4.3. Common Matrices

● Translation (free space): T(d)=[1d01]T(d) = \begin{bmatrix} 1 & d \ 0 & 1 \end{bmatrix}

In geometric optics, when light travels through free space, we can use a matrix to describe how it moves. The matrix for translation in free space, denoted T(d), shows how the ray is translated a distance 'd'. Here, '1' in the first row and first column indicates there are no changes to the slope of the ray as it travels through free space. The 'd' represents the distance of translation, meaning the ray moves forward without changing angle.

Imagine a car moving straight along a road. The distance it covers is similar to 'd' in our translation matrix. Just as the car moves forward without changing direction, the ray does the same in free space.

● Refraction at spherical surface: R(n1,n2,R)=[10(n1−n2)n2Rn1n2]R(n_1, n_2, R) = \begin{bmatrix} 1 & 0 \ \frac{(n_1 - n_2)}{n_2 R} & \frac{n_1}{n_2} \end{bmatrix}

When a ray of light passes through a spherical surface (like a lens), it bends due to refraction. The matrix R(n1, n2, R) describes this behavior. The surface's curvature and the indices of refraction (n1 and n2) of the two media affect how much the light bends. The first row shows that the light’s horizontal position does not change immediately. The second row calculates the new angle of the ray based on the difference in refractive indexes divided by the radius of curvature 'R'.

Think about how a straw looks bent when placed in a glass of water. Just like the straw appears bent because of the different refractive indexes of air and water, the matrix shows how light is refracted at the spherical interface.

● Reflection and lenses have corresponding matrices.

For lenses and reflective surfaces (like mirrors), specific matrices are defined to describe how rays behave when they encounter these surfaces. Each type of matrix alters parameters of the ray, like its height and angle. For example, a matrix for a concave mirror will differ from that of a convex lens in how it manipulates the light's path. While the details may vary, they essentially serve the same purpose of predicting how a ray will change direction following interaction with the surface.

Imagine looking into a spoon. When you look at your reflection, the spoon's curved surface acts like a mirror, bending light rays to create an image. Similarly, a matrix calculates how these light rays interact with the spoon’s surface, just like it would for a lens used in glasses.

● System Matrix: For a multi-element system, multiply all individual matrices in reverse order of light travel: M=MN⋅MN−1⋅…⋅M1M = M_N \cdot M_{N-1} \cdot \ldots \cdot M_1 Use the result to find the final position and angle of the emerging ray.

When light passes through a series of lenses or reflective surfaces, we can analyze this complex optical system using a single matrix, known as the system matrix. To create this matrix, we multiply the individual matrices corresponding to each optical element, but we do so in reverse order because we need to account for the path of light as it travels. The resulting matrix helps us determine where the light ray exits the system and at what angle.

Consider a multi-lane highway where cars enter at different points. To calculate where a specific car exits after passing through various intersections (like lenses or mirrors), you can diagram the route. Each intersection corresponds to a matrix, and by combining them in reverse order of travel, you can predict the car's exit point effectively, just as matrices help us analyze light rays in an optical system.

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