Lens Maker’s Formula and Combination of Thin Lenses

Lens Maker’s Formula (Refraction by a Lens)

A relation between the focal length of a lens, radii of curvature of two surfaces and the refractive index of the material is called lens maker’s formula.

Consider a thin convex lens of focal length f and refractive index µ. Suppose a ray OP parallel to the principle axis incident on the lens at a small height, h above it. After refraction, the ray will pass through the focus F as shown in the figure and deviates through an angled. The angle of deviation is given by

\begin{align*} \tan \delta &= \frac hf \\\: \text {Since} \delta \: \text {is small,} \tan \delta = \delta, \text {and} \\ \delta &= \frac hf \dots (i) \\ \end{align*}

The portion PQ of the lens is a small angle prism which is formed by two tangent planes to the lens surfaces at P and Q. Since the angle of deviation in small angle prism is independent of the angle of incident, equation (i) is equal to the angle of deviation in such prism,
\begin{align*} \delta = A(\mu -1) \delta \dots (ii) \\ \text {where A is the small angle of the prism.} \\ \text {From equation} \: (ii) \text {and} \: (iii), \text {we have} \\ \frac hf &= A (\mu -1) \\ \text {or,} \: \frac 1f &= \frac Ah (\mu -1) \dots (iii) \\ \end{align*}

Let C₁ and C₂ be the centre of curvature of the two spherical surfaces. In figure PC₁ = R₁ is normal at P and OC₂ = R₂ normal at Q where C₁ and C₂ are centre of curvature of lens surfaces. Let Ï´ and Ф be the angles made by R₁ and R₂ with the principal axis. Since the angle between two tangents forming a prism is equal to the angle between two radii, so we have

\begin{align*} \angle PKC = \angle QKC_1 = A \\ \text {From the geometry, we have} \\ A &= \theta + \phi \\ \text {and for small angles,} \theta = \frac {h}{R_2} \: \text {and} \: \phi = \frac {h}{R_1} , \text {and then} \\ A &= \frac {h}{R_2} + \frac {h}{R_1} \\ \text {or,} \: \frac Ah &= \frac {1}{R_1} + \frac {1}{R_2} \dots (iv) \\ \text {Substituting equation} \: (iv) \text {in equation} \: (II), \: \text {we get} \\ \frac 1f &= \frac Ah (\mu -1) \\ \text {or,} \: \frac 1f &= (\frac {1}{R_1} + \frac {1}{R_2}) (\mu – 1) \\ \text {or,} \: \frac 1f &= (\mu – 1) (\frac {1}{R_1} + \frac {1}{R_2}) \dots (v) \\ \end{align*}

This is lens maker’s formula. Here µ is refractive index of lens material to the medium outside. So, focal length of a lens increases when it is immersed in water.

Combination of Thin Lenses

Consider two thin convex lenses L₁ and L₂ of focal length f₁ and F₂ placed coaxially in contact with each other. A point object O is placed on the principal axis at distance u from the lens L₁. In the absence of lens L₂, as the rays of light incident on the lens L₁, this lens L₁ converges the rays and thus, image is formed at point I’. So for lens L₁, we have

\begin{align*} \text {Object distance} = u \\ \text {Image distance} = v’ \\ \text {So, from lens formula,} \\ \frac {1}{f_1} &= \frac 1u + \frac {1}{v’} \dots (i) \end{align*}

When lens L₂ is placed in contact, the image at I’ acts as virtual object for lens L₂ since the converging beam is incident on L₂ as shown in the figure

\begin{align*} \text {Object distance} = -v’ \\ \text {Image distance} = v \\ \text {So, from lens formula,} \\ \frac {1}{f_2} &= \frac {1}{-v’} + \frac {1}{v} \\ \frac {1}{f_2} &= \frac {1}{v} - \frac {1}{-v} \dots (ii) \\ \text {Adding equation} \: (i) \: \text {and} \: (ii) \\ \frac {1}{f_1} + \frac {1}{f_2} &= \frac {1}{u} + \frac {1}{v’} - \frac {1}{v’} + \frac {1}{v} \\ \frac {1}{f_1} + \frac {1}{f_2} &= \frac {1}{u} + \frac {1}{v} \dots (iii) \\ \end{align*}If F is the combined foal length of two thin lenses placed in contact having object distance u and image distance v, then\begin{align*} \frac 1F &= \frac {1}{u} + \frac {1}{v} \\ \text {From equation} \: (iii) \: \text {and} \: (iv) \\ \frac {1}{f_1} + \frac {1}{f_2} &= \frac 1F \\ \text {or,} \: \frac {1}{f_1} + \frac {1}{f_2} &= \frac 1F \\ \end{align*}

This formula is applicable for any two lenses, both concave or convex or concave and convex lenses in contact. The proper sign of focal length must be inserted in the formula.