Aberration

The aberration is the deviation from ideal imaging in the electron optical system. In the case of ideal imaging, all the electron beams that exit from a single point on the object plane and pass through a lens meet at a single point on the image plane. However, in the real lens, those electron beams do not meet at the single point but form an image blur. The aberrations are classified into “geometrical aberrations” and “chromatic aberrations”.

The geometrical aberrations depend on the position and the angle of the electron beam exiting from the object plane, and are expressed as a function of r (distance of the electron beam from the optical axis) and α (angle of the electron beam against the optical axis). Among those aberrations, the aberrations which depend only on the angle α are called the axial geometrical aberrations, whereas the aberrations which depend on r, and on both r and α, are called the off-axial geometrical aberrations.
The chromatic aberration is produced when the electron beams do not have a single energy but have an energy spread. That is, the electron beams having different energies are deflected in different angles when passing through the electro-magnetic lens, resulting in the chromatic aberration.

Table 1 lists the aberrations. In usual electron microscopes, magnetic-field lenses are used. These lenses possess a symmetric magnetic field with respect to the optical axis, and cause the aberrations enclosed by red frames. In particular, the aberrations indicated by green colors are called “Five Seidel aberrations”, which are the geometrical aberrations inevitably arising from the principle of optics. Furthermore, since the magnetic-field lens rotates the image, “S-shaped distortion” (proportional to r³) is produced, where the image rotates at the periphery of the field of view. The other aberrations in Table 1 are called “parasitic aberrations” and are caused by magnetic non-uniformity of the pole-piece material, precision of pole-piece machining and disagreement of the optical axes between lenses.
An electrostatic lens utilizing electric field is used in the electron gun. The electrostatic lens possesses an electric field symmetric to the optical axis and generates the aberrations similar to those of the magnetic-field lens. However, the electrostatic lens does not rotate the electron beam around the optical axis and thus, the S-shaped distortion and azimuth components of the off-axial geometrical aberrations are not produced.

Magnetic field multipoles are used for the correction of the aberrations. A magnetic dipole is used in the deflector because it can deflect the electron beam in a certain direction and can compensate the misalignment between the lenses. A magnetic quadrupole is used in astigmatism corrector (stigmator) because it generates a magnetic field with a two-fold symmetry or a two-fold astigmatism. By setting the two quadrupoles at different angles, a two-fold astigmatism in any direction can be created, with which the astigmatism of the lens can be canceled. Magnetic quadrupoles, hexapoles, and octupoles are used for the third-order spherical aberration correctors. In the case of hexapoles, the thick hexapoles produce a third-order spherical aberration as a combination aberration effect (Refer to Term “combination aberration”). This spherical aberration has negative polarity and is used to cancel the spherical aberration with positive polarity of the objective lens. The hexapole itself mainly produces a three-fold astigmatism, but this unnecessary three-fold astigmatism is cancelled out by using two hexapoles of opposite polarity serially.

Geometrical aberrations

In the case of the observation of a high-resolution image at a high magnification, the observation area is small or the distance r from the optical axis of the observation area is small. Thus, in high-resolution imaging, the axial aberrations (not off-axial aberrations) are important for the image quality. In particular, the third-order spherical aberration (one of Five Seidel aberrations) of the objective lens located closest to the specimen has a significant effect. (Refer to Term “axial geometrical aberration”) The amount of the third-order spherical aberration is given by Csα³ on the specimen plane. Here, Cs is the spherical aberration coefficient, the value of which is specific to each polepiece of the objective lens. Thus, it is important to design the objective-lens polepiece with a small Cs. A very-well designed objective-lens polepiece achieves a Cs of ~0.5 mm. For example, when the convergence semi-angle α is taken to be 10 mrad (actually used), the image (or probe) blur or the size of least confusion (1/2) Csα³ becomes 0.25 nm. When the angle is assumed to be 25 mrad, the blur due to the spherical aberration reaches a value of 4 nm, though the diffraction aberration decreases to be less than 0.1 nm.
If a Cs corrector is installed to an electron microscope, Cs of the objective lens can be canceled out to be zero. Once spherical aberration is removed, the next important step is to remove the axial parasitic aberrations. The Cs corrector can correct two-fold astigmatism, axial coma aberration, three-fold astigmatism, star aberration, and four-fold astigmatism, in order of contribution among axial parasitic aberrations. Furthermore, modern Cs correctors can also correct a parasitic aberration called six-fold astigmatism. As a result, electron beams with a large opening angle α (~60 mrad) can be focused onto a single point on the image plane.

