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# Aberration Correctors, Monochromators, Spectrometers

## Abstract

After four decades of attempts to correct the primary spherical and chromatic aberrations of electron lenses that led to no improvement in resolution, success was at last achieved in the 1990s with both quadrupole-octupole and sextupole correctors. The successful correctors focused on three aspects of aberration correction: primary aberrations, parasitic aberrations , and overall stability. They quickly demonstrated resolution improvement in the microscopes they were built into, and in the early 2000s, they advanced the attainable resolution to \(<{\mathrm{1}}\,{\mathrm{\AA{}}}\)—a level not achievable by uncorrected electron microscopes. Subsequent generations of correctors included further multipoles and corrected aberrations up to the fifth order, enabling resolution of better than \({\mathrm{0.5}}\,{\mathrm{\AA{}}}\) to be reached at \({\mathrm{300}}\,{\mathrm{kV}}\) primary voltage, and around \({\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{30}}\,{\mathrm{kV}}\). The effect of chromatic aberration was reduced by the use of hybrid quadrupoles or by incorporating a monochromator in the microscope column.

After a brief summary of the optics of multipoles, the various types of correctors are examined in detail: quadrupole–octopole correctors , which first improved the performance of a scanning electron microscope and, soon after, that of scanning transmission electron microscopes; and sextupole correctors , which first increased the resolving power of conventional (fixed-beam) transmission electron microscopes, and were later used in scanning transmission electron microscopes as well. Ways of combating chromatic aberration are then described, including mirror correctors employed in low-energy-electron and photoemission microscopes ( and PEEM ). A section is devoted to studies of aberrations beyond the third order and of parasitic aberrations.

Electron spectrometers and imaging filters are routine accessories of electron microscopes, and they too must be carefully designed, especially when attached to aberration-corrected instruments. A section covers these devices, and much of the reasoning also applies to monochromators. Separate paragraphs are devoted to post-column and in-column spectrometers and monochromators, and the attainable energy resolution is discussed. Practical aspects of the correction process are described, notably autotuning and aberration measurement. We conclude with a survey of current performance limits and comments on the problems to be overcome if further progress is to be made.

## Keywords

quadrupole correctors sextupole correctors monochromators in-column spectrometers post-column spectrometers geometrical aberrations chromatic aberrations parasitic aberrationsIt has been known since the early days of electron optics that the rotationally symmetric lenses employed in electron microscopes and similar instruments suffer from severe aberrations that cannot be eliminated by skillful lens design [13.1]. Immense effort has been devoted to finding lenses with small aberrations and devising aberration correctors . The theoretical proof that the two most important aberrations, the spherical and chromatic aberrations , cannot be eliminated required that several conditions be satisfied and, by relaxing one or another of these conditions, correctors could be designed. A near-exhaustive list was published by *Scherzer* [13.2] and reviews charting trends in thinking about aberration correction and progress in implementing correctors are to be found in [13.3], reprinted in [13.4], [13.5, 13.6]. These contain very full accounts of earlier attempts to correct aberrations with extensive reference lists, and the material presented there is not always reproduced here. In particular, a survey of attempts to build apochromats and aplanatic lenses by O. Scherzer, H. Rose and colleagues in Darmstadt is to be found in the article by *Marko* and *Rose* [13.7].

The types of correctors that seem most promising today are examined below but first, we describe the various kinds of aberrations and explain why they are important. We then look more closely at the aberration coefficients themselves, which leads naturally to a study of the correctors.

Some familiarity with basic electron optics is assumed here. In particular, the reader is expected to be acquainted with the paraxial properties of lenses and the cardinal elements that characterize them. The *Handbook of Charged Particle Optics* edited by *J. Orloff* [13.8] is recommended for readers who wish to brush up their knowledge of electron lenses, notably the chapters by *Munro* [13.9] (only in the first edition, 1997), *Tsuno* [13.10], *Lencová* [13.11] and *Hawkes* [13.12]. For very full accounts, see [13.13, 13.6].

## 13.1 Types of Aberrations

Geometrical aberrations act on electrons of the same energy whereas chromatic aberrations act on electrons of different energies. Mixed geometrical-chromatic aberrations arise beyond the lowest order of aberrations. The aberrations may be intrinsic, inseparable from the optics of the system as designed, or parasitic, arising from imperfections in machining, homogeneity of materials, alignment, etc.

### 13.1.1 Geometrical Aberrations

An ideal lens would provide point-to-point mapping of the structure of an object into an image, and such imaging is indeed predicted by the simplest approximate description of the effect of a lens on an electron beam, the *paraxial approximation* . This is valid provided that the electrons remain close to the optic axis (the axis of symmetry in the case of a round lens) and that the electron trajectory remains inclined at a small angle to this axis. When the electrons depart too far from the axis or the trajectories are inclined at a steeper angle, the paraxial approximation is perturbed by geometrical aberrations.

*Rose*[13.14] is a very relevant reference to much of the material presented here, the relation between his notation and that adopted here is also given. The relativistically corrected potential \(\hat{\phi}\) is given by

*coma-free plane*, the exact position of which is determined by the relative magnitudes of the spherical aberration and coma coefficients. In practice, for magnetic lenses, it falls within the lens field, upstream from the image focus (the

*diffraction plane*) . Specifically, let us suppose that the aberrations are expressed in terms of ray position in the object plane, \(z=z_{\mathrm{o}}\), and some aperture plane, \(z=z_{\mathrm{a}}\). The paraxial solutions appearing in the aberration integrals will then be \(s(z)\) and \(t(z)\), which satisfy the conditions

*two*sets of cardinal elements, one set for the \(x{-}z\) plane and a second set for the \(y{-}z\) plane. These equations take the form

Since the cardinal elements are different in the two planes, the planes conjugate to a given object plane will generally not coincide. The system is astigmatic, and if we operate two quadrupoles in tandem, the*object* of the second member will be astigmatic. Clearly, if we require quadrupoles to produce a stigmatic image of an object, we must use quadrupole multiplets and must somehow arrange that the cardinal elements of the multiplet are the same in the two planes. One multiplet configuration is of particular importance. It can be shown by symmetry arguments that if the multiplet is geometrically symmetric and electrically antisymmetric about its center plane, the focal lengths in the \(x{-}z\) and \(y{-}z\) planes will be equal. It is then only necessary to satisfy one condition (coincidence of the focal planes) in order to render the multiplet stigmatic. Quadruplets with this property have been extensively studied and are known as Russian quadruplets , since their properties were first investigated by *Dymnikov* and *Yavor* [13.17] translated as [13.18], in the Ioffe Institute in Saint Petersburg (Fig. 13.2a-d).

The aberrations of quadrupoles are more numerous than those of round lenses but they fall into the same categories: aperture aberrations (the analogues of the spherical aberration), comas, field astigmatisms and curvatures, and distortions. Here we consider only the aperture aberrations, since it is they that will be exploited in aberration correctors. At this point, we note that octopoles , which have eight electrodes or magnetic poles, also have quadrupole symmetry and should hence be included in the formalism for the aberrations. They have no linear effect and hence have no effect on the paraxial behavior.

*quadratic*: the trajectories depend on products of two of the object coordinates \((x_{\mathrm{o}},y_{\mathrm{o}},x_{\mathrm{o}}^{\prime},y_{\mathrm{o}}^{\prime})\). Their primary aberrations are third order, like round lenses and quadrupoles, and their aperture aberration, the aberration that depends only on gradient and is independent of off-axis distance, has the same nature as the spherical aberration of round lenses. It is for this reason that sextupoles are potential correctors of \(C_{\mathrm{s}}\), but some way of eliminating the quadratic effects must be devised. We shall see how this is achieved in a later section. For extensive discussion of their optical properties, see [13.19].

*Sextupole Correctors*, be used to cancel the spherical aberration of an adjoining round lens.

### 13.1.2 Chromatic Aberrations

*chromatic*effect, characterized by chromatic aberration coefficients, that blurs the image. We can include this in the paraxial formalism by writing

We thus have two types of aberrations: the chromatic aberration \(C_{\mathrm{c}}\), which is linear in gradient and hence does not vanish for object points close to the axis, just like the spherical aberration; and the complex chromatic aberration of magnification (\(C_{\mathrm{D}}+\mathrm{i}C_{\theta}\)), analogous to the distortion in that it is independent of gradient. We can therefore expect that the chromatic aberration, like the spherical aberration, will impose a limit on the resolution attainable in very high-resolution work. In practice, it defines an *information limit*, which, in instruments that have not been corrected for spherical aberration, is usually less severe than the limit imposed by spherical aberration . With the arrival of spherical aberration correctors, however, the situation is reversed, and it becomes imperative to correct the chromatic aberration as well, or find some way of rendering it innocuous. Incidentally, this is a return to the situation in the early days of electron microscopy, when chromatic effects dominated as a result of the relatively poor stabilization circuitry of the time.

### 13.1.3 Parasitic Aberrations

Parasitic aberrations are, as their name suggests, not intrinsic defects of electron lenses. They result from imperfections of the construction or alignment of the lenses or of the entire instrument in question. The most serious is a (first-order) astigmatism , traditionally associated with imperfect circularity of the bore of the lens: in fact, Fourier analysis shows that this astigmatism will usually be the dominant parasitic effect, whatever the origin of the problem. It is as though a very weak quadrupole had been superimposed on the lens and the effect can be canceled by adding a weak quadrupole with the opposite strength. Such a device, which in practice has a more complex structure, is known as a stigmator, and several such correctors are routinely incorporated in commercial instruments. In high-resolution operation, other parasitic effects become noticeable, and ways of correcting or compensating for these are now known, as we shall see in Sect. 13.3.1. Attention has been concentrated on axial parasitic aberrations, and several classifications of these are in use. We list these in Table 13.1.

*Uhlemann*and

*Haider*[13.20] and

*Haider*et al [13.21, 13.22, 13.23, 13.24] on angle can be recognized from the expression for the wave aberration

*Krivanek*et al [13.25, 13.26] are as follows

*Sawada*et al [13.27, 13.28, 13.29, 13.30] is similar to that of Uhlemann and Haider but not identical. The starting point is again

Comas are labeled \(P_{m}:P_{3}\) is an axial coma, arising from \(\Re(P_{3}\theta\theta^{\ast{2}}/3)\), and \(P_{5}\) is a fourth-order axial coma, \(\Re(P_{5}\theta^{2}\theta^{\ast 3}/5)\).

