On the optical conductivity

doi:10.1016/j.ijleo.2019.163067

Optik

Volume 194, October 2019, 163067

https://doi.org/10.1016/j.ijleo.2019.163067 Get rights and content

Abstract

The quantum electrodynamics of a single photon in introduced. Using the quantized Maxwell's equations, an interesting optical conductivity of a dielectric is derived. It is found to be complementary to the electric conductivity derived that was obtained using wave nature of light. While the former varies quadratically with the refractive index of the material, the latter varies linearly. Both conductivities vary inversely with light wavelength.

Introduction

Several of the electromagnetic properties of matter are classically described by Maxwell's theory [1], [2]. However, at the quantum level, the photon energy and momentum are quantized so that they were not included in Maxwell's equation. For instance, when an electromagnetic field incident on bounded electrons, they interact with the electromagnetic field by oscillating about their equilibrium positions. In a quantum mechanical picture, these electrons can interact with the photon comprising the electromagnetic field. In the event of the electromagnetic field interaction, these electrons collide with the photons, and exchange energy and momentum with them. As a result an electric current can be created. The physical significance of this current has to be thoroughly investigated. And since the photon exchanges angular momentum and energy with the medium electrons, a possible photonic current can be established (eclectic or magnetic). One way to bring the quantum description of the electromagnetic field (including light) is made by the quantum electrodynamics (QED), where the coupling between free particles (electron) and photons is made by invoking the principle of minimal coupling. An analogous principle is also employed to bring the interaction of particles with gravity (the gravitational field) that is termed the general covariance principle. In this formalism light is considered to be massless, since the presence of the photon mass in the theory destroys the gauge symmetry.

We will study in this work how one can incorporate the photon quantum behavior in Maxwell's equations without destroying the gauge symmetry. In a recent paper we found that the electric and magnetic fields of a photon, as a particle, are defined in terms of its angular momentum [3], [4]. These fields give rise to additional (quantum) contribution in Maxwell's equations. They add electric and magnetic charge and currents densities. We aim here to investigate some of these contributions. Of these contributions is the optical conductivity the photon can impart. The optical conductivity of a dielectric was obtained in the framework of the classical Maxwell's equation where the light behavior was attributed mainly to its wave aspect [1], [2]. We derive here the optical conductivity as due to the photon particle nature. A conductivity associated with a current perpendicular to the polarization and momentum vectors is found. We call this conductivity, the transverse optical conductivity.

Section snippets

Quantized Maxwell's equations

An attempt to include the particle (quantum) behavior of the photon was made by quantizing Maxwell's equations, where additional terms containing the Planck's constant, were found to appear. Maxwell's equations describing the classical electromagnetic fields were recently modified to read [5] $- {\tilde{\nabla}}^{*} {\tilde{F}}^{*} = μ_{0} \tilde{J} \tilde{J} = (ic ρ, \vec{J}),$ which yields $\vec{\nabla} \cdot \vec{E} = \frac{ρ}{ε_{0}} + \frac{\partial Λ}{\partial t}, \vec{\nabla} \cdot \vec{B} = 0,$ and $\vec{\nabla} \times \vec{B} = μ_{0} \vec{J} + \frac{1}{c^{2}} \frac{\partial \vec{E}}{\partial t} - \vec{\nabla} Λ, \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} .$ where $- Λ = \frac{1}{c^{2}} \frac{\partial φ}{\partial t} + \vec{\nabla} \cdot \vec{A} = \partial_{μ} A^{μ},$ The inclusion of the Λ term is tantamount to introducing the photon mass. While

Standard optical conductivity

When light incident on a material it gets attenuated. To see how this happened we expressed the refractive index as a complex number as $\tilde{n} = n + i κ$ , where n is the real(ordinary) refractive index, and κ is known as the extinction coefficient [1], [2]. A propagating plane wave along the z-direction can be described by the electric field as $\vec{E} = {\vec{E}}_{0} e^{i (kz - ω t)}$ , where ω is the wave frequency and k = ωn/c is its wave vector. A complex wave number can be expressed as $\tilde{k} = ω \tilde{n} / c$ . The electric field inside a

Concluding remarks

The way in which light interacts with matter is determined by its optical properties. But light has two behaviors; one due to its matter nature and the second due to its wave nature. An optical conductivity based on the particle nature of the photon is derived. The optical conductivity is found to depend quadratically on the refractive index, and linearly for the wave nature. The two conductivities are equal for a refractive index of n_c = 3.3. The deviation of the refractive index from this

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Short noteOn the optical conductivity

Abstract

Introduction

Section snippets

Quantized Maxwell's equations

Standard optical conductivity

Concluding remarks

Optik

Optik

Optik

Introduction to Electrodynamics

Classical Electrodynamics

Extended electrodynamics and its consequences

Mod. Phys. Lett. B

Multiferroicity: the coupling between magnetic and polarization orders

Adv. Phys.

Short note
On the optical conductivity