Introduction to Quality Factor

The quality factor (Q-factor) is used in many disciplines of physics and engineering. It was originally developed for electronic circuits and microwave cavities. The Q factor is a dimensionless number that indicates how much energy is stored versus how much energy is lost in a resonator. A resonator can be an electrical circuit or in our case an optical resonator with light bouncing between two reflective surfaces. The Q factor is defined as \(2 \pi \) times the energy stored divided by the energy lost in each oscillation (Cory and Chaniotakis, 2006),(Kao & Santosa, 2008) , .A high Q factor means less energy is lost so it is a measure of the resonator’s quality. The Q factor of a resonator can be calculated using equation 1.

$$ Q = 2 \pi \frac{Energy\_stored}{Energy\_lost\_per\_cycle} $$

The Q-factor can also be expressed as a function of the central frequency (f₀) and the bandwidth of the resonator (\(\Delta f_c\)) (equation 2). Figure 1 shows the bandwidth and center frequency of a resonator.

$$ Q= \frac{f_o}{\Delta f_c} $$

For optical resonators the bandwidth \( \Delta f_c \) is linked to the lifetime (\(\tau_c\)) of the photons in a cavity as \( \Delta f_c = 1/(2 \pi \tau_c) \) and therefore the Q factor can be calculated as \(Q = 2 \pi f_o \tau_c \). Furhtermore, the energy stored in the resonator is proportional to the amount of photons resonating inside the cavity, and the energy loss is proportional to the amount of photons lost from the cavity (change in photon number) (“Chapter 3 Passive Variable-Pitch Design Concepts," n.d.).

Additionally the Q-factor can be calculated from the light’s frequency, the fraction of power loss per round trip (usually \(<< 1\) ) and the time of the trip. The formula to calculate the Q factor for those resonators is (Paschotta, 2008):

$$ Q = f_o \cdot \tau_{rt} \cdot \frac{2*\pi}{l}$$ $$Q = \frac{2 \pi f_o E}{P}$$

Where \(f_o\) is the optical frequency, \( \tau_{rt} \) the oscillation time, l is the fractional power loss per roundtrip, E is the Energy stored and P is the Power Loss. We can manipulate the Q-factor of an optical resonator in three ways. Firstly, we can increase the Q-factor by increasing the frequency (decreasing the wavelength). Secondly, when the size of thecavity increases the Q factor increases as well. Finally, in order to achieve very high Q factor, the fractional power loss per roundtrip needs to be reduced.

A very useful way to determine the Q factor of an optical resonator is from the eigenmodes of the on the resonator ( Kao & Santosa, 2008). The eigenmodes of a resonator give its resonation frequencies. The reason this method is useful is that we don’t need to know how much energy is stored and lost, the path of the light or the fractional power loss per circle. Those parameters are easy to determine in two parallel plates with light moving between them but for more complex structures like ours, where whispering gallery modes take place (Q1), and the light path is not so straight forward,these parameters are very hard to determine. By finding the modes we can extract their corresponding eigenfrequencies and use the following formulas to calculate the Q factor:

$$ Q = \frac{Total\_Energy\_stored\_at\_beginning\_of\_the\_cycle}{|Energy\_lost\_during\_a\_circle|}$$ $$ Q \cong 2 \pi \frac{E(0)}{E(0) – E(T)} = 2 \pi \frac{1}{1-\exp{-2 \gamma \frac{2 \pi}{\omega_1}}} \cong \frac{\omega_1}{2 \gamma}$$

Where \( \omega = \pm \omega_1 + i \gamma \)

So the Q-factor can be calculated by:

$$ Q \cong \frac{1}{2} \frac{|Real\_part\_of\_\omega|}{|Imaginary\_part\_of\_\omega|}$$

Modelling of the Q factor

In order to determine the Q-factor for a biolaser we implemented a model in COMSOL Multiphysics® to determine the eigenmodes of our structure. By determining the modes we can calculate the Q factor using the formula \( Q \cong \frac{1}{2} \frac{|Real\_part\_of\_\omega|}{|Imaginary\_part\_of\_\omega |} \) that was described earlier.

Model description

Parametric method of modelling 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 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, representing the world away from the structure, the intensity of the intended electromagnetic field and the material parameters epsilon defined later. The values of those parameters are shown in the table below.

We selected wavelength of 500 nm (frequency about 6 THz) because we are mainly using GFP and fluorophores derived from GFP, so we wanted to search for lasing modes close to green light radiation. Additionally, the size of the cell was set to \(1 \mu m\) in diameter because it is the same length scale of the bacteria we are using. Finally, the thickness of the polysilicate layer was set to 80 nm as an arbitrary size of a small layer that we are likely to obtain.

According to the aforementioned parameters a two dimensional model of a layered circle, representing our structure, was created. The 2D model can be seen in figure 2. In this model the inner part is the r₀, representing the cell, 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 surrounding further away from the structure, called Perfectly Matched Layer (thickness t_pml). Here we can see that the layer of the medium is very big, this practice can generate problems in finite element models due to the increased resources needed for the simulations. In this case the risk is lower because we are modeling in two dimensions, meaning that while increasing the size of the domain, the number of nodes will increase but slower than in the 3D models.