Terms in \(R_{m}\) correspond to threefold symmetry, \(\Re(R_{5}\theta\theta^{\ast 4}/5),\dots\), and terms in \(S_{m}\) to fourfold symmetry, \(\Re(S_{6}\theta\theta^{\ast 5}/6),\dots\)

We note that Krivanek's notation covers all possible axial aberrations, whereas a new symbol has to be introduced for each new order of aberrations in the notations of Haider and Sawada, rather like the Roman numbering system (I, X, C, M and V, L, D). The difference between the two types of notation becomes especially apparent when combination aberrations generated by the interaction of aberrations occurring in different parts of a multi-element system are considered.

A study of parasitic aberrations has been made by *Krivanek* [13.31], see also [13.32, 13.33, 13.34, 13.35, 13.36]. For measurement techniques, see *Saxton* [13.37, 13.38, 13.39, 13.40], *Saxton* et al [13.41] and *Chand* et al [13.42], as well as *Ishizuka* [13.43], *Meyer* et al [13.44, 13.45], *Lupini* [13.46], and *Lupini* et al [13.47]. The use of the Ronchigram is described in Sect. 13.3.2. The work of *Zemlin* on alignment is also relevant here [13.48, 13.49]; see also *Krivanek* [13.50]. Among the earlier literature on parasitic aberrations, we draw attention to the work of *Glaser* [13.51], *Sturrock* [13.52, 13.53], *Archard* [13.54], *Glaser* and *Schiske* [13.55], *Der*-*Shvarts* [13.56], *Stoyanov* [13.57] and *Herrmann* et al [13.58]. A long survey has been written by *Yavor* [13.59], and other references are listed in *Hawkes* [13.12]. The many publications on the stigmator are also relevant; see Chapter 31 of *Hawkes* and *Kasper* [13.6] for numerous references to these.

Various notations for axial aberration coefficients

Aberration | Haider | Krivanek | Sawada |
---|---|---|---|

Shift | \(A_{0}\) | \(C_{0,1}\) | |

Defocus | \(C_{1}\) | \(C_{1,0}\) | \(O_{2}\) |

Twofold astigmatism | \(A_{1}\) | \(C_{1,2}\) | \(A_{2}\) |

Second-order axial coma | \(B_{2}\) | \((1/3)C_{2,1}\) | \(P_{3}\) |

Threefold astigmatism | \(A_{2}\) | \(C_{2,3}\) | \(A_{3}\) |

Third-order spherical aberration | \(C_{3}\) | \(C_{3,0}\) | \(O_{4}\) |

Third-order star aberration | \(S_{3}\) | \((1/4)C_{3,2}\) | \(Q_{4}\) |

Fourfold astigmatism | \(A_{3}\) | \(C_{3,4}\) | \(A_{4}\) |

Fourth-order axial coma | \(B_{4}\) | \((1/5)C_{4,1}\) | \(P_{5}\) |

Fourth-order three-lobe aberration | \(D_{4}\) | \((1/5)C_{4,3}\) | \(R_{5}\) |

Fivefold astigmatism | \(A_{4}\) | \(C_{4,5}\) | \(A_{5}\) |

Fifth-order spherical aberration | \(C_{5}\) | \(C_{5,0}\) | \(O_{6}\) |

Fifth-order star aberration | \(S_{5}\) | \((1/6)C_{5,2}\) | \(Q_{6}\) |

Fifth-order rosette aberration | \(R_{5}\) | \((1/6)C_{5,4}\) | |

Sixfold astigmatism | \(A_{5}\) | \(C_{5,6}\) | \(A_{6}\) |

Sixth-order axial coma | \(B_{6}\) | \((1/7)C_{6,1}\) | \(P_{7}\) |

Sixth-order three-lobe aberration | \(D_{6}\) | \((1/7)C_{6,3}\) | \(R_{7}\) |

Sixth-order pentacle aberration | \(F_{6}\) | \((1/7)C_{6,5}\) | |

Sevenfold astigmatism | \(A_{6}\) | \(C_{6,7}\) | \(A_{7}\) |

Seventh-order spherical aberration | \(C_{7}\) | \(C_{7,0}\) | \(O_{8}\) |

Seventh-order star aberration | \(S_{7}\) | \((1/8)C_{7,2}\) | \(Q_{8}\) |

Seventh-order rosette aberration | \(R_{7}\) | \((1/8)C_{7,4}\) | |

Seventh-order chaplet aberration | \(G_{7}\) | \((1/8)C_{7,6}\) | |

Eightfold astigmatism | \(A_{7}\) | \(C_{7,8}\) | \(A_{8}\) |

## 13.2 Aberration Correction

In this section, we provide the theoretical basis for aberration correction and then describe correctors that have been successful in correcting spherical and chromatic aberration, as well as more complex correctors that correct additional aberrations. We also discuss practical aspects of aberration correction such as aberration autotuning, and its application to other electron-optical instruments such as monochromators and energy-loss spectrometers. We focus on correctors that made key advances in the aberration correction field and on correctors that led to practical success: those that provided better spatial resolution than uncorrected instruments. We stress the importance of combination aberrations, employed for correction of aberrations for which no direct controls are available, and we provide several examples of present-day performance made possible by aberration correction. A wider-ranging survey which included correctors that did not improve the resolution of the microscope they were built into was provided in the aberration correction chapter in *Science of Microscopy*, the predecessor of this volume [13.60]. Aberration correction has also been reviewed extensively by *Hawkes* [13.12, 13.61, 13.62] and *Rose* [13.63], with the last *Hawkes* publication providing a comprehensive list of papers published on aberration correction up to 2015; some later publications are listed in Chapter 41 of [13.6]. A general introduction to aberration-corrected transmission electron microscopy has been provided by *Erni* [13.64], and volume 153 in the *Advances in Imaging and Electron Physics* series has been dedicated to it [13.65]; the collection edited by *Brydson* [13.66] and the chapter by *Sawada* [13.27] are also relevant.

The integrand \(f[B(z),h(z)]\) can be written in different ways (see [13.6] for many of these and for a general formula from which all the others can be generated; forms particularly useful for programming are given by *Lencová* and *Lenc* [13.67] (for magnetic lenses) and [13.68, 13.69] as well as *Lencová* [13.11] (for electrostatic lenses)). In 1936, Otto Scherzer derived a nonnegative-definite form of the integrand, a sum of squared terms, from which it is clear that the sign of the coefficient cannot change. Scherzer’s formula was nonrelativistic, but *Rose* [13.70] (see corrigendum in [13.71]) has established a relativistic version, which essentially confirms Scherzer’s conclusion. Efforts to find field or potential distributions for which the integrand vanishes [13.72, 13.73] failed, as they were sure to do given the form of the integrand found by Scherzer. (Attempts to find a loophole nevertheless continue; see *Nomura* [13.74, 13.75], for example, refuted by *Hawkes* [13.76].) *Tretner* [13.77] later established bounds on the coefficient.

Prior to successful aberration correction, the quest for higher resolution primarily took the direction of reducing \(\lambda\) by going to higher primary energies. In \(1{-}1.5\,{\mathrm{MeV}}\) electron microscopes, this allowed \({\mathrm{1}}\,{\mathrm{\AA{}}}\) resolution to be reached [13.82]. But improving the performance in this way required large and expensive electron microscopes, and typically also produced unacceptably high radiation damage. Successful aberration correction provided a way round these difficulties.

The correction relies on the fact that Scherzer’s formula does not apply when any of the assumptions made in its derivation are not fulfilled. The assumptions are that the optical system must possess rotational symmetry, the excitation must be static, only dioptric operation is permissible (excluding an electron mirror mode), and in the case of electrostatic lenses, the potential distribution and its derivatives must be continuous. Object and image must both be real (not virtual). *Scherzer* himself showed [13.2] that by relaxing one or other of these conditions, a corrector could be devised.

Throughout the second half of the twentieth century, efforts were made to build and test the various types of correctors. These are described in detail in *Septier* [13.3] and *Hawkes* [13.5]; more recent information is included in *Hawkes* [13.12, 13.61, 13.65, 13.83] and *Hawkes* and *Kasper* [13.6, Chap. 41]. Departure from rotational symmetry has always seemed a promising approach, and as early as the 1950s, *Seeliger* [13.84, 13.85] attempted to put Scherzer's suggestion for exploiting the idea into practice; his work was pursued by *Möllenstedt* [13.86]. *Archard* [13.87] showed that relatively simple configurations would create the desired field distribution, and *Burfoot* [13.88] found a three-electrode geometry that would in principle be aberration-free. Numerous further experiments were conducted over the years, including a reassuring proof-of-principle \(C_{\mathrm{s}}\)-correction experiment by *Deltrap* [13.89] and a \(C_{\mathrm{c}}\)-correction experiment by *Hardy* [13.90]. However, subsequent attempts to use aberration correctors to improve the resolution beyond what was attainable in the best uncorrected electron microscopes were unsuccessful, and in the 1980s, aberration correction temporarily acquired an aura of an impossibly difficult problem.

Quadrupole correctors that can improve a microscope's resolution by correcting its geometrical aberrations consist of four or more quadrupoles and three or more octopoles, and sextupole correctors consist of two or more sextupoles as well as round lenses. The correctors principally correct the spherical aberration of the objective lens. Substantial contributions to the system’s total \(C_{\mathrm{s}}\) can also come from other parts of the optical system—for instance, the electron gun and condenser lenses in a probe-forming system using a large beam current, and hence a wide angular range beam extracted from the electron gun—and these contributions must be corrected as well. In addition to the strict stability requirements, every aberration-corrected system must also address parasitic aberrations described in Sect. 13.1.3 above, where the notation used later in this section is defined. The corrected optical system is invariably more complicated than the corresponding uncorrected system, and this opens up many new avenues for parasitic aberrations to be produced. At the same time, when aiming for higher resolution, the precision with which the aberrations must be nulled increases considerably.

Successful aberration-corrected optical systems must therefore accomplish three separate tasks: measure and correct the principal aberrations affecting the resolution, diagnose and eliminate parasitic aberrations, and improve the instrumental stabilities. This stage of development was first reached in the 1990s by the Darmstadt/Heidelberg group led by Harald Rose, Max Haider and Joachim Zach, and the Cambridge/Seattle group led by Ondrej Krivanek and Niklas Dellby.

As we shall see in the following sections, both types of multipole correctors are now in common use. The quadrupole–octopole corrector is used in scanning transmission electron microscopes ( s) and in conventional (fixed-beam) transmission electron microscopes ( s) when chromatic aberrations are corrected together with geometrical ones. The sextupole arrangement is incorporated in both CTEMs and STEMs. A valuable study of the performance of sextupole correctors was carried out by *Haider* et al [13.91] and more recently by *Müller* et al [13.92], and a description of various aspects of quadrupole–octopole corrector construction, including stability requirements, has been provided by *Krivanek* et al [13.25].

