Review of Fundamental Relations for Optical Phenomena

Abstract

In this chapter is reviewed some fundamental relations based on Maxwell’s equations which are used for studying optical phenomena in solid state materials.

Problems

15.1

Optical constants and attenuated field amplitudes.

• (a) Starting from (15.11) show that the optical constants $\tilde{n}$ and $\tilde{k}$ (real (Re($\tilde{N}_\mathrm{complex}$)) and imaginary (Im($\tilde{N}_\mathrm{complex}$)) part of the complex propagation constant) are given by $\tilde{n}= c\sqrt{\frac{\varepsilon \mu }{2}}\left[ 1+\sqrt{1+\left( \frac{\sigma }{\varepsilon \omega } \right) ^2}\right] ^{1/2}$ and $\tilde{k}= c\sqrt{\frac{\varepsilon \mu }{2}}\left[ -1+\sqrt{1+\left( \frac{ \sigma }{\varepsilon \omega } \right) ^2}\right] ^{1/2}$.

• (b) Consider a plane wave whose electric field $\mathbf {E}$ is polarized along x and propagating along z is given by $\mathbf {E}(z,t)=E_0e^{- \frac{\omega }{c}\tilde{k} z}e^{i( \frac{\omega }{c}\tilde{n} z-\omega t)}\hat{\mathbf {i}}$. Use Maxwell’s equations to demonstrate that the respective magnetic field is given by $\mathbf {H}(z,t)=\frac{1}{c}\frac{\tilde{N}}{\mu }E_0e^{- \frac{\omega }{c}\tilde{k} z}e^{i( \frac{\omega }{c}\tilde{n} z - \omega t)} \hat{\mathbf {j}}$.

• (c) Show that the ratio between the amplitudes of the fields is given by $\frac{H_0}{E_0}= \sqrt{\varepsilon / \mu \sqrt{1+\left( \frac{\sigma }{\varepsilon \omega } \right) ^2 }}$.

• (d) Plot the fields $\mathbf {E}(z,t)$ and $\mathbf {H}(z,t)$ inside the solid material.

15.2

• (a) Show that for a very good conductor the phase difference between the magnetic field $\mathbf {H}$ and the electric field $\mathbf {E}$ is $\pi /4$. (Hint: Use the result obtained in the previous problem and write the complex quantity $\tilde{N}=\vert \tilde{N} \vert e^{i\phi }$.)

• (b) Verify the result obtained in (a) for a metal with a conductivity $\sigma \approx 10^7$ $\Omega \mathrm{m}^{-1}$.

15.3

• (a) Show that the time averaged electromagnetic energy density of a plane wave in a conducting medium is given by $(K^2/2\mu \omega ^2)E_0^2e^{-2 \frac{\omega }{c}\tilde{k} z}$.

• (b) Show that the magnetic contribution to the electromagnetic energy density is given by $(\tilde{n}^2/c^2 \mu )E_0^2e^{-2 \frac{\omega }{c}\tilde{k} z}$.

• (c) Plot the result derived in (a) and (b) as a function of z and $\omega$ thereby showing that the magnetic contribution always dominates the energy in this situation.

15.4

• (a) Show that for insulating materials, i.e., $\sigma \ll \omega \varepsilon$, the skin depth is independent of frequency and given by $\delta = \frac{c}{2 \omega \tilde{k}}=\frac{1}{\sigma }\sqrt{\frac{\varepsilon }{\mu }}$.

• (b) Estimate the skin depth in diamond and water, and give a numerical value in each case.

• (c) What is the difference in absorption at a depth of 1000 m in the ocean and in an inland lake.

15.5

When a sample consists of more than one material and is non-homogeneous, the optical properties are modified in a non-trivial manner. One approach to approximate the optical response of a heterogeneous material in terms of its microstructure is by using an effective medium theory (EMT) . EMT relates the dielectric function of a composite material with the dielectric function of the constituents materials, where the complex dielectric function, $\varepsilon _1+i\varepsilon _2$ , is that used in Maxwell’s equations and is defined as

$$D = \varepsilon E = E + 4\pi P.$$

The dielectric function of a composite material can be easily solved for two situations given by (a) and (b) below.

• (a) Show that if the internal boundaries are parallel to the applied electric field, the situation is analogous to capacitors in parallel, and the effective dielectric function is related to the dielectric function of the composites by the following equation:

  $$\varepsilon _{composite} = \sum _j f_j\varepsilon _j$$

  where $\varepsilon _j$ and $f_j$ are the complex dielectric function and volume fraction of each constituent material, j.

• (b) Show that in the opposite limit, i.e., the boundaries are perpendicular to the applied electric field, the situation is analogous to capacitors in series and the composite’s dielectric function is given by:

  $$ \varepsilon _{composite}^{-1}=\sum _j f_j \varepsilon _j^{-1} $$

These two cases define the absolute bounds to $\varepsilon$. The dielectric function of all composite materials lie on or within the region defined in the complex plane of $\varepsilon$.

15.6

• (a) Derive a formula for the normal incidence reflectivity for a thin film of thickness t and the optical constants $\tilde{n}$ and $\tilde{k}$. Assume that $\tilde{n} \gg \tilde{k}$ and t is within a factor of 2 of the wavelength of light $\lambda$.

• (b) Consider explicitly the case of light from a CO$_2$ laser ($\lambda = 10.6\,\upmu$m) and a sample thickness $t = 5\,\upmu$m and $t = 20\,\upmu$m.

• (c) Suppose that you have a superlattice of alternating thin films of dielectric constants $\varepsilon _1$ and $\varepsilon _2$, and thickness $t_1$ and $t_2$ respectively. Find the normal incidence reflectivity, neglecting optical losses (i.e., take the optical constants $k_1 = k_2 =0$).