iGEM TU Delft

Question 4:

What is the quality factor of our resonator?

COMSOL modeling

Quality Factor

The quality factor (Q) 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 resonator with an RLC circuit or in our case an optical resonator with light bouncing between two reflective surfaces. The Q factor is defined as the energy stored divided by the energy lost in each circle times $2 \pi $ (Cory_and_Chaniotakis, 2006), (Kao & Santosa, 2008), (“Chapter 3 Passive Variable-Pitch Design Concepts," n.d.), (“Quality Factor / Q Factor Tutorial," n.d.) so a higher Q factor means less energy is lost, and that’s why it is a measure of the resonator’s quality. The formula is:

$$ Q = 2 \pi \frac{Energy\_stored}{Energy lost per cycle} $$

And as a function of the frequency and bandwidth of the resonator:

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

Where $ f_o $ is the central frequency of the resonance and $ \Delta f_c $ is the bandwidth of the resonance (“Chapter 3 Passive Variable-Pitch Design Concepts," n.d.),(“Quality Factor / Q Factor Tutorial," n.d.). Figure 1 below shows the bandwidth and center frequency of a resonator.

model — Figure 1: Frequency versus response of a resonator, $\Delta f_c$ is the bandwidth and $f_0$ the center frequency.

For optical resonators the bandwidth $ \Delta f_c $ is linked to the lifetime of the photons in a cavity with the formula $ \Delta f_c = 1/(2 \pi \tau_c) $ and therefore the Q factor can be calculated as $Q = 2 \pi f_o \tau_c $. Also for an optical resonator the energy is proportional to the amount of photons resonating and the energy loss to the amount of photons lost (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 oscillators is(Paschotta, 2008):

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

Where $f_o$ is the optical frequency, $ \tau_{rt} $ the time for a roundtrip and l is the fractional power loss per roundtrip. It can be seen that there are three ways to manipulate the Q factor of an optical resonator. First is by changing the frequency of light, increasing the frequency (decreasing the wavelength) corresponds to increased Q factor. Additionally, increase of the resonator’s size means increases in the Q factor. Nevertheless, in order to achieve very high Q factor, the fractional power loss per roundtrip needs to be reduced.

Finally 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 reason this method is useful is that by calculating the modes of an optical resonator we can determine the quality factor for those frequencies. The formulas used are presented below:

$$ Q = \frac{Total-Energy-stored-at-beginning-of-the-circle}{|Energy-lost-during-a-circle|}$$ $$ Q \cong 2 \pi \frac{E(0)}{E(0) – E(T)} = 2 \pi \frac{1}{1-\exp{-2 \gamma \frac{2 \pi}{\omega_1}}} \cong \frac{\omega_1}{2 \gamma}$$

Where $ \omega = \pm \omega_1 + i \gamma $

So Q can be calculated by:

$$ Q \cong \frac{1}{2} \frac{|Real-part-of-\omega|}{|Imaginary-part-of-\omega|}$$

Modelling for the Q factor

In order to determine the Q factor for a bio laser 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-pf-\omega |} $ that was described earlier.

COMSOL model description

COMSOL Multiphysics was used to find the eigenmodes of our structure. 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, 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.

Table 1: Values of parameters used in the model.

Parameter	Value	Description
r0	5*10-7 [m]	Radius of the cell
lda	$5 \times 10^7$ [m]	Wavelength
k0	$1.2566 \times 10^7$ [1/m]	Wavenumber in vacuum
f0	$5.9958\times 10^14 $ [1/s]	Frequency
t_medium	$2.5\times10^{-7}$ [m]	Thickness of air layer
t_pml	$6 \times 2.5 \times 10^{-7}$ [m]	Thickness of Perfectly Matched Layer
t_sil	$ 8\times 10^{-8}$ [m]	Thickness of silicate layer
E0	1 [V/m]	Intended electromagnetic field

According to the aforementioned parameters a two dimensional model of a layered circle, representing our structure, was created in COMSOL Multiphysics. The 2D 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 (thickness of t_sil), then the other two layers are the medium (thickness t_medium) with the outermost representing the world 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 is not such a big issue here because we are modeling in two dimensions meaning that while increasing the size of the domain, the number of nodes will increase but way slower that the 3D models.

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 in the optical 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 and the relative permittivity can be calculated from the refractive index (n) using the following formula (Griffiths, 1999):

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

So it is:

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

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:

$$\epsilon = n^2$$

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 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.

The intended electric field is E0*exp(-j*k0*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 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$). And the radius was set to that big of a size 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. As seen in figure 4 we can see that we have nice modes for radius $1 \mu m$ 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 $10^3$.

