We describe several families of exact unbounded solutions of Maxwell's equations in vacuum. These solutions depict one-͑or 1 1 2 -͒ cycle electromagnetic pulses whose fields are of either transverse magnetic or transverse electric character and are confined to toroidal wave packets that converge to a focus and then diverge in a manner that is expected from familiar rules of diffraction. These ''focused doughnut'' pulses constitute a subset of the ''modified power spectrum'' pulse solutions discovered by Ziolkowski ͓Phys. Rev. A 39, 2005 ͑1989͔͒. We derive the total energy, the energy spectrum, the ability to accelerate an electron, and other properties of these focused doughnut pulse solutions. ͓S1063-651X͑96͒07407-7͔PACS number͑s͒: 41.20.Bt, 41.20.Jb

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We have found simple analytical solutions for the coupled differential equations that are commonly used to describe unipolar photoconduction in a homogeneous material with shallow traps. Our solutions cover transient photoconduction in the dark after an initial low-intensity pulse of light creates some spatial pattern of photoexcited carriers. We assume that there is no conduction in equilibrium in the dark.Our solutions introduce two parameters to describe the shallow traps: a mean time between trapping events in the conduction band and a mean time spent in a shallow trap. When used to analyze extensive experimental data on a particular sensitive crystal of n-type cubic Bi»Si02O, our solutions predict an observed, but unexpected, dependence of carrier grating decays (in the dark) on applied electric field. The solutions also correlate transient photoconductivity results with grating-decay behavior both in the dark and under uniform illumination, giving a mean time between shallow traps in the conduction band of 26+3 ns, and a mean time of 800+300 ns spent in a shallow trap. A value p, =5+3 cm /(V s) is inferred for the electron mobility in the conduction band between trapping events. Also, the electron diffusion length inferred from our solutions matches that observed previously by light-induced grating decays.

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