I'd like to discuss the photon density of states. What you're looking at here is you're looking at essentially the photon analogy to what we just did for electrons and holes. So, here, this is three-dimensions on looking at the mode density versus frequency, and here you can see a case of where the photons are only free to move in two dimensions and what the mode density looks like as a function of frequency. So, what I'd like to do now is actually derive the 3-dimensional photon density of states. So, it depends as we said on the dimensionality of the cavity. So, we know that rho of Omega is the photon density of states, and this is going to be nothing more than one over V, and then here, this is going to be dN, d omega. So, essentially, this rho of omega is the density of photon modes per volume, per frequency. So, let's consider the case of a cube. So, that we're going to have a cube where essentially L is much greater than lambda. What does that mean? That essentially means that we are in the case where we're looking at bulk or free space. So, we can go ahead and write down an expression for the allowed ki, these are just equal to two pi mi over L. So, these modes are closely spaced because L is large and so mi can be large. We can now think about, thinking about an infinite sinking about d cubed k. So, this is going to be equal to dkxdky, and dkz. So, this is essentially just equal to two pi cubed over V delta N. If we want, we can write this down in the case of spherical coordinates, and so this is going to be equal to k squared dk, and then here, d capital omega. So, omega here is the solid angle. We also know the dispersion relationship for photons. Omega equals c times k. So, using this, we can now write down an expression for delta N. So, delta N is going to be equal to V over two pi cubed, d cubed k, and so this is equal to V over two pi cubed k squared dk d omega. So, if we want, we can now substitute N for this dispersion relation. So, we get V over two pi cubed, omega squared over c squared dk d omega, and then d little omega, d capital omega. So, now we can go ahead and write down rho of omega. Rho of omega is equal to one over V, and then dN, d omega. So, this is equal to one over V times V over two pi cubed, and then omega squared over c squared dk d omega, and then d omega, d capital omega over d little omega. So, remember, this is equal to now one over two pi cubed and then omega squared over c cubed d capital omega. There's no dependence on beta or phi until we can integrate over d omega and essentially we end up with four pi. So, for three-dimensions, the bottom line here is that for L much greater than lambda, rho of omega equals omega squared over two pi squared and then C cubed.