80
Chapter 2. Geometrical optics
probe-forming systems such as a scanning electron microscope.
It was first shown by Scherzer [77] that spherical aberration cannot
be eliminated in static systems with axial symmetry in the absence
of space charge, and excluding particle mirrors. This is equivalent
to the coefficient B defined in (2.214) always being positive definite. This fact can be proven by successive partial integrations of
the expression for B, in which the integrand can be expressed by a
sum of positive definite terms. The reader is referred to Rose [75]
for details.
Spherical aberration imposes a fundamental limit on the resolution of electron microscopes. Substantial correction of spherical
aberration has been successfully demonstrated using multipole elements [67, 75]. Corrected electron microscopes are now commercially available which achieve resolution better than 0.1 nm. This
is sufficient to view single atoms.
2.5.7 Field aberrations
The aberrations represented by the terms proportional to C, D, E,
F, e, c, f in (2.216, 2.217) represent isotropic astigmatism, field
curvature, isotropic distortion, isotropic coma, anisotropic distortion, anisotropic astigmatism, and anisotropic coma, respectively.
Unlike spherical aberration, all of these aberrations depend on
field position, as represented by the object coordinates (x O , y O ).
We therefore call them field aberrations. Isotropic aberrations
are characterized by having no dependence on azimuthal coordinate, while anisotropic aberrations have an azimuthal dependence.
The isotropic aberrations arise in both electrostatic and magnetic
lenses, while anisotropic aberrations only occur in magnetic lenses.
The field aberrations can be best appreciated by simply plotting
the geometric figure, which in general is a function of the object
coordinates (x O , y O ) and the aperture coordinates (x A , y A ). In this
Chapter 2. Geometrical optics
probe-forming systems such as a scanning electron microscope.
It was first shown by Scherzer [77] that spherical aberration cannot
be eliminated in static systems with axial symmetry in the absence
of space charge, and excluding particle mirrors. This is equivalent
to the coefficient B defined in (2.214) always being positive definite. This fact can be proven by successive partial integrations of
the expression for B, in which the integrand can be expressed by a
sum of positive definite terms. The reader is referred to Rose [75]
for details.
Spherical aberration imposes a fundamental limit on the resolution of electron microscopes. Substantial correction of spherical
aberration has been successfully demonstrated using multipole elements [67, 75]. Corrected electron microscopes are now commercially available which achieve resolution better than 0.1 nm. This
is sufficient to view single atoms.
2.5.7 Field aberrations
The aberrations represented by the terms proportional to C, D, E,
F, e, c, f in (2.216, 2.217) represent isotropic astigmatism, field
curvature, isotropic distortion, isotropic coma, anisotropic distortion, anisotropic astigmatism, and anisotropic coma, respectively.
Unlike spherical aberration, all of these aberrations depend on
field position, as represented by the object coordinates (x O , y O ).
We therefore call them field aberrations. Isotropic aberrations
are characterized by having no dependence on azimuthal coordinate, while anisotropic aberrations have an azimuthal dependence.
The isotropic aberrations arise in both electrostatic and magnetic
lenses, while anisotropic aberrations only occur in magnetic lenses.
The field aberrations can be best appreciated by simply plotting
the geometric figure, which in general is a function of the object
coordinates (x O , y O ) and the aperture coordinates (x A , y A ). In this