Introduction

In the second part of our project we created polysilicate covered E. coli cells with the intention to use them as biological microlences. In order to determine if our cells act as lenses when covered by polisilicate we modeled their interaction with light. Since we are making lenses which at lenghtscales close to the wavelength of light, simple ray optics does not apply, therefore we need to work approach light with its wave form properties. In the sections bellow we present different studies, models and software we used. Initially we modeled a spherical model and then a more accurate representation of actual E. coli bacteria, a rod shaped model. Finally we are comparing the optical properties of a spherical and rod shaped cell.

Short intro about COMSOL Multiphysics:

The software COMSOL Multiphysics was used to model the electromagnetic field’s interaction with our structure, a polysilicate layer covered cell. COMSOL Multiphysics is a CAE (Computed Aided Engineering) 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 incident wavelength (in the visible spectrum). Figure 1 below demonstrates the different options for different structure size and wavelengths (COMSOL, 2013).

3D model – Sphere

For the in teractions of the Electromagnetic field with the structure interaction there are two models built. The first and simplest module assumes that the shape of our structure is a simple sphere. Even though the spherical model is an oversimplification of the actual E.coli shape we used it as initial studies. We realize that the shape of E.coli is very different than a sphere so we also created more accurate representation of E.coli bacteria, the rod shaped models. Using the simple sphere model we have symmetry allowing to model only a quarter of the sphere, so we need smaller domain compared to a rod shaped structure, resulting in a computational less expensive model and faster conversions.

A parametric method of modeling was used, meaning that the most important parameters for the model were defined and then used for the model. The parameters used for this model are: the radius of the sphere, the wavelength, wavenumber and frequency of the incidentincident light, the thickness of the polysilicate layer, air layer and Perfectly Matched Layer , which is basically the representation of the outside world, the intensity of the incidentincident electromagnetic field, and the material parameters epsilon defined later. The values of those parameters are shown in table 1.

According to the aforementioned parameters a three dimensional model of a layered sphere, representing our structure, was created (Figure 2). In this model the inner part is the cell radius, the first layer is the polysilicate layer covering the cell (thickness of t_sil), then the other two layers are the medium (thickness t_medium) with the outermost representing the surroundings further away from the structure, called Perfectly Matched Layer (thickness t_pml). 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:

The material parameters used for this model are summarized in Table 2. 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.

Electromagnetic wave

The incidentincident electric field is \(E_0 \cdot e^{j\cdot k_0\cdot x}\) and passes through the whole structure. Figure 3 demonstrates how EM radiation, in this case almost green light as the wavelength is 500 nm 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 method 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 polysilicate layer between the cell and the medium. Keep in mind that when changing the mesh, the number of the Degrees of Freedom of the model increases in the power 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 4.

3D - Rod Shaped model

After constructing the spherical model, a rod shaped model was made, which resembles the shape of E. coli in a better way. 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 figure 5. The length of the middle part was set to (0.5 µm). The meshing was created with the same rule as for the spherical model, where the maximum mesh size selected was 50 nm.

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 \( 80 nm\) polysilicate. The medium selected was water and the domain (volume used for the calculations) was circular for better use of symmetry. The domain in this study was \( 1 \mu m\). 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 in this figure we can clearly see the propagation of the light as plane wave until it reaches our structure. 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. We can see that the focal point of the lens is further away from the cell surface than the \(1 \mu m\) inthis simulation and it is somewhere in the Perfectly Matched Layer (PML) where we cannot see it. In order to investigate if this is the reason we cannot 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. We run a number of simulations with larger domains but for more demanding models we used the CST Studio Suit and servers from the Imaging Physics department of TU Delft.

One additional comment on this model is that the incident 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 . This shows us that the simulations work properly and we don’t have parasitic scattering due to errors in PML or wrong symmetry definitions.

Simulations with larger domain

The domain was increased to a radius of \(2.1 \mu m\). The result of the second simulation with greater domain can be seen in figure 9 below. Here the full focusing area can be seen , around \( 0.5 -1.5 \mu m \) behind 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 we have scattering. This is due to the fact that the layer of polysilicate is thin compared to the rest of the system and because the cell has similar refractive index to the medium (which simplified to water). Also due to the spherical shape of the structure there are some aberrations observed above the focal point. Spheres in general don’t have nice focal points as traditional lenses so aberrations in the scattering light is a common effect of spherical structures. Again the propagation of light can be seen in figure 10.

Discussion

This set of simulations shows that for the circular structure of the polysilicate covered cells in water acts as a lens and it focuses the light about \( 0.5-1 \mu m\) away from the lens. Also, the focusing point is not an actual point but rather a broad area of focusing, and this can be partially explained due to the differences in refractive indices of the materials.

