## Animated Ray Diagrams

Optics illustrated with GIF animation

### Introduction

In this page animated ray diagrams are used to make graphic the results of varying a parameter in some classic examples of geometrical optics. For a thin lens, the object is brought from infinity up to contact with the lens. For an equilateral prism, the rays are drawn for all the angles of incidence for which the ray enters a face and leaves without internal reflection, passing through the position of minimum deviation. Ray diagrams show how waves propagate in a way that is easy to grasp. For the thin lens, the rays are drawn by special rules for paraxial rays using cardinal points. For the prism, the rays are traced trigonometrically, using Snell's Law. The individual frames were assembled with the aid of Microsoft GIF Animator. This page is intended to be viewed with an Internet browser.

### The Thin Lens

The principal planes (unit lateral magnification) and nodal planes (unit angular magnification) of a thin lens coincide at the vertex V. This is an approximation, but not a bad one if the object and image distances are large compared to the thickness of the lens. If the lens is in air, the focal points are equal distances from the vertex. This distance is f, the focal length of the lens. The focal length is given approximately by 1/f = (n - 1)(1/r' - 1/r"), where n is the index of refraction, and r', r" are the front and back radii, positive when convex to the left, negative otherwise. The centres of curvature of these surfaces define the optic axis of the lens.

A point a distance s to the left of the vertex and a height h above the axis is imaged by paraxial rays in a point a distance s' to the right of the vertex and a height h' above the axis, where h'/h = -s'/s is the linear magnification M and 1/f = 1/s + 1/s'. These two points are conjugate, so that one is the image of the other, depending on the direction of the rays. In the diagrams, rays are considered to proceed from left to right, but are, in fact, reversible.

The animation shows an object starting an infinite distance from the lens, and approaching until it reaches the lens. The object and image are represented by black arrows (when at finite distances), rays sufficient to locate them by blue lines. One ray usually passes through the vertex V as representing the nodal points, and the other through one or the other of the focal points, using the vertex as representing the principal points. A virtual image is shown dashed. In this case, the rays appear to pass through the image when seen from the right of the lens (in the image space), but do not physically pass through this point.

With the object at infinity, the image is at the focal point, with zero magnification. As the object approaches, the image is inverted, M<0, and grows larger as it recedes from the focal point. The image is the same size as the object when object and image are symmetrically placed with respect to the lens, at a separation equal to four times the focal length, which is the minimum distance between object and image. When the object reaches the focal point, the image is at infinity and M is infinite. As the object passes the focal point, the virtual image comes in from negative infinity with M>0, catching up with the object as it reaches the vertex.

### The Triangular Prism

The rays are shown that pass through the prism in a plane perpendicular to the refracting edge of the prism, and the prism angle is assumed to be 60°, which is often the case. Snell's Law is applied at the two faces crossed by the ray, sin i = n sin r, n sin i' = sin r, which are connected by the relation r + i' = A, the prism angle. Together with the geometry of the prism, this is enough to determine the ray path.

When the incident ray approaches the first side at grazing incidence, the angle of refraction inside the prism will be about 42° for n = 1.5. At the second face, the angle of incidence will be about 18° for a 60° prism, and the angle of refraction about 27.6°, making a total deviation of the ray direction of 57.6°. If we consider the ray reversed, then any smaller angle of incidence than 27.6° will result in total reflection at the other face.

When the ray passes symmetrically through the prism, so that it is parallel to the base of an isosceles prism, the total deviation of the ray will be stationary with respect to a change in the angle of incidence, and must be a minimum. In this case the deviation is given by 2 sin-1[n sin(A/2)] - A. For a 60° prism with n = 1.5, this is 37.2°.

In the animation, the angle of incidence is varied from the smallest angle that does not produce total reflection at the other face up to 90°, passing through minimum deviation. It can be shown that at minimum deviation, a homocentric bundle of rays (one which radiates from a single point) remains homocentric on refraction through the prism. This is not true away from minimum deviation. Minimum deviation in a prism explains several of the halo phenomena caused by ice crystals in the atmosphere, since the intensity of rays refracted in randomly oriented prisms is a maximum near minimum deviation.