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

                

            

• 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

                    
            

                            ...(1)

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

            
            

                            ...(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

            
            

            
            

            
        

            

• 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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