Other possibilities for correcting spherical aberration include the use of space charge [13.93] or of phase plates , with both devices changing the phase of the electron wavefront so that \(C_{\mathrm{s}}\) is eliminated, or an optimal phase-contrast transfer function is obtained. Unfortunately, the space charge method has not proven to be reliable, and practical difficulties have so far prevented the phase plate method from reaching its theoretical potential [13.94]. Holography and wavefront reconstruction employing bright-field through-focus series data can also lead to the elimination of the influence of spherical aberration by post-processing of experimental data. In this chapter, however, we concentrate on electron-optical methods of correcting aberrations. The introduction of an electrode on the optic axis, and hence the use of conical beams , has been revived recently. We refer to *Khursheed* and *Ang* [13.95, 13.96] and *Kawasaki* et al [13.97, 13.98] for details.

Chromatic aberration correction also has a long history, though less effort has been devoted to it than to spherical aberration correction because, in the high-resolution imaging mode, developments in microscope design soon rendered the adverse effect of spherical aberration greater than that of chromatic aberration. This is obvious from an examination of the phase-contrast transfer function, which is a sinusoidal curve in the absence of any energy spread (chromatic effects or temporal partial coherence) and neglecting the nonvanishing source size. The sinusoidal curve is damped by an envelope function, representing the effect of energy spread, but the first zero of the sinusoidal curve (a measure of the limit of resolution determined by the spherical aberration) occurs well before the damping curve reduces it to an unacceptably small value (the so-called information limit). With the arrival of \(C_{\mathrm{s}}\) correctors, however, the situation has changed dramatically, and it is now of interest, indeed essential, to improve the information limit as well by reducing the undesirable effects of energy spread.

for electrostatic lenses. As in the case of\(C_{\mathrm{s}}\) (and, indeed, of all the aberration coefficients), the integrands in (13.28a) and (13.28b) can be written in different ways. The ones given here show immediately that \(C_{\mathrm{c}}\) is positive definite, and the best that can be hoped for in a round lens is a design for which \(C_{\mathrm{c}}\) is small.

Over the years, two approaches to the problem of avoiding the limitations imposed by the chromatic aberration of round lenses have emerged. One is a natural continuation of the efforts to reduce the energy spread of the beam emitted by the electron gun; by introducing a monochromator into the column, electrons with energies outside the chosen range can be excluded. There is of course a loss of beam current, but since the energy spread of the filtered beam can be made appreciably narrower than that of the original beam, this reduction in current may be acceptable. This solution is attractive not only in the imaging mode but also for electron energy-loss spectroscopy ( ) and energy-filtered transmission electron microscopy ( ), since it improves the energy resolution and hence the information content of EELS data.

From 2004 onward, monochromators have enabled substantially improved spatial resolution to be reached in aberration-corrected CTEM axial bright-field imaging (which is highly sensitive to resolution loss resulting from \(C_{\mathrm{c}}\) effects). More recently, a resolution improvement through the use of monochromator has also been demonstrated in the STEM, which is less sensitive to chromatic effects. Both developments are discussed in Sect. 13.3.3.

The alternative to using a monochromator is to devise a corrector of chromatic aberration. There are several ways in which such correctors can be conceived, involving the use of superimposed round lens and quadrupole fields, hybrid electrostatic–magnetic quadrupoles, or electron mirrors. An early suggestion by *Scherzer* [13.2] involved combining an electrostatic round lens and an electrostatic quadrupole in such a way that the overall chromatic aberration coefficient of the combination is negative. Such a device could then be used to correct the chromatic aberration of a round lens acting as an objective. This suggestion was taken up by *Archard* [13.99] and has subsequently been carefully investigated.

In 1961, Kel'man and Yavor showed that the chromatic aberration coefficient of a hybrid electrostatic–magnetic quadrupole can have either sign, depending on the relative strengths of the component quadrupoles, and hence that such hybrid lenses could be used to correct chromatic aberration [13.100], translated as [13.101]. The result was rediscovered by *Septier* [13.102] and generalized by *Hawkes* [13.103, 13.104]. Such hybrid quadrupoles appear in the latest designs of correctors, intended for the correction of both geometrical and chromatic aberrations.

Another possibility, pointed out by *Rose* [13.105], involves the use of a long Wien filter . This too has been studied carefully [13.106]. Attempts have also been made to redistribute the energy of the beam electrons in such a way that the energy spread is reduced. For details, see [13.107, 13.108, 13.109].

Finally, an electron mirror can be coupled to a round lens in such a way that the chromatic aberration is canceled. This system has led to important resolution improvements in low-energy-electron microscopy ( ) and photoemission electron microscopy ( ) of solid surfaces, see [13.110, 13.111, 13.112, 13.113, 13.114, 13.115, 13.116, 13.117, 13.118, 13.119, 13.120, 13.121, 13.122, 13.123]. Two designs including twin mirrors to correct the aberrations of a SEM are proposed by *Dohi* and *Kruit* [13.124].

### 13.2.1 Spherical Aberration Correctors

#### Quadrupole–Octopole Correctors

For the correction of spherical aberration, Scherzer proposed a sequence of cylindrical lenses and octopoles in his seminal paper of 1947 on ways of avoiding the consequences of his 1936 proof. Cylindrical lenses are the electron optical counterparts of glass lenses with cylindrical (as opposed to spherical) faces and are characterized by a round lens and a quadrupole potential distribution. It was soon realized [13.87] that quadrupole lenses could be used to advantage instead of cylindrical lenses, and the basic corrector configuration, which has remained essentially unaltered, soon emerged (Fig. 13.4): a sequence of three or preferably four quadrupole lenses, with an octopole situated at each of the line foci to cancel or over-correct the aperture aberrations in the \(x{-}z\) and \(y{-}z\) planes, together with a third octopole to complete the task of correction.

An important step forward was the introduction of the Russian quadruplet, which has geometrical symmetry and electrical antisymmetry about its midplane, as we have already mentioned. In common with all multiplets possessing these symmetry properties, such quadruplets have the same focal length in the \(x{-}z\) and \(y{-}z\) planes. For a given geometry, the positions of the foci in these planes can then be made to coincide by suitable choice of the two excitations, whereupon the quadruplet has the same overall paraxial behavior as a round lens. Sets of load curves , showing the appropriate excitations as a function of geometry, are available (see [13.15] for many such curves and [13.16]). Another interesting early contribution was made by *Burfoot* [13.88], who sought the (electrostatic) configuration with the smallest number of electrodes that would be free of spherical aberration. He established suitably shaped apertures in a three-electrode lens (a remarkable achievement in precomputer times), but concluded that the necessary tolerances could not be achieved in practice; a simpler way of attaining the same objective was proposed by *Archard* [13.125].

In 1964, Deltrap showed that the spherical aberration of a test lens could be reduced by means of a quadrupole–octopole corrector, and thus confirmed that the principle of correction was sound. However, for the next three decades, all attempts to make a corrector capable of improving the performance of a well-designed objective failed; with hindsight, we can see that these disappointing failures were due to the natural complexity of the system and hence the need to control parasitic aberrations, as well as to the difficulty in satisfying the increased stability requirements. Progress was made in the Darmstadt project [13.126, 13.127, 13.128, 13.129, 13.130, 13.131, 13.132, 13.133, 13.134, 13.135, 13.136, 13.137, 13.138, 13.139, 13.140, 13.141, 13.142, 13.143] (see *Scherzer* [13.144] for a summary, and *Marko* and *Rose* [13.7] for a later account) and in the Chicago project [13.145, 13.146, 13.147, 13.148, 13.149, 13.150, 13.151]. The Darmstadt project demonstrated aberration correction including manual control of parasitic aberrations, but it did not demonstrate resolution improvement relative to the best uncorrected electron microscopes available at the time. The Chicago project introduced new concepts in aberration correction, such as the sextupole corrector, but it did not succeed in demonstrating aberration correction in practice. The tools necessary for the operation of such devices, including autotuning and electronics of sufficient stability, were not yet available.

In the early 1990s, *Zach* et al showed that a quadrupole/octopole \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector could improve the performance of scanning electron microscopes, and this finding continues to be exploited in commercial instruments [13.152, 13.153, 13.154, 13.155, 13.156, 13.157, 13.158, 13.159, 13.160, 13.161, 13.162]. For the transmission electron microscope, success came in 1997, when *Krivanek* and colleagues, working in the Cavendish Laboratory in Cambridge, built a corrector equipped with computer control, capable of making the many necessary adjustments rapidly and systematically [13.163, 13.164, 13.165]. This corrector was fitted to a STEM and enabled the size of the illumination aperture to be increased, which led to either an electron probe of decreased size with no loss of beam current, or an increase in the beam current with no loss of resolution.

The Cambridge corrector consisted of the basic quadrupoles and octopoles, all under computer control, together with other multipole fields designed to compensate for misalignments and parasitic aberrations in general. In a follow-up first-generation Nion aberration corrector , the quadrupoles and the strong octopoles were separated so that there would be no magnetic cross talk between them, in a 7-multipole arrangement. The correction half-angle of this corrector was limited to \({\mathrm{25}}\,{\mathrm{mrad}}\) by the magnitude of \(C_{5,4}\) (fourfold astigmatism of the fifth order), about \({\mathrm{10}}\,{\mathrm{cm}}\), and this gave a spatial resolution of \(\approx{\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{100}}\,{\mathrm{keV}}\) [13.166, 13.167, 13.168, 13.169, 13.170, 13.26] and \({\mathrm{0.78}}\,{\mathrm{\AA{}}}\) at \({\mathrm{300}}\,{\mathrm{keV}}\) [13.171, 13.172]. A second-generation Nion corrector employs 16 quadrupoles and three combined quadrupole–octopole elements, with additional multipoles to make all parasitic aberrations up to \(C_{5,6}\) adjustable [13.173, 13.174, 13.175, 13.176, 13.177]. The corrector itself consists of an alternating sequence of quadrupole quadruplets and quadrupole–octopole elements (Fig. 13.5). With this arrangement, the center-planes of the quadrupole–octopole elements are all conjugates, and four further quadrupoles situated between the corrector and the probe-forming lens transfer the image of the correcting planes to the vicinity of the coma-free plane of the probe-forming lens. In this way, the fifth-order geometrical aberrations of the combination of corrector and probe-forming lens can be eliminated. Imaging the correction planes into each other is also a feature of the Rose superaplanator and the related transmission electron aberration-corrected microscope (TEAM ) corrector, described in more detail in Sect. 13.2.3. Software adjusts the various components systematically. Another quadrupole–octopole corrector designed for an FEI STEM/TEM was described by *Mentink* et al [13.178].

