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Optical aberration can be described by the socalled wave aberration function. For the ideal cases without aberrations, when an object is imaged using a perfect lens the wave at the back focal plane of a lens is spherical and is converged in the image plane (I_{g}, or called I_{k} for expression convenience). In actual lenses as shown in Figure 3752a, the wave aberration function is defined as the phase difference (χ_{x,y}) or the distance difference (W_{x,y}) in the optical path from the perfect (reference) spherical wave to the actual wavefront for the given lens. Since the area on the sample that is viewed at most of the time in EMs (especially in TEMs) is very small the wave aberration function’s dependence on the location of the point source in the object plane can be usually ignored, which is known as the isoplanatic approximation. The relationship between the phase difference (χ_{x,y}) and the distance difference (W_{x,y}) can be given by,
 [3752a]
where ,
λ  The wavelength of the electron wave
Figure 3752a. Schematic illustration of wave aberration function (W_{x,y} or χ_{x,y}) and ray aberration (δ_{r}).
In some cases for the convenience of theoretical expressions, the wave aberration function is simply given by the distance between the intersecting points (on the image plane) of the ideal ray and the aberrated ray indicated in in Figure 3752a. The distance (also called ray aberration), δ_{r} is given by,
 [3752b]
The ray aberration is proportional to the gradient of the wave aberration W_{x,y}.
In general, the wave aberration function is determined by the following factors:
i) The complex reciprocal space vector k = k_{x} + i k_{y}.
ii) The position in the image plane r.
However, in the case of small field of view (e.g. at high magnifications), the dependence of W_{x,y} on the image plane can be negligible based on the isoplanatic approximation. The dependence of W_{x,y} on the image plane can also be neglected in STEM mode because of the small probe dimensions.
iii) The energy of the electrons (E_{0}).
However, for a wellaligned field emission electron source, based on the monochromatic approximation the energydependent terms can be ignored.
In polar coordinate system, this twodimensional (2D) surface can be expressed as a function of the angle to the optic axis (θ) and azimuthal angle around the optic axis (ϕ) as shown in Figure 3752a. For a perfect lens, the actual wavefront is the same as the perfect spherical wavefront and χ(θ, ϕ) would be zero everywhere, but for a practical lens the actual wavefront is a function that is close to zero near the optical axis when a microscope is in the wellaligned (best imaging) condition and it increases rapidly when the highorder aberrations become predominant away from the optical axis. Furthermore, as shown in Figure 3752b the aberrated wavefront can be irregular so that the aberrations can be more complex and the wave aberration function can be both positive and negative.
Figure 3752b. Schematic illustration of irregularly aberrated wavefront.
Note that the aberration behavior is described as a firstorder aberration if the image aberration for defocus is linear in angle, while it is described as a thirdorder aberration if the image aberration for spherical aberration is cubic in angle. Optical aberrations negatively affect the image quality, imaging system transfer characteristics, and reduce precision of the relevant measurements. However, the key electron optical component affecting the formation of the images in CTEM and STEM is the objective lens so that in many cases we only need to consider the wave aberration function in the objective lens.
In practice, the imaging conditions in EMs are that the lens aberrations tend to zero for rays very close to the optic axis. In most cases, only the two lowest order terms of the wave aberration function (i.e., defocus (C_{1,0} or C_{1}) and spherical aberration coefficients (C_{3,0} or C_{3})) are considered with the expression below,
 [3752c] where,
Δf  The defocus, or C_{1,0} (or C_{1}) with the notation of aberration coefficients in page3740.
C_{S}  The thirdorder spherical aberration coefficient C_{3,0} (or C_{3}) in page3740.
g  The spatial frequency.
λ  The wavelength of incident electrons.
The phasecontrast term isin2πχ(g) in the contrast transfer function becomes zero if C_{S}=0 and Δf=0, while the amplitudecontrast term cos2πχ(g) is maximum, 1. Therefore, for aberrationcorrected microscope, the TEM images present atomic structures by amplitudecontrast rather than by phasecontrast, and thus in HRTEM images the projected atom column are imaged in bright contrast on a dark background.
