The conditions of the radiance increase are analyzed in detail on the base of the equation of radiance transfer and requirements of thermodynamics. It is shown, that the most important of these conditions are the nonreversability of the beam propagation process and inappropriateness of the beam approach.
Andrew V. Shepelev,
"Radiance increase in nonimaging optics", Proc. SPIE 3139, Nonimaging Optics: Maximum Efficiency Light Transfer IV, (3 October 1997); doi: 10.1117/12.279223; https://doi.org/10.1117/12.279223