#### Sextupole Correctors

Sextupoles were not among the correctors envisaged by Scherzer in his 1947 paper. In 1965, it was pointed out that the third-order aberrations, including of course the spherical aberration, of sextupoles have the same dependence on gradient in the object plane as that of a round lens [13.179]. However, the fact that the principal optical effect of sextupoles is not linear, as it is in round lenses and quadrupoles, but quadratic (second-order) seemed to rule out any hope of using them for aberration correction. It was not until 1979 that combinations of sextupoles and round lenses from which the quadratic effects had been eliminated by compensation were proposed [13.180], and subsequent developments have confirmed that such correctors are suitable for incorporation into transmission electron microscopes. As we have seen, the second-order effect of a sextupole is characterized by four terms of the form \(\int S(z)h^{3-n}k^{n}\mathrm{d}z\), in which \(S(z)\) represents the field distribution in the (electrostatic or magnetic) sextupole, and \(h(z)\), \(k(z)\) are two linearly independent solutions of the familiar paraxial equation for round lenses (these solutions collapse to straight lines in the absence of any round lens component). The integer \(n\) takes the four values 0, 1, 2 and 3. All four terms can be made to vanish by suitable choice of the symmetry of the configuration; the simplest is shown in Fig. 13.6. Before coupling such a device to a microscope objective, we must however ensure that the coma-free condition is satisfied. The (isotropic) coma-free plane of an objective is situated within the lens field and must hence be imaged onto the front focal plane of the round-lens doublet in the corrector by means of another doublet (Fig. 13.7). If it should be necessary to eliminate the anisotropic coma as well as the isotropic coma, an objective design in which two coils are used in tandem would have to be adopted [13.129]. Sextupole correction may be traced in the following articles (in addition to the early publications already cited) [13.181, 13.182, 13.183, 13.184, 13.185, 13.186, 13.187, 13.188, 13.189, 13.19, 13.190, 13.191, 13.192, 13.193, 13.194, 13.195, 13.196, 13.197, 13.198, 13.199, 13.200, 13.201, 13.202, 13.203, 13.204, 13.205, 13.206, 13.207, 13.208, 13.209, 13.21, 13.210, 13.211, 13.212, 13.213, 13.214, 13.215, 13.216, 13.217, 13.218, 13.219, 13.22, 13.220, 13.221, 13.222, 13.223, 13.224, 13.225, 13.226, 13.227, 13.228, 13.229, 13.23, 13.27, 13.30, 13.91, 13.92].

Two alternative ways of generating the sextupole fields are also of interest. Following a suggestion of *Nishi* et al [13.230], *Hoque* et al [13.231] have designed a corrector in which the fields are generated by wires that follow the optic axis; see *Nishi* et al [13.232] and *Hoque* et al [13.233] for details. Janzen has observed that the regular sextupole correctors are too bulky for use in miniature microscopes and multicolumn arrays and has therefore inquired whether the necessary fields could be generated by noncircular apertures such as octagons, hexagons and squares [13.234]. A promising design is described by *Janzen* et al [13.235].

### 13.2.2 Chromatic Aberration Correctors

#### All-Electrostatic Correctors

Multipole correctors consisting of electrostatic elements only can correct both the chromatic and spherical aberration, provided that their fields satisfy the so-called Scherzer condition [13.2]. Configurations in which this condition is satisfied have been found by *Weißbäcker* and *Rose* [13.236, 13.237, 13.238] and by *Maas* and coworkers [13.239, 13.240, 13.241, 13.242]. In the studies of Weißbäcker and Rose, several configurations were examined, in which the complexity increased with the practical usefulness of the corrector. The configurations were described in detail in the corresponding chapter of *Science of Microscopy* [13.60]. The symmetry conditions are arranged in such a way that the chromatic aberration and the coma vanish, while the spherical aberration is corrected by means of octopoles. In the complementary investigations of Henstra, Maas, and Mentink and coworkers, a configuration consisting of nine elements was explored.

Although the correction principles are sound, electrostatic-only multipole correctors have not yet found practical application in electron optics. They are, however, highly promising for ion-optical applications, as electrostatic focusing is more suitable for slow ions than magnetic focusing, and their use has been demonstrated by *Bajo* et al [13.243].

#### Quadrupole \(C_{\mathrm{s}}/C_{\mathrm{c}}\) Correctors

Quadrupole lenses consisting of four electrodes and four magnetic poles situated midway between the electrodes have the power of correcting the chromatic aberration of a round lens. They must of course be part of a suitable configuration, such as the one used by *Hardy* [13.90], the Darmstadt \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector [13.127, 13.135], and the *Zach* and *Haider* \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector for a SEM [13.154, 13.155]. The simpler configurations produce large field and fifth-order combination aberrations, but this can be overcome by two types of more complex arrangements. The first of these relies on precise imaging of the correction planes into each other and then into the coma-free plane of the objective lens, which avoids combination fifth-order spherical aberration \(C_{5}\) and decreases field aberrations. This is the arrangement used by the superaplanator corrector due to Rose, whose ideas have led to the TEAM and PICO correctors , [13.21, 13.22, 13.222] and by the Nion \(C_{3}/C_{5}\) corrector [13.175, 13.176, 13.177]. In the TEAM corrector [13.13], two symmetric quadrupole quintuplets and three (or more) octopoles correct spherical and chromatic aberration as well as \(C_{5}\). The quadrupole fields are symmetric with respect to the center-plane of each quintuplet; conversely, the whole (double-quintuplet) unit exhibits antisymmetry about its midplane (Fig. 13.8).

The second type of arrangement distributes the correction action into several planes and makes sure that higher-order combination aberrations produced by incorrectly imaged correction elements cancel each other. This can be accomplished by using pairs of identically excited correcting elements (identical in theory, small deviations are likely to be needed in practice); these elements are located at equal distances from the corrector midplane, with trajectories that are symmetric or antisymmetric about this plane. This maps all the effective correction planes, encapsulating the summed effects of the correcting elements, into the midplane of the corrector, and the midplane is then imaged into the vicinity of the coma-free plane of the objective lens by coupling lenses. Three types of aberrations are corrected internally in a quadrupole–octopole corrector (\(C_{3,0,}C_{3,2}\) and \(C_{3,4}\)), and the effective correction planes for all of them are projected into the corrector midplane. This type of corrector was also proposed by Rose, in versions with five, seven or eight multipoles consisting of quadrupoles and octopoles and trajectories that are symmetric or antisymmetric about the central plane [13.128, 13.129, 13.192]. A modified eight-multipole version has been implemented as the SALVE corrector [13.244, 13.245], with additional controls for parasitic aberrations.

The external appearance of the SALVE corrector and the ray diagram of its multipole part (with round coupling lenses not included) are shown in Fig. 13.9a,b. Multipoles MO3 and M13 contain the hybrid electrostatic-magnetic quadrupoles, plus octopoles and other multipoles. The corrector’s height and weight are about \({\mathrm{50}}\%\) of those of the PICO \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector, but it reaches the same goals, and is less affected by instabilities and Johnson–Nyquist noise (discussed in Sect. 13.2.5).

Very high stability of the excitations is essential for all correctors, especially \(C_{\mathrm{s}}/C_{\mathrm{c}}\) ones. When aiming for \({\mathrm{0.5}}\,{\mathrm{\AA{}}}\) resolution at \(200{-}300\,{\mathrm{keV}}\), or \({\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) and below, better than one part in \(\mathrm{10^{7}}\) stability is needed for the regular optical elements, and about two parts in \(\mathrm{10^{8}}\) or better for the hybrid quadrupoles that perform the \(C_{\mathrm{c}}\) correction [13.21]. Fortunately, such stabilities are now available with modern electronics. Attention must also be paid to Johnson–Nyquist noise , which can affect the performance of \(C_{\mathrm{s}}/C_{\mathrm{c}}\) correctors.

#### An All-magnetic Corrector

An \(\Upomega\)-shaped all-magnetic corrector has been developed for use at high voltages [13.246].

#### Wien Filters and Correction

In an attempt to design a corrector that is reasonably easy to align and consists of as few separate elements as possible, *Rose* [13.105] has also examined the properties of an inhomogeneous Wien filter, and his ideas were followed up by *Mentink* et al [13.247] and *Steffen* et al [13.248]. No practical corrector based on these ideas appears to have been built to date, and the theoretical designs were reviewed in detail in [13.60].

#### Mirror Correctors

Several schemes have been proposed to compensate for the aberrations of round lenses by introducing an electron mirror (see [13.249, 13.250, 13.251] for early work on electron mirrors) into the optical system [13.252, 13.253, 13.254]. Aberration correction has now reached the stage at which simultaneous correction of all the aberrations that are likely to impair the performance of the instrument in question must be envisaged: correction of individual aberrations without considering the effect of the remainder is no longer sufficient. For this reason, we focus here on the scheme devised by *Preikszas* and *Rose* [13.255] and surveyed by *Hartel* et al [13.116], intended for the SMART project at BESSY II and also adopted for PEEM3 at the Lawrence Berkeley National Laboratory [13.256]. This differs from earlier schemes, notably that of *Rempfer* [13.110], *Shao* and *Wu* [13.257], *Rempfer* and *Mauck* [13.111] and *Rempfer* et al [13.258], in that the beam splitter, the role of which is to separate the beam incident on the mirror from the beam emerging from it, is now nondispersive. A four-electrode mirror, such as that shown in Fig. 13.10a, offers enough degrees of freedom to adjust the focal length and the spherical and chromatic aberration coefficients satisfactorily [13.259, 13.260]. As an example, *Preikszas* and *Rose* [13.255] show that the spherical and chromatic aberration coefficients can be chosen anywhere inside the shaded region in Fig. 13.10b for a fixed position of the Gaussian image plane. Such a mirror could be combined with a dispersion-free magnetic beam splitter as shown in Fig. 13.11a,ba. Figure 13.11a,bb shows the device incorporated in the spectromicroscope for all relevant techniques ( ). In practice, the SMART system has demonstrated \({\mathrm{2.6}}\,{\mathrm{nm}}\) spatial resolution in LEEM mode, using a landing energy of \({\mathrm{15}}\,{\mathrm{eV}}\) [13.261]. Another very successful mirror-type LEEM/PEEM corrector has been built at IBM T.J. Watson laboratories [13.120, 13.121, 13.122] and is shown in Fig. 13.12. It allows a variety of operating modes that include energy-filtered imaging, and it has achieved \({\mathrm{2}}\,{\mathrm{nm}}\) edge definition. A similar design has been described by *Mankos* and *Shadman* [13.262]. *Dohi* and *Kruit* [13.124] have suggested that, by using microelectromechanical systems (MEMS) technology to fabricate very small mirrors, a compact SEM corrector could be built.

