Achromatic phase matching (APM), a favorable method in increasing phase matching bandwidth, is explored for second harmonic generation of ultrashort pulses based on a typical grating telescope system. A set of coupled equations incorporating angular dispersion is constructed in the space-time domain. An analytic solution with a pump of tilting pulsed Gaussian beam to the equations is given under the undepleted pump approximation. With the aid of matrix formalism, some properties of the conversion are demonstrated. Though a maximal phase matching bandwidth is obtained, angular dispersion makes the conversion active only around the geometrical focus. Theory shows that APM does not require an overall pulse-front matching in the conversion process.

The typical phase correction term introduced in a diffraction grating-pair is rediscussed. It shows that the correction causes a conceptual difficulty in geometrical optics. A study reveals that Fraunhofer diffraction explains the correction and only mean-phase light rays are allowed for diffraction analysis. Besides, an equivalent phase formulation without correction is recommended.

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A theoretical analysis of the transformation of pulsed Gaussian beams through a grating is presented under paraxial scalar approximation. By introducing the concepts of local carrier frequency and stationary phase frequency, the angular dispersive process is effectively modelled by two formulas for pulses with only single-or few-cycle duration. The expressions predict some spatiotemporal coupling phenomena, which are found to be associated with high-order angular dispersion, which distorts the electric field with varied pulse width, chirp, pulse front and energy distribution. Related results are confirmed by a rigorous numerical simulation at last. The method proposed here can be generalized to other dispersive systems with the aid of the theory of paraxial optics.