Dielectric refraction

Abstract

In this work we study and analyze in detail the characteristics of the modulus and phase of the reflection and transmission coefficients in interfaces between isotropic media, when the incident electromagnetic wave is propagating from a transparent medium towards an active one. We also demonstrate analytically that Amplified Reflection is impossible if semi-infinite media are involved. Due to these coefficients, the oscillatory or monotonic character of the phase difference between p and s modes is shown as a function of the angle of incidence for different active media. A qualitative and quantitative comparison between our own results and those obtained by many authors on absorbing media is made. We consider that this work can clarify some aspects that can contribute in the use of ellipsometric techniques for the determination of optical properties of active media. The index of refraction tells you how much the speed of light is reduced through a material, compared to the vacuum speed.
The dielectric constant tells you how much the electric field is attenuated in a substance, compared to vacuum.
Light is an electromagnetic wave, therefore we conclude (without any other knowledge of mathematics) that there has to be some connection between the two.

Mathematically, n, the dielectric constant, is the ratio c/v, where c is the vacuum speed of light and v is the speed of light in the material. Or, since $c = \sqrt{(} ϵ_{0} μ_{0})$ ϵ0μ0), $n = \sqrt{(} ϵ μ / (ϵ_{0} μ_{0}))$ ϵμ/(ϵ0μ0)). Here $ϵ a n d μ$ are the electric permittivity magnetic permeability, respectively, while the subscript of 0 indicates that those are the values in vacuum.
The dielectric constant is defined as $κ = ϵ / ϵ_{0}$ ϵ/ϵ0.

So one can rewrite $n = \sqrt{(} κ μ / μ_{0})$

Dielectrics and Optics

Basics

	· This subchapter can easily be turned into a whole lecture course, so it is impossible to derive all the interesting relations and to cover anything in depth. This subchapter therefore just tries to give a strong flavor of the topic.
	· We know, of course, that the index of refraction n of a non-magnetic material is linked to the dielectric constant å_r via a simple relation, which is a rather direct result of the Maxwell equations.

But in learning about the origin of the dielectric constant, we have progressed from a simple constant år to a complex dielectric function with frequency dependent real and imaginary parts.

1. What happens to n then? How do we transfer the wealth of additional information contained in the dielectric function to optical properties, which are to a large degreee encoded in the index of refraction?

· Well, you probably guessed it: We switch to a complex index of refraction!

2. But before we do this, let's ask ourselves what we actually want to find out. What are the optical properties that we like to know and that are not contained in a simple index of refraction?

3. Lets look at the paradigmatic experiment in optics and see what we should know, what we already know, and what we do not yet know.

What we have is an electromagnetic wave, an incident beam (traveling in vacuum to keep things easy), which impinges on our dielectric material. As a result we obtain a reflected beam traveling in vacuum and a refracted beam which travels through the material. What do we know about the three beams?

· The incident beam is characterized by its wavelength ë_i, its frequency í_i and its velocity c₀, the direction of its polarization in some coordinate system of our choice, and the arbitrary angle of incidence á. We know, it is hoped, the simple dispersion relation for vacuum.

c₀

í_i · ë_i

Ø The incident beam also has a certain amplitude of the electric field (and of the magnetic field, of course) which we call E₀. The intensity I_i of the light that the incident beams embodies, i.e. the energy flow, is proportional to E₀² - never mix up the two!

Ø The reflected beam follows one of the basic laws of optics, i.e. angle of incidence = angle of emergence, and its wavelength, frequency and magnitude of velocity are identical to that of the incident beam.

A bit more involved is another basic relations coming from the Maxwell equations. It is the equation linking c, the speed of light in a material to the material "constants" å_r and the corresponding magnetic permeability ì₀ of vacuum and ì_r of the material via

· Since most optical materials are not magnetic, i.e. ì_r = 1, we obtain for the index of refraction of a dielectric material our relation from above.

c₀

(ì₀ · ì_r · å₀ · å_r)^1/2

(ì₀ · å₀)^1/2

å_r^1/2

· Consult the basic optics module if you have problems so far.