DK> “I don’t see how they are violating the second law of thermodynamics”
Take a large body C, and a small body H. Collect the thermal radiation from C in some manner and deposit that energy on H. The power density emitted from C grows with temperature. The temperature of H grows with the power density deposited. If, without adding external energy, we concentrate the power density from the large body C to a higher power density on the small body H, H gets hotter than C. We may then use a heat engine between H an C to make free energy. This is not possible, therefore we cannot do the concentration.
The Etendue argument is just a special case where the concentration is attempted with mirrors or lenses. Changing the method to involve photovoltaic/microwave/rectenna power concentration doesn’t fix the issue, because the argument from the second law is broader, and encompasses any method of concentrating the power density as shown above.
When we extrapolate exponential growth, we must take care to look for where the extrapolation fails. Nothing in real life grows exponentially without bounds. “Eternity in Six Hours” relies on power which is 9 orders of magnitude greater than the limit of fundamental physical law.
How much extra energy external energy is required to get an energy flux on Mercury of a billion times that leaving the sun? I have an idea, but my statmech is rusty. (the fourth root of a billion?)
And do we have to receive the energy and convert it to useful work with 99.999999999% efficiency to avoid melting the apparatus on Mercury?