Use of the round-trip operator technique for optical resonators with dispersion elements

The round-trip operator technique is widely used for dispersionless optical resonators beginning from pioneering studies of Fox and Li. Resonator modes are determined as the eigen-functions of the round-trip operator and may be calculated by means of numerical linear algebra. Corresponding complex eigen-values determine the wavelength shifts relative to reference value and damping factors. Dispersion elements, for example, Bragg mirrors in VCSEL result in dependence of propagation operator on the wavelength and damping factor. We can determine the round-trip operator in this case also, but unknown values of the wavelength and damping factor enter into the operator by a complicated manner. Trial-and-error method for determination of wavelength shifts and damping factors is possible but it is very laborious method. The proposed proximate numerical method for calculation of resonator modes is based on solution of linear eigen-problem for the round-trip operator with reference wavelength and zero damping. The wavelength shifts and damping factors can be calculated by simple formulae using eigenvalues obtained and previously computed effective length of the resonator. Calculations for anti-guided VCSELs were executed for model verification