin, 2012) the relative permittivity is 6.58.

Table 2: Material parameters used in the model

Electromagnetic wave

The intended electric field is \(E0*\exp(-j*k0*x)\) and passes through the whole structure. Figure 3 below demonstrates how EM radiation propagates in space, we have set the propagation direction as x, the electric field oscillation as z and of course the magnetic field oscillation as y.

Mesh

A very important part of finite element modelling (FEM) is the meshing of the design. The important part is that there should be enough nodes that the structure is well represented from the FEM but not so many that the system runs out of memory and the simulation never finishes. The meshing is very important for our model because we have a very thin silicate layer between the cell and the air. We cannot model it with coarse element size as there wouldn’t be enough detail and the simulation will never converge and on the other hand we can’t mesh the whole structure finely because it is unnecessary. Keep in mind that when changing the mesh, the number of the Degrees of Freedom of the model increase in the number of 3 because we are using a 3 dimensional domain. The general rule of thumb for meshing in RF simulations is to use as maximum element size about one tenth of the wavelength, in our case this is about 50 nm. The meshed structure can be seen in Figure 3.

3D - Rod Shaped model

Additionally to the spherical model, a rod shaped model that resembles the shape of bacteria closer was used. The modeling method, parameters and materials used to create the rod shaped structure are the same as the spherical one. The way this was modeled is with a layered cylinder and two half spheres in each end of the cylinder. The 3D model of the rod can be seen in the figure below. The length of the middle part was set to \({0.5} \mu m\)

The meshing was created with the same rule as well, maximum mesh size selected was again 50 nm for this structure.

First simulations with small domain

The aforementioned models were used to predict the behavior of light when it meets our biolenses. The first simulation was that of a circular cell of diameter \( 0.5 \mu m \) covered with a thin film of \( 80 nm\) silica. The domain selected was water and the domain was circular for better use of symmetry. The domain in this study was very small at \( 1 \mu m\). The reason a small domain was selected is the small computational time during the troubleshooting period. The first results of those simulations can be seen below.

Figure 7 shows the z component of the electric field. It can be seen that some focusing is present in the structure. This can be seen better in figure 8, the normalized electric field. The focusing here can be seen in the edge of the cell. One reason that we see that can be that the focal point of the lens is further than the \(1 \mu m\) of this simulation and it is somewhere in the Perfectly Matched Layer (PML) where we can’t see it. In order to investigate if this is the reason we can’t see a focal point additional simulations were performed with larger domain. Here needs to be noted again that while the domain is enlarged the number of nodes increase in the power of 3 so the increase of domain for example to \(1 mm\) is impossible at least with the equipment we have.

One additional comment on this model is that the intended electric field can be seen propagating with no change except from the part that interacts with the sphere and a little around it due to some scattering.

Simulations with bigger domain

The domain was increased to \(2.1 \mu m\). The result of the second simulation with greater domain can be seen in figure 9 below. Here the focusing area is better seen around \(1 -2 \mu m\) and because the sphere’s radius is \( 0.5 \mu m\) the focal area is around \( 0.5 -1.5 \mu m \) from the structure. Note here that we are talking about a focal area not a focal point as expected compared to traditional lenses. One of the reasons that that we have focal area and not focal point is that most of the scattering is happening due to the thin layer of silica because the cell has similar refractive index as the medium, water. Also due to the spherical shape of the structure there are some aberrations observed.

Again the propagation of light can be seen in figure 10 below.

Discussion

This set of simulations show that for the circular structure of the silica covered cells of \( 1 \mu m\)$ in water acts as a lens and it focuses the light about \( 0.5-1 \mu m\) from the lens. Also the focusing point is not an actual point but rather a broad area of focusing and this can be explained partially due to the fact that the refractive index of the cell, 1.401 (Jericho et al., 2012), and the refractive index of water, 1.33 (Daimon & Masumura, 2007), are almost equal so most of the reflection is happening doe to the silicate layer with refractive index between 2.33 and 2.8 for 500 nm (Baco et al., 2012).

In order to double check the findings of those simulations we decided to use an additional software and run simulations for the same models. The second software used is the CST STUDIO SUITE that is more specialized for RF and Microwave simulations.

The same simulation was run with the same material parameters but in bigger domain. Figure 5 shows that the focal point is at about \(0.5 – 1.5 \mu m\) from the lenses as well. It can be seen more clearly in figure 11 that there is not a focal point but rather a focus area. This is also shown in figure 12 where the intensity vs the distance is plotted for the yz plane when x is 0 .