Aplanatic points of a spherical refracting surface

Sacademy May 19, 2023 1 min read

Aplanatic points of a spherical refracting surface

$\mu _{1}h_{1}sin\theta _{1}=\mu _{2}h_{2}sin\theta _{2}$

$\therefore\;\;\;\;\; \frac{sin\theta _{2}}{sin\theta _{1}}= \frac{\mu _{1}h_{1}}{\mu _{2}h_{2}}$

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.

$\frac{sin\theta _{1}}{PC}= \frac{sini}{OC}$

$or\;\;\;\;\; \frac{sin\theta _{1}}{R}= \frac{sini}{R/\mu }$

$or\;\;\;\;\;sin\theta _{1}=\mu{sini}$ …(1)

$\frac{sini}{sinr}=\frac{1}{\mu }$

$\therefore\;\;\;\;\;sinr=\mu sini$ …(2)

sin θ₁ = sin r ⇒ θ₁ = r

$\frac{CO}{CP}=\frac{CP}{CI}$
$or\;\;\;\;\;CI=\frac{(CP)^{2}}{CO}$
$or\;\;\;\;\;CI=\frac{R^{2}}{R/\mu }$
$or\;\;\;\;\;CI=\mu R$

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.

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