Materials

Next we will determine the materials used.The RF module, which is the COMSOL module best suited for modeling wavelengths close to the length scales of the structures was used. This module uses three important material parameters for its calculations: relative permeability (\( \mu_r \)), electrical conductivity (\( \sigma \)) and relative permittivity (\( \epsilon_r \)). The relative permeability approaches unity for most real materials in the frequency range that concern us (visible spectrum of the EM field) (Mcintyre, Laboratories, & Hill, 1971). The values for the electrical conductivity were obtained from the material library of COMSOL Multiphysics and from literature (Banyamin et al., 2014, Castellarnau et al., 2006). The relative permittivity can be calculated from the real part of the refractive index (n) and the imaginary part (k) of the refractive index using the following formula where j is the complex number (\(j^2=-1\)) (Griffiths, 1999):

$$\epsilon = \epsilon' – j \epsilon''$$ $$ \epsilon' = \frac{n^2 – k^2}{\mu} = n^2 – k^2$$ $$\epsilon'' = 2 \cdot n \cdot \frac{k}{\mu} = 2 \cdot n \cdot k$$

This adds up to:

$$\epsilon = n^2 – k^2 – j \cdot 2 \cdot n \cdot k $$

Because we assume that we have non absorbing materials the complex part of the refractive index is zero (\( k=0 \)), this is a safe assumption while working on the THz frequencies and it was verified from the spectroscopy measurements where we didn’t find large amounts of light lost. The relative permittivity can be calculated from the refractive index as:

$$\epsilon = n^2$$

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 furmula 12. 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. All the material parameters used in this model are summarized in table 2 .

Parameters/Materials	Medium (water)	Glass Layer	Tin dioxide (SnO2)	Cell
Relative permeability (\(\mu_r\))	1	1	1	1
Electrical conductivity (\(\sigma\))	0.05 [S/m]	\(10^{-14}\) [S/m]	0.025 (Banyamin, et al., 2014)	0.48 [S/m] (Castellarnau, et al., 2006)
Relative permittivity (\(\epsilon_r\))	1.77	2.09	6.58	1.96

The intended electric field is \(E_0=e^{-j\times k_0 \times x}\) and passes through the whole structure.

Simulations performed

A common practice while searching for modes of an optical resonator is to start from simple structures with well-defined modes. This practice helps also to validate the model before investigating more complex structures such as our cell covered in polysilicate or tin dioxide.

The initial, simplified model investigated that had good chances to present eigenmodes was a solid sphere from tin dioxide with radius \( 1.5 \mu m\). Tin dioxide was selected instead of polysilicate because it has higher refractive index (\(\epsilon_r\)). Initially, the radius has been set to a large size (\(1.5 \mu m\))because it is more likely to find modes on bigger resonators. Figure 3 shows one of the modes of this structure, the Q factor is around \(10^3\) for this size and material.

The next step was decreasing the size of the sphere to \(1 \mu m\) and \(0.5 \mu m\) in radius. In figure 4 we can see that we have nice modes for radius of \(1 \mu m\) as well and the Q factor is in the order of \(10^4\) and in figure 5 for radius \(0.5 \mu m\) as well with Q factor in the order of \(10^3\).

Using a series of models helps us to veify that the models work properly and determine if there are modes. Now the last step is to add the thin tin dioxide and polysilicate layer and the cell inside.

For the tin dioxide covered cell there were two main modes found the first with eigenfrequency with Real part \(5.67 \cdot 10^{14}\) and imaginary part \( 4.2 \cdot 10^{11}\) \( \omega_1 = 5.67 \cdot 10^{14} + i \cdot 4.2 \cdot 10^{11}\) and the second with eigenfrequency with Real part \(6.03 \cdot 10^{14}\) and Imaginary part \(2.3 \cdot 10^{11}\) (\( \omega_2 = 6.03 \cdot 10^{14} + i \cdot 2.3 \cdot 10^{11}\). Figures 6 and 7 demonstrate those modes for the tin dioxide covered cell. Interesting here is to note that the modes can be seen in the tin dioxide layer and not in the cell.

The Q factor from this resonator can be calculated using \(\omega_1\) and \(\omega_2\) from the aforementioned formula (equation…):

$$ Q \cong \frac{1}{2} \frac{|Real\_part\_of\_\omega|}{|Imaginary\_part\_of\_\omega|}$$

And the Q factors for the two modes are:

$$ Q_1 \cong \frac{1}{2} \frac{5.67 \cdot 10^{14}}{4.2 \cdot 10^{11}} =675$$ $$ Q_2 \cong \frac{1}{2} \frac{6.03 \cdot 10^{14}}{2.3 \cdot 10^{11}} =1.3109\times 10^{03} $$

The last investigation is for the modes of the silica covered cell. The only difference from the last model is the material of the second layer was changed from tin dioxide to silica. For this structure we did not find any modes of interest so we concluded that there are no useful modes of the silica covered cell.

Finally we investigated if there are any modes only in the cell but as expected there were no modes of interest there either.

Conclusion

We created a model that investigates the eigenfrequencies of our structures. We first verified that the model is working properly by using the common method for this kind of modeling, starting from simple structures and moving forward to the more complicated as in our biolasers. The silicate model didn’t show any useful modes but the tin dioxide covered cell did, for \( \omega_1 = 5.67 \cdot 10^{14} + i \cdot 4.2 \cdot 10^{11} \) and (\( \omega_2 = 6.03 \cdot 10^{14} + i \cdot 2.3 \cdot 10^{11}\) with \(Q_1=675\) and \(Q_2=1.3109\times 10^3 \) respectively. Finally, important is to note that the modes on the tin dioxide covered cell were in the tin dioxide layer and not in the cell, when we investigated for modes only in the cell no modes were found.

The COMSOL Models used in this study can be found bellow:

Download Model File 1

Download Model File 2

Download Model File 3

Download Model File 4

Download Model File 5