With these series of models we say that there are modes and that our model works properly. 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 is with Real part $5.67 \times 10^{14}$ and Imaginary part $ 4.2 \times 10^{11}$ ($ \omega_1 = 5.67 \times 10^{14} + i \times 4.2 \times 10^{11}$) and the second with Real part $6.03 \times 10^{14}$ and Imaginary part $2.3 \times 10^{11}$ }\) ($ \omega_2 = 6.03 \times 10^{14} + i \times 2.3 \times 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.

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The Q factor from this resonator can be calculated using $\omega_1$ and $\omega_2$ from the aforementioned formula:

$$ Q \cong \frac{1}{2} \frac{|Real-part-of-\omega|}{|Imaginary-part-of-\omega|}$$

And the Q fctor for the two modes are:

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

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 didn’t 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. The silicate model didn’t show any useful modes but the tin dioxide covered cell did, for $ \omega_1 = 5.67 \times 10^{14} + i \times 4.2 \times 10^{11} $ and ($ \omega_2 = 6.03 \times 10^{14} + i \times 2.3 \times 10^{11}$ with $Q_1=675$ and $Q_2=1.3109e+03 $ 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.

Baco, S., Chik, A., & Md. Yassin, F. (2012). Study on Optical Properties of Tin Oxide Thin Film at Different Annealing Temperature. J. Sci. Technol., 4, 61–72. Retrieved from http://penerbit.uthm.edu.my/ojs/index.php/JST/article/view/468
Chapter 3 Passive Variable-Pitch Design Concepts. (n.d.), 47–79.
Cory_and_Chaniotakis. (2006). Frequency response: Resonance, Bandwidth, Q factor Resonance. 1–11. Retrieved from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/resonance_qfactr.pdf
Daimon, M., & Masumura, A. (2007). Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region. Applied Optics, 46(18), 3811–3820. http://doi.org/10.1364/AO.46.003811
Griffiths, D. J. (1999). Introduction To Electrodynamics.
Jericho, M. H., Kreuzer, H. J., Kanka, M., & Riesenberg, R. (2012). Quantitative phase and refractive index measurements with point-source digital in-line holographic microscopy. Appl. Opt., 51(10), 1503–1515. http://doi.org/10.1364/AO.51.001503
Kao, C. Y., & Santosa, F. (2008). Maximization of the quality factor of an optical resonator. Wave Motion, 45(4), 412–427. http://doi.org/10.1016/j.wavemoti.2007.07.012
Mcintyre, J. D. E., Laboratories, B. T., & Hill, M. (1971). J.D. E. MCINTYRE and D., 24, 417–434.
Paschotta, R. (2008). Q factor. Retrieved October 9, 2016, from https://www.rp-photonics.com/q_factor.html?s=ak
Quality Factor / Q Factor Tutorial. (n.d.). Retrieved October 9, 2016, from http://www.radio-electronics.com/info/formulae/q-quality-factor/basics-tutorial.php

[1] Baco, S., Chik, A., & Md. Yassin, F. (2012). Study on Optical Properties of Tin Oxide Thin Film at Different Annealing Temperature. J. Sci. Technol., 4, 61–72. Retrieved from http://penerbit.uthm.edu.my/ojs/index.php/JST/article/view/468

[2] Chapter 3 Passive Variable-Pitch Design Concepts. (n.d.), 47–79.

[3] Cory_and_Chaniotakis. (2006). Frequency response: Resonance, Bandwidth, Q factor Resonance. 1–11. Retrieved from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/resonance_qfactr.pdf

[4] Daimon, M., & Masumura, A. (2007). Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region. Applied Optics, 46(18), 3811–3820. http://doi.org/10.1364/AO.46.003811

[5] Griffiths, D. J. (1999). Introduction To Electrodynamics.

[6] Jericho, M. H., Kreuzer, H. J., Kanka, M., & Riesenberg, R. (2012). Quantitative phase and refractive index measurements with point-source digital in-line holographic microscopy. Appl. Opt., 51(10), 1503–1515. http://doi.org/10.1364/AO.51.001503

[7] Kao, C. Y., & Santosa, F. (2008). Maximization of the quality factor of an optical resonator. Wave Motion, 45(4), 412–427. http://doi.org/10.1016/j.wavemoti.2007.07.012

[8] Mcintyre, J. D. E., Laboratories, B. T., & Hill, M. (1971). J.D. E. MCINTYRE and D., 24, 417–434.

[9] Paschotta, R. (2008). Q factor. Retrieved October 9, 2016, from https://www.rp-photonics.com/q_factor.html?s=ak

[10] Quality Factor / Q Factor Tutorial. (n.d.). Retrieved October 9, 2016, from http://www.radio-electronics.com/info/formulae/q-quality-factor/basics-tutorial.php

Team:TU Delft/Model/Q4

Modeling