Limits to Concentration in Resonant Cavities

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Abstract
Various limits to concentration are derivable from the proposition that the specific intensity of light cannot be increased by a passive optical system, the comparison being made at two points with the same refractive index. ‘Specific intensity’ is here taken to mean power per unit area, per unit solid angle, per unit angular frequency range, in any one state of polarization. There are circumstances in which the specific intensity is not well defined, for example in a cavity at an excited resonant frequency. There is a limit to concentration, applicable in such circumstances, which takes the form of an upper bound on the ratio of the energy density U v at a given point to the maximum value I max of the incident specific intensity. The incident light is assumed to be describable as an incoherent superposition of plane waves. Let the ambient refractive index n be 1, and let the given point be separated by at least the distance R from any point at which np 1. Let the incident light have wavenumber approximately k 0, and let the range i k of incident wavenumbers satisfy the conditions i kR 1 and i k/k 0 1. Then U v r U v.max, , where F(v) = 1 and for v > 2 ~, F(v) does not differ from 1 by more than 3 per cent. An analogous upper bound is obtained, in the wave mechanics of a spinless particle, for the ratio of the particle density to the maximum incident density in phase space.

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