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Lens Aberrations Theory

To understand satisfactorily the theory of lens aberrations, it is necessary to start with the expansion of the sines of angles into a power series. According to Maclaurin’s theorem the expansion of sin θ is given by

When the value of θ is small, the series is a rapidly converging one i.e. the value of any term is smaller than the preceding one. In case the slope angle is small, sin θ = θ, approximately. The equations developed on the basis that the sines of the angles are equal to the angles form the basis of the first order theory.

In fig. for small values of θ, the height of the perpendicular AC can be taken approximately equal to the length of the arc AB.

Table below gives the variation of sin θ with increasing angle.

The differences in the values of sin θ and θ – θ³/(3 !) is much smaller than sin θ and θ.