Strong focusing uses alternating gradient focusing to converge particle beams in accelerators. It uses alternating field gradients that cause the net effect of focusing the beam, as opposed to weak focusing which only intersects particle orbits once per revolution in a uniform magnetic field. Modern particle accelerators employ multipole magnets like quadrupoles and sextupoles arranged with alternating focusing gradients to strongly focus beams down after each deflection section, as deflection sections have a defocusing effect. The interaction of charged particles with linear magnets can be expressed mathematically using ray transfer matrix analysis.

3. Strong focusing
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In accelerator physics strong focusing or alternating-gradient focusing is
the principle that the net effect on a particle beam of charged particles
passing through alternating field gradients is to make the beam
converge. By contrast "Weak focusing" is the principle that nearby
circles, described by charged particles moving in a uniform magnetic
field, only intersect once per revolution.
Earnshaw's theorem shows that simultaneous focusing in two directions
at once is impossible. However, ridged poles of a cyclotron or two or
more spaced quadrupole magnets (arranged in quadrature[disambiguation
needed]) alternately focus horizontally and vertically.[1][2]

4. Strong focusing was first conceived by Nicholas
Christofilos in 1949 but not published (Christofilos opted
instead to patent his idea),[3] In 1952, the strong focusing
principle was independently developed by Ernest
Courant, M. Stanley Livingston, Hartland Snyder and J.

5. Multipole magnets

6. Modern systems often use multipole magnets,
such as quadrupole and sextupole magnets, to
focus the beam down, as magnets give a more
powerful deflection effect than earlier
electrostatic systems at high beam kinetic
energies. The multipole magnets refocus the
beam after each deflection section, as deflection
sections have a defocusing effect that can be
countered with a convergent magnet 'lens'.

7. Mathematical modelling
The action upon a set of charged particles by a set of
linear magnets (i.e. only dipoles, quadrupoles and the
field-free drift regions between them) can be expressed
as matrices which can be multiplied together to give their
net effect, using ray transfer matrix analysis.[7] Higher-
order terms such as sextupoles, octupoles etc. may be
treated by a variety of methods, depending on the
phenomena of interest.