# Microscope-Based Optical Frequency Comb Generator for 200 GHz Spaced WDM Data Transmission System
## 2 Methods  
### Calculation of Microresonator Characteristics  
FSR is defined as \(c/(2\pi Rn_{\textit{eff}})\), where \(c\) is the speed of light, \(R\) is the radius of a microresonator, and \(n_{\textit{eff}}\) is the effective refractive index for an operating family of WGMs, so, it is essential to choose the appropriate size and take into account that not only the material but also the waveguide component gives a contribution to \(n_{\mathit{eff}}\). To find an optimal size of a silica microsphere for obtaining FSR = 200 GHz, we solve the characteristic equation for different radii [34]:  
\[n^{m}\frac{\left[\left(kR\right)^{1/2}\!I_{l+1/2}(kR)\right]^{\prime}}{\left(kR \right)^{1/2}\!I_{l+1/2}(kR)}=\frac{\left[\left(k_{0}R\right)^{1/2}\!H_{l+1/2} ^{(1)}\!\left(k_{0}R\right)\right]^{\prime}}{\left(k_{0}R\right)^{1/2}\!H_{l+ 1/2}^{(1)}\!\left(k_{0}R\right)}\,, \tag{1}\]  
where \(m=-1\) for transverse magnetic (TM) and \(m=1\) for transverse electric (TE) modes; the prime denotes the derivative with respect to the argument in brackets; \(\!J_{l+1/2}\) is the Bessel function of the order of \((l+1/2)\) with the azimuthal index \(l\); \(\!H_{l+1/2}\!\left(1\right)\) is the Hankel function of the 1 st kind of the order of \((l+1/2)\); \(\!k_{0}=2\pi\
u/c\) is the propagation constant in vacuum; \(\
u\) is a frequency; \(k=n\!\cdot\!k_{0}\) and \(n\) is the refractive index of the silica glass given by the Sellmeier formula [35]:  
\[n^{2}=1+\sum_{m=1}^{3}\frac{C_{m}\omega_{m}^{2}}{\omega_{m}^{2}-\omega^{2}}, \tag{2}\]  
with constants \(C_{1}=0.6961663\), \(C_{2}=0.4079426\), \(C_{3}=0.8974794\); \(\lambda_{1}=0.0684043\)\(\upmu\)m, \(\lambda_{2}=0.1162414\)\(\upmu\)m, \(\lambda_{3}=9.896161\)\(\upmu\)m [35], here \(\lambda_{m}=2\pi c/\omega_{m}\). To solve numerically Equation (1) and find the \"cold\" eigenfrequencies \(\
u_{l}\), we use home-made software. We consider only a fundamental mode family corresponding to the first roots. The roots are localized by using approximation formulas for the eigenfrequencies \(\
u_{l}\)\({}^{approx}\) given, for example, in [34]:  
\[\
u_{l}^{approx}=\frac{c}{2\pi Rn}\Bigg{[}(l+1/2)+1.85576(l+1/2)^{1/3}-n^{m} \bigg{(}\frac{1}{n^{2}-1}\bigg{)}^{1/2}\Bigg{]}. \tag{3}\]  
These values \(\
u_{l}\)\({}^{approx}\) are used as the initiators of the algorithm for searching the roots of Equation (1) by the modified Powell method [36]. The iterative algorithm is implemented with allowance for silica glass dispersion given by Equation (2). After finding eigenfrequencies, we calculate the second-order dispersion [30]:  
\[\beta_{2}=-\frac{1}{4\pi^{2}R^{2}}\frac{\Delta(\Delta\
u_{l})}{(\Delta\
u_{l} )^{3}}, \tag{4}\]  
where  
\[\Delta\
u_{l}=\frac{\
u_{l+1}-\
u_{l-1}}{2};\quad\Delta(\Delta\
u_{l})=\
u_{l +1}-2\
u_{l}+\
u_{l-1}. \tag{5}\]  
FSR is calculated as the difference between two neighboring eigenfrequencies near 193.1 THz. The nonlinear Kerr coefficients \(\gamma\) are also estimated [33]:  
\[\gamma=\frac{2\pi n_{2}}{\lambda}\frac{2\pi R}{V_{eff}}\approx\frac{2\pi n_{2 }}{\lambda}\frac{2\pi R}{3.4\pi^{3/2}\Big{(}\frac{\lambda}{2\pi n}\Big{)}^{3} l11/6}. \tag{6}\]  
Expressions for fields corresponding to eigenmodes are very cumbersome, therefore they are not written here but can be found in [34]. We also estimate the contribution of thermo-optical effects on eigenfrequencies and FSR using the approach developed and applied previously for tellurite glass microspheres [37]. This approach is based on the numerical simulation of steady-state temperature distribution within the heat equation framework by the finite-element method with the COMSOL software [37]. In the model, the heat source is set near the microsphere surface in the equatorial area and originates due to partially dissipated pump energy. The volume of the heat source is equal to the effective WGM volume at the pump frequency. We use the boundary conditions \"external natural convection\" implemented in COMSOL.