Quality Factor of a Resonant Cavity

The quality factor $Q$ of a resonant cavity is defined

$$Q = 2\pi\,\frac{\mbox{energy stored in cavity}}{\mbox{ energy lost per cycle to walls}}.$$ (1320)

For a specific normal mode of the cavity, this quantity is independent of the mode amplitude. By conservation of energy, the power dissipated via ohmic losses is minus the rate of change of the stored energy, $U$ . We can thus write a differential equation for the variation of $U$ as a function of time:

$$\frac{dU}{dt} = -\frac{\omega_0}{Q}\,U,$$ (1321)

where $\omega_0$ is the oscillation frequency of the normal mode in question. The solution to the above equation is

$$U(t) = U(0) \,{\rm e}^{-\omega_0\, t/Q}.$$ (1322)

This time dependence of the stored energy suggests that the oscillations of the electromagnetic fields inside the cavity are damped as follows:

$$E(t) = E_0 \,{\rm e}^{-\omega_0\, t/2\,Q} \,{\rm e}^{-{\rm i}\,(\omega_0+{\mit\Delta}\omega)\,t},$$ (1323)

where we have allowed for a shift ${\mit\Delta}\omega$ of the resonant frequency, as well as for the damping. A damped oscillation such as that specified above does not consist of a pure frequency. Instead, it is made up of a superposition of frequencies centered on $\omega=\omega_0+{\mit\Delta}\omega$ . Standard Fourier analysis yields

$$E(t)= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} E(\omega) \,{\rm e}^{-{\rm i}\,\omega\, t}\,d\omega,$$ (1324)

where

$$E(\omega) = \frac{1}{\sqrt{2\pi}}\int_0^\infty E_0 \,{\rm e}^{-\o... ...0\, t/2\,Q} \,{\rm e}^{\,{\rm i}\,(\omega-\omega_0-{\mit\Delta}\omega)\,t}\,dt.$$ (1325)

It follows that

$$\vert E(\omega)\vert^{\,2} \propto \frac{1}{(\omega-\omega_0-{\mit\Delta}\omega)^{\,2} +(\omega_0/2\,Q)^{\,2}}.$$ (1326)

The above resonance curve has a full width at half-maximum equal to $\omega_0/Q$ . For a constant input voltage, the energy of oscillation within the cavity as a function of frequency follows this curve in the neighborhood of a particular resonant frequency. It can be seen that the ohmic losses, which determine $Q$ for a particular mode, also determine the maximum amplitude of the oscillation when the resonance condition is exactly satisfied, as well as the width of the resonance (i.e., how far off the resonant frequency the system can be driven, and still yield a significant oscillation amplitude).