Abstract

Beginning with the free-space scalar wave equation, a Green's function formulation of diffraction is presented in space and time. Depending on the choice of the modified Green's function, the formulation requires specification of the field and is time derivative or specification of its space derivative in the aperture plane. Of interest in optical computing is a space-time pulse with an identifiable carrier frequency, v₀. For this special case, if the logarithmic time derivative of the pulse envelope is much smaller than 2πv₀, then the diffraction formula may be approximated by an integral in which the time derivative of the pulse envelope does not appear explicitly. Two examples are presented. In the first example, the pulse is assumed to have a factored form with a Gaussian spatial envelope multiplied by a Gaussian temporal envelope. The condition of validity for the approximation dictates a distance (4/π)λ₀(v₀/Δv)² over which pulse factorization is maintained. Beyond this distance the spatial and temporal envelopes influence each other significantly. For a picosecond pulse the distance is about 25 cm, with λ₀ = 500 nm; for a nanosecond pulse it is 106 times larger. In the second example, temporal distortion after passage through a grating is studied. For a grating of 103 lines, a narrow pulse broadens to about 3 ps. The approximate formula enables explicit calculation of the pulse's space-time structure.

© 1990 Optical Society of America