### 13.2.3 Correction of Aberrations Beyond Third Order

Two complex correctors were proposed by *Rose* in the early 2000s that are capable of correcting the spherical and chromatic aberrations, as well as other primary aberrations such as field curvature and astigmatism that can be harmful once the axial aberrations have been brought under control [13.13, 13.14, 13.263, 13.264, 13.265]. These are called *superaplanator* and *ultracorrector* . The first of these, discussed briefly in the context of \(C_{\mathrm{c}}/C_{\mathrm{s}}\) correctors above, is suitable for transmission electron microscopes, while the second is intended for lithography where a wide field of view is required. A simplified version of the superaplanator , called an *achroplanator* [13.222], has been used in the \(C_{\mathrm{s}}/C_{\mathrm{c}}\) TEAM and PICO projects [13.63]. The results from these projects and from SALVE can be found in several publications [13.244, 13.245, 13.266, 13.267, 13.268, 13.269, 13.270, 13.271, 13.272].

It has been known since the work of *Shao* [13.273] that when threefold astigmatism \(C_{2,3}\) is canceled by opposing the astigmatisms created by two sextupoles, sixfold astigmatism \(C_{5,6}\) remains as the strongest uncorrected aberration. The magnitude of \(C_{5,6}\) can be reduced by decreasing the length of the two sextupoles and increasing their excitation [13.274], but a complete cancellation of \(C_{\mathrm{5,6}}\) is not possible without adding a 12-pole \(C_{5,6}\) stigmator to the two-sextupole design. Another way around this difficulty is possible with three-sextupole corrector designs, as these provide an additional degree of freedom that allows both \(C_{2,3}\) and \(C_{5,6}\) to be nulled simultaneously [13.218].

Attaining optimal performance with correctors of higher-order aberrations means that parasitic aberrations of all orders up to the highest one being corrected must be kept under strict control. We discuss the work in this field in Sect. 13.3.1 below.

### 13.2.4 Aberration Correction in Monochromators, Electron Spectrometers and Imaging Filters

The task of a monochromator is to reduce the energy spread of the illuminating beam, typically by producing an energy spectrum of electrons emitted by the electron source and projecting the spectrum onto an energy-selecting slit which defines the passband of energies that is admitted into the rest of the electron microscope column. The task of an electron energy-loss spectrometer is to produce an energy spectrum and project it onto a detector, typically with adjustable magnification, and to read it out efficiently. An imaging filter performs two related tasks: (a) it projects an energy-loss spectrum onto an energy-selecting slit, admits a passband of energies through the slit, and transforms the selected spectrum into an achromatic and largely distortion-free image on the final detector, or (b) it projects the spectrum onto the final detector, the slit having been withdrawn or opened wide. The three types of instruments are therefore closely related, and the same electron-optical principles apply to all of them.

- 1.
Image coupling , in which a very small image of the illuminated area of the sample is projected into the

*spectrometer entrance object plane*, where a small crossover is needed, with a magnification (relative to the magnification on the sample) of typically around 10\(\times\), or - 2.
Diffraction coupling , in which a very small diffraction pattern, with a camera length of the order of a few micrometers, is projected into the spectrometer entrance object plane.

The spectrometer energy-disperses the entrance object into a spectrum and images it onto the final detector, whereas the imaging filter energy-disperses the entrance object into a spectrum and images it onto the energy-selecting slit. Monochromators use the image-coupling mode, with a small image of the source dispersed into an energy spectrum imaged onto the monochromator slit.

It is useful to remember that a diffraction pattern appears on the final viewing screen as well as at the spectrometer (or imaging filter) entrance aperture in every CTEM operating in the *image-coupling* mode, and an image of the sample appears in these places in the *diffraction-coupling* mode. Planes containing reciprocally related images succeeding each other is the usual order of the day in electron microscopes, and it is important not to lose sight of the reciprocal relationship.

Only the image-coupling mode is used to couple to EEL spectrometers in STEMs. Good coupling efficiency for energy-loss events is obtained with EELS entrance apertures slightly larger than the STEM bright-field cone , and this means that electrons scattered elastically to higher angles (typically greater than about \({\mathrm{80}}\,{\mathrm{mrad}}\)) can be directed at the same time to a high-angle annular dark-field ( ) detector. Structural information on the sample can thus be obtained from the HAADF detector at the same time as the EEL spectrum, and simultaneous collection of the two types of signals has become the standard procedure in STEM. Alternatively, a pixelated two-dimensional () detector can be inserted in the pre-spectrometer plane, temporarily obscuring the EELS entrance. This is used for recording convergent-beam Ronchigrams for STEM autotuning, as well as diffraction patterns obtained with a more parallel beam.

Figure 13.13 shows a schematic cross section of a STEM column containing a ground-potential monochromator, probe-forming optics, and a spectrometer. In all three of these microscope components, the object being imaged is the electron source, i. e., the virtual crossover that is located inside the field emission tip and from which the electrons appear to emanate. The correction of aberrations affecting the monochromator spectrum, sample-level probe and the final EEL spectrum has the same aim: decreasing the size of the imaged object as much as possible. The correction inside the monochromator compensates for aberrations introduced by the electron gun, the monochromator coupling lenses and the monochromator optics; the probe-formation corrector compensates for the aberrations of the objective lens, the condenser lenses and any remaining aberrations of the monochromator; and the spectrometer-level correction compensates for aberrations of the spectrometer as well as aberrations of post-sample optics including the post-sample part of the objective lens.

A major difference between the correction carried out in the monochromator and the spectrometer and the correction of the sample-level probe is that electron spectrometers, imaging filters and monochromators use prisms that have inherent second-order aberrations that are not present in optical systems with round symmetry. The prisms employ transverse magnetic or electrostatic fields to disperse electrons in energy, and they *bend* the electron beam. (Wien filters constitute an important exception: they disperse the beam using combined transverse magnetic and electrostatic fields working in opposition in such a way that there is no overall *bend*.)

Optical systems with transverse fields can be mirror-symmetric about their \(x{-}z\) (dispersion) plane, but cannot be rotationally symmetric. Second-order aberrations are usually very strong in uncorrected versions of such systems, and sextupoles or curved prism faces, which produce sextupole effects, have been employed to correct second-order aberrations of prisms for some time [13.275, 13.276, 13.277]. Inclining the prism entrance and exit faces so that they are not exactly normal to the beam produces quadrupole moments, and these too have been used to adjust the prism’s first-order focusing properties.

The quadrupole and sextupole strengths produced by modifying the prism faces are not adjustable, and in more recent designs, inclined and curved prism faces have been abandoned in favor of flat prism faces at \(90^{\circ}\) combined with multipoles next to them that produce adjustable quadrupole and sextupole moments [13.278]. This makes it possible to fine-tune the aberrations exactly as needed. It also avoids the danger of the deviations from the ideal trajectory growing catastrophically on passing through a system containing many prisms with inclined and curved faces and no way of fine-tuning the quadrupole and sextupole strengths, a phenomenon that has been known to render an imaging filter using four prisms—making a total of eight inclined and curved prism faces—inoperable. Extension to higher-order correction through the use of higher-order multipoles is straightforward, and recent versions of spectrometers and imaging filters include correction of aberrations up to the fifth order.

- 1.
In imaging filters, the

*aberrations*of the spectrum and*distortions*of the energy-filtered image typically need to be corrected - 2.
When the electron beam becomes dispersed in energy, mixed chromatic–geometrical aberrations arise and need to be corrected

- 3.
Aberrations in the spectrum in the direction perpendicular to the energy dispersion are typically of limited interest and are often not corrected.

*is*required, notably when a spectrum has to be squeezed onto a narrow detector. Furthermore, the standard notation for the aberrations of spectrometers and imaging filters is tailored to a matrix algebra approach for calculating them (e. g.,

*Egerton*[13.279], where a full account of spectrometers, imaging filters and monochromators is to be found); it is not the same as the notation typically used for systems with a straight optic axis.

We now describe the various families of these devices in turn. Spectrometers and imaging filters can be situated either at the end of the microscope column, or incorporated in the column itself, and we consider the two types separately.

#### Post-Column Spectrometers and Imaging Filters

Whereas in-column filters use multiple prisms to create energy dispersion, post-column spectrometers typically use a single magnetic prism for this purpose. Post-column spectrometer configurations using multiple prisms are also possible and can provide important advantages [13.280]. But the benefits of using multiple prisms in post-column spectrometers must be weighed against the extra complexity, and the multiple-prism solution has not gained wide acceptance.

The single prism deflects the electron beam typically through \(90^{\circ}\), although other angles can be used as well. A single prism with straight faces without extra multipoles provides no aberration correction and cannot deliver a sufficiently large range of scattering angles to the spectrometer. It also fails to provide spectra with sufficiently high dispersion to be well matched to the channel size of a parallel detector. The optics of post-column spectrometers has therefore been improved through the addition of various optical elements. Adding a quadrupole-based magnification system to a second-order-corrected prism allowed spectra to be detected in parallel with independently adjustable dispersion and width of the spectrum [13.281]. Adding a second prism and three flexible dodecapole elements, in which rotatable dipole, quadrupole, sextupole and octopole fields plus a nonrotatable 12-pole field could be generated under computer control, allowed spectrum aberrations up to third order to be canceled [13.280]. Adding sextupoles to a quadrupole-based imaging filter allowed second-order distortions of energy-filtered images (or diffraction patterns) to be canceled [13.282, 13.283]; adding octopoles to this system allowed third-order aberrations to be canceled as well [13.284], and adding further multipoles allowed full third-order and partial fourth-order correction [13.285].

