Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping

We present a theory of ultrashort-pulse second-harmonic generation (SHG) in materials with longitudinally nonuniform quasi-phase-matching (QPM) gratings. We derive an expression for the output second-harmonic field generated in an arbitrary QPM grating from an arbitrary fundamental field, valid for arbitrary material dispersion in the undepleted-pump approximation. In the case when group-velocity dispersion can be neglected, a simple transfer-function relationship describes the SHG process. This SHG transfer function de- pends only on material properties and on the QPM grating design. We use this SHG transfer function to show that nonuniform QPM gratings can be designed to generate nearly arbitrarily shaped second-harmonic output pulses. We analyze in detail a technologically important example of pulse shaping: the generation of com- pressed second-harmonic pulses from linearly chirped fundamental input pulses. The efficiency of these interactions as well as the limits imposed by higher-order material dispersion are discussed.