Refraction at Spherical Surfaces

# Lens Maker’s Formula

Consider a thin lens whose optical centre is $C$ and is made of material of refractive index $μ$. Let $f$ be the focal length of the lens and $R_1$ and $R_2$ be the radii of curvature of its two surfaces. Lens Maker’s formula gives a relation between the focal length, radii of curvature and the refractive index of the lens.

Then, the thin lens behaves as a prism of small angle. We know, the deviation $(δ)$ produced by a prism of small angle (say $A$) is independent of the angle of incidence and is given by, $δ=A(μ-1)\text{ __(1)}$

Let a ray of light $OP$ parallel to the principal axis is incident on the lens at a height $h$ above the principal axis. Then, the ray gets refracted and the refracted ray cuts the principal axis at its focus $F$.

​If deviation produced is $δ$, then, $\tan δ=\frac{PC}{CF}$ $∴\tan δ=\frac{h}{f}$

Since the lens is thin, $δ$ will be very small and hence, $\tan δ ≈ δ$ $∴δ=\frac{h}{f}\text{ __(2)}$

From $(1)$ and $(2)$, $\frac{h}{f}=A(μ-1)$ $∴\frac{1}{f}=\frac{A}{h}(μ-1)\text{ __(3)}$

Let $C_1$ and $C_2$ be the centres of curvature of the two surfaces of the lens. Then, $XC_2$ and $YC_1$ are drawn which are normal to the lens at the point $X$ and $Y$ respectively.

​In quad. $TXPY$, $∠XTY+∠XPY=180 \text{ [∵∠PXT+∠PYT=180]}$ $∴A+∠XPY=180\text{ __(4)}$

Also, $∠XPY+∠YPC_2=180\text{ __(5) [Being straight angle}]$ From $(4)$ and $(5)$, $∠YPC_2=A$ Similarly, $∠XPC_1=A$

Now, $A=α+β\text{ __(6)}$ and, $\tan α=\frac{PC}{C_1C}=\frac{h}{R_1}$ $\tan β=\frac{PC}{CC_2}=\frac{h}{R_2}$

Since the angle of $α$ and $β$ are very small, $α ≈ \tan α$ and $β≈\tan β$. $∴α=\frac{h}{R_1} \text{ and, } β=\frac{h}{R_2}$

From $(6)$, $A=\frac{h}{R_1}+\frac{h}{R_2}$ $\frac{A}{h}=\frac{1}{R_1}+\frac{1}{R_2}$

From $(3)$ and $(7)$, $\frac{1}{f}=(μ-1)\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$ This is the Lens Maker’s formula.

[Also See: Dispersion]