# Widely-Tunable Ring-Resonator Semiconductor Lasers
## 2 Basic Principle of Wide-Tunability Utilizing Ring Resonators
Due to the slight difference in circumference of the two rings, their resonances are mostly not aligned,and allow for single-mode operation in a much wider bandwidth that the FSR of an individual ring, as shown in Figure 1b. It is common to define tuning enhancement parameter \(M\):  
\[M=L_{1}/(L_{2}-L_{1}), \tag{2}\]  
where \(L_{1}\) and \(L_{2}\) are the circumferences of Ring 1 and Ring 2, as shown in Figure 1b. The FSR of the combined two ring structure utilizing the Vernier effect then equals FSR\({}_{1}\cdot M\), where FSR\({}_{1}\) is the FSR of Ring 1. We call the two ring structure that utilizes the Vernier effect to select single longitudinal mode a single wavelength filter (SWF). A typical widely-tunable ring-resonator semiconductor laser architecture using the SWF placed inside the Fabry-Perot cavity, with a gain element and phase tuner to align the longitudinal mode to the resonance of the SWF, is shown in Figure 2. Fabry-Perot mirrors could be realized as cleaved facets, but that would limit the integration potential of such lasers, so we utilize Sagnac loop mirrors, in which two same-side outputs of a coupler (directional, MMI or any other kind) are connected together to form a loop. Such a structure acts as a mirror whose reflectivity can be controlled by the coupling strength; from theoretically 100% for a 50:50 coupler, all the way to very low values, which are important for higher output powers. The analysis of the complex laser cavity is simplified by using the effective mirror theory \(R_{\mathit{eff}}\)[21]. In the case of the laser cavity shown in Figure 2, \(R_{\mathit{eff}}\) is a complex wavelength dependent mirror reflectivity, that includes the passive phase section, both ring resonators, and one of the loop mirrors (\(R_{2}\) in the case of Figure 2). Figure 1: (**a**) Add-drop response of single ring resonator (\(L_{1}\) = 200 \(\upmu\)m, FSR = 3.31 nm); (**b**) Vernier of two rings showing 60 nm of free-spectral range (FSR) (\(L_{1}\) = 200 \(\upmu\)m, \(L_{2}\) = 211 \(\upmu\)m, \(M\) = 18.1); Propagation loss is assumed to be 1 dB/cm and all power coupling coefficients are equal to 0.15. Figure 2: Typical laser cavity comprising of two ring resonators with slightly different radii forming a single frequency filter (SWF) in between two loop mirrors (\(R_{1}\) and \(R_{2}\)) forming a Fabry–[PERSON] cavity. We define effective reflectivity plane \(R_{\mathit{eff}}\) to simplify the analysis. The complex amplitude reflectivity formula is given by the following expressions:  
\[r_{eff}=\big{(}S_{21,passive}\big{)}^{2}\cdot\big{(}S_{21,ring1}\big{)}^{2}\cdot \big{(}S_{21,ring2}\big{)}^{2}\cdot r_{2}, \tag{3}\]  
\[S_{21,passive}=e^{-\alpha_{p}L_{p}}e^{-j\beta_{p}L_{p}}, \tag{4}\]  
\[S_{21,ringX}=\frac{|\kappa|^{2}X_{X}}{1-\tau^{2}X_{X}^{2}}, \tag{5}\]  
\[X_{x}=e^{-\alpha_{p}\frac{|X}{2}}e^{-j\beta_{p}\frac{|X}{2}}, \tag{6}\]  
where \(\alpha_{p}\) and \(\beta_{p}\) are the waveguide electric field propagation loss, and the effective propagation constant in waveguides, respectively, \(r_{2}\) is the amplitude reflectivity of the back-side mirror, \(L_{x}\) is the circumference of the ring, and the subscript \(X\) takes values 1 and 2 for the two rings in the SWF section. A number of useful laser parameters, including effective cavity length, mode spacing, and the adiabatic chirp reduction factor, can be determined using \(r_{eff}\)[22]. The chirp reduction is given by factor \(F=1+A+B\), and the Lorentzian linewidth (\(\Delta v_{0}\)) is reduced by \(F^{2}\), where \(A\) and \(B\) are defined as  
\[\Delta v=\frac{\Delta v_{0}}{F^{2}}, \tag{7}\]  
\[M=L_{1}/(L_{2}-L_{1}), \tag{2}\]0  
\[M=L_{1}/(L_{2}-L_{1}), \tag{2}\]1  
where \(\alpha_{H}\) is the linewidth enhancement factor, \(\tau_{in}=2n_{eff}L_{a}/c\) is the roundtrip time in the active part of the cavity, where \(n_{eff}\) is the effective index of the gain section, and \(L_{a}\) is the length of the gain section.