scholarly journals Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media

2019 ◽  
Author(s):  
Er'el Granot
Keyword(s):  
Dispersive Medium ◽  
Ultrashort Pulses ◽  
Analytical Approximation ◽  
Dispersive Media ◽  
Physical Realization ◽  
Technical Literature ◽  
Short Pulses ◽  
New Findings ◽  
Analysis Of Dispersion ◽  
Analytical Expressions

Ultrashort pulses are severely distorted even by low dispersive media. While the mathematical analysis of dispersion is well known, the technical literature focuses on pulses, which keep their shape: Gaussian and Airy pulses. However, the cases where the shape of the pulse is unaffected by dispersion is the exception rather than the norm. It is the object of this chapter to present a variety of pulses profiles, which have analytical expressions but can simulate real-physical pulses with greater accuracy. In particular, the dynamics of smooth rectangular pulses, physical Nyquist-Sinc pulses, and slowly rising but sharply decaying ones (and vice-versa) are presented. Besides the usage of this chapter as a handbook of analytical expressions for pulses' propagations in a dispersive medium, there are several new findings. The main ones are: Analytical expressions for the propagation of chirped rectangular pulses, which converge to extremely short pulses; analytical approximation for the propagation of Super-Gaussian pulses; the propagation of Nyquist Sinc Pulse with smooth spectral boundaries and an analytical expression for a physical realization of an attenuation compensating Airy pulse.

scholarly journals Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media

2019 ◽  
Vol 9 (3) ◽  
pp. 527 ◽  
Author(s):  
Er'el Granot
Keyword(s):  
Dispersive Medium ◽  
Ultrashort Pulses ◽  
Analytical Approximation ◽  
Great Accuracy ◽  
Dispersive Media ◽  
Technical Literature ◽  
Short Pulses ◽  
New Findings ◽  
Analysis Of Dispersion ◽  
Analytical Expressions

Ultrashort pulses are severely distorted even by low dispersive media. While the mathematical analysis of dispersion is well known, the technical literature focuses on pulses, Gaussian and Airy pulses, which keep their shape. However, the cases where the shape of the pulse is unaffected by dispersion is the exception rather than the norm. It is the objective of this paper to present a variety of pulse profiles, which have analytical expressions but can simulate real-physical pulses with great accuracy. In particular, the dynamics of smooth rectangular pulses, physical Nyquist-Sinc pulses, and slowly rising but sharply decaying ones (and vice versa) are presented. Besides the usage of this paper as a handbook of analytical expressions for pulse propagations in a dispersive medium, there are several new findings. The main findings are the analytical expressions for the propagation of chirped rectangular pulses, which converge to extremely short pulses; an analytical approximation for the propagation of super-Gaussian pulses; the propagation of the Nyquist-Sinc Pulse with smooth spectral boundaries; and an analytical expression for a physical realization of an attenuation compensating Airy pulse.

High-frequency nonlinear optics: from the nonlinear Schrödinger approximation to ultrashort-pulses equations

2011 ◽  
Vol 141 (2) ◽  
pp. 253-286 ◽  
Author(s):  
David Lannes
Keyword(s):  
High Frequency ◽  
Hyperbolic Systems ◽  
Ultrashort Pulses ◽  
Frequency Dispersion ◽  
Dispersive Media ◽  
Nls Equation ◽  
Short Pulses ◽  
Nonlinear Schrödinger ◽  
Striking Fact ◽  
Large Time Scale

After a brief introduction and physical motivation, we show how the nonlinear Schrödinger (NLS) equation can be derived from a general class of nonlinear hyperbolic systems. Its purpose is to describe the behaviour of high-frequency oscillating wave packets over a large time-scale that requires us to take into account diffractive effects. We then show that the NLS approximation fails for short pulses and propose some alternative models, including a modified Schrödinger equation with improved frequency dispersion. It turns out that these models have better properties and are quite accurate for short pulses. For ultrashort pulses, however, they must also be abandoned for more complex approaches. We give the main steps for such an analysis and explain one striking fact about ultrashort pulses: their dynamics in dispersive media is linear.

