Three methods of multimode fiber interferometer signal averaging are investigated theoretically and experimentally: the ensemble averaging, the averaging over a long realization, and the averaging over transverse coordinates of an output speckle pattern. The analysis is performed for two wavelengths and two launch conditions of the graded-index multimode fiber: few mode and multimode. Applicability of methods for practical applications is discussed.
The empirical models of the lower ionosphere are used for fast prediction of VLF-LF propagation properties, for the initialization in the inverse problem solvers and as a climatological testbed for new numerical models. We used two widely used empirical models and verify them on the experimental VLF data from Mikhnevo geophysical observatory for 2014 year. Numerical results were obtained by parabolic equation method. The presented results prove the severe limitations of the current empirical models. The main bottlenecks are formulated.
KEYWORDS: Solar radiation models, Solar processes, Ionization, Transmitters, Numerical simulations, X-rays, Data modeling, Monte Carlo methods, Physics
The progress in the physics and chemistry of the lower ionosphere depends on the verification of the numerical models on the experimental data. We establish the framework, that the lower ionosphere model can be considered as a valid one, only if the prediction for the VLF-LF radiowave propagation coincides with evidence both in amplitude and phase temporal dynamics. The extremely strong X-flares 06 and 10 September 2017 were chosen as a testbed for the empirical and theoretical models of the midlatitude lower ionosphere. Both models used GOES-15 X-ray flux measurements. Empirical model captures only the time moment of disturbance. Theoretical model captures the main feature in VLF response. We summarize the observed problems in simulation and prospective solutions as well.
Mode structure and nonlinear dynamics of the chirped pulse are studied in the graded-index optical fiber with a
longitudinal inhomogeneity of the refractive index. Chirps are classified with respect to the relationship between the
depth of the linear frequency modulation and the width of the pulse spectrum. Considered in the paper are the regimes:
(1) the modulation depth essentially less than the spectrum width - chirped pulses; (2) the depth of modulation is
commensurate with the width ofthe pulse spectrum - strongly chirped pulses. The pulse propagation is modelled with a
nonlinear wave equation in which the refractive index depends quadratically on the wave field. This equation is solved
asymptotically with two different ansatzes for chirp and strong chirp regimes. The mode structure ofthe pulse is shown
to differ for chirped and strongly chirped pulses, and in both cases relationships are stated confining the coefficient of
the linear frequency modulation with the phases of high-frequency carrier and envelope. Consequent asymptotic
procedure leads to the nonlinear equations governing the dynamics of the envelopes of chirped and strongly chirped
pulses. Studied in more details is the envelope of the chirped pulse, in this case some additional assumptions on the
longitudinal inhomogeneity of the optical fiber enable to reduce the equation for the envelope to the second Painleve equation. Comparison with sech-soliton of the nonlinear Schroedinger equation is carried out and important features
conditioned by the linear frequency modulation are ascertained.
Short optical pulse propagation is investigated in the light guide characterized with a strong dependence of the fiber material refractive index on the radial coordinate and a weak dependence on the longitudinal coordinate, with a weak spatial bending of the light guide axis being allowed as well. A three-dimensional nonlinear wave equation used in modeling the process is solved asymptotically with respect to a small parameter setting the order of magnitude of the pulse amplitude. A relationship between the propagating modes and the eigenvalues and eigenfunctions of a singular Sturm-Liouville problem is elucidated. The pulse propagation is shown to be three-scale: the high-frequency carrier is modulated with the envelope which evolves in a two-scale manner and is described with a nonlinear Schroedinger equation with coefficients depending on the longitudinal coordinate. For several types of the transverse and longitudinal inhomogeneities, expressions through elementary functions are obtained for the transverse distribution of the wave field and the envelope soliton. The possibility is stated for managing pulse parameters by means of varying the transverse and longitudinal inhomogeneities of the light guide. A formula for amplitude modulation of the pulse due to the third-order dispersion and self-steepening is obtained and it is shown that under a certain relationship between these quantities the envelope can propagate without distortions.
A mathematical approach for modeling the subpicosecond pulse transmission in inhomogeneous optical fibers is suggested. Propagation of subpicosecond pulses is described under the assumption that they are affected by third-order dispersion and self-steepening. The pulse envelope is shown to obey a generalized nonlinear Schrodinger equation with additional terms characterizing the third-order dispersion and self- steepening of the pulse, and the Cauchy's problem for this equation with the initial values of the soliton type is investigated. Asymptotic solutions to this equation are constructed for the cases of (1) small third-order dispersion and self-steepening, and (2) short distances in the fiber for finite third-order dispersion and self- steepening. Phase distortions are considered for both cases as well.
Propagation of optical pulses in graded-index light guides is described under the assumption of the combined influence of large power, short duration and inhomogeneity which is supposed to be strong in the transverse and weak in the longitudinal direction. The propagation is treated as a weak nonlinear process modeled with a nonlinear wave equation. This equation is solved by means of a consistent asymptotic procedure with respect to a small parameter related to the pulse amplitude, with the second power of the parameter characterizing the weak longitudinal inhomogeneity of the graded-index light guide. A Sturm-Lionville problem, is obtained for the principal approximation to the asymptotic solution. The transverse distribution of the wave field and the phase of the high-frequency carrier are expressed through eigenfunctions and eigenvalues of this problem. For the pulse envelope the Nonlinear Schroedinger equation is derived, its coefficients depending on the longitudinal coordinate. The Cauchy's problem for this equation with the initial values of the soliton type is investigated. The conditions of persistence of the pulse shape are ascertained and for a large class of weak longitudinal inhomogeneities a distance is estimated where the pulse remains to be a solitary wave. A series of additional pulses with diminishing amplitudes where the pulse remains to be a solitary wave. A series of additional pulses with diminishing amplitudes arises as an effect of the weak longitudinal inhomogeneity. Formulae for these distortions are derived and it is shown that this effect may be eliminated for the case of a special relationship between the characteristics of inhomogeneity.
Weak nonlinear process of propagation of short optical pulses in graded-index light guides with the quadratic dependence of the refractive index on the transverse coordinate and a slight dependence on the longitudinal coordinate is modelled with the nonlinear wave equation. Transverse distribution of the wave field is shown to be characterized with the parabolic cylinder functions, and the nonlinear Schroedinger equation with variable coefficients is obtained for the pulse envelope. A solution of this equation describing a soliton pulse is found for a class of longitudinal inhomogeneities and formulae are presented for variations of amplitude, shape and velocity of the pulse during its propagation.
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