CHAPTER 2 

 GEOMETRICAL ELECTRON OPTICS 



ALTHOUGH the subject of electron optics has been ex- 

 haustively treated in several well-known textbooks,^' ^' ^' ^'^ 

 it will be useful to recapitulate its principles to an extent 

 sufficient to the understanding of the operation of electron lenses. 

 This can be done much more simply now than at Busch's time, 

 since we have been reminded of the Hamiltonian analogy. 



Hamilton discovered that the trajectory of a material particle 

 in a conservative field, that is to say, in a field in which the force 

 can be derived from an ordinary, scalar, potential can be inter- 

 preted as the path of a light ray in a medium with suitable 

 refractive properties. This follows at once from comparing the 

 principle of least action of dynamics with the principle of least 

 time in optics. According to the dynamical principle, the action 

 connected with the trajectory along which a particle will travel 

 from one given point to another will be less than for any other 

 geometrically, but not dynamically, possible path.* The action 

 W is defined as the time integral of the kinetic energy, that is, 



W= C hnvHt 



I" 



ds 

 and as the velocity v is -77, where ds is the length of a path ele 



ment, this can be also written 



Let us compare this with Fermat's principle of least time, the 

 foundation of geometrical optics. In a medium of refractive 



index n, the velocity of light is — if c is the velocity in vacuo. 



W = ^m I V as 



n 



* With the silent clause that the two given points must be sufficiently 

 close together, so that the trajectories which can start from the first do not 

 come to a focus before they reach the second. This does not restrict the 

 analogy, as the requirements are precisely the same in optics. 



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