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

Elementary arguments for presumed standing wave -dependence of: A kz + B kz such that the solution has nodes or antinodes at both ends lead one to conclude that only: k = p&pi#pi;d for are supported by the cavity. For TM modes must vanish on the caps because the nonzero field must be the only E field component sustained, hence: E_z = &psi#psi;(x,y)(p &pi#pi;zd)

For TE modes must vanish as the only permitted field component is a non-zero , hence: H_z = &psi#psi;(x,y)(p &pi#pi;zd)

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.:
_&perp#perp;& = & -p&pi#pi;d(&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2)(
p &pi#pi;zd) _&perp#perp;&psi#psi;

_&perp#perp;& = & -i&epsi#epsilon;&omega#omega;&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2
(p &pi#pi;zd) (
×_&perp#perp;&psi#psi;)
for TM fields and
_&perp#perp;& = & -i&mu#mu;&omega#omega;&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2(
p &pi#pi;zd) (
×_&perp#perp;&psi#psi;)

_&perp#perp;& = & p&pi#pi;d(&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2)
(p &pi#pi;zd) _&perp#perp;&psi#psi;
for TE fields, with
determined as before for cavities.

However, now
is *doubly* determined as a function of *both*
and
*and* as a function of
and
. The only
frequencies that lead to acceptable solutions are ones where the two
*match*, where the resonant
in the
direction corresponds to
a permitted
associated with a waveguide mode.

I leave you to read about the definition of : Q = &omega#omega;_0&Delta#Delta;&omega#omega; 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 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 -
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
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.