The quality factor of a resonant cavity

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

$$Q = 2\pi\,\frac{\mbox{energy stored in cavity}}{\mbox{ energy lost per cycle to walls}}.$$ (1060)

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

$$\frac{dU}{dt} = -\frac{\omega_0}{Q}\,U,$$ (1061)

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

$$U(t) = U(0) \,{\rm e}^{-\omega_0 t/Q}.$$ (1062)

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

$$E(t) = E_0 \,{\rm e}^{-\omega_0 t/2Q} \,{\rm e}^{-{\rm i}\,(\omega_0+\Delta\omega)t},$$ (1063)

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

$$E(t)= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} E(\omega) \,{\rm e}^{-{\rm i}\,\omega t}\,d\omega,$$ (1064)

where

$$E(\omega) = \frac{1}{\sqrt{2\pi}}\int_0^\infty E_0 \,{\rm e}... .../2Q} \,{\rm e}^{{\rm i}\,(\omega-\omega_0-\Delta\omega)t}\,dt.$$ (1065)

It follows that

$$\vert E(\omega)\vert^2 \propto \frac{1}{(\omega-\omega_0-\Delta\omega)^2 +(\omega_0/2Q)^2}.$$ (1066)

The resonance shape has a full width $\Gamma$ at half-maximum equal to $\omega_0/Q$. For a constant input voltage, the energy of oscillation in the cavity as a function of frequency follows the resonance curve in the neighbourhood 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).