Full width at half maximum

The full width at half maximum (FWHM) is a measure of the sharpness of the resonance. As the name suggests, it is given as the width of the resonance peak where the power drops to half of the resonance value.

A configuration that drops about half of the maximum power is realized for a propagation constant $\beta + \delta \beta$ with $1 /( 1 + \vert{\sf {S}}_{\mbox{\scriptsize bb}}\vert^4 \mbox{e}^{\displaystyle... ...e}^{\displaystyle -\alpha L} \cos{(\beta L_{\mbox{\scriptsize cav}} - 2 \phi)})$ $=2 / (1 + \vert{\sf {S}}_{\mbox{\scriptsize bb}}\vert^4 \mbox{e}^{\displaystyl... ..._{\mbox{\scriptsize cav}} + \delta \beta L_{\mbox{\scriptsize cav}} - 2 \phi)})$. Using the second order approximation of the cosine terms around a resonant cavity propagation constant, one obtains

$$\delta \beta = \pm \frac{1}{L_{\mbox{\scriptsize cav}}} \left( \f... ... {S}}_{\mbox{\scriptsize bb}}\vert \mbox{e}^{\displaystyle -\alpha L/2} \right)$$ (1.14)

for the shift in the propagation constants that distinguishes configurations with the maximum and half of the maximum dropped power.

Using an approximation $\delta \beta \approx - (\beta_{m}/\lambda) \delta \lambda$, analogous to Equations (1.9),(1.10), Equation (1.14) leads to an expression

$$2\delta \lambda = \left. \frac{\lambda^2}{\pi L_{\mbox{\scriptsiz... ... {S}}_{\mbox{\scriptsize bb}}\vert \mbox{e}^{\displaystyle -\alpha L/2} \right)$$ (1.15)

which gives the full width at half maximum $2 \delta \lambda$ of the resonance of order $m$.