The advancement in technology of fast communication, signal processing, optical computing, and other fields, necessitates the improvement of our ability to control light[1]. Nothing can exceed the speed of light. So being able to manipulate light waves, the quantum of which is known as a photon, we can reach the limits of highspeed information processing. In electronics, understanding the crystal structure led to miniaturization and multifunctionality of devices like smart phones, etc. This is one of the motivations to explore periodic structures for the manipulation of photons. The structure may be a periodic dielectric material in 1D, 2D, or 3D. Such structures are known as photonic crystals. Discrete translational symmetry and quantum mechanics play a fundamental role in understanding the electrons in a crystal. In a periodic crystal, we shall use electromagnetism as the symmetry of discrete translation to study the behavior of photon in it. In the next section, we shall discuss the analogy between the Schr¨odinger equation,
which governs the quantum particles like electrons, and Maxwell’s equations which describe how electromagnetic waves behave. The following section describes the effect of translational symmetry. We shall justify why Bloch’s theorem, which is a statement about particles in a periodic potential, can be applied to electromagnetic waves.Then we shall try to understand what happens when plane waves are incident on linear periodic stacks of two dielectric media, often known as Distributed Bragg Reflector (DBR). We shall use the transfer matrix formalism and Bloch’s theorem to justify that unit reflectivity can be achieved. We shall show that the band gaps in a crystal are analogous to the band of unit reflectivity in a DBR.