scholarly journals Newmark-FDTD Formulation for Modified Lorentz Dispersive Medium and Its Equivalence to Auxiliary Differential Equation-FDTD with Bilinear Transformation

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Hongjin Choi ◽  
Jeahoon Cho ◽  
Yong Bae Park ◽  
Kyung-Young Jung
Keyword(s):  
Differential Equation ◽  
Numerical Stability ◽  
Dispersive Medium ◽  
Fdtd Method ◽  
Dispersive Media ◽  
Complex Conjugate ◽  
Dispersion Models ◽  
Bilinear Transformation ◽  
Quadratic Complex ◽  
Difference Time

The finite-difference time-domain (FDTD) method has been popularly utilized to analyze the electromagnetic (EM) wave propagation in dispersive media. Various dispersion models were introduced to consider the frequency-dependent permittivity, including Debye, Drude, Lorentz, quadratic complex rational function, complex-conjugate pole-residue, and critical point models. The Newmark-FDTD method was recently proposed for the EM analysis of dispersive media and it was shown that the proposed Newmark-FDTD method can give higher stability and better accuracy compared to the conventional auxiliary differential equation- (ADE-) FDTD method. In this work, we extend the Newmark-FDTD method to modified Lorentz medium, which can simply unify aforementioned dispersion models. Moreover, it is found that the ADE-FDTD formulation based on the bilinear transformation is exactly the same as the Newmark-FDTD formulation which can have higher stability and better accuracy compared to the conventional ADE-FDTD. Numerical stability, numerical permittivity, and numerical examples are employed to validate our work.

scholarly journals Coherent soliton propagation through doped optical fibers: cloning, breakup, and soliton interactions

2001 ◽  
Vol 73 (2) ◽  
pp. 197-209
Author(s):  
SOLANGE B. CAVALCANTI ◽  
EDUARDO J. DA S. FONSECA ◽  
DILSON P. CAETANO ◽  
JANDIR M. HICKMANN
Keyword(s):  
Optical Fibers ◽  
Dispersive Medium ◽  
Coherent Population Trapping ◽  
Dispersive Media ◽  
Level System ◽  
Optical Pulses ◽  
Bloch Equations ◽  
Soliton Propagation ◽  
Soliton Interactions ◽  
Population Trapping

The simultaneous propagation of two optical pulses through a doped nonlinear dispersive medium modelled by a resonant three-level system was investigated numerically, within the framework of a pair of coupled extended nonlinear Schrödinger equations. These included the contribution of the dopant resonances whose dynamics is governed by Bloch equations. In this work, we review the interesting possibilities on the manipulation of fields such as cloning, breakup and soliton interactions, that the combination of coherent population trapping with nonlinear dispersive media offers.

scholarly journals Statistics for Ratios of Rayleigh, Rician, Nakagami-m, and Weibull Distributed Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Dragana Č. Pavlović ◽  
Nikola M. Sekulović ◽  
Gradimir V. Milovanović ◽  
Aleksandra S. Panajotović ◽  
Mihajlo Č. Stefanović ◽  
...  
Keyword(s):  
Random Variables ◽  
Joint Probability ◽  
Wireless Systems ◽  
Technical Literature ◽  
Product Moments ◽  
Outdoor Environments ◽  
Indoor And Outdoor Environments ◽  
Analytical Expressions ◽  
Joint Probability Density ◽  
Indoor And Outdoor

The distributions of ratios of random variables are of interest in many areas of the sciences. In this brief paper, we present the joint probability density function (PDF) and PDF of maximum of ratiosμ1=R1/r1andμ2=R2/r2for the cases whereR1,R2,r1, andr2are Rayleigh, Rician, Nakagami-m, and Weibull distributed random variables. Random variablesR1andR2, as well as random variablesr1andr2, are correlated. Ascertaining on the suitability of the Weibull distribution to describe fading in both indoor and outdoor environments, special attention is dedicated to the case of Weibull random variables. For this case, analytical expressions for the joint PDF, PDF of maximum, PDF of minimum, and product moments of arbitrary number of ratiosμi=Ri/ri,i=1,…,Lare obtained. Random variables in numerator,Ri, as well as random variables in denominator,ri, are exponentially correlated. To the best of the authors' knowledge, analytical expressions for the PDF of minimum and product moments of{μi}i=1Lare novel in the open technical literature. The proposed mathematical analysis is complemented by various numerical results. An application of presented theoretical results is illustrated with respect to performance assessment of wireless systems.

