Derivation of the fundamental lens equation

Putting this expansion into the Laplace equation 

For charged particles travelling in a potential, energy conservation requires 

In the present context the total energy E can be identified with the kinetic 

With the help of this last relation, the time derivatives in equ. (10.32a) can be replaced by derivatives with respect to the z coordinate, thus eliminating the time dependences. Together with equ. (10.32a) (equ. (10.32b) is taken care of by implementing energy conservation which leads to equ. (10.35)) this gives 

An equivalent form of this differential equation is presented in equ. (4.38), and important aspects which can be deduced from this equation are discussed there. 

In order to derive the optical properties of an electrostatic lens, one has to establish and solve the equations of motion for a transmitted particle of mass m and charge q. Since the particle moves in the lens under the action of an electric field which can be derived from a potential p, this potential will be considered first. Assuming cylindrical symmetry around the optical axis (z-axis) and treating paraxial rays only, the potential p(p, z) depends on the cylindrical coordinates p and z. It can be expanded as a power series in p with z-dependent coefficients. Due to the rotational symmetry, only even powers of p appear in the expansion, and one has the ansatz 

The result means that the longitudinal momentum pz is given by pi w qlmffJ — O(z)] = —q2m O(z), 

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