The paper is devoted to optimization of resonances in a 1-D open optical cavity. The
cavity’s structure is represented by its dielectric permittivity function
ε(s). It is assumed that
ε(s) takes values in the range
1 ≤ ε1 ≤ ε(s) ≤ ε2.
The problem is to design, for a given (real) frequency α, a cavity having
a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a
given frequency α, we define cavities and resonant modes with locally
extremal decay rate, and then study their properties. We show that such locally extremal
cavities are 1-D photonic crystals consisting of alternating layers of two materials with
extreme allowed dielectric permittivities ε1 and
ε2. To find thicknesses of these layers, a nonlinear
eigenvalue problem for locally extremal resonant modes is derived. It occurs that
coordinates of interface planes between the layers can be expressed via arg-function of
corresponding modes. As a result, the question of minimization of the decay rate is
reduced to a four-dimensional problem of finding the zeroes of a function of two
variables.