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.
In this paper, the linear propagation of x rays in laser produced
plasma is studied theoretically, with a quantum mechanical
technique and the so-called V representation, where the
representation transformation is made by using the potential
Hamiltonian V. A modified expression relating the phase difference
between the probe and the reference x ray light to the plasma
electron density is derived, in which the electron density
gradient and its higher-order effects are taken into account. The
coupling relation between phase and amplitude of x-ray is derived,
and the solutions with higher-order corrections are given. An
important parameter is given, which is related to the errors of
the electron density measurement using x-ray interferometry. It is
depicted that providing the parameter is less than one, the x-ray
interferometry can be used for the measurement of the electron
density, while, the greater value of the parameter, the higher
order modifications need to make.
Starting from nonparaxial pulsed-beam propagation equation in free
space in temporal frequency domain and making use of spatial
Fourier transform, the nonparaxial pulsed-beam solution is derived
based on the paraxial pulsed beam solution, where the nonparaxiality is evaluated by a series of expansion. Specifically, the general lowest-order correction field is given in an integral form. Due to the complexity of complex analytical signal (CAS) theory treatment, by using a different initial value from the previous CAS treatment, the lowest-order correction to the paraxial approximation of a fundamental Gaussian beam, uniformly driven by a Gaussian pulse, whose waist plane has a parallel shift from the z=0 plane, are presented. Correspondingly, numerical simulation shows that our lowest-order correction agrees well with the exact paraxial solution(CAS). Apparently, the larger the order is, the more accurate the obtained approximated solution is, and, of course, the more complicated the obtained approximation solution is also.
With the help of the Fast Fourier Transform (FFT) method, a formula is given to study the pulsed spherical beams with the truncated cross section propagating in free space, where the paraxial approximation are also adopted in the derivation. As an application, a theoretical calculation of convergent isodiffracting pulsed beam propagation is presented as an approximation of spherical pulsed beam propagation. Simulation is also shown to illustrate the characteristics of focusing pulsed beam through several kinds of converging lens.