The Gatan imaging filter (GIF ) *Quantum* [13.286] uses multipole elements consisting of 12-poles in an arrangement similar to the scheme of independent computer control of individual pole excitations introduced by *Haider* [13.280, 13.287]. It provides rotatable dipole, quadrupole, sextupole, octopole and 10-pole fields, plus a nonrotatable dodecapole field. A 12-pole element capable of generating these multipoles precedes the prism, and seven further such elements are situated after the prism, followed by fast deflectors and a fiber-optically coupled CCD or CMOS, or a direct-exposure CMOS 2-D detector. The disposition of these components and of the energy-selecting slit is shown in Fig. 13.14. The ability to use very large entrance apertures—up to \({\mathrm{9}}\,{\mathrm{mm}}\) diameter—for good energy resolution has been demonstrated for this spectrometer [13.286], as well as an energy resolution of \({\mathrm{12}}\,{\mathrm{meV}}\), measured as the full width at half-maximum ( ) of the zero-loss peak in a spectrum acquired in \({\mathrm{2}}\,{\mathrm{ms}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) primary energy [13.288].

The Nion *Iris* spectrometer , designed for ultrahigh-energy resolution [13.289], has three layers of multipoles of up to 16-poles in front of the prism and nine layers of multipoles after the prism. Used together with the Nion ground-potential monochromator described later in this section, and employing two linkage schemes that make the spectrometer insensitive to instabilities in the microscope's high voltage and in the prism current, it has been able to attain energy resolution (FWHM of the zero-loss peak) of \({\mathrm{6}}\,{\mathrm{meV}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) primary energy [13.290], and \({\mathrm{4.2}}\,{\mathrm{meV}}\) at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.291].

*Iris*spectrometer is that it produces the magnetic field distributions needed for dipoles, quadrupoles, sextupoles, octopoles, etc. by serially connecting coil windings distributed over different poles. Here, the number of turns on each pole ensures that the desired multipole is created. For a multipole \(m\) (\(m=1\) for a dipole, \(m=2\) for a quadrupole, \(m=3\) for a sextupole, \(m=4\) for an octopole, etc.), and an angle \(\varphi\) between the mirror symmetry plane and the \(j\)-th pole, we have

#### In-Column Analyzers

In-column imaging filters are placed just before the final projector lens (or lenses) in an electron microscope. They typically perform \(1:1\) imaging of an entrance object into an exit image, and they also produce an energy spectrum in a plane containing a removable energy-selecting slit. Round lenses placed between the objective lens and the filter allow it to operate either in a diffraction-coupled mode, in which there is a diffraction pattern in the energy-selecting plane and an image of the sample in an achromatic plane typically within the filter, or an image-coupled mode, in which the energy-selecting plane contains an energy-dispersed image and the achromatic plane a diffraction pattern. The post-filter lenses image either an energy-filtered image (or diffraction pattern) or the energy-loss spectrum onto the viewing screen or the camera of the microscope.

The first in-column imaging filter was the *Castaing*–*Henry* analyzer [13.292], in which a prism first deflects the electron beam through \(90^{\circ}\), an electrostatic mirror reflects the beam, and the beam re-enters the prism and is deflected through a further \(90^{\circ}\) and thus continues along the original optic axis. The optics of this device is well understood and set out in full by *Metherell* [13.293], reprinted in [13.294]. For accelerating voltages of a hundred kilovolts or more, it was clearly desirable to avoid using a mirror; by introducing extra prisms, all-magnetic devices were designed by *Senoussi* [13.295, 13.296]. These led to the family of \(\Upomega\)-filters and \(\upalpha\)-filters (so called from their resemblance to the Greek letters, e. g., [13.297, 13.298, 13.299, 13.300, 13.301]). Such filters suffer from the usual aberrations of prisms and the primitive models have evolved into more sophisticated devices, in which numerous supplementary multipoles cancel harmful intrinsic and parasitic aberrations.

Imaging filters are closely related to monochromators, the principal difference arising from the type of symmetry employed. The filters need to be able to display the spectrum on the final detector, and this means that the energy-deviating ray cannot be symmetric about the filter midplane, since this would result in the energy dispersion in the midplane being canceled at the filter's exit. (This type of symmetry is very suitable for monochromators, but not for imaging filters.) The energy-deviating ray in imaging filters is therefore neither symmetric nor antisymmetric about the midplane, but arranged such that the dispersion keeps on growing as the ray progresses through the filter. At the same time, axial and field rays are typically arranged to be either mirror-symmetric or point-symmetric (also called antisymmetric) about the midplane, and this is used for canceling key aberrations.

Both \(\upalpha\) and \(\Upomega\) configurations are possible, and there are a great many different possible embodiments—Table 1 in [13.301] lists more than \(\mathrm{1000}\) different solutions! Filters of comparable complexity have comparable properties, and they can be greatly improved by incorporating extra aberration-canceling multipoles [13.299]. Two broad classes of both the \(\upalpha\) and \(\Upomega\) filters can be distinguished: A-type filters, in which the filter midplane contains a stigmatic (doubly focused) image, and B-type filters, in which the midplane contains an astigmatic image—actually an image in one direction and a diffraction pattern in the perpendicular direction.

\(\Upomega\) and \(\upalpha\) in-column filters are shown schematically in Fig. 13.15a-c. In each case, the device is mechanically symmetric about the midplane, and the axial trajectories (blue) and field trajectories (green) are either mirror-symmetric or point-symmetric about the midplane. It is readily seen that the polarity of the magnets in \(\upalpha\) filters alternates in each half of the device so that the beam returns to the optic axis with no overall deflection. In the \(\upalpha\) filters, the polarity is the same throughout and the beam has undergone a deflection of \(2\uppi\) when it returns to the optic axis.

Aberration correction is accomplished by multipoles located in planes in which the first-order properties of the beam are substantially different: different widths in the dispersion and nondispersion directions, different amounts of energy dispersion. For the best performance, it can be advantageous to add an extra symmetry element to the basic \(\Upomega\) design. The result is the *mandoline* filter [13.303, 13.304], in which the 60 and \(120^{\circ}\) planes are also symmetry planes. In reality, the symmetries described above are never perfectly respected and one role of the correction elements is to compensate for the ensuing parasitic aberrations. Figure 13.16 shows the mandoline filter in detail, including the numerous correction multipoles. Here, the first and fourth prisms of the original \(\Upomega\) configuration collapse to a single prism, as in the Castaing–Henry arrangement. The action of the prisms and multipoles is explained in great detail by *Rose* [13.13].

#### Monochromators

- 1.
Dispersing-only monochromators that do not cancel the energy dispersion at their exit, leading to a larger (energy-dispersed) source image and a consequent loss of brightness [13.305].

- 2.
Dispersing–undispersing monochromators that use more complicated optical arrangements, in which the energy selection is typically accomplished in the midplane of the instrument, and the energy dispersion in the beam leaving the monochromator is canceled [13.304, 13.306, 13.307, 13.308].

A useful way to view a monochromator is as a spectrometer in which the detector is replaced by an energy-selecting slit. The slit selects a range of pass energies (as it does in imaging filters and did in earlier *serial* spectrometers), and the post-slit optics sends the monochromatized beam into the rest of the electron-optical column. Simple monochromators that do not cancel the energy dispersion at their exit are equivalent to a single spectrometer; dispersing–undispersing monochromators are equivalent to two spectrometers arranged back-to-back, the first one producing the energy dispersion and the second one canceling it.

An early electrostatic monochromator design was described by *Plies* [13.309] and *Huber* et al [13.310]. Other early monochromator models are described in Chapter 52 of [13.6]. Although many monochromators use the \(\Upomega\) or \(\upalpha\) configuration, several designs based on the Wien filter have also been proposed and developed. The principle is easily understood: in an ideal Wien filter, transverse electrostatic and magnetic fields with the same axial distribution deflect all electrons except those whose energy satisfies the Wien condition. Those that do not approximately satisfy this condition can easily be intercepted, leaving a beam with a narrowed energy range. Practical studies have been presented by *Tiemeijer* and colleagues, who designed the monochromators for the TEAM instruments [13.311, 13.312, 13.313, 13.314, 13.315]. All these designs lengthen the microscope column and, in an attempt to overcome this inconvenience, *Mook* and *Kruit* [13.316] have tested a fringe-field design, in which the fringing fields are dominant, the main filter being only \({\mathrm{4}}\,{\mathrm{mm}}\) long. A double Wien filter arrangement is employed in aberration-corrected JEOL microscopes [13.308, 13.317, 13.318]. The optics of the four designs in practical use is illustrated in Fig. 13.17a-d.

An important decision in monochromator design concerns the placement of the monochromator in the microscope column. Most monochromators are located inside the electron gun, before the accelerator. They act on electrons of a few keV in energy, which helps them achieve energy dispersion values of the order of \(10{-}30\,{\mathrm{\upmu{}m/eV}}\) in a compact design. A recently introduced ground-potential monochromator placed in the probe-forming column after a set of round magnetic lenses coupling the source image produced by the gun into the monochromator [13.288, 13.307] is an exception to this rule. A cross section of this monochromator is shown in Fig. 13.18. The monochromator employs an \(\alpha\) design, and has three magnetic prisms and 16 multipoles. The energy-selecting slit is situated in the midplane. The energy dispersion it achieves just beyond its primary prism is only \({\mathrm{2}}\,{\mathrm{\upmu{}m/eV}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) primary energy, but it magnifies the spectrum with quadrupole optics to give dispersions between \(\mathrm{30}\) and \({\mathrm{100}}\,{\mathrm{\upmu{}m/eV}}\) at its slit. Aberration correction up to third order is performed by sextupoles and octopoles in the multipole layers indicated in the figure. Higher-order correction is at present not needed—the placement of the monochromator in the optical column is such that the imaging properties demanded from it are similar to the requirements made on round condenser lenses, which are not aberration-corrected. Moreover, in a well-aligned STEM, small aberration contributions that originate in the condenser lenses and the monochromator are readily corrected by the probe corrector. Another factor that makes aberration correction in monochromators less demanding than in spectrometers is that the beam passing through the monochromator has not yet been broadened and shifted to higher angles by scattering in the sample.

Yet another design has been investigated by *Mankos* et al [13.320]. Here the beam is deflected towards an electron mirror and then returned to the original optic axis, as in the Castaing–Henry device. A knife edge intercepts some of the slower electrons when they approach the mirror and some of the faster ones as they leave it.

