Optics Problem #0064


614.

Problem 44.15 (RHK)

We have to show that the focal length f for a thin lens whose index of refraction is n and which is immersed in a fluid whose index of refraction is is given by

Solution: Click For PDF Version
We will derive the result by considering refraction at two curved surfaces of radius of curvature

and

, respectively. For determining the focal length of the lens we consider a ray parallel to the optic axis and find the distance from the lens where this ray after undergoing two refractions crosses the optic axis. We will calculate the focal length f in thin lens approximation. Without loss of generality, we assume that the index of refraction of the glass n is less than the index of refraction

of the medium in which the lens is immersed.

For showing refractions at the two curved surfaces we will draw enlarged diagrams, as shown below.

Let the angle of incidence of the ray at A be . By Snell’s law the angle of refraction and the angle of incidence are related as
.
For a paraxial ray and in thin lens approximation we will assume that both angles and are small, and we can use the approximations

In the triangle the exterior angle will be equal to the sum of angles and . We have
.
From Snell’s law relation, we note that
.
We therefore have the relation
.
As
,
we get the relation
.
We consider next the refraction at the second curved surface, whose centre of curvature is and the radius of curvature is .

In this case Snell’s law connects angle of incidence and the angle of refraction . We have
.
In the small angle approximation, we get
.
As is the exterior angle of , we have
.
And, as is the exterior angle of triangle , we have
.
Using the result , we have

In the small angle approximation, we thus get

We use the sign convention as shown in the following figure:

As is in the R-side, is positive and ; and as is in the V-side, is negative and . Also, by definition, , the focal length of the thin lens. We therefore have the result