Nonlinear phase shifts that result from cascading of second-order nonlinear processes have been investigated theoretically. In the nonlinear processes such as second harmonic generation(SHG), the intensities and the phases of the incident waves undergo significant changes. The origin and the properties of the phase shifts in the second-order processes are different from those in the third-order processes, so called intensity dependent refractive index or Kerr effect. Phase control using second-order nonlinearity has many advantages over the third-order, such as large nonlinear effect, low pump intensity, and fast responses.
In general a phase mismatch between the fundamental and the second harmonic waves is necessary to obtain the phase shifts accumulated through second-order cascading processes, and to some extent the larger the phase mismatch the larger the phase shifts. It was found that even in the case of exact phase matching, there can be phase shifts if the SH field is initially nonzero in type I SHG. Both the amplitude of the initial SH wave and the phase difference between the fundamental and the SH waves are important factors to govern the resulting phase shifts. The energy exchange period and the magnitude of the phase shifts depend upon the intensities of the initial second harmonic wave and the phase difference.
The group velocity and the dispersion effect of the ultra short pulses were taken into consideration by solving the parabolic equations. The group velocity effect limits the interaction length in the interaction of several picosecond pulses, and for the highly efficient conversion there is a need of group velocity matching in addition to the phase matching. However the phase matching and the group velocity matching are in general not satisfied simultaneously. A technique such as pre-delay of a pulse can be used for the efficient conversion in type II SHG. There also appears asymmetry in the intensity and phase profile of the pulses because of differences in their group velocity. The dispersion effect of group velocity is very high in subpico second pulses. Besides the changes from the second-order cascading process, there are additional effects such as pulse broadening and acquired chirp.
Optical rectification(OR) is also the second-order process as the SHG, but the phase shifts through OR was considered rarely. Nonlinear phase shifts through OR have been included in this study. Nonlinear phase shift of the fundamental beam caused by the cascading process was found to be proportional to the input intensity and the propagation length as in the Kerr effect. It is notable that the depletion of the fundamental optical pulse energy is negligibly small even for the case of large nonlinear phase shifts. The negligible depletion in the fundamental is because of the facts that the nonlinear coupling is small and the phase matching is not satisfied in general.
