Ray Optics

What is ray optics?

Ray optics, commonly known as geometrical optics, are considered light as a ray rather than a wave. This gives us an advantage in the calculation regarding optics. By considering light as a ray, we can use simple geometry to calculate certain properties of light. Using ray optics, we can explain certain properties of light such as reflection, refraction, absorption, etc.

The ray that we use to represent light here could be a straight line or a curve with some direction perpendicular to the wavefront of the light.

History

The study of light, known as optics, began with the development of lenses in ancient Egypt, followed by the development of geometrical optics in the ancient Roman empire. The term optics comes from the Greek term ‘optica’, which loosely translates to appearance or look. The early study of optics is commonly referred to as classical optics, and the current study of optics, including wave and quantum optics, is called modern optics.

Reflection

Reflection is one of the main chapters of ray optics. Reflection happens when light touches a surfaces. For example, the reflection of light by a plane mirror. The properties of reflection of light are governed by the law of reflection. The law of reflection states the following:

1. The angle of incidence and the angle of reflection with the normal to the surface are always equal.

2. Both incident and reflected rays will always lie in the same plane.

This can be mathematically represented using simple geometry. But in the case of irregular or curved surfaces, we have to use ray tracing and the above law to model their reflective properties. 

Refraction

Refraction is the phenomenon of bending of light when light travels into a medium with a change in refractive index. It can be explained using the Snell’s law, which states that if a ray of light passes from a medium of refractive index n1 to a medium of refractive index n2 at angle i, then the angle of refraction r is given by the relation  

i sin (n1) = r sin (n2)

Fermat’s principle

One of the main pillars of Ray optics is the concept of Fermat’s principle, which states that the optical path length traversed by a ray of light to travel from one point to another is an extremum (either maximum, minimum, or stationary ) in comparison to paths in the certain regular neighbourhood of it. Here the condition for the regular neighbourhood is that the path difference must be negligible compared to the path length itself.

The framework of  Fermat’s principle comes from the variational principle in mechanics.

One important thing about Fermat’s principle is that we can use Fermat’s principle to prove other theorems, such as Snell’s theorem.

Consider a light beam travelling from one medium to another medium. Here we have a ray of light passing from one medium to another and is incident on the surface at an angle i with the normal to the surface. This ray of light gets refracted at an angle r. Let us now consider two points A and B, at distance h1 and h2, respectively, from the surface on the ray of light. Now let n1 and n2 be the refractive index of the two mediums.

Now the optical path length between A and B can be written as

L = n1(AO) + n2(OB)

Therefore from the figure, using Pythagoras theorem, we can write,

L = n1(h12 + x2)1/2 + n2(h22 + (d-x)2)½

Now according to Fermat’s principle, the value of L as a function of x is an extremum. So if we differentiate L with respect to x we get zero.

d Ld x = 0

⇒   d Ld x = (n1x/(h12 + x2)½) – (n2(d-x)/(h22 + (d-x)2)½) = 0 

⇒ n1sin(i) -n2sin(r) = 0

⇒ n1sin(i) = n2sin(r)

This is the equation for snell’s law.

Note: The relation between the speed of light in a medium,v and its refractive index, n, is given by

v = c / n

where, c is the speed of light in free space. 

Matrix method 

In ray optics, the matrix method is one of the most efficient methods used to analyse the properties of translation, reflection, and refraction of a ray of light. Using this method, we can derive many important equations related to ray optics.

The general form of the matrix method will be,

( output ray ) = (system matrix) (input matrix ) 

Conclusion 

Ray optics helps us to understand the nature of light without going into the realm of quantum mechanics and waves. We can use geometry to understand the path of the optical ray. Theories like Fermat’s principle, Snell’s law etc., helps us in understanding the nature of light on interaction with surfaces. Snell’s law can be derived using Fermat’s principle. Snell’s law can be used to study the lenses and their properties. The speed of the light in a medium is governed by the refractive index of the material.