  In this paper, an analytical solution to the problem of optimal dielectric
coating design of mirrors for gravitational wave detectors is found. The
technique used to solve this problem is based on Herpin's equivalent layers,
which provide a simple, constructive, and analytical solution. The performance
of the Herpin-type design exceeds that of the periodic design and is almost
equal to the performance of the numerical, non-constructive optimized design
obtained by brute force. Note that the existence of explicit analytic
constructive solutions of a constrained optimization problem is not guaranteed
in general, when such a solution is found, we speak of turbo optimal solutions.
