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# Ermakov equation arising from electromagnetic fields propagating in 1D inhomogeneous media

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## Abstract

The Ermakov equation is derived from Maxwell’s equations for inhomogeneous transparent media in one dimension. The general properties of this equation and its associated invariants are discussed. Numerical results are presented for refractive index changes, which take place in the order of fractions of the wavelength.
... Nonlinear superposition relationships have been obtained invoking Noether's theorem [5], Lie symmetry methods [6] and differential equations manipulation [7,8]. These methods have been successfully applied to different problems such as the time dependent harmonic oscillator [9][10][11], nonlinear acoustic oscillators [12], wave propagation in stratified media [13], reflectivity from rugate films [14,15] and various quantum mechanical problems [16][17][18]. The nonlinear counterpart to a second order linear ODE is the Ermakov-Pinney equation [19]. ...
... Add Eqs. (13) and (14) to evaluate the Ermakov invariant (4), expressed solely in terms of the ρ variable, ...
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Floquet’s theorem applied to Hill’s equation, is translated to its Ermakov pair, namely, the nonlinear amplitude differential equation with periodic parameter. The nonlinear version states that if $$\rho \left( z\right)$$ is a solution within one period d, to the nonlinear differential equation $$d^{2}\rho /dz^{2}+\rho \varOmega ^{2}-Q^{2}\rho ^{-3}=0$$, with periodic parameter $$\varOmega ^{2}\left( z+Nd\right) =\varOmega ^{2}\left( z\right)$$, the solution after N periods is given by $$\rho \left( z+Nd\right) =\rho \left( z\right) \rho _{d}^{-N}\left[ 1+\left( \rho _{d}^{4N}-1\right) \cos ^{2}\left( \int Q/\rho ^{2}\left( z\right) dz+N\phi _{d}\right) \right] ^{\frac{1}{2}}$$. This proposition is proved and a physical interpretation to the Floquet solution is given in terms of counter-propagating waves when the formalism describes one dimensional wave propagation.
... However, the rise and fall times are then also greatly increased. Another alternative is to synthesize each step with an exponential function with real argument [6] or a hyperbolic tangent function [8]. This approach embraces various possibilities because it permits the control of the duty cycle as well as different rise/fall times. ...
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A function that is well suited to describe a square wave as well as a sequence of delta functions is presented. The squdel function has a closed rational form instead of a series approximation. Bandwidth limitations are readily incorporated in this function without producing undesirable ringing artifacts. The squdel function is infinitely differentiable and analytic for a squareness parameter as close as required to the square function provided that the limit is not taken. Even its Fourier series decomposition does not exhibit overshooting when truncated. Two-dimensional soft pixel structures are shown to be economically modeled with this function.
... Este procedimiento se ha utilizado tanto en mecánica cuántica [4] como en problemas del oscilador armónico con parámetro dependiente del tiempo [5]. En electromagnetismo, la AmF se ha utilizado para analizar la propagación de una capa con variación de tangente hiperbólica en el índice de refracción [6] y la reflectividad de distintas funciones a incidencia normal. Con este formalismo se ha predicho la reflectividad aumentada debido a las discontinuidades en las derivadas en el índice de refracción (aunque n sea constante) [7]. ...
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La representación de amplitud y fase permite obtener ecuaciones diferenciales para la amplitud y el vector de onda. Éstas ecuaciones pueden desacoplarse utilizando el invariante de Ermakov. Para medios estratificados transparentes, se muestra que la componente del vector de onda en la dirección de estratificación , cumple con la relación kz tan teta = constante, donde teta es el ángulo de inclinación de propagación. Para variaciones suaves del índice de refracción comparadas con la longitud de onda, la relación anterior deviene en la relación de Snell generalizada. En el caso de interfase abrupta entre dos medios homogéneos, se recupera la relación usual de Snell.
... La intención aquí es modelar una interfase dieléctrica común, entre materiales no magnéticos µ = µ 0 . Con ese objeto se elige un perfil de índice de refracción n(z) real, que sea monotónico y tienda asintóticamente por ambos extremos a n i y n t , en particular un perfil de tangente hiperbólica [52] El parámetro D es una medida del grosor de la interfase, es la distancia en la que sucede el 90 % del cambio en el índice de refracción. Este perfil no se debe confundir con el de Epstein 2.2.8.3, que es muy parecido. ...
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Electromagnetic wave propagation through media with a stratified refractive index, the amplitude and phase representation.
... Este procedimiento se ha utilizado tanto en mecánica cuántica [4] como en problemas del oscilador armónico con parámetro dependiente del tiempo [5]. En electromagnetismo, la AmF se ha utilizado para analizar la propagación de una capa con variación de tangente hiperbólica en el índice de refracción [6] y la reflectividad de distintas funciones a incidencia normal. Con este formalismo se ha predicho la reflectividad aumentada debido a las discontinuidades en las derivadas en el índice de refracción (aunque n sea constante) [7]. ...
