Resonant Cavities

We will consider a resonant cavity to be a waveguide of length $d$ with caps at both ends. As before, we must satisfy TE or TM boundary conditions on the cap surfaces, either Dirichlet in $E_z$ or Neumann in $B_z$. In between, we expect to find harmonic standing waves instead of travelling waves.

Elementary arguments for presumed standing wave $z$-dependence of:

$$A \sin kz + B \cos kz$$ (10.82)

such that the solution has nodes or antinodes at both ends lead one to conclude that only:

$$k = p\frac{\pi}{d}$$ (10.83)

for $p = 0,1,2...$ are supported by the cavity. For TM modes $\mbox{\boldmath$E$}_\perp$ must vanish on the caps because the nonzero $E_z$ field must be the only E field component sustained, hence:

$$E_z = \psi(x,y)\cos\left(\frac{p \pi z}{d}\right)$$ (10.84)

For TE modes $H_z$ must vanish as the only permitted field component is a non-zero $\mbox{\boldmath$H$}_\perp$, hence:

$$H_z = \psi(x,y)\sin\left(\frac{p \pi z}{d}\right)$$ (10.85)

Given these forms and the relations already derived for e.g. a rectangular cavity, one can easily find the formulae for the permitted transverse fields, e.g.:

$$\mbox{\boldmath$E$}_\perp = -\frac{p\pi}{d(\mu\epsilon\omega^2 - k^2)}\sin\left( \frac{p \pi z}{d}\right) \mbox{\boldmath$\nabla$}_\perp \psi$$ (10.86)

$$\mbox{\boldmath$H$}_\perp = -\frac{i\epsilon\omega}{\mu\epsilon\omega^2 - k^2} \cos\left(\fra... ...d}\right) (\hat{\mbox{\boldmath$z$}}\times \mbox{\boldmath$\nabla$}_\perp \psi)$$ (10.87)

for TM fields and

$$\mbox{\boldmath$E$}_\perp = -\frac{i\mu\omega}{\mu\epsilon\omega^2 - k^2}\sin\left( \frac{p \... ...}\right) (\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$\nabla$}_\perp \psi)$$ (10.88)

$$\mbox{\boldmath$H$}_\perp = \frac{p\pi}{d(\mu\epsilon\omega^2 - k^2)} \cos\left(\frac{p \pi z}{d}\right) \mbox{\boldmath$\nabla$}_\perp \psi$$ (10.89)

for TE fields, with $\psi(x,y)$ determined as before for cavities.

However, now $k$ is doubly determined as a function of both $p$ and $d$ and as a function of $m$ and $n$. The only frequencies that lead to acceptable solutions are ones where the two match, where the resonant $k$ in the $z$ direction corresponds to a permitted $k(\omega)$ associated with a waveguide mode.

I leave you to read about the definition of $Q$:

$$Q = \frac{\omega_0}{\Delta\omega}$$ (10.90)

or the fractional energy loss per cycle of the cavity oscillator in the limit where this quantity is small compared to the total energy. Note that $\Delta \omega$ is the full width at half maximum of the presumed resonant form (basically the same as was presumed in our discussions of dispersion, but for energy instead of field).

I strongly advise that you go over this on your own - $Q$ describes the damping of energy stored in a cavity mode due to e.g. the finite conductivity of the walls or the partial transparency of the end caps to energy (as might exist in the case of a laser cavity). If you go into laser physics, you will very much need this. If not, you'll need to understand the general idea of $Q$ to teach introductory physics and e.g. LRC circuits or damped driven harmonic oscillators, where it also occurs and should know it at least qualitatively for e.g. qualifiers. I added an optional problem for resonant cavities to the homework assignment in case you wanted something specific to work on while studying this.