Spherical Aberration Single Surface

Fig. illustrates the effect of spherical aberration produced by refraction at a convex spherical surface. XY is the spherical surface of radius of curvature R. O is a point object on the axis and I’ and I” are the images formed by the axial and marginal rays respectively. C is the centre of curvature of the spherical surface. OA and OD are two incident rays. AI” and DI’ are the refracted rays. AB is perpendicular to the axis and the measures the radius of the some on the refracting surface with reference to the point A.

In the Δ OAC



Similarly, in the Δ ACI”



Dividing (i) by (ii)



In the Δ OAI”,



In the Δ OAC,

OA2 = AC2 + OC2 – 2AC.OC cos θ

OA2 = R2 + (u + R)2 – 2R(u + R) cos θ       …(v)

From the third order theory



From equation (v)

Applying the sign convention,

u is –ve, R is +ve



Similarly the length of the refracted ray,



Applying the sign convention in equation (iv)



Dividing by uvR





Substituting the values of x and y



According to the first order theory

Substituting this value of 1/v in equation (ix)



For an object at infinity and for marginal rays ( u = ∞)

For paraxial rays,



From equations (xii) and (xiii), v for marginal rays is less than the paraxial rays. Hence, the marginal rays meet the axis at points nearer the surface as compared to the paraxial rays. Further, the second expression on the right hand side of equation (xi) measures the spherical aberration of the refracting surface.

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