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The 2D wave-envelope equations for the beat-wave-cascade system are studied analytically and numerically. An expression for the mean square width of the cascade envelope is obtained, and is used to predict the long-term behavior of the waves. The amplitude of a resonantly driven plasma wave falls significantly over a stage length because of enhanced diffraction of the cascade envelope. Conversely, detuning the pumps from the plasma frequency can lead to focusing of the envelope and a corresponding increase in plasmon amplitude of up to 200% over the same distance.
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Gibbon et al. (1988) studied this question.
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