**Table of Contents:**

**ELECTROMAGNETIC FORMULATION**

In a periodic medium, the dielectric constant and susceptibility become periodic functions of x, given as:

\begin{equation} \epsilon(x) = \epsilon(x+a) \label{period_di} \end{equation}

\begin{equation} \mu(x) = \mu(x+a) \end{equation}

where “$a$” is a lattice vector. Recalling the definition of the index of refraction $n$ as $n = \frac{\epsilon}{\epsion_0}$, we can see then, using Equation \eqref{period_di}, how the index of refraction is modulated in a periodic medium. The physical interpretation of these equations is that the crystal medium “looks” the same at points $x$ and $x+a$.

As we know from the “Classical Optics” section, the propagation of a monochromatic beam of light in a medium is described by the wave equation,

\begin{equation} \nabla \times (\nabla \times E) – \omega^2\mu\epsilon{E} = 0 \end{equation}

In periodic media, the normal modes of this equation are given by:

\begin{equation} E = E_K(x)e^{-iK\cdot{x}} \label{period_E}\end{equation}

\begin{equation} H = H_K(x)e^{-iK\cdot{x}} \label{period_H}\end{equation}

where the coefficients $E_K(x)$ and $H_K(x)$ are both periodic functions of $x$, given as:

\begin{equation} E_K(x) = E_K(x+a) \end{equation}

\begin{equation} H_K(x) = H_K(x+a) \end{equation}

This form of solution in periodic media is known as the Bloch theorem, and “$K$” represents the Bloch wave vector. The Bloch wave vector is essentially a reciprocal lattice vector. Notice that each equation consists of the plane wave solution from homogeneous media multiplied by a periodic function that modulates the plane wave. This solution is commonly used in solid state and condensed matter physics to describe the concept of electronic band structures. As such, if you would like to further understand what these equations mean for the atomic scale, I would suggest finding a good book on either of these subjects. We will be going into further detail and proving Equations \eqref{period_E} and \eqref{period_H} in the following section.

**ONE-DIMENSIONAL PERIODIC MEDIA**

In modern optics, we can often make the simplifying assumption that we are dealing with a one-dimensional periodic medium such that:

\begin{equation} \epsilon(z) = \epsilon(z+l\Lambda) \end{equation}

where “$\epsilon$” is the permittivity tensor, “$\Lambda$” is the period, and “$l$” is an integer number that keeps track of the period number. For the following analysis, assume one-dimensional periodicity in a manner similar to the below figure.

As can be seen from the figure above, the one-dimensional direction of the periodicity is labeled as “$\Lambda$”. Now, if we have a beam of laser light incident on this medium, it will be reflected and refracted at each interface as described by the Bragg condition. Fun fact: the Nobel Prize in 1915 was given to a father-son duo with the last name of Bragg who came up with this condition after unexpected experimental observation of strong reflection peaks. Lawrence Bragg, the son, holds the record for the youngest person to win the Nobel Prize in Physics as 25 (and what are you doing with your life?). Back to the Bragg condition, the condition describes the conditions necessary for constructive interference between reflected waves. It is given as:

\begin{equation} n\lambda =2dsin(\theta) \label{Bragg}\end{equation}

or, alternatively in the notation we have used previously,

\begin{equation} m\lambda = 2\Lambda{cos(\theta)} \end{equation}

We will derive Equation \eqref{Bragg} here so you can get a sense of what this physically means, but we will be actually applying it a little later on. To derive the Bragg condition heuristically, we can use the following figure.