# Translation Matrix

The matrix Tm is called Translation Matrix.

The matrices can be used to find the position of the final image. The method of writing linear equations for the coordinates of the paraxial rays and then transforming them into the matrices for the purpose of optical analysis is called Matrix method in optics.

When a ray travel in a homogeneous medium, the height (y) of the ray varies but the angle (α) with the line parallel to the optical axis or principal axis remains the same. It is known as the translation of the ray.

## Derivation of Translation Matrix

Let ray AB be traveling in a homogeneous medium of refractive index n. Let (y1, α1) be the coordinates of input ray at A and (y2, α2) be the coordinates of output ray at B. Let x1 and x2 be the position of point A and B on the x-axis.

As the ray is moving in a homogeneous medium.

From the given figure, we find that

Substituting this in equation (2), we get

Equation (3) and (1) can be rewritten respectively as

Translating these equations into matrix form, we get

This matrix equation represents the translation of the ray from coordinates (y1, α1) to (y2, α2).

### Output Matrix

This matrix is called output matrix. It gives the final location of the light ray.

### Input Matrix

This matrix is called the input matrix. It gives the initial location of the light ray.

This matrix Tm is called Translation Matrix. It specifies the translation of the light rays.

Hence equation (6) can be written as

Output matrix = Translation matrix × Input matrix

O.M = Tm × I.M

Determinant of Tm is given by

## FAQ on Translation Matrix

### What is meant by transformation matrix?

When a ray travel in a homogeneous medium, the height (y) of the ray varies but the angle (α) with the line parallel to the optical axis or principal axis remains the same. It is known as the translation of the ray.

### Why do we need rotation matrix?

The matrices can be used to find the position of the final image.