In recent years, the use of joint time-frequency representations to characterize and interpret shaped femtosecond laser pulses has proven to be very useful. However, the number of points in a joint time-frequency representation is daunting as compared with those in either the frequency or time representation. In this article we introduce the use of the von Neumann representation, in which a femtosecond pulse is represented on a discrete lattice of evenly spaced time-frequency points using a non-orthogonal Gaussian basis. We show that the information content in the von Neumann representation using a lattice of radicalN points in time and radicalN points in frequency is exactly the same as in a frequency (or time) array of N points. Explicit formulas are given for the forward and reverse transformation between an N-point frequency signal and the von Neumann representation. We provide numerical examples of the forward and reverse transformation between the two representations for a variety of different pulse shapes; in all cases the original pulse is reconstructed with excellent precision. The von Neumann representation has the interpretational advantages of the Husimi representation but requires a bare minimum number of points and is stably and conveniently inverted; moreover, it avoids the periodic boundary conditions of the Fourier representation.

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We experimentally demonstrate the generation and characterization of polarization-shaped femtosecond laser pulses in the ultraviolet at a central wavelength of 400 nm. Near-infrared laser pulses are first polarization shaped and then frequency doubled in an interferometrically stable setup that employs two perpendicularly oriented nonlinear crystals. A new pulse shaper design involving volume phase holographic gratings reduces losses and hence leads to an increase in pulse energy.