The most commonly used focusing elements are magnetic quadrupoles. At low energies electric quadrupoles are also very widely used. Figure III.6 shows the geometric configurations of electric and magnetic quadrupoles.

The general features of quadrupolar focusing elements can be sum­marized by the pattern of the forces acting on the beam perpendicular to

its direction. If Oz is the direction of the beam, a quadrupolar pattern is obtained with the force components:

Fx = — f0x (III.49)

Fy = fy. (III. 50)

With such a pattern, particles which are off axis in the Ox direction receive an acceleration towards the beam axis and are, therefore, focused. In contrast, particles which are off axis in the Oy direction receive an acceleration away from the beam axis and are, therefore, defocused. More quantitatively the equations of motion of the particle, assuming no force in the Oz direction, read

equations (III.52) are separable. The solution of equations (III.52) follows, for an initial value of z, z0 = 0, with x = dx/dz:

r, sin^/Kz)

x(z) = X0 cos( KZ)+ X0———— —=

K

x’(z) = —x0yrK sin f/Kz) + x0 cos ([Kz)

. p, ’ sinh(—Kz)

y(z) = У0cosh( Kz)+ У0 —

K

y (z) = y^/Ksinh^/Kz)+y0 cosh^/Kz)

so that, after a field region of length L, the initial values of x and x are modified as:

yL = y0 cosh( VkL) + y0

K

yL = y^/K sinh^/KL)+y0 cosh^/KL). Defining the vectors

X0 =

one sees that X0 transforms into XL via the matrix multiplication

A focusing device should be such that a particle with initial velocity parallel to Oz (x0 = y0 = 0) exhibits a velocity change towards the beam. This happens when x’Lx0 < 0. Thus equations (III.60) describe a system which is focusing along x and defocusing along y. This is a general feature of quadrupolar configurations: they are focusing in one direction and defocus­ing in the other orthogonal one. Note that when к = 0 one gets the free motion solution.

The small-angle approximation of the focusing element reads

rn -1 kl2) l

F — kL (1 — 1 kL2)

while that of the defocusing element reads

Q Г (1 +1 kL2 ) l

D kL (1 +1 kL2)

The focusing or defocusing character of the element is therefore given by the sign of matrix element Q21. The association of a focusing device with a defocusing one allows one to get focusing in all cases. Qualitatively, this results from the fact that, after a first defocusing, particles reach out from the beam, leading to a stronger focusing field in the next optical element, while the reverse is true when focusing happens first. At the lowest approxima­tion, the transfer matrix for the defocusing-focusing (DF) arrangement reads

+ kL2 2L

k2 L3 1 — kL2

which is focusing. Similarly the FD arrangement leads to the transfer matrix

which is also focusing. Other quadrupole arrangements have interesting properties such as the triplet which allows stigmatic focusing. In the triplet case the transfer matrix reads

A focusing device can be characterized by its focal length f. It can be shown that a quadrupole can be assimilated to a thin lens in the centre of a drift

space of length l. The focal distance of the lens is related to the focusing parameter of the quadrupole by

f = ±;k	(III.66)
K = ±fL.	(III:67)
This formula is useful if one wants a simple estimate of к of the focusing array is known. For doublets [183],	when the geometry
/Fd 2 3 k2L3	(III:68)
K = v/fdL3 .	(III.69)