On the other hand, the off-axial geometrical aberrations are important when observing a large field of view (approx. 10 µm × 10 µm or more) at a low magnification, or the peripheral region of the observation field distant from the optical axis. The off-axis geometrical aberrations at the lower part of the imaging lens system, i.e. the intermediate lens and the projection lens, have a significant effect on image degradation (distortion and blurring). This is because the more the image is magnified by the lenses, the greater the distance r at the periphery of the magnified image (Refer to Term “off-axial geometrical aberration”).
The off-axis geometrical aberrations include “Distortion” (proportional to r³), “curvature of image field” (proportional to r²α), “off-axial astigmatism” (proportional to r²α) and off-axial coma aberration (proportional to rα²), which are the aberrations in “Five Seidel aberrations”, and “S-shaped distortion” (proportional to r³) which is specific to the magnetic-field lens and rotates the image at the periphery of the field of view. The contributions of the “parasitic” off-axial geometrical aberrations are small and are neglected for practical purposes.

It should be noted that the geometrical aberration is expressed as the deviation of the actual image point from the ideal image point from the deference of the actual electron beam path from the path of ideal imaging. On the other hand, the wave aberration describes aberrations as a deviation of the wavefront in actual imaging with aberrations from the wavefront of ideal imaging with no aberrations (Gaussian imaging). The geometrical aberration is easier to intuitively understand the aberration, but the wave aberrations are expressed by simpler mathematical formulas. It is noted that the geometrical aberration and wave aberration express the same aberration in different orders. (The geometrical aberration is expressed as the derivative of the wave aberration by α. Thus, the order of α differs by 1. For example, the spherical aberration is a third-order aberration proportional to α³ in terms of the geometrical aberrations, but is a fourth-order aberration proportional to α⁴ in terms of the wave aberrations. In many cases, the spherical aberration is expressed as the third-order aberration according to the order of the geometrical aberration.)

Chromatic aberrations

Chromatic aberration is an aberration that depends on the energy of the electron beam. Since the difference in the electron energy (frequency/wavelength) corresponds to the difference in the color (frequency/wavelength) of light, the aberration due to the difference in energy is called chromatic aberration in the electron optics as in the light optics. Like the geometrical aberrations, the chromatic aberrations include the axial chromatic aberration (not dependent on the distance r of the electron beam from the optical axis) and the off-axial chromatic aberration (dependent on the distance r of the electron beam from the optical axis). Chromatic defocus (change of focus due to difference in electron energy), which is one type of the axial chromatic aberration, is usually called “chromatic aberration” (chromatic aberration in the narrow sense). The magnitude of the chromatic aberration due to the focus change is expressed as CcΔE/E on the specimen plane. Here, Cc is the chromatic aberration coefficient. In the case of a thin lens, the value of Cc is nearly the same as its focal length. Since the electrons with different energies are incoherent to each other, the chromatic aberration (the focus spread due to the energy spread) causes a reduction in image contrast. The chromatic aberration can be reduced by decreasing the energy spread ΔE. To achieve this, it is important to improve the stability of the acceleration voltage and use an electron source with a small energy spread. (Modern TEMs achieve the stability of the accelerating voltage ΔE/E to be as high as <1 × 10^-6. The energy spreads of the electron sources are ~2 eV for the LaB₆ thermionic-emission electron gun, ~0.7 eV for the Shottky-type electron gun, and ~0.4 eV for the cold field-emission electron gun. If a monochromator is used, the energy spread can be reduced to 0.1 eV or less, though the electron intensity is greatly decreased because the electrons exceeding the required energy spread are blocked by a slit. The design and fabrication of lenses with small chromatic aberration coefficients is also important. As in the case of the spherical aberration coefficient Cs, the chromatic aberration coefficient Cc is specific to the pole piece of the objective lens and is Cc ~ 1 mm for a well-designed objective lens pole piece. As a method of correcting chromatic aberration, a Cc corrector has been developed to cancel out the positive chromatic aberration of the objective lens with the negative chromatic aberration of the corrector to zero.
As in the case of the geometrical aberration, the off-axial chromatic aberration of the intermediate and projection lenses located at the lower part of the imaging lens system can cause image blur at the periphery of the field of view of a low magnification image.

Table 1  List of aberrations

Related Term(s)