Reducing Spherical Aberration
Spherical aberration produced by lenses is minimized or eliminated by the following methods.
1. Spherical aberration can be minimized by using stops, which reduce the effective lens aperture. The stop used can be such as to permit either the axial rays of light or the marginal rays of light. However, as the amount of light passing through the lens is reduced, corresponding the image appears less bright.
2. The longitudinal spherical aberration produced by a thin lens for a parallel incident beam is given by
where x is the longitudinal spherical aberration, ρ is the radius of the lens aperture and ƒ2 is the second principal focal length.
where R1 and R2 are the radii of the curvature. For a given values of 
, ƒ2 and ρ, the condition for minimum spherical aberration is
dx/dk = 0,
Differentiating equation (i) and equating the result to zero
From equation (ii), for a lens whose material has a refractive index = 1.5, k = –1/6. Thus the lens which produces minimum spherical aberration is biconvex or biconcave and the radius of curvature of the surface facing the incident light is one sixth the radius of curvature of the other face. In general, the more curved surface of the lens should face the incident or emergent beam o flight whichever is more parallel to the axis. A lens whose R1/R2 = –1/6 is called a crossed lens. This process in which the shape of the lens is changed without changing the focal length of the lens is called bending of the lens for minimum spherical aberration. A crossed lens (R1/R2 = –1/6) is the deviation produced by the two surfaces is the same and the axial and marginal rays of light come to focus with minimum of spherical aberration. However, it should be noted that the spherical aberration cannot be completely eliminated in a lens with spherical surfaces. For a lens of refractive index 1.5, focal length 100 cm and radius of the lens aperture of 10 cm, the longitudinal spherical aberration is 1.07 cm for k = –1/6. For the same values of focal length and radius of the lens aperture, if the values of and k are 2 and + 1/5, the longitudinal spherical aberration reduces to 0.44 cm.
3. Plane convex lenses are used in optical instruments so as to reduce the spherical aberration. When the curved surface of the lens faces the incident or emergent light whichever is more parallel to the axis, the spherical aberration is minimum. The spherical aberration in a crossed lens (R1/R2 = –1/6) in only 8% less than that of a plano-convex lens having the same focal length and radius of the lens aperture. This is the reason why plano-convex lenses are generally used in place of crossed lenses without increasing the spherical aberration appreciably. This represents the variation of longitudinal spherical aberration with the radius of the lens aperture for lenses of the same focal length and refractive index.
The spherical aberration will, however, be very large if the plane surface faces the incident light. The spherical aberration is a result of larger deviation of the marginal rays than the paraxial rays. If the deviation of the marginal rays of light is made minimum, the focus ƒm for a parallel incident beam will shift towards ƒp the focus for the paraxial rays of light and the spherical aberration will be minimum.
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