#### Attainable Energy Resolution

The fundamental limit on the attainable energy resolution in the spectrum produced at the final detector in spectrometers or at the energy-selecting slit in imaging filters and monochromators is given by the size of the beam crossover produced in the detector (slit) plane, divided by the energy dispersion. This means that for the best energy resolution, the dispersion should be made as large as possible, and the size of the crossover as small as possible. The dispersion needs to be maximized for the primary spectrum produced by the prism—increasing the dispersion by post-prism magnifying lenses increases the size of the final crossover just as much as it increases the dispersion, and therefore does not change the attainable energy resolution. (Practical exceptions to this rule arise when the spectrum magnification needs to be increased so that the resolution is not limited by the point spread function of a parallel detector, or unsharp edges of an energy-selecting slit.)

The primary energy dispersion can be made larger by using a larger-bending-radius prism, or a multiple-prism arrangement. However, practical considerations such as the prism weight, needed space, mechanical stability and sensitivity to stray magnetic fields typically place an upper limit on the optimal prism size. This often leaves making the crossover smaller as the most practical path to improved energy resolution.

The size of the spectrum crossover is limited by three principal influences, just like the size of the sample-level electron probe: the source size, the diffraction limit imposed by the angles used in the crossover formation, and instabilities. In nonmonochromatized electron microscopes, the energy resolution in a well-functioning spectrometer is typically dictated by the energy spread of the electron source. The addition of a monochromator makes it possible to reduce the width of the energy distribution of the source considerably. The next important limit on the energy resolution is then typically posed by instabilities, especially instabilities of the high voltage of the microscope and of the prism supply current.

- 1.
By putting the monochromator and spectrometer on the same high voltage, as was done by

*Boersch*et al [13.321], eventually producing an energy resolution of \({\mathrm{3}}\,{\mathrm{meV}}\) at \({\mathrm{30}}\,{\mathrm{keV}}\) (while analyzing a sample area about \({\mathrm{30}}\,{\mathrm{\upmu{}m}}\) large; [13.322]), and by*Terauchi*et al [13.323], producing an energy resolution of \({\mathrm{12}}\,{\mathrm{meV}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) (with the beam in vacuum, \({\mathrm{25}}\,{\mathrm{meV}}\) when the beam passed through a sample),or - 2.
By putting an all-magnetic monochromator and an all-magnetic spectrometer both at ground potential and running the same current through all the prisms of the system by linking them in series [13.25, 13.288, 13.307]. This solution has given \({\mathrm{4.2}}\,{\mathrm{meV}}\) energy resolution at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.291] compatible with a \({\mathrm{1}}\,{\mathrm{\AA{}}}\) diameter monochromatized electron probe.

### 13.2.5 Johnson–Nyquist Noise

*Uhlemann*et al [13.324]. The underlying cause is random electron motions [13.325, 13.326] that occur at finite temperatures in conductors and cause fluctuations in the background magnetic field. The fluctuations change with time and produce random deflections of electrons traveling through the instrument, and can be strong enough to worsen the attainable spatial resolution. For the image broadening (image spread) \(\langle\sigma^{2}\rangle\), we have [13.324]

*originating*(or

*arriving*, in the STEM) at the same place on the sample and corresponding to different angles.) The broadening is added to the usual resolution-limiting factors such as the diffraction limit and the source size in the STEM case or detector point spread function in the TEM case. Equation (13.30) shows that the effect is especially important when there is a beam with wide axial trajectories propagating in extended lengths of narrow drift tubes, or, in the absence of a drift tube, narrow openings surrounded by the poles of multipoles, or small-bore round lenses. These kinds of trajectories are difficult to avoid in \(C_{\mathrm{c}}\) correctors such as the TEAM/PICO one, in which the electrostatic fields present in the hybrid electrostatic–magnetic quadrupoles are weak and require wide axial trajectories extending through the length of the quadrupoles, as well as narrow openings through the quadrupoles. In the SALVE corrector, the resolution loss has been reduced by a careful choice of axial trajectories.

In magnetic-only correctors of geometrical aberrations, in which strong multipole fields allow the diameter of the axial trajectories to be smaller than in \(C_{\mathrm{c}}\) correctors, Johnson–Nyquist noise has not yet emerged as a problem. But the noise may raise its head once more if the spatial resolution is improved significantly beyond present levels, and it may also become important in ultrahigh-energy resolution spectrometers and monochromators, in which broad beams are needed to achieve useful energy dispersion.

## 13.3 Practical Aspects of Corrector Operation

### 13.3.1 Correction of Parasitic Aberrations

When the primary aberrations have been corrected and instabilities brought under sufficient degree of control, the next important resolution-limiting influence is likely to be parasitic aberrations. It is important to remember that the tolerance to higher-order aberrations increases as \(1/\alpha^{(n+1)}\), where \(\alpha\) is the illumination half-angle (in the STEM) or collection half-angle (in the TEM), and \(n\) is the order of the aberration [13.23, 13.25, 13.327], and also that higher-order aberrations can be partially compensated by lower-order aberrations of the same azimuthal multiplicity [13.25, 13.327]. The angle \(\alpha\) is typically about \({\mathrm{0.05}}\,{\mathrm{rad}}\) or less, and this means that whereas first-order aberrations (defocus and twofold astigmatism) need to be set with a precision of a few nanometers or less for better than \({\mathrm{1}}\,{\mathrm{\AA{}}}\) resolution, fourth-order aberrations of around \({\mathrm{100}}\,{\mathrm{\upmu{}m}}\) and fifth-order aberrations of \({\mathrm{1}}\,{\mathrm{mm}}\) are not likely to spoil even resolution of \({\mathrm{0.5}}\,{\mathrm{\AA{}}}\) at \({\mathrm{200}}\,{\mathrm{keV}}\) primary energy [13.25]. These numbers apply to the STEM case, in which high spatial frequencies are transferred into the electron probe via pairwise interference of electron wave subcomponents. In the CTEM case, three different beams (the unscattered beam and \(\pm g\) beams) have to be combined with the phase differences accurately preserved, which means that the objective aperture angles need to be 1.6\(\times\) as large to attain the same spatial resolution as in the STEM. This means that the tolerances to higher-order aberrations are \((n+1)^{1.6}\) times more strict in CTEM [13.23] than in STEM.

Nulling parasitic aberrations with precision sufficient for the increasingly large aperture angles used by present-day aberration correction remains a substantial challenge, both for the autotuning algorithms that measure them and for the controls that eliminate them. One useful strategy is to align the corrector so that the beam passes close to the optical centers of all its stages, which usually minimizes the parasitic aberrations. But if a thorough alignment of this kind does not reduce one or more parasitic aberrations as much as needed, or the alignment that minimizes one particular aberration causes other aberrations to rise unacceptably, then a procedure able to change that particular aberration separately from all the other aberrations is needed. This kind of situation typically arises when the correction half-angles are pushed beyond \(40{-}50\,{\mathrm{mrad}}\), as becomes possible when principal aberrations up to the fifth order are nulled, and chromatic aberration effects are rendered unimportant by means of a \(C_{3}{-}C_{5}{-}C_{\mathrm{c}}\) corrector or a \(C_{3}{-}C_{5}\) corrector used together with a monochromator set to produce a sufficiently narrow energy passband.

with the second aberration appearing only when \(mu+mv\leq nu+nv\). Table 13.2 shows how different aberrations produced in separate multipoles of an aberration corrector combine to produce new aberrations. Some combination aberrations are of limited interest; e. g., combining defocus \(C_{1,0}\) with \(C_{n,m}\) produces \(C_{n,m}\) as the combination aberration, and these have been omitted from the table. Line 2 shows that displacing a round beam (changing \(C_{0,1}\) of element \(u\)) in a sextupole \(v\) producing \(C_{2,3}\) leads to twofold astigmatism \(C_{1,2}\), and this is typically used for twofold stigmation in sextupole correctors. Line 3 shows that displacing the beam in a \(C_{3,0}\)-producing element such as the objective lens allows axial coma \(C_{2,1}\) to be adjusted, and line 5 shows that displacing a round beam in an excited octopole leads to adjustable threefold astigmatism \(C_{2,3}\), which is used to adjust \(C_{2,3}\) in quadrupole-octopole correctors not containing any sextupoles.

Selected combination aberrations which can be used for controlling parasitic aberrations. The combination aberrations listed on lines shown in bold (2, 3, 5, 15 and 19) are in common use. Lines 16–21 and 23, shown in italics (or bold for line 19), are used in the improved Nion corrector for complete control of parasitic aberrations of fourth and fifth order

Line | \(nu\) | \(mu\) | \(nv\) | \(mv\) | \(N1\) | \(M1\) | \(N2\) | \(M2\) | Principal effect |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 1 | 2 | 1 | 1 | 0 | 1 | 2 | \(C_{1,0}\); \(C_{1,2}\) |

