Team:TU Delft/Model/Q4

iGEM TU Delft

Modeling

Question 4:

What is the quality factor of the cavity?

Cavity Quality Factor

Introduction to 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 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 circle (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 is 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.

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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):

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

Where \(f_o\) is the optical frequency, \( \tau_{rt} \) the 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. Firstly,by increasing the frequency (decreasing the wavelength) increases Q factor. Secondly, increase of the resonator’s size means increase of the Q factor. 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 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 the light path is not so straight forward we it is harder. By finding the modes we can extract their eigenfrequencies and use the following formulas to calculate the Q factor:

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

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 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.

Table 1: Values of parameters used in the model.

ParameterValueDescription
r05*10-7 [m]Radius of the cell
lda\(5 \cdot 10^7\) [m]Wavelength
k0\(1.2566 \cdot 10^7\) [1/m]Wavenumber in vacuum
f0 \(5.9958\cdot 10^14 \) [1/s]Frequency
t_medium \(2.5\cdot 10^{-7}\) [m]Thickness of air layer
t_pml\(6 \cdot 2.5 \cdot 10^{-7}\) [m]Thickness of Perfectly Matched Layer
t_sil \( 8\cdot 10^{-8}\) [m]Thickness of silicate layer
E01 [V/m]Intended electromagnetic field

We selected frequency of 500 nm because we are mainly using GFP, so we wanted something close to green light radiation. Additionally the size of the cell was set to \(1 /micro m\) in diameter because it is the same length scale of the bacteria we are using. Finally, the layer thicknss 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 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 practice can generate problems in finite element models due to the increased resources needed for the simulations but in this case it does not 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.

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Figure 2: 2D design of the structure.

Materials

Next the materials need to be determined. The RF module, whick is the COMSOL module best suited in this kind of wavelengths and structure size, 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 thefrequency 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 refractive index (n) using the following formula (Griffiths, 1999):

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

This adds up to:

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

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 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 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 .

Table 2:Material parameters used for the eigenmodes study.
Parameters/MaterialsMedium (water)Glass LayerTin dioxide (SnO2)Cell
Relative permeability (\(\mu_r\))1111
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 (\(\epsilon_r\))1.772.096.581.96

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 (\(\epsilon_r\)). 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.

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Figure 3: modes on \(1.5 \mu m\) circular structure.

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 \(1 \mu m\) 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 \(10^3\).

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Figure 4: modes on \(1 \mu m\) circular structure.
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Figure 5: Modes on \(0.5 \mu m\) circular structure.

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 is 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 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.

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Figure 6: Mode 1, \( \omega_1 = 5.67 \cdot 10^{14} + i \cdot 4.2 \cdot 10^{11} \).
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Figure 7: Mode 2, (\( \omega_2 = 6.03 \cdot 10^{14} + i \cdot 2.3 \cdot 10^{11}\).

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 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. 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 \(Q_2=1.3109e+03 \) 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.

Our COMSOL Multiphysics models were to large 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 modes models here.

References

  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
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