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Electron optics is a mathematical framework for the calculation of electron trajectories in the presence of
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
s. The term ''optics'' is used because
magnetic Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, m ...
and
electrostatic Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), mean ...
lenses act upon a charged particle beam similarly to
optical lens A lens is a transmissive optics, optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a #Compound lenses, compound lens consists of several simple ...
es upon a
light beam A light beam or beam of light is a directional projection of light energy radiating from a light source. Sunlight forms a light beam (a sunbeam) when filtered through media such as clouds, foliage, or windows. To artificially produce a li ...
. Electron optics calculations are crucial for the design of
electron microscopes An electron microscope is a microscope that uses a beam of electrons as a source of illumination. It uses electron optics that are analogous to the glass lenses of an optical light microscope to control the electron beam, for instance focusing i ...
and
particle accelerators A particle accelerator is a machine that uses electromagnetic fields to propel electric charge, charged particles to very high speeds and energies to contain them in well-defined particle beam, beams. Small accelerators are used for fundamental ...
. In the
paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray that makes a small angle (''θ'') to the optica ...
, trajectory calculations can be carried out using
ray transfer matrix analysis Ray transfer matrix analysis (also known as ABCD matrix analysis) is a mathematical form for performing ray tracing calculations in sufficiently simple problems which can be solved considering only paraxial rays. Each optical element (surface, ...
.


Electron properties

Electrons are charged particles ( point charges with
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
) with spin 1/2 (hence they are
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s). Electrons can be accelerated by suitable
electric Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
fields, thereby acquiring
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects. According to wave particle duality, electrons can also be considered as
matter waves Matter waves are a central part of the theory of quantum mechanics, being half of wave–particle duality. At all scales where measurements have been practical, matter exhibits wave-like behavior. For example, a beam of electrons can be diffract ...
with properties such as
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
,
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
and
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
.


Geometric electron optics

The Hamilton's optico-mechanical analogy shows that electron beams can be modeled using concepts and mathematical formula of light beams. The electron particle trajectory formula matches the formula for
geometrical optics Geometrical optics, or ray optics, is a model of optics that describes light Wave propagation, propagation in terms of ''ray (optics), rays''. The ray in geometrical optics is an abstract object, abstraction useful for approximating the paths along ...
with a suitable electron-optical index of refraction. This
index of refraction In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
functions like the material properties of glass in altering the direction ray propagation. In light optics, the refractive index changes abruptly at a surface between regions of constant index: the rays are controlled with the shape of the interface. In the electron-optics, the index varies throughout space and is controlled by electromagnetic fields created outside the electron trajectories.


Magnetic fields

Electrons interact with magnetic fields according to the second term of the Lorentz force: a
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
between the magnetic field and the electron velocity. In an infinite uniform field this results in a
circular motion In physics, circular motion is movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate ...
of the electron around the field direction with a radius given by: : r = \frac where ''r'' is the orbit radius, ''m'' is the mass of an electron, v_\perp is the component of the electron velocity perpendicular to the field, ''e'' is the electron charge and ''B'' is the magnitude of the applied magnetic field. Electrons that have a velocity component parallel to the magnetic field will proceed along helical trajectories.


Electric fields

In the case of an applied electrostatic field, an electron will deflect towards the positive gradient of the field. Notably, this crossing of electrostatic field lines means that electrons, as they move through electrostatic fields change the magnitude of their velocity, whereas in magnetic fields, only the velocity direction is modified.


Relativistic theory

At relativistic electron velocity the geometrical electron optical equations rely on an index of refraction that includes both the ratio of electron velocity to light v/c =\beta and \mathbf\cdot\mathbf, the component of the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...
along the electron direction: n(\mathbf) = \frac + \frac\mathbf\cdot\mathbf where m, e, and c are the electron mass, electron charge, and the speed of light. The first term is controlled by electrostatic lens while the second one by magnetic lens. Although not very common, it is also possible to derive effects of magnetic structures to charged particles starting from the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
.


