Aplanatic points of a spherical refracting surface | Optics | General theory of image formation

Aplanatic points of a spherical refracting surface

From Abbe’s sine condition

$https://latex.codecogs.com/svg.image?\mu _{1}h_{1}sin\theta _{1}=\mu _{2}h_{2}sin\theta _{2}$

$https://latex.codecogs.com/svg.image?\because \frac{sin\theta _{2}}{sin\theta _{1}}= \frac{\mu _{1}h_{1}}{\mu _{2}h_{2}}$

If this ratio is constant for a particular surface, then the surface is known as aplanatic surface.
An aplanatic surface is a surface which forms a point image of a point object situated on its axis.
The image formed by aplanatic surface is free from optical aberrations.

Using sine law in △OPC

$https://latex.codecogs.com/svg.image?\frac{sin\theta _{1}}{PC}= \frac{sini}{OC}$ $https://latex.codecogs.com/svg.image?\Rightarrow \frac{sin\theta _{1}}{R}= \frac{sini}{R/\mu }$

$https://latex.codecogs.com/svg.image?sin\theta _{1}=\mu{sini}$ ...(1)

Since refraction is taking place from denser to rarer, so from Snell's law

$https://latex.codecogs.com/svg.image?\frac{sini}{sinr}=\frac{1}{\mu }$

$https://latex.codecogs.com/svg.image?sinr=\mu sini$ ...(2)

Now from above two equations, we get

sin θ₁ = sin r ⇒ θ₁ = r

In ΔIOP,

θ₁ = θ₂ + (r - i) ⇒ θ₂ = i

From ΔOCP and ΔICP

$https://latex.codecogs.com/svg.image?\frac{CO}{CP}=\frac{CP}{CI}$

$https://latex.codecogs.com/svg.image?CI=\frac{(CP)^{2}}{CO}$

$https://latex.codecogs.com/svg.image?CI=\frac{R^{2}}{R/\mu }$

$https://latex.codecogs.com/svg.image?CI=\mu R$

This relation does not depends on θ₁ and θ₂.
The beam divergent from O at any angle is definitely converged at I.
A point object O situated at a distance R/µ from C will be imaged on I at µR distance from C.
Since image is free from θ₁ and θ₂, so the image will be free from optical defects.

To know more about Aplanatic points of a spherical refracting surface click on https://youtu.be/IFrvMhwgQbQ

Search This Blog

खनन और खनिज उद्योगों में पर्यावरणीय स्थिरता विषय पर विशेषज्ञों का मंथन