Theory of double-chirped mirrors

A theory of double-chirped mirrors (DCMs) for dispersion compensation in ultrashort pulse laser sources is presented. We describe the multilayer interference coating by exact coupled-mode equations. They show that the analysis and synthesis of a coating with a slowly varying chirp in the layer thicknesses can be mapped onto a weakly inhomogeneous transmission line problem. Solutions of the transmission line equations are given using the WKB-method. Analytic expressions for reflectivity and group delay are derived. The solutions show that the main problem in chirped mirror design is the avoidance of spurious reflections, that lead to Gires-Tournois-like interference effects responsible for the oscillations in the group delay. These oscillations are due to an impedance matching problem of the equivalent transmission line. The impedance matching can be achieved by simultaneously chirping the strength of the coupling coefficient and the Bragg wavenumber of the mirror. An adiabatic increase in the coupling coefficient removes the typical oscillations in the group delay and results in broad-band mirrors with a controlled dispersion. Finally, the mirror is matched to air with a broadband antireflection coating. We discuss a complete design of a laser mirror with a reflectivity larger than 99.8% and a controlled dispersion over 300-nm bandwidth. Using such mirrors in a Ti:sapphire laser, we have demonstrated ≈30-fs pulses, tunable over 300 nm, as well as 8-fs pulses from the same setup. A different design resulted in 6.5-fs pulses