Free spectral range

The free spectral range (FSR) is defined as the wavelength difference $\Delta \lambda$ between two successive maxima of the dropped power (or minima of the through power).

The resonant configuration next to a resonance found for $\beta_{m}$ is approximated as

$$\beta_{m-1} = \frac{2 (m-1) \pi + \phi}{L_{\mbox{\scriptsize cav}... ... \left. \frac{\partial \beta}{\partial \lambda} \right\vert _{m} \Delta \lambda$$ (1.8)

where the right hand side is obtained as a first order Taylor series expansion for the propagation constant around the $m$'th resonance wavelength; $\Delta \lambda$ is the difference between the vacuum wavelengths corresponding to the two resonant configurations.

By virtue of homogeneity arguments [36] for the propagation constants $\beta(\lambda, q_{j})$, viewed as a function of the wavelength $\lambda$ and all geometrical parameters $q_{j}$ that define the cavity waveguide cross section, one finds

$$\frac{\partial \beta}{\partial \lambda} = -\frac{1}{\lambda} \lef... ...} \frac{\partial \beta}{\partial q_{j}} \right) \approx - \frac{\beta}{\lambda}$$ (1.9)

for the wavelength dependence of the propagation constants in the cavity loop. The same (crude) approximation can be obtained if one writes the propagation constant in terms of vacuum wavenumber and effective mode index as $\beta = 2 \pi n_{\mbox{\scriptsize eff}} / \lambda$ and neglects the wavelength dependence of the effective index:

$$\frac{\partial \beta}{\partial \lambda} = - \frac{\beta}{\lambda}... ..._{\mbox{\scriptsize eff}}}{\partial \lambda~~} \approx - \frac{\beta}{\lambda}.$$ (1.10)

This leads to the expression

$$\Delta \lambda = - \frac{2 \pi}{L_{\mbox{\scriptsize cav}}} \left... ...mbda^2}{n_{\mbox{\scriptsize eff}} L_{\mbox{\scriptsize cav}}} \right\vert _{m}$$ (1.11)

for the free spectral range (FSR) $\Delta \lambda$ of the resonator around the resonance of order $m$ that is associated with the wavelength $\lambda$ and the effective mode index $n_{\mbox{\scriptsize eff}} = \lambda \beta_{m} / 2 \pi$ of the cavity waveguide.

A more accurate and still simple expression can be obtained if one does not introduce the approximations (1.9), (1.10), i.e. if the wavelength dependence of $\beta$ or $n_{\mbox{\scriptsize eff}}$ is explicitly incorporated. Customarily one can write

$$\frac{\partial \beta}{\partial \lambda} = - \frac{k}{\lambda} n_{... ...eff}} - \lambda \frac{\partial n_{\mbox{\scriptsize eff}}}{\partial \lambda~~},$$ (1.12)

where $n_{\mbox{\scriptsize eff, g}}$ is the group effective index of the cavity mode [35]. Then the free spectral range is given by

$$\Delta \lambda = \frac{\lambda^2}{n_{\mbox{\scriptsize eff, g}} L_{\mbox{\scriptsize cav}}}.$$ (1.13)