Refraction by a Lens
Figure (a)
Figure (b)
Figure (c)
The above figure shows the image formation by a convex lens.
Assumptions made in the derivation:

The lens is thin so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens.

The aperture of the lens is small.

The object consists only of a point lying on the principle axis of the lens.

The incident ray and refracted ray make small angles with the principle axis of the lens.
A convex lens is made up of two convex spherical refracting surfaces.
The first refracting surface forms image I of the object O [figure (b)].
Image I_{1} acts as virtual object for the second surface that forms the image at I [figure (c)]. Applying the equation for spherical refracting surface to the first interface ABC, we obtain
A similar procedure applied to the second interface ADC gives
For a thin lens, BI_{1} = DI_{1}
Adding equations (i) and (ii), we obtain
Suppose the object is at infinity i.e.,
OB → ∞ and DI → f
Equation (iii) gives
The point where image of an object placed at infinity is formed is called the focus (F) of the lens and the distance f gives its focal length. A lens has two foci, F and, on either side of it by the sign convention.
BC_{1} = R _{1}
CD_{2} = −R _{2}
Therefore, equation (iv) can be written as
Equation (v) is known as the lens maker’s formula.
From equations (iii) and (iv), we obtain
As B and D both are close to the optical centre of the lens,
BO = − u, DI = + v, we obtain
Equation (vii) is known as thin lens formula.