CST STUDIO SUITE simulations

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

The same simulation was run with the same material parameters but in larger domain. The materials and the parameters used for this model were exactly the same as the COMSOL model. The design parameters can be seen in table 1 and the materials in table 2.

Figure 7 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 focus 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.

Conclusion for the spherical model

The refraction of a spherical cell was simulated with two different software, COMSOL Multiphysics and CST STUDIO SUIT. Both of those models showed that there is a focusing area about \(0.5 – 1.5 \mu m\) from the lens. This leads to the conclusion that the polysilicate covered spherical cells in water can act as lenses.

As discussed in the presentation of the models a second 3D model, resembling the shape of actual bacteria, was created: the rod shaped model. This model, due to the larger domain needed, was much harder to implement than the simpler spherical model. Figure 13 and figure 14 show the interaction of the electromagnetic field with the rod shaped structure

The same phenomena as with the spherical model in small domain are observed here. The PML absorbs the field so we cannot observe the correct focal point. This model requires way bigger domain in order the focal point to be seen. Considering that this is the same phenomenon as in the spherical model we can assume that the focal point of the rod shaped model is in the area \( 1 – 2 \mu m\) further from the structure

This assumption was tested by using the CST software. Again the reason we are using two different software packages is that we want to double check the results and with the CST we have access to better hardware equipment meaning that we can simulate bigger domains with better resolution. Figure 15 shows the rod–shaped structure design. In this orientation the field is propagating along z axis with polarization in x axis , we investigated the polarization in y axes as well but the results were the same.

Again we investigated for the scattering fieled to see if there is a focusing effect with this shape. Figure 16 shows the scattering of the rod shaped structure. In this structure we observe a focusing area with almost the same focal area around \( 0.5 – 1 \mu m \) from the cell as the spherical model.

The focusing is better visualized in figure 17 where a 2D graph of the field in a cut that passes exactly in the center of the structure in the z-direction is shown. From this graph we can see that the maximum electric field is at about 1500 nm from the center of the structure, this means that the focal point is about 700 nm behind the cell.

Orientation of the rod-shaped cells

The most important difference between the two models, the spherical and the rod-shaped, is that the spherical has a higher order of symmetry. We call those cells orientation independent because the scattering will be the same no matter what is the orientation of the filed compared to the cell. For the rod shaped cells this is not so trivial and so we modified our model to see what is the scattering of a field coming vertical to its long edge. Figure 18 shows that model, it is the same model as earlier but with different initial conditions, now the electromagnetic field propagates through the z direction, the electric field at x and magnetic field at y.

Figure 19 shows the interaction of EM field under this angle and figure 20 the 2D plot of electric field passing through the middle plane. We can see that the focusing area is more spread out and less defined than the other orientation or the spherical model. Additionally, some aberrations can be seen in the electromagnetic field. From figure 20 we can also see that the maximum of the scattered field can be found at the position 2000 nm, so about 1200 nm behind the cell. This focal length is higher than the other direction, wider and the intensity of the field is smaller, if we compare figure 20 and figure 17 (intensity shown in the y axis of those graphs).

From this study we can conclude that for the rod shape the focal area is dependent from its orientation compared to the incident radiation. Additionally, comparing the two extreme conditions, that is the light coming perpendicular to the small side or the light coming perpendicular to the ling side we can see that we can obtain better focusing effect in the first case.

We have modeled the interaction of electromagnetic field with bacteria covered with a layer of polysilicate. The goal of this part of the modeling was to see if those structures can act as microlenses focusing light on the other side. In order to verify the results of the models we used two types modeling software, COMSOL Multiphysics^® and the CST STUDIO SUIT. The models clearly show that the light is focused in about \(0.5 – 1 \mu m\) away from the cell. We cannot observe a traditional focus point but rather a focus area where the intensity of the light is stronger than the incoming. This was explained from the fact that the cell and the medium have similar refractive indices and the light bending is happening mostly due to the very thin polysilicate layer so we cannot expect a very well defined focusing. Additionally, in spherical lenses spherical aberration take place. Concluding, these models have shown that the polysilicate covered cells can act as biological microlenses and that the preferred structure is the spherical as it is orientation independent. Therefore we recommended the lab team to investigate the possibilities of making cells spherical. They managed to use the bolA gene to create spherical lenses.

Our COMSOL Multiphysics and CST Studio Suit models were too large and unsupported to upload to the wiki servers. After a discussion with the Headquarters we were advised to upload them on Google drive. You can find the COMSOL models here. and the CST models here.