The 2D programs have an option for evaluating lens properties. These programs output the focal lengths f1 and f2 (for the object and image spaces respectively) and the mid-focal lengths F1 and F2 (which are the distances of the principal focal points from the position z=0). Spherical and chromatic aberration coefficients can also be requested, together with the Petzval integral, to give a complete specification of all the third-order aberrations of an electrostatic lens.
All the programs have an option for obtaining lens or spectrometer coefficients by direct ray tracing. In the 2D programs the dependence of 4 final ray parameters (r, z, r_angle and z_angle) on 5 initial parameters (r,z, r_angle, z_angle and kinetic energy, where r is rho or x, for cylindrical or planar symmetry respectively) can be requested. The 'optic axis' is defined by a median ray, and so can be straight or curved. The User also specifies the `order' of the coefficients.
For example if the 'order' is 1 for rho and 3 for z_angle, the program will find the dependence of each of the 4 final parameters r, z, r_angle and z_angle on rho_initial and on the cube of z_angle_initial. In other words the program would give the coefficients in the expansion
p - p_median = (p/r)*dr + (p/aaa)*da*da*da
where p represents a final parameter, and dr and da are the changes in the initial values of rho_initial and z_angle. In this example, the coefficient r/r is the linear magnification, and r/aaa gives the spherical aberration coefficient. A third term (p/raaa)*dr*da*da*da is excluded because it is a fourth order term (and the maximum order is 3).
As another example the order 1 could be chosen for z_angle and for the energy. Then the program would give the coefficients in the expansion
p - p_median = (p/a)*da + (p/e)*de + (p/ae)*da*de
where de is the change in the initial value of the energy. In this example the coefficient r/ae is the Cc*M, where Cc is the chromatic aberration coefficient.
The program can be asked to give any first order coefficient p/u, second order coefficient p/uv or third order coefficient p/uvw, except that the energy is treated only to first order.
In the 3D programs the coefficients give the dependence of 6 final ray parameters (x, y, z, cosx, cosy, cosz) on 7 initial parameters (x, y, z, cosx, cosy, cosz, en), where cosx etc are the direction cosines and en is the initial kinetic energy.