What are the equations of Bragg mirror?

Bragg Mirror Equations and Explanation

A Bragg mirror, also known as a distributed Bragg reflector (DBR), is a structure that reflects particular wavelengths of light and transmits others. This is achieved through the interference of light waves in a periodic structure of alternating layers of different refractive indices. The fundamental equations governing the operation of Bragg mirrors are derived from the principles of wave interference and optical path difference.

Key Equations

1. Bragg's Condition: nλ = 2d sin(θ)
   Where n is the order of reflection (an integer), λ is the wavelength of light in vacuum, d is the period of the Bragg mirror (sum of the thicknesses of one pair of layers), and θ is the angle of incidence of the light wave.
2. Reflectance: The reflectance of a Bragg mirror for light at the Bragg wavelength can approach 100%, especially for mirrors with a large number of periods. The exact reflectance depends on the number of layers, the contrast in refractive indices between the layers, and the wavelength of the incident light.

Explanation

The operation of a Bragg mirror is based on constructive interference of light waves reflected at the interfaces of different layers. When light of certain wavelengths meets the Bragg condition, it is reflected efficiently, leading to high reflectance at those wavelengths. The periodic structure creates a photonic bandgap for certain wavelengths, which cannot propagate through the mirror, thus being reflected. The specific wavelengths that are reflected are determined by the Bragg condition, which depends on the periodicity of the structure and the angle of incidence of the light.

The design of a Bragg mirror involves choosing materials with different refractive indices and layer thicknesses such that the desired wavelengths are reflected. By adjusting these parameters, Bragg mirrors can be tailored for specific applications in optical devices, such as lasers, sensors, and reflective coatings.