Generation of pulse train

A train of ultrashort pulses are naturally generated as high harmonic generation from atoms and molecules, which may be used as an optical source to excite another target. In our consideration here, however, a pulse train is meant to be a series of pulses formed from well designed combination of multiples of a phase controlled continuous external field. To be precise, we assume a field formed of multiples of the base frequency u , 

All the phases are taken (f n =0 to generate a pulse train. Inclusion only of odd multiples of the base frequency generates a field with an envelope symmetric with respect to the sign of the field, with the separation between the pulses in the pulse train Atp given by the period of the lowest frequency component 

The number of pulses within the envelope is given by t /Atp, where the duration of the pulse train Te is 

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