Analysis of Dispersion of Small Spherical Particles in a Random Velocity Field

1990 ◽  
Vol 112 (1) ◽  
pp. 114-120 ◽  
Author(s):  
H. Ounis ◽  
G. Ahmadi
Keyword(s):  
Particle Velocity ◽  
Digital Simulation ◽  
Density Ratio ◽  
Spherical Particles ◽  
Theoretical Predictions ◽  
The Mean ◽  
Analysis Of Dispersion ◽  
Gradient Effects ◽  
Analytical Expressions ◽  
Good Agreement

The equation of motion of a small spherical rigid particle in a turbulent flow field, including the Stokes drag, the Basset force, and the virtual mass effects, is considered. For an isotropic field, the lift force and the velocity gradient effects are neglected. Using the spectral method, responses of the resulting constant coefficient stochastic integrao-differential equation are studied. Analytical expressions relating the Lagrangian energy spectra of particle velocity to that of the fluid are developed and the results are used to evaluate various response statistics. Variations of the mean-square particle velocity and particle diffusivity with size, density ratio and response time are studied. The theoretical predictions are compared with the digital simulation results and the available data and good agreement is observed.

scholarly journals An analytical approximation of the transient response of a voltage dependent supercapacitor model

2019 ◽  
Vol 39 (3) ◽  
pp. 310-319
Author(s):  
Tomislav Barić ◽  
Hrvoje Glavaš ◽  
Ružica Kljajić
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Nonlinear Differential Equation ◽  
Exact Analytical Solution ◽  
Initial Response ◽  
Analytical Approximation ◽  
Weight Functions ◽  
Transient Phenomenon ◽  
Voltage Dependent ◽  
Analytical Expressions

Supercapacitors are well known for their voltage dependent capacity. Due to this, it is not possible to obtain the exact analytical solution of the nonlinear differential equation which describes the transient charging and discharging. For this reason, approximations of differential equations must be carried out in order to obtain an approximate analytical solution. The focus of this paper is on a different approach. Instead of approximating the differential equation and obtaining analytical expressions for such approximations, an intuitive approach is chosen. This approach is based on the separation of the initial response from the rest of the transient phenomenon. Both parts of the transient phenomenon are described with adequate functions. Using appropriate weight functions, both functions are combined into a single function that describes the whole transient phenomenon. As shown in the paper, such an approach gives an excellent description of the whole transient. Also, it provides simpler expressions compared to those obtained by approximation of the nonlinear differential equation. With respect to their accuracy, these expressions do not lag behind the aforementioned approach. The validity of the presented analytical expressions was confirmed by comparing their results with those obtained by numerically solving the nonlinear differential equation.

scholarly journals Analytical approximation for oscillator strengths of H-like ions

1990 ◽  
Vol 8 (4) ◽  
pp. 709-714 ◽  
Author(s):  
E. Minguez
Keyword(s):  
Oscillator Strength ◽  
Analytical Approximation ◽  
Oscillator Strengths ◽  
Analytical Formulas ◽  
Analytical Expressions

The main objective of this work is to find analytical formulas for the oscillator strength f of hydrogen-like ions. It is well known that f is proportional to the energy of the transitions between eigenstates (ΔE), and to the square of the R matrix. Therefore, the problem of calculating f can be reduced to finding analytical expressions for both parameters, ΔE and R. Hence these expressions would be in accordance with quantum results based on more sophisticated calculations.

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