When a pulsed, few-cycle electromagnetic wave is focused by optics with f-number smaller than two, the frequency components it contains are focused to different regions of space, building up a complex electromagnetic field structure. Accurate numerical computation of this structure is essential for many applications such as the analysis, diagnostics, and control of high-intensity laser-matter interactions. However, straightforward use of finite-difference methods can impose unacceptably high demands on computational resources, owing to the necessity of resolving far-field and near-field zones at sufficiently high resolution to overcome numerical dispersion effects. Here, we present a procedure for fast computation of tight focusing by mapping a spherically curved far-field region to periodic space, where the field can be advanced by a dispersion-free spectral solver. In many cases of interest, the mapping reduces both run time and memory requirements by a factor of order 10, making it possible to carry out simulations on a desktop machine or a single node of a supercomputer. We provide an open-source C++ implementation with Python bindings and demonstrate its use for a desktop machine, where the routine provides the opportunity to use the resolution sufficient for handling the pulses with spectra spanning over several octaves. The described approach can facilitate the stability analysis of theoretical proposals, the studies based on statistical inferences, as well as the overall development and analysis of experiments with tightly-focused short laser pulses.
The degree of optimization of the process of guniting of the linings of metallurgical equipment is determined to a great degree by the possibility of regulating and controlling the working characteristics of the guniting stream. In connection with this there is much interest in investigation of the aerodynamics and heat exchange in two-phase streams (jets) with a high concentration of solid polydispersed guniting particles, particularly in a mathematical description of the guniting stream and an investigation of its aerodynamic characteristics.The guniting jet is formed in the following manner. Through the two coaxial channels to the mouth of the torch are supplied oxygen (through the outer orifice) and the guniting mixture by a flow of compressed air (through the inner orifice).In the inner stream the concentration of the addition is very high and the flow rate of the air and the rate of movement of the gas suspension is much less than the flowrate and rate of movement of the oxygen at the mouth of the torch.It is known that at a certain distance from the orifices the streams are mixed to a sufficient degree and in the main portion they may be considered as a single gas stream with a mixture.Therefore in development of the calculation method a free axially symmetric nonisothermal subsonic stream with a polydispersed solid addition was considered. In the general case, the guniting mixture may contain particles of materials differing in size, densities, and thermophysical characteristics.The basis of the proposed method of calculation of the guniting stream was [1-3].One of the methods of an analytical description of two-phase jet flows used at present is the method of integral relationships of the boundary layer with specified universal functions of distribution of the parameters in the transverse sections of the streams.In jet flow the rule of preservation of the longitudinal component of the full impulse Io is fulfilled;where R(x) is the radius of the stream at a distance of x from the orifice in m, p_ is the density of the gaseous phase in kg/m 3, ug is the velocity of the gaseous phase in ~/sec, r is the current value of the radius of the ss in m, S is the number of fractions of the solid particles, n. is the concentration of particles of the i-th fraction in a unit of volume in l i/m ~, u i is the velocity of movement of the particles of the i-th fraction in m/sec, m. is the average weight of a particle of the i-th fraction in kg, m i = (4/3)~6~pi, Pi is th~ density of the particles of the i-th fraction in kg/m ~, 6. is the average diameter of the particles of the i-th fraction in m, and i is the number of theZfraction of the guniting mixture.Here and subsequently, 0, m, and ~ designate the values of the parameters at the nozzle edge, on the axis, and on the outer edge of the stream.For the i-th fraction the rule of the preservation of its mass flow rate Qi,o is preserved:
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