By Peter W. Hawkes (Ed.)

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**Extra info for Advances in Electronics and Electron Physics, Vol. 76**

**Example text**

They permit two approaches to the calculation of aberration characteristics: calculation of aberration coefficients in terms of the theory described above and direct integration of the exact equation of motion. The latter has the advantage of including all the aberrations present in the system. The aberration value is defined as the difference between the exactly calculated nonparaxial trajectory and the trajectory derived from the paraxial equation. However, this involves errors resulting from the subtraction of nearly equal quantities.

The quadratures in the polynomial coefficientsdo not depend on the object position but are determined only by the lens parameters. The dependence on the object position is introduced in the aberration coefficients in terms of magnifications. , 1974). Since in practical applications lens systems are usually preferred to single lenses, it is important to know the principle of aberration summation. Down to the third-order accuracy, the total aberration of the system is a sum, each term of which is equal to the product of a single lens aberration and linear magnification of the subsequent part of the system.

If x,(z) becomes zero at z = zix, at this point x(zix)= 0 for any xb. Therefore, the projections of all trajectories outgoing at a point on the axis z = zo will intercept after passing through the lens at the same point z = zix (Fig. 5). A planar beam emerging from the point z = zo and lying in the x0z-plane will converge at the point z = zix,so we can speak of its focusing at this point. Consider a similar beam emerging from the point P, which is at a distance xo from the lens axis. From Eq. (42) we get It can be seen from Eqs.