Diffractive electron optics

As electrons can exhibit non-particle (wave-like) effects such as
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...
and
diffraction Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
, a full analysis of electron paths must go beyond geometrical optics. Free electron propagation (in
vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
) can be accurately described as a de Broglie
matter wave Matter waves are a central part of the theory of quantum mechanics, being half of wave–particle duality. At all scales where measurements have been practical, matter exhibits wave-like behavior. For example, a beam of electrons can be diffract ...
with a wavelength inversely proportional to its longitudinal ( possibly relativistic) momentum. Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. The essential character of this approach is to use geometrical ray tracing but to keep track of the wave phase along each path to compute the intensity in the diffraction pattern. As a result of the charge carried by the electron, electric fields, magnetic fields, or the electrostatic mean inner potential of thin, weakly interacting materials can impart a phase shift to the wavefront of an electron. Thickness-modulated
silicon nitride Silicon nitride is a chemical compound of the elements silicon and nitrogen. (''Trisilicon tetranitride'') is the most thermodynamically stable and commercially important of the silicon nitrides, and the term ″''Silicon nitride''″ commonly re ...
membranes and programmable phase shift devices have exploited these properties to apply spatially varying phase shifts to control the far-field spatial intensity and phase of the electron wave. Devices like these have been applied to arbitrarily shape the electron wavefront, correct the aberrations inherent to
electron microscopes An electron microscope is a microscope that uses a beam of electrons as a source of illumination. It uses electron optics that are analogous to the glass lenses of an optical light microscope to control the electron beam, for instance focusing i ...
, resolve the orbital angular momentum of a free electron, and to measure
dichroism In optics, a dichroic material is either one which causes visible light to be split up into distinct beams of different wavelengths (colours) (not to be confused with Dispersion (optics), dispersion), or one in which light rays having different P ...
in the interaction between free electrons and magnetic materials or plasmonic nanostructures.


Limitations of applying light optics techniques

Electrons interact strongly with matter as they are sensitive to not only the nucleus, but also the matter's electron charge cloud. Therefore, electrons require
vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
to propagate any reasonable distance, such as would be desirable in electron optic system. Penetration in vacuum is dictated by
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
, a measure of the probability of collision between electrons and matter, approximate values for which can be derived from Poisson statistics.


See also

* Charged particle beam *
Strong focusing In accelerator physics strong focusing or alternating-gradient focusing is the principle that, using sets of multiple electromagnets, it is possible to make a particle beam simultaneously converge in both directions perpendicular to the direction ...
* Electron beam technology *
Electron microscope An electron microscope is a microscope that uses a beam of electrons as a source of illumination. It uses electron optics that are analogous to the glass lenses of an optical light microscope to control the electron beam, for instance focusing it ...
* Beam emittance *
Ernst Ruska Ernst August Friedrich Ruska (; 25 December 1906 – 27 May 1988) was a German physicist who won the Nobel Prize in Physics in 1986 for his work in electron optics, including the design of the first electron microscope. Life and career Ernst R ...
* Hemispherical electron energy analyzer


Further reading

* P. Grivet, P.W. Hawkes, A.Septier (1972). ''Electron Optics, 2nd edition''. Pergamon Press. . * A.Septier (ed.) (1980). ''Applied Charged Particle Optics. Part A.''. Academic Press. . * A.Septier (ed.) (1967). ''Focusing of Charged Particles. Volume 1.''. Academic Press. * D. W. O. Heddle (2000). ''Electrostatic Lens Systems, 2nd edition''. CRC Press. . * A.B El-Kareh, J.C.J. El-Kareh (1970).''Electron Beams, Lenses, and Optics Vol. 1''. Academic Press. * Hawkes, P. W. & Kasper, E. (1994). ''Principles of Electron Optics''. Academic Press. . * Pozzi, G. (2016). ''Particles and Waves in Electron Optics and Microscopy''. Academic Press. . * Jon Orloff et al., (2008). ''Handbook of Charged Particle Optics. Second Edition''. CRC Press. . * Bohdan Paszkowski. (1968). ''Electron Optics'', Iliffe Books Ltd. * Miklos Szilagyi (1988). ''Electron and Ion Optics'', Springer New York, NY. . * Helmut Liebl (2008). ''Applied Charged Particle Optics ''. Springer Berlin. . * Erwin Kasper (2001). ''Advances in Imaging and Electron Physics, Vol. 116 , Numerical Field Calculation for Charged Particle Optics''. Academic Press. . * Harald Rose (2012). ''Geometrical Charged-Particle Optics ''. Springer Berlin, Heidelberg. .


Electron Optics Simulation Software

Commercial programs * SIMION (Ion and Electron Optics Simulator) * EOD (Electron Optical Design) * CPO (electronoptics.com) * MEBS (Munro's Electron Beams Software) * Field Precision LLC Free Software * IBSIMU (by Taneli Kalvas) (ibsimu.SourceForge.net)


References

{{DEFAULTSORT:Electron Optics Electromagnetism Accelerator physics