
50 4. Introduction to Aberrations
Because the surface equation (3.5.5) is quadratic in z, the solution for z contains a
square root, as will Eq. (4.1.3) when z is eliminated. We proceed, therefore, by
expanding the square root as a power series in small quantities before substituting
into Eq. (4.1.3). Solving Eq. (3.5.5) for z gives
Z :
^ + (i+/:)^+(i+«=^+-. (4.1.4)
Substituting Eq. (4.1.4) into Eq. (4.1.3) gives
R il+K)r' {\+K){3+K)r'
^"2 4R 16R' '"' ^ ^^^
Examination of Eq. (4.1.5) shows that / = i?/2 for K =—\, a paraboloid.
Although higher power terms are not included in Eq. (4.1.5), this statement
about a paraboloid is true when all term ...