Matrix formulation in geometrical optics: Translation refers to ray tracing in a single homogeneous medium. Ray coordinates are transformed from one point to another through a powerful matrix formulation of optical systems. Photo Credit: mdashf.org

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Matrix formulation in geometrical optics

Topics covered in this lecture

Ray tracing
a. Translation matrix

b. Refraction matrix

c. System matrix 

In this lecture, we will discuss about one of the most interesting and powerful methods in Geometrical Optics. As we have discussed here (https://mdashf.org/2017/02/25/fermats-principle-a-lecture-in-optics/), geometrical optics is that segment of optics in which we are limited to a situation when the wavelength of light is negligible eg λ is insignificant compared to the size of the objects light interacts with.

As a consequence light can be considered as rays or geometrical straight lines and the nuances of light as wave undulations can be postponed to a happy hour.

Ray Tracing

Any general optical system has a ray which can be traced through two basic types of traversal of the ray: Translation and Refraction. The law of refraction is thus the central tool for ray-tracing.A ray can be described in an optical system by its coordinates which we will define soon.

Our goal is to find the matrix which governs the displacement of the ray from one coordinate to another coordinate of the ray, as the ray travels from one geometric point to another. This will enable us to study simple as well as much more complicated systems in the most effective and powerful way as we will see.

Lets discuss the basic matrices available for ray tracing when the  ray travels from one coordinate to another in two cases.

I. Translation Matrix for simple straight line motion in a homogeneous medium.

II. Refraction Matrix for refraction at the interfaces of two different media.

In general therefore the total traversals of the ray can constitute of any number of translations or refraction. A reflection would merely be two translations and a general refraction might be construed from refraction as well as translations.

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