Tuning

The realization and actual performance of the resonators are constrained by several factors, e.g. an accurate definition of the resonance wavelengths requires a high degree control of the geometrical parameters, temperature dependant changes in the material parameters detune the spectral response. Active (e.g. electro-optical, thermo-optical) tuning of the resonators greatly relaxes these constraints. This is quite essential for stable performance of the devices. See Ref. [38] and the references quoted there for further details. Here we outline a framework for the modeling of tuning.

As explained in Section 1.4.3, at resonance the condition $\beta = (2 m \pi + \phi) / L_{\mbox{\scriptsize cav}} = \beta_{m}$ holds for the cavity mode propagation constant, where the integer $m$ gives the order of the resonance. Assume that the wavelength dependence of the propagation constant $\beta = \beta(\lambda)$ is given. Then one can write $\beta(\lambda_{m}) = \beta_{m}$, where $\lambda_{m}$ is the resonance wavelength associated with the resonant cavity mode propagation constant $\beta_{m}$.

Disregarding its influence on the response of the (short) couplers as a first approximation, a tuning mechanism is modeled by a parameter $p$, which affects mainly the field propagation along the cavity. Thus now, besides the wavelength, the cavity mode propagation constant also depends on the tuning parameter, i.e. $\beta = \beta(p, \lambda)$, with $p=0$ representing the original state: $\beta(0, \lambda_{m})=\beta_{m}$.

With the tuning applied, the resonance of order $m$ is shifted towards a new wavelength $\tilde{\lambda}_{m}$, such that $\beta(p,\tilde{\lambda}_{m}) = (2 m \pi + \phi) / L_{\mbox{\scriptsize cav}} \overset{!}{=} \beta_{m}$ is satisfied again. A linear approximation in the tuning parameter and in the wavelength differences

$$\beta(p,\tilde{\lambda}_{m}) \approx \beta(0, \lambda_{m}) + p \l... ... \beta}{\partial \lambda}\right\vert _{0, \lambda_{m}} \overset{!}{=} \beta_{m}$$ (1.19)

leads to an expression for the shift in the wavelength $\Delta_{p} \lambda_{m} = \tilde{\lambda}_{m} -\lambda_{m}$ that is affected by the tuning mechanism

$$\Delta_{p} \lambda_{m} = -\left. p \left( \frac{\partial \beta}{\... ...c{\partial \beta}{\partial \lambda} \right)^{-1} \right\vert _{0, \lambda_{m}},$$ (1.20)

which on simplification by using Equation (1.9) leads to

$$\Delta_{p} \lambda_{m} = p \frac{\partial \beta}{\partial p} \frac{\lambda_{m}}{\beta_{m}} \hspace{0.2cm} \Delta_{p} \lambda_{m} = p \frac{\partial \beta}{\partial p} \frac{\lambda_{m}^2}{2 \pi n_{\mbox{\scriptsize eff}, m}},$$ (1.21)

i.e. the wavelength shift compensates the change in the cavity mode propagation constant due to a nonzero perturbation strength $p$. Note that the wavelength shift does not depend on the length of the cavity. If available, the effective group index $n_{\mbox{\scriptsize eff}, g}$ according to Equation (1.12) can replace the effective index $n_{\mbox{\scriptsize eff}}$ in Equation (1.21).

We are specifically interested in tuning effected by the change of the cavity core permittivity (i.e. refractive index). For further details about this, see Section 2.5 and Section 4.6.