mavc said:
Dickfore, what is a mode? What do the eigenvalues represent? I know that it's the solution to A x = \lambda xbut what does it mean?
Modes are the frequencies of standing electromagnetic waves, where \lambda labels the two possible polarization for a given
k-vector. To avoid certain divergencies arrising from the infinite volume of space, it is simplest to consider periodic boundary conditions, where each component of the electromagnetic field is periodic:
<br />
f(x + L_{x}, y, z) = f(x, y + L_{y}, z) = f(x, y, z + L_{z}) = f(x, y, z)<br />
and can be expanded in a Fourier series:
<br />
f(\mathbf{x}) = \frac{1}{V} \, \sum_{\mathbf{k}}{f_{\mathbf{k}} \, \exp{(i \mathbf{k} \cdot \mathbf{x})}}, \ f_{\mathbf{k}} = \int{d\mathbf{x} \, f(\mathbf{x}) \, \exp{(-i \, \mathbf{k} \cdot \mathbf{x})}}<br />
where the possible
k-vectors are of the form:
<br />
\mathbf{k} = 2 \pi \, \left(\frac{n_{x}}{L_{x}} \, \mathbf{e}_{x} + \frac{n_{y}}{L_{y}} \, \mathbf{e}_{y} + \frac{n_{z}}{L_{z}} \, \mathbf{e}_{z}\right), \; n_{x}, n_{y}, n_{z} = 0, 1, \ldots<br />
Then, the wave equation:
<br />
\frac{1}{c^{2}} \, \frac{\partial^{2} f}{\partial t^{2}} - \nabla^{2} \, f = 0<br />
becomes:
<br />
\left(\frac{\omega^{2}}{c^{2}} - \mathbf{k}^{2} \right) \, f_{\mathbf{k}}(\omega) = 0<br />
The only non-zero Fourier components are for \omega = \pm \omega_{\mathbf{k}}, \; \omega_{\mathbf{k}} = c |\mathbf{k}|.
These are the classical modes of the electromagnetic field. They correspond to the degrees of freedom of the field and are equivalent to a system of independent linear harmonic oscillators and are quantized accordingly.
