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# 物理代写|电动力学代考Electrodynamics代写|PHYS1002 Electromagetic waves in metamaterial

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## 物理代写|电动力学代考Electrodynamics代写|On dispersion in 1D metamaterial

A concept of a metamaterial from a formal point of view is directly based on dispersion relation [1], that develops on the Drude model, see chapter 6 , subsection 6.3.2 as a particular case of Lorentz model. The term ‘metamaterial’ itself has an interesting history [1]. One of the important applications of such a model relates to metamaterials that are characterized by negative values of the parameters $\varepsilon$ and $\mu$, that must be anomalous dispersive, i.e. their permittivity and permeability must be frequency dependent, otherwise they would not be causal [2]. The two-time derivative Lorentz material (2TDLM) model encompasses the metamaterial models most commonly discussed; it has the frequency domain susceptibility [3]:
$$\chi_e=\frac{\omega_p^2 \chi_\alpha+i \omega_p \chi_\beta \omega-\chi_\gamma \omega^2}{\omega_0^2+i \omega \Gamma-\omega^2},$$
its particular case is the 2TDLM model, which produces a resonant response at $\omega=\omega_0$ when $\Gamma=0$. The natural $1 \mathrm{D}$ metamaterials construction originated from layered medium $[4]$

The nonlinear behavior of electromagnetic (EM) wave propagation depends on relations between the field and induced polarization. It is obvious that it is necessary to use either a numerical scheme or approximations to obtain an analytical solution of a nonlinear problem. The first successful approach of such a reduction was the use of a set of slowly varying envelopes. The simplest model scalar equation for a directed wave propagation, based on this approach, have a form of the nonlinear Schrödinger equation, derived by Zakharov in 1968 [5, 6]. Its integrability [7] made the model very attractive because of the rich ‘zoo’ of the equation explicit solutions [8].

## 物理代写|电动力学代考Electrodynamics代写|Maxwell’s equations for matter inside a waveguide

For such a problem, for points inside a waveguide, more close to engineering, we would present the Maxwell system in SI units as:
$$\begin{gathered} \operatorname{div} \vec{D}=0, \ \operatorname{div} \vec{B}=0, \ \operatorname{rot} \vec{E}=-\frac{\partial \vec{B}}{\partial t}, \ \operatorname{rot} \vec{H}=\frac{\partial \vec{D}}{\partial t}, \end{gathered}$$
that is formulated for the case of absence of charges and currents. The material operator relations for an isotropic dispersive medium in a waveguide we write as in section 13.2 from [25] in SI units:
$$\vec{D}=\varepsilon_0 \hat{\varepsilon} \vec{E}, \vec{H}=\frac{1}{\mu_0} \hat{\mu}^{-1} \vec{B}$$

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|On dispersion in 1D metamaterial

$$\chi_e=\frac{\omega_p^2 \chi_\alpha+i \omega_p \chi_\beta \omega-\chi_\gamma \omega^2}{\omega_0^2+i \omega \Gamma-\omega^2}$$

## 物理代写|电动力学代考Electrodynamics代写|Maxwell’s equations for matter inside a waveguide

$$\operatorname{div} \vec{D}=0, \operatorname{div} \vec{B}=0, \operatorname{rot} \vec{E}=-\frac{\partial \vec{B}}{\partial t}, \operatorname{rot} \vec{H}=\frac{\partial \vec{D}}{\partial t},$$

$$\vec{D}=\varepsilon_0 \hat{\varepsilon} \vec{E}, \vec{H}=\frac{1}{\mu_0} \hat{\mu}^{-1} \vec{B}$$

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