Wave Guides

Conductive tubes that guide light waves

Topics

• Rectangular Wave Guides

$$\{\begin{array}{l}{\vec{E}}^{\parallel }=\vec{0}\\ B^{\perp }=0\end{array}$$

• Boundary conditions for a hollow wave guide

\partial_{x}E_{y}-\partial_{y}E_{x}=i\omega B_{z} \\ \partial_{y}E_{z}-ikE_{y}=i\omega B_{x} \\ ikE_{x}-\partial_{x}E_{z}=i\omega B_{y} \\ \partial_{x}B_{y}-\partial_{y}B_{x}=-\frac{i\omega}{c^{2}}E_{z} \\ \partial_{y}B_{z}-ikB_{y}=-\frac{i\omega}{c^{2}}E_{x} \\ ikB_{x}-\partial_{x}B_{z}=-\frac{i\omega}{c^{2}}E_{y} \end{cases}

• Equations of electric fields obtained by Maxwell’s equations (Griffith’s pg. 426)

$$\{\begin{array}{l}{E}_x=\frac{i}{{\left(\frac{\omega }{c}\right)}^2-k^2}(k{\partial }_x{E}_z+\omega {\partial }_y{B}_z)\\ E_y=\frac{i}{{\left(\frac{\omega }{c}\right)}^2-k^2}(k{\partial }_y{E}_z-\omega {\partial }_x{B}_z)\\ B_x=\frac{i}{{\left(\frac{\omega }{c}\right)}^2-k^2}\left(k{\partial }_x{B}_z-\frac{\omega }{c^2}{\partial }_x{E}_z\right)\\ B_y=\frac{i}{{\left(\frac{\omega }{c}\right)}^2-k^2}\left(k{\partial }_y{B}_z-\frac{\omega }{c^2}{\partial }_x{E}_x\right)\end{array}$$

• Transverse components of EM waves in wave guides

$$\{\begin{array}{l}\left[{\partial }_x^2+{\partial }_y^2+{\left(\frac{\omega }{c}\right)}^2-k^2\right]E_z=0\\ \left[{\partial }_x^2+{\partial }_y^2+{\left(\frac{\omega }{c}\right)}^2-k^2\right]B_z=0\end{array}$$

• Longitudinal components of waves in wave guides

$!(E_z=B_z=0)$

• You cannot have transverse EM waves in hollow wave guides