Orthogonality properties of transverse eigenmodes of phase conjugate optical resonators

The orthogonality properties of the transverse eigenmodes of optical resonators which have phase conjugate mirrors at both ends are derived. As in conventional resonators and also resonators with only one phase conjugate mirror, it is shown that the transverse eigenmodes are essentially biorthogonal, a relation which is satisfied between the set of modes propagating in one direction around the resonator and the adjoint set of modes propagating in the reverse direction.