The propagation of paraxial rays through a series of lenses can be conveniently approximated by a matrix approach.
To take a simple case, assume that each particle in a 2D cylindrically symmetric lens system can be uniquely described at an axial position z by a vector , where r is the radial distance of the particle from the axis and r’ is the slope of the ray (, for time t and velocity v). The states of the particle and respectively at planes and near the entrance and exit of a lens respectively can be related by a matrix equation:
where the matrix is referred to as the transfer matrix and is specific to the plane locations and lens. The matrix coefficients can depend on parameters such as focal lengths. If you partition your optic axis by a series of (transverse) planes, then the transfer matrix between the outermost planes is the product of all the transfer matrices between successive planes.
The transfer matrix can have larger dimensions if motion in more than one direction (here, r) and/or aberations are taken into account.
Many books such as those cited below     describe the transfer matrix approach in more detail.  briefly provides matrix coefficients in terms of cardinal points in 2D cylindrical lens. A more general and detailed treatment is in .  makes extensive use of transfer matrices throughout in its derivations.  has a description in the context of various components in mass spectrometry. The analogous concept is used in light optics (ray transfer matrix analysis or ABCD matrix analysis–see BYU tutorial). The concept is also related to Hamiltonian mechanics (see also Emittance).
Transfer matrices are not normally used directly by SIMION because SIMION is equipped to take the more general, more accurate, and more computationally intensive approach of computing the entire trajectory of each particle (position as a function of time) from the fully known (or calculated) electromagnetic field; it does this by integrating the Lorentz Force Law. You may, however, wish to use SIMION to compute the lens properties (and hence transfer matrices) of some arbitrarily shaped lenses (SIMION Example: lens_properties) in the first place.
|||(1, 2) Douglas W. O. Heddle. Electrostatic lens systems. G:ixmUMS47QfwC. 13 December 2000. CRC Press. ISBN:978-0-7503-0697-3.|
|||(1, 2) Mikhail Yavor. Advances in Imaging and Electron Physics: Optics of Charged Particle Analyzers. Volume 157 of Advances in Imaging and Electron Physics. G:ntYf2V-pbIgC. 24 July 2009. Academic Press. ISBN:978-0-12-374768-6.|
|||(1, 2) Helmut Liebl. Applied charged particle optics. November 2007. Springer. G:XqRFZX0ssRMC ISBN:978-3-540-71924-3 springer|
|||Peter W. Hawkes. E. Kasper. Principles of Electron Optics: Basic Geometrical Optics G:XHEDruIilgEC. 24 April 1996. Academic Press. ISBN:978-0-12-333341-4.|
|||Thomas W. Burgoyne and Gary M. Hieftje. Department of Chemistry, Indiana University. Mass Spectrometry Reviews. Volume 15, Issue 4, pp 241-259, 1996. An introduction to ion optics for the mass spectrograph. link online|