Lens Maker’s Formula

The Lens Maker's Formula is a fundamental equation in optics that describes the relationship between the focal length of a lens, the curvature of its surfaces, and the refractive index of the material from which the lens is made. It provides a way to calculate the focal length of both convex and concave lenses based on their geometry and material properties. This formula is crucial in the design and manufacturing of optical instruments, including eyeglasses, cameras, microscopes, and telescopes, as it helps determine how a lens will bend and focus light.

1.0Definition Lens Maker’s Formula

• This formula connects the focal length of a lens with the refractive index of the lens material and the radius of curvature of its two surfaces.
• This formula is referred to as such because it is used by manufacturers to design lenses with a specific focal length, using glass with a given refractive index.

2.0Cartesian Sign Conventions For Spherical Lenses

1. All distances are measured from the lens's optical center.
2. The distance measured in the direction of the incident light is considered positive.
3. The distance measured opposite to the direction of the incident light is considered negative.

Note: (1) For converging Lens (convex Lens in air), focal length is positive.

          (2) For diverging Lens (concave Lens in air), focal length is negative.

3.0Key Points to Derive Lens Maker’s Formula

1. The lens is considered thin, allowing the distances measured from its surfaces to be taken as equal to those measured from its optical center.
2. The object is a point located along the principal axis.
3. The lens has a small aperture.
4. All the rays are paraxial, meaning they make very small angles with both the normals to the lens surfaces and the principal axis.
5. $R_1$ is the radius of curvature of that surface on which light rays incident initially and R2  is the radius of curvature of that surface on which light rays incident after refraction from the first surface.
6. $v,u,R_1,R_2,f$ (whatever is given in question) should be put along with a sign.

4.0Derivation of Lens Maker’s Formula

Spherical Refraction from 1st surface,

$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$

${\mu }_2={\mu }_l,{\mu }_1={\mu }_s,u=u,v-v_1,R=R_1$

$\frac{{\mu }_l}{v_1}-\frac{{\mu }_s}{u}=\frac{{\mu }_l-{\mu }_s}{R_1}$…………(1)

Spherical Refraction from 2ndsurface,

$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$

${\mu }_2={\mu }_s,{\mu }_1={\mu }_l,u=v_1,v=v,R=R_2$

$\frac{{\mu }_s}{v}-\frac{{\mu }_l}{v_1}=\frac{{\mu }_s-{\mu }_l}{R_2}$………..(2)

Add equations (1) and (2)

$\frac{{\mu }_s}{v}-\frac{{\mu }_s}{u}=\frac{{\mu }_l-{\mu }_s}{R_1}+\frac{{\mu }_s-{\mu }_l}{R_2}$

$\frac{{\mu }_s}{v}-\frac{{\mu }_s}{u}=({\mu }_l-{\mu }_s)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

$\frac{1}{v}-\frac{1}{u}=\left(\frac{{\mu }_l}{{\mu }_s}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

If object placed at infinity i.e. u = −∞ then v = f

$\frac{1}{f}-\frac{1}{-\infty }=\left(\frac{{\mu }_l}{{\mu }_s}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

$\frac{1}{f}=\left(\frac{{\mu }_l}{{\mu }_s}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\Rightarrow$Lens Maker's Formula

5.0Special Case for Lens Maker’s Formula

1. Sun-glasses or goggles: radii of curvature of two surfaces are equal with centres of curvatures on the same side of the lens.

$R_1=R_2=+R$, So,$\frac{1}{f}=(\mu -1)[\frac{1}{R}-\frac{1}{R}]$

$\Rightarrow \frac{1}{f}=0\Rightarrow f=\infty$and P=0 $(\therefore P=\frac{1}{f})$

Sun-glasses or goggles have no power.

2. If refractive index of medium < refractive index of lens

If ${\mu }_s<{\mu }_l$ then $\frac{{\mu }_l}{{\mu }_s}>1$or $(\frac{{\mu }_l}{{\mu }_s}-1)>0$

A convex lens behaves as a convergent lens. While concave lens behaves as a divergent lens.

3. Refractive index of medium = Refractive index of lens $({\mu }_s={\mu }_L)$

$\frac{1}{f}=(\frac{{\mu }_1}{{\mu }_s}-1)(\frac{1}{R_1}-\frac{1}{R_2});\frac{1}{f}=0\Rightarrow \infty$and P=0

Lens will behave as a plane transparent plate

4. Refractive index of surrounding medium > Refractive index of lens

If ${\mu }_s>{\mu }_l$ then $\frac{{\mu }_l}{{\mu }_s}<1$or $(\frac{{\mu }_l}{{\mu }_s}-1)<0$

Convex lens will behave as a divergent lens and concave lens will behave as a convergent lens. An air bubble in water behaves as a concave lens.

6.0Optical Power In Thin Lens

• Optical power refers to the ability of a lens, mirror, or optical system to converge or diverge light.
• Optical power is also referred to as dioptric power, refractive power, focusing power, or convergence power.
• The unit of optical power is a diopter ($m^{-1}$).

    $P=\frac{{\mu }_s}{f}$

   $\frac{1}{f}=\left(\frac{{\mu }_l}{{\mu }_s}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

$P=({\mu }_l-{\mu }_s)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Example-1. Find out optical power of equiconvex lens of refractive index  and radius of curvature R.

Solution:

$\frac{1}{f}=\left(\frac{{\mu }_{\ell }}{{\mu }_s}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)=\left[\frac{\mu }{1}-1\right]\left(\frac{1}{R}-\frac{1}{-R}\right)$

$f=\frac{R}{2(\mu -1)}$

$P=\frac{2(\mu -1)}{R}\Rightarrow$Power is +ve, so the lens is converging.

Example-2.Find out optical power of the lens (l= 1.5).

Solution:

$\frac{1}{f}=\left(\frac{{\mu }_{\ell }}{{\mu }_s}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

$\frac{1}{f}=\left(\frac{1.5}{1}-1\right)\left(\frac{1}{20}-(-\frac{1}{30})\right)$

$f=24cm$

$P=\frac{1}{f}=\frac{100}{24}=\frac{25}{6}D$

7.0Sample Questions On Lens Maker’s Formula

Q-1.If focal length of convex Lens $({\mu }_l=\frac{3}{2})$in air is 20 cm . Find the focal length of the same lens in water. $({\mu }_w=\frac{4}{3})$

Solution:

$\frac{1}{f}=\left(\frac{{\mu }_{\ell }}{{\mu }_s}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

In Air: $\frac{1}{20}=(\frac{3}{2\times 1}-1)(\frac{1}{R_1}-\frac{1}{R_2})$……….(1)

In water:$\frac{1}{f_w}=(\frac{3}{2\times \frac{4}{3}}-1)(\frac{1}{R_1}-\frac{1}{R_2})$……..(2)

by dividing equation (1) by (2)

$\frac{f_w}{20}=\frac{\frac{1}{2}}{\frac{1}{8}}\Rightarrow f_w=80cm$