In accelerator physicsstrong 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 focussing in two directions at once is impossible. However, ridged poles of a cyclotron or two or more spaced quadrupole magnets (arranged in quadrature) alternately focus horizontally and vertically.[1]
Strong focusing was first invented at Brookhaven National Laboratory and deployed on the Alternating Gradient Synchrotron there. The theory was developed in 1957 by E. D. Courant and H. S. Snyder.[2] The theory revolutionised cyclotron design, and permitted very high field strengths to be employed, while massively reducing the size of the magnets needed by minimising the size of the beam. All current cyclotrons use strong focusing.
Four bar magnets configured to produce a quadrupole.
Modern systems often use quadrupole magnets to focus the beam down as magnets give a more powerful deflection effect than earlier electrostatic systems at high beam kinetic energies. The quadrupole magnets refocus the beam after each deflection section as deflection sections have a defocusing effect that can be countered with a convergent magnet 'lens'.
This can be shown schematically as a sequence of divergent and convergent lenses.
The quadrupoles are often laid out in what are called FODO patterns (where F focusses vertically and defocusses horizontally, and D focusses horizontally and defocusses vertically and O is a space or deflection magnet):
Following the beam particles in their trajectories through the focussing arrangement, an oscillating pattern would be seen.
Mathematical modelling
The effect of a set of focusing magnets can be expressed as a matrix.[3]
