Third order optical limiting utilizes nonlinear absorption or scatter to preferentially reduce the optical throughput of intense light through a medium. While these processes are dependent on intensity, it is often the transmitted energy or power that needs to be limited to avoid eye, sensor or detector damage. This is problematic when the input is a vortex mode or other higher order Gaussian distributions such as Bessel-Gaussian modes where the relationship between optical power and the on axis intensity is more complicated than for the TEM 00 Gaussian mode. Here we show that for higher order Bessel-Gaussian modes that beam power transmission is not equivalent to peak intensity transmission through intensity dependent absorption or scatter based optical limiters. We present a theoretical model and experimental data that demonstrates that the power throughput in these media, increases by 4 and 7 times for n = 1 and 2 Bessel-Gaussian modes, respectively, over that observed when a TEM 0...