Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings

We consider diffraction of a three-dimensional (3D) spatiotemporal optical pulse by a phase-shifted Bragg grating (PSBG). The pulse diffraction is described in terms of signal propagation through a linear system with its transfer function determined by the reflection or transmission coefficient of the PSBG. It is shown that a PSBG can perform temporal differentiation in reflection and temporal integration in transmission of a 3D spatiotemporal optical pulse envelope. Second order differentiation of the incident pulse envelope is achieved using two differentiating PSBGs separated by a vacuum layer with the optical thickness of a quarter of the Bragg wavelength. The possibility of performing the said operations are confirmed by rigorous simulation results.