More comprehensively, under the isoplanatic approximation as well as Taylor expansion, the wave aberration function, W, using the notation provided by Krivanek [2], can be concisely expanded to sixth order in k about the origin of zero scattering angle in terms of the coherent aberration coefficients, given by,
 [3752d]
Here, factors 1/(n+1) are used for all contributions to the nth order. [3] The first subscript refers to the order of the coefficient in terms of real space displacement, while the second subscript describes the angular symmetry. If the vector k is replaced by a complex angle variable ω = α·e^{iθ }= ω_{x} + iω_{y}= λ(k_{x} + ik_{y}), the wave aberration function can be rewriten by,
 [3752e]
or in a different format of,
 [3752f.a]
 [3752f.b]
where,
ω  (= λg) the related complex scattering angle,
ϖ  The complex conjugate of ω,
c.c.  A complex conjugate.
ω = α·e^{iθ } represents that the incident
ray has a tilt angle α and azimuth θ.
The wave aberration function can also be expanded in polar notation. The following equation shows the terms to fourth order,
 [3752g]
The other way of presenting the wave aberration function is based on polar angular coordinates θ and φ, given by,
 [3752h]
The sum over m is taken from 0 (or 1) to n+1 for each order n with the additional constraint that m+n is odd.
The wave aberration function can also be given as a function of scattering angle of the incident electrons and defocus of the objective lens, [2]
 [3752i]
where,
α  The scattering angle,
z  The defocus (the difference of TEM specimen height from the standard focus height),
A_{2}  The twofold astigmatism coefficients,
A_{3}  The threefold astigmatism coefficients,
B  The coefficients corresponding to coma displacement,
D  The coefficients corresponding to image displacement,
Φ_{2}, Φ_{b}, Φ_{3} and Φ_{d}  The initial phases of corresponding aberrations.
Table 3740 lists the aberration coefficient nomenclature, their order (of both the ray deviation (N) and the wavefront), and radial (azimuthal) symmetry.
The wave
front deviation due to the phase shift between the
wave function in the object plane ψ_{obj}(g) and the wave
function in the image plane ψ_{im}(g) is given by,
ψ_{im}(g) = ψ_{obj}(g)exp(2πiχ(g))  [3752j]
The ray aberration is linked to the wave aberration by, [4]
 [3752k]
For perfect aberration correction, because the aberration function is zero, all the ray displacements are eliminated, resulting in a zero contrast delocalization.
The wave aberrations (χ) produce a scrambling of amplitude and phase information in the EM (electron microscopy) images. Figure 3752c shows the schematic illustration of the image of a weak object formed by the objective lens. Sin(χ) indicates the crosstalk between amplitude and phase and cos(χ) presents the direct transfer of amplitude or phase information. For instance, in the conventional TEM, the detector only records the image amplitude, while in holography, both the amplitude and phase of the image wave are recorded in the interference fringes of the hologram.
Figure 3752c. Schematic illustration of imaging a weak object. χ represents coherent wave aberration.
Furthermore, a small misalignment of the direction of the incident electron beam introduces wave aberrations and thus affects the quality of the images [ 5,6] so that comafree alignment is essential for HRTEM.
[1] Hawkes, P., and Kasper, E. (eds.). (1996). “Principles of Electron Optics: Wave Optics.” Academic Press, London.
[2] K. Ishizuka, Ultramicroscopy 55 (1994) 407.
[3] Saxton,W.O. (2000). A new way of measuring microscope aberrations. Ultramicroscopy 81, 41–45.
[4] Lichte, H. (1991). Optimum focus for taking electron holograms. Ultramicroscopy 38, 13–22.
[5] K. Ishizuka and S. Iijima, Proc. 39th EMSA Annual Meeting,
Atlanta (1981) 96.
[6] D.J. Smith. W.O. Saxton, M.A. O’Keefe, G.J. Wood and
W.M. Stobbs. Ultramicroscopy I1 (1983) 263.
