Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics

We present a theory for the analytical design of double-chirped mirrors with special dispersion characteristics. A simple analytical equation takes an arbitrarily desired group delay dispersion (that also includes possible higher order dispersion) as an input function and gives the chirp law as an output. The chirp law determines the local Bragg wavelengths in the mirror. It allows the calculation of the thicknesses of the high- and low-index layers if the double chirp of the layers in the front part of the mirror is taken into account. We use this method to design a highly dispersive double-chirped semiconductor Bragg mirror and a double-chirped TiO₂-SiO₂ mirror for higher order dispersion compensation in optical parametric oscillators operating in the visible spectral range. The design formulas are applicable to general chirped Bragg gratings and provide insight into the reasons why certain dispersion characteristics might be impossible to achieve