| | | | | | | – | – | \(\mathbf{C_{1,2}}\) |

| | | | | | | – | – | \(\mathbf{C_{2,1}}\) |

4 | 0 | 1 | 3 | 2 | 2 | 1 | 2 | 3 | \(C_{2,1}\); \(C_{2,3}\) |

| | | | | | | – | – | \(\mathbf{C_{2,3}}\) |

6 | 0 | 1 | 4 | 1 | 3 | 0 | 3 | 2 | \(C_{3,0}\); \(C_{3,2}\) |

7 | 0 | 1 | 4 | 3 | 3 | 2 | 3 | 4 | \(C_{3,2}\); \(C_{3,4}\) |

8 | 0 | 1 | 4 | 5 | 3 | 4 | – | – | \(C_{3,4}\) |

9 | 0 | 1 | 5 | 0 | 4 | 1 | – | – | \(C_{4,1}\) |

10 | 2 | 1 | 2 | 1 | 3 | 0 | 3 | 2 | \(C_{3,0}\); \(C_{3,2}\) |

11 | 2 | 1 | 2 | 3 | 3 | 2 | 3 | 4 | \(C_{3,2}\); \(C_{3,4}\) |

12 | 2 | 1 | 3 | 0 | 4 | 1 | – | – | \(C_{4,1}\) |

13 | 2 | 1 | 3 | 2 | 4 | 1 | 4 | 3 | \(C_{4,1}\); \(C_{4,3}\) |

14 | 2 | 1 | 3 | 4 | 4 | 3 | 4 | 5 | \(C_{4,3}\); \(C_{4,5}\) |

| | | | | | | – | – | \(\mathbf{C_{3,0}}\) |

| | | | | | | – | – | \(C_{4,3}\) |

| | | | | | | | | \(C_{4,1}\); \(C_{4,5}\) |

| | | | | | | – | – | \(C_{4,1}\) |

| | | | | | | – | – | \(\mathbf{C_{5,0}}\) |

| | | | | | | – | – | \(C_{5,2}\) |

| | | | | | | – | – | \(C_{5,4}\) |

22 | 3 | 2 | 3 | 2 | 5 | 0 | 5 | 4 | \(C_{5,0}\); \(C_{5,4}\) |

| | | | | | | | | \(C_{5,2}\); \(C_{5,6}\) |

24 | 3 | 4 | 3 | 4 | 5 | 0 | – | – | \(C_{5,0}\) |

Using the above principles to eliminate second- and third-order parasitic aberrations is straightforward, but the situation grows more complicated for nulling parasitic fourth- and fifth-order aberrations. Lines 17 and 18 of the table show that combining a \(C_{2,3}\)-producing sextupole with a \(C_{3,2}\)-producing octopole (an octopole acting on an elliptical beam) gives adjustable \(C_{4,1}\) and \(C_{4,5}\), whereas combining \(C_{2,3}\) with a \(C_{3,4}\)-producing octopole (one acting on a round beam) gives adjustable \(C_{4,1}\). Both controls become possible when weak sextupoles are incorporated in quadrupole stages of a \(C_{3}/C_{5}\) quadrupole–octopole corrector, and this has been done in the improved Nion \(C_{3}/C_{5}\) corrector. The corrector is therefore able to null \(C_{4,1}\) and \(C_{4,5}\) independently. In similar ways, independent controls for all parasitic aberrations up to \(C_{5,6}\) (and higher) can be devised. In the case of the Nion corrector, this approach has increased the maximum half-angle of correction to beyond \({\mathrm{50}}\,{\mathrm{mrad}}\) ([13.177] and Sect. 13.3.3). Correction angles greater than \({\mathrm{50}}\,{\mathrm{mrad}}\) have also been achieved by the JEOL Delta corrector [13.328] and the CEOS SALVE corrector [13.244, 13.245].

### 13.3.2 Autotuning

In the STEM, aberrations up to third order produce distinct signatures in Ronchigrams (shadow images or far-field convergent-beam diffraction patterns) of amorphous materials [13.25, 13.46]. Figure 13.19a-d shows how twofold astigmatism, axial coma and threefold astigmatism affect the appearance of the Ronchigram. Skilled operators are therefore able to tune correctors up to third order by optimizing the appearance of Ronchigrams, and this can be used on third-order corrected STEMs in lieu of autotuning. However, requiring operators to perform nontrivial tuning of the microscope manually would restrict the number of users, and automated microscope tuning (autotuning) is therefore nearly always provided. Furthermore, when the correction progresses up to fifth-order axial aberrations, 25 different aberrations (\(C_{1,0}\) to \(C_{5,6b}\)) need to be measured and set to acceptably small values. Many of them produce effects that are difficult to distinguish visually in a Ronchigram: second- and fourth-order threefold astigmatisms (\(C_{2,3}\) and \(C_{4,3}\)) both produce a *star of Mercedes* similar to Fig. 13.19a-dd. Tuning such aberrations is therefore not a task that can be done manually, and precise and fast autotuning is essential.

Different autotuning strategies have been reviewed by *Erni* [13.64], and we provide only a brief account here. The tuning methods can be divided into two broad classes: image-based and Ronchigram-based. Image-based methods developed prior to practical aberration correction, for instance as reviewed by *Krivanek* and *Fan* [13.33], can be readily employed in aberration-corrected transmission electron microscopes as well. They are based on either beam-tilt-induced image shift or beam-tilt-induced changes in apparent defocus and astigmatism. The second method analyzes diffractogram tableaux of amorphous samples recorded for different values of beam tilt [13.48, 13.50]. Unlike shift-based methods, the diffractogram method is not greatly affected by changes in image appearance for different beam tilts or by small sample drift. This results in better tuning precision, and the method has become the most widely used for CTEM aberration correctors. Its usual implementation divides each experimental diffractogram into azimuthal segments, cross-correlates the intensity profile of the segments with theoretical profiles computed for different defocus values and thereby determines defocus values in different azimuthal directions, and then analyzes the azimuthal variation in each diffractogram to determine its apparent defocus and astigmatism [13.32]. The variation in these values with the induced beam tilt is then used to determine the aberration coefficients. In principle, any number of aberrations can be quantified, provided that there is a sufficient number of experimental diffractograms corresponding to different beam tilts.

For the STEM, the above methods are applicable as well, thanks to the principle of reciprocity linking STEM and TEM imaging [13.329, 13.330, 13.331, 13.332]. The image-based techniques simply record bright-field images for different detector tilts [13.164, 13.26]. Images for different beam tilt values can be collected in the STEM in parallel by recording images corresponding to different detection angles in the Ronchigram plane. By reciprocity, this corresponds to collecting CTEM images for different beam tilts of the incident illumination. The parallel recording can be implemented using multiple detectors (or a pixelated detector with fast readout) to monitor the intensities at different detection angles as the probe is scanned across the sample. This can lead to a substantial increase in collection efficiency and hence shorter collection time for the required data than in CTEM.

An even more efficient approach is analyzing experimental Ronchigrams directly, without acquiring whole scanned images. An out-of-focus Ronchigram is a shadow image of the sample that is distorted in characteristic ways by different aberrations, and the distortion can be quantified by analyzing how the local magnification varies from place to place in the Ronchigram [13.25, 13.46]. This can be seen in Fig. 13.19a-d, especially in Fig. 13.19a-dc and Fig. 13.19a-dd.

To determine the local magnification experimentally on a general sample whose features are not known a priori, the probe is moved by a small amount in the \(x\) and \(y\) directions, and Ronchigrams are recorded for each probe displacement ([13.168], see Chap. 17). Cross-correlating small Ronchigram subareas then shows how far that part of the Ronchigram moved as a result of the probe shift, and this allows the local magnification of the Ronchigram to be determined. In practice, more Ronchigrams than the minimum of three (no probe displacement, displacement in \(x\), displacement in \(y\)) are recorded for each autotuning run, so as to be able to account for spurious effects such as sample drift and other instabilities. But even so, high-quality Ronchigram data can be typically acquired in \(<{\mathrm{5}}\,{\mathrm{s}}\), and complete geometrical aberration analysis up to the fifth order can be performed in \(<{\mathrm{10}}\,{\mathrm{s}}\). The aberrations are then tuned using various controls whose effects on the aberrations have been calibrated, and a new aberration diagnosis is performed. The procedure is repeated until all aberration coefficients being tuned are within acceptable ranges. When amorphous samples giving suitable contrast are used in a well-performing STEM, fifth-order tuning can typically be completed in about \({\mathrm{30}}\,{\mathrm{s}}\). In practice, day-to-day changes in fourth- and fifth-order aberrations are much smaller than tuning tolerances, and fourth- and fifth-order tuning is therefore not performed as a daily adjustment. Third-order tuning is typically performed at the start of the day, followed by second-order tuning, which can be done in \({\mathrm{10}}\,{\mathrm{s}}\), whenever a substantially new sample area is selected. Related Ronchigram-based autotuning techniques have been developed by *Sawada* et al [13.29, 13.333], and image-based autotuning for adjustment of defocus and other low-order aberrations using crystalline samples has been demonstrated by *Lazar* et al [13.334].

### 13.3.3 Current Performance Limits

When spherical aberration is corrected in an electron microscope, other aberrations begin to limit the resolution. The resolution that becomes attainable in the STEM as different types of aberrations are corrected is depicted in Fig. 13.20 as a function of the primary energy of the microscope. The theoretical curves are computed for probe currents equal to \({\mathrm{10}}\%\) (solid lines) and \({\mathrm{50}}\%\) (dashed lines) of the coherent current \(I_{\mathrm{c}}\), using the expressions given in *Krivanek* et al [13.81]. \(I_{\mathrm{c}}\) is the probe current value for which the probe is \({\mathrm{50}}\%\) coherent and \({\mathrm{50}}\%\) incoherent, and it is a property of the gun only: it does not depend on the aberrations of the system, or on its primary energy. It is also is described more fully in *Krivanek* et al [13.81].

The purple curves in the upper right part of the graph correspond to the uncorrected, low-\(C_{\mathrm{s}}\) STEMs marketed by several manufacturers (FEI, Hitachi, JEOL and VG) before aberration correction became commercially available. This type of microscope was able to form \({\mathrm{2}}\,{\mathrm{\AA{}}}\) probes at \({\mathrm{100}}\,{\mathrm{keV}}\), \({\mathrm{1.4}}\,{\mathrm{\AA{}}}\) probes at \({\mathrm{200}}\,{\mathrm{kV}}\), and smaller probes still at \({\mathrm{300}}\,{\mathrm{keV}}\). Third-order aberration correction improved the performance to the blue curves, and allowed \({\mathrm{1}}\,{\mathrm{\AA{}}}\) probes to be reached at \({\mathrm{100}}\,{\mathrm{keV}}\). Further improvements came when fifth-order aberrations were either corrected or sufficiently minimized, with probes about \({\mathrm{0.5}}\,{\mathrm{\AA{}}}\) in size becoming possible at \({\mathrm{200}}\,{\mathrm{keV}}\), \({\mathrm{0.47}}\,{\mathrm{\AA{}}}\) (and later \({\mathrm{0.405}}\,{\mathrm{\AA{}}}\)) at \({\mathrm{300}}\,{\mathrm{keV}}\) [13.219, 13.335, 13.336], and \({\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) [13.337]. Chromatic aberration \(C_{\mathrm{c}}\) then became the new limit, especially at lower primary energies, and it could be overcome either by correcting \(C_{\mathrm{c}}\) or decreasing the energy spread of the beam \(\updelta E\). Only moderate monochromatization is needed in the STEM: for instance, \({\mathrm{1}}\,{\mathrm{\AA{}}}\) spatial resolution can be achieved at \({\mathrm{30}}\,{\mathrm{keV}}\) with an energy spread of \(\approx{\mathrm{100}}\,{\mathrm{meV}}\), as is shown below.

It is useful to mention that in the CTEM, the unscattered beam parallel to the optic axis interferes with scattered beams passing some distance from the axis, and this makes the phase changes that are imparted to the interfering beams by defocus spread (caused by \(C_{\mathrm{c}}\) combined with energy spread) very different. As a result, CTEM imaging is more sensitive to \(C_{\mathrm{c}}\) effects than STEM imaging, in which much of the interference is between partial beams passing at equal angles to the optic axis, and therefore insensitive to defocus spread ([13.338], Chap. 2). The heightened \(C_{\mathrm{c}}\) sensitivity and the relatively wide energy spread of Schottky guns used in CTEM \((\updelta E\approx{\mathrm{700}