Assume a 2D square array of masses with mass $m$ connected by springs with constant $\kappa$. Then the equation of motion for small perturbations in the plane of the array to the mass at $(l,n)$ is
$$ \ddot{\vec{\psi}}_{l,n} = \Omega^2(\vec{\psi}_{l+1,n}-2\vec{\psi}_{l,n} + \vec{\psi}_{l-1,n})+\Omega^2(\vec{\psi}_{l,n+1}-2\vec{\psi}_{l,n} + \vec{\psi}_{l,n-1}) $$ where $\Omega^2=\kappa/m$. The normal modes are $$ \vec{\psi}_{l,n,\vec{k}}=\vec{A}_{\vec{k}}e^{i\left[a(k_xl+k_yn)-\omega t\right]} $$ where $a$ is the separation between masses at equilibrium.
Perturbations have 2 polarizations here, for example $\hat{x}$ and $\hat{y}$.
How can I make this system exhibit birefringince? I.e., how can I make waves moving with polarization $\hat{x}$ move at a different velocity that with polarization $\hat{y}$?
First guess would be to change the spring constant of let's say the springs along the $y$ direction to $\kappa'$. Then the equation of motion would be
$$ \ddot{\vec{\psi}}_{l,n} = \Omega^2(\vec{\psi}_{l+1,n}-2\vec{\psi}_{l,n} + \vec{\psi}_{l-1,n})+{\Omega'}^2(\vec{\psi}_{l,n+1}-2\vec{\psi}_{l,n} + \vec{\psi}_{l,n-1}) $$
However, a plane wave of the form $\vec{\psi}=\vec{A}e^{iak_xl-\omega t}$, i.e. moving at the $\hat{x}$ direction, will only feel the springs with constant $\kappa$, no matter if the polarization is along the $\hat{x}$ or $\hat{y}$ direction. In other words, the dispersion relation does not depend on the polarization.
How can I make it sensitive to the polarization (and not the propagation direction)?
