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 $F$.

Lens Maker's Formula Figure (1)

​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.

Lens Maker's Formula Figure (2)

​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]