Is Photon BEC possible in planar cavity? It is well known result that Photon BEC was achieved in slightly curved cavity, because curved mirrors provide trapping potential (https://www.nature.com/articles/nature09567).
We can see that this is possible from the following dispersion relation for energy:
$$ E(r, k_{||}) = \hbar c \sqrt{k_z+k_{||}^2}  =\hbar c \sqrt{\frac{j^2\pi^2}{L^2(r)}+k_{||}^2} \approx \frac{\pi \hbar c j}{L_0}  + \frac{ \hbar c k_{||}^2}{2 \pi j} + \frac{\pi \hbar c j r^2}{2 L_0^2 R}, $$
where $j$ is integer for the resonance condition, $c$ is the speed of light, $R$ is mirror curvature, $L$ is cavity length at lateral distance $r$ from the optical axis, $L_0$ is the cavity length at optical axis.
However, would Photon BEC be possible in cavity with flat mirrors?
The dispersion relation now would be for flat mirrors ($R \to \infty $) in approximation when $k_{||} << k_z$:
$$ E(k_{||}) = \hbar c \sqrt{\frac{j^2\pi^2}{L_0^2}+k_{||}^2} \approx \frac{\pi \hbar c j}{L_0} + \frac{ \hbar c k_{||}^2}{2 \pi j}.  $$ We still have the effective photon mass, so condensation should happen, right? (I do not know, but I think it should, I need a reason). The thing is to properly quantize $k_{||}$. How should it be quantized? Or is Photon BEC possible at all with flat mirrors?
 A: In the thermodynamic limit, the BEC would not be possible with a planar cavity. With flat mirrors there is no light focussing, no spatially varying intensity, and hence no trapping & confinement. The cavity modes are then continuous - in the sense that, for a spherical cavity in the middle of which you can approximate the potential as harmonic, the states are the Hermite-Gauss polynomials. Each has a definite energy, with energy gaps.
The flat mirrors do not give you any confinement and you would hence be in a homogeneous system.
In a homogenous system in 2D, BEC does not occur. At least not in the thermodynamic limit, because the integral giving you $1/T_c$ diverges and hence $T_c=0$ is the only acceptable solution.  But in a finite system the integral becomes a sum and you can define a critical particle number, then giving you a (quasi)condensate. In your case the finite size of the system is determined by the size of the pumping laser beam.
A: Planar cavity (aka Fabry-Perot interferometer) is so frequently used in theoretical discussions (for the sake of simplicity) that its disadvantages are easily overlooked.
Its main disadvantage is that most of its modes are propagating (!!!)... in the direction parallel to the mirrors, i.e. most theoretical analyses are dealing only with the modes that have $k_\parallel = 0$. The cavity where most of the light escapes (since the mirrors have finite size) is a non-starter for practical applications, and creating modes where $k_\parallel$ is exactly $0$ is nearly impossible. This is the major reason why one uses either closed cavities (resonators) or curved mirrors, to assure that the modes are either non-propagating or that most of light comes back.
Another disadvantage of the Fabry-Perot resonator is the existence of multiple interference minima/maxima along the light path, i.e. there are multiple points where efficient coupling between BEC atoms and the light is possible. Again, it is more efficient to have coupling in one place, i.e. by focusing the light to one point.
