next up previous contents
Next: Wave Guides Assignment Up: Wave Guides Previous: Rectangular Waveguides   Contents

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 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 $ p = 0,1,2...$ are supported by the cavity. For TM modes $ \Vec{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#psi;(x,y)(p &pi#pi;zd)

For TE modes $ H_z$ must vanish as the only permitted field component is a non-zero $ \Vec{H}_\perp$ , 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.: \Vec{E} _&perp#perp;& = & -p&pi#pi;d(&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2)( p &pi#pi;zd) _&perp#perp;&psi#psi;
\Vec{H} _&perp#perp;& = & -i&epsi#epsilon;&omega#omega;&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2 (p &pi#pi;zd) (\Hat{z} ×_&perp#perp;&psi#psi;) for TM fields and \Vec{E} _&perp#perp;& = & -i&mu#mu;&omega#omega;&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2( p &pi#pi;zd) (\Hat{z} ×_&perp#perp;&psi#psi;)
\Vec{H} _&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 $ \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 = &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 $ \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.

next up previous contents
Next: Wave Guides Assignment Up: Wave Guides Previous: Rectangular Waveguides   Contents
Robert G. Brown 2017-07-11