Article
The representation of waves in amplitude and phase variables can be decoupled using the Ermakov invariant. The wave vector component in the direction of stratification kz, satisfies the relationship kztanθ = constant, where θ is the angle of propagation. This component must fulfill the nonlinear differential equation kz k̈z - 3/2 k̇ z2 + 2[kz2 - (n2 - α2)k02, where n is the refractive index, α is a constant and k0 the wave vector magnitude in vacuum. For soft variations of the refractive index compared with the wavelength, this relationship becomes the so called generalized Snell relationship n2(z)sin2θ(z). For an abrupt interface, the usual Snell equation n1sinθ1 = n2sinθ2 is recovered.
... The Ermakov invariant allows to construct a nonlinear superposition law linking the solutions of the equations of motion composing the Ermakov system [4]. Ermakov systems have recently been of interest in diverse scenarios, such as accelerator physics [5], dielectric planar waveguides [6], cosmological models [7,8], analysis of supersymmetric families of Newtonian free damping modes [9], study of open fermionic systems [10], analysis of the propagation of electromagnetic waves in one-dimensional inhomogeneous media [11], algebraic approach to integrability of nonlinear systems [12], coupled linear oscillators [13], the semiclassical limit of quantum mechanics [14], supersymmetric quantum mechanics [15], computation of geometrical angles and phases for nonlinear systems [16]- [18], search for Noether [19,20] and Lie [21,22] symmetries, the possible linearization of the system [23,24], extension of the Ermakov system concept [21], [25]- [27], the search for additional constants of motion [28] and some discretizations of Ermakov systems [29,30]. ...
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We construct Poisson structures for Ermakov systems, using the Ermakov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them degenerate, in which case we derive the Casimir functions. In some situations, the existence of Casimir functions can give rise to superintegrable Ermakov systems. Finally, we characterize the cases where linearization of the equations of motion is possible.
... In recent papers, a method to find numerical solutions for the electric field equation by solving the associated amplitude equation was proposed [1][2][3]. After solving the amplitude equation for numerous n(z) profiles, with different types of derivative discontinuity, a consistent behavior was found [1]. ...
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Surfaces within a dielectric material, where the derivatives of a continuous and real refractive index profile are discontinuous, are shown to enhance reflection. To this end, the amplitude and phase representation of electromagnetic waves is used to model light propagating normally through a transparent medium with a continuous refractive index profile that varies only in one spatial direction. The amplitude equation is solved under the slowly varying refractive index approximation (SVRI). To isolate the effect of a single surface where the refractive index derivatives are discontinuous, an n(z) profile is proposed that is analytical, smooth and slowly varying except for a single piece wise junction. At this junction, n(z) is continuous but some of the m th order derivatives are not. Two different SVRI approximated solutions are joined at the discontinuity plane and by demanding that boundary conditions are satisfied, a general complex reflection coefficient is obtained. By categorizing profiles according to the lowest order discontinuous derivative at the junction, a simple expression for the reflection coefficient can be written. Results are compared favorably with previous numerical solutions. Furthermore, a conjecture by the authors in a previous paper: “For a C^(m-1) refractive index profile type, where m stands as the order of the lowest order discontinuous derivative, phase change upon reflection at the discontinuity plane is (2-m)*pi/2 for an increasing lowest order discontinuous derivative and -m*pi/2 for the decreasing case”, is proved here.
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We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati-type systems, we obtain invariants of Abel-type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs.
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A new general expression of a solution to Maxwell׳s equations derived recently has been applied to a 1-D inhomogeneous medium. It is shown that it solves any inhomogeneous refractive index profile. Its main advantage is that it does not require integration of either the differential wave equation or the refractive index profile, as it is the case with other methods. The solution is expressed in a closed form. The obtained numerical results compare favorably with both the slow-varying envelope method as well as with the exact numerical method.
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The amplitude and phase representation of classic electromagnetic waves is used to model light propagating through a stratified medium, with a continuous refractive index profile. This medium is assumed to be isotropic, non magnetic, electrically neutral, transparent, dielectric, with a linear response, but letting the electric permittivity vary along the z direction. No approximations regarding a slowly or strongly varying refractive index, compared to the wavelength, are made. This is particularly convenient when the refractive index varies on a wavelength scale. The case of a thin film at normal incidence is studied for various thicknesses and interface ranges. The corresponding nonlinear amplitude differential equation is numerically solved. The amplitude oscillations are construed in terms of counter-propagating waves and the film's reflectivity is evaluated.
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The ray formalism is reviewed and a first integration of the ray equation is obtained without writing explicity the refractive index function. This result is used to solve exactly the Gaussian index profile. Confinement criteria are discussed within this framework.
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Noether's theorem is used to derive the Lewis invariant for the time-dependent harmonic oscillator. The application of the method to non-linear dynamical systems is discussed.
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A new approach to the problem of scattering is presented via the exact invariants of quantum hydrodynamics, where the solution preserves information about the quantum potential which accounts for quantum-wave features such as interference and diffraction effects. Within this formalism, a formula for the transmission coefficient of a steady flux of fluid-particles scattered by an arbitrarily shaped potential well is derived and discussed for some exact and approximate solutions.
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Ermakov systems are pairs of coupled, time-dependent nonlinear dynamical equations possessing a joint constant of motion. We show how to derive the Ermakov system from nonharmonic oscillators. We present a detailed study of Ermakov systems from a classical and quantum point of view. Finally the nonadiabatic Hannay's angle and Berry's phase for the system are calculated along with its adiabatic limit.