Abstract
In this paper, an integral solutions for the transverse and longitudinal components of the laser field for ultrashort pulsed-beam propagation in free space are derived, applying Fourier transform and the paraxial approximation in frequency domain. Furthermore, using the complex analytical signal (CAS) theory, the solutions for the components of a pulsed Gaussian-like beam in rectangular coordinate are derived and contrasted with those derived with slowly-varying-envelop-approximations (SVEA). Meanwhile, the results show that if the pulses as short as sub-cycle, SVEA will cause the spatial singularity for both transverse and longitudinal components of the laser field and CAS must be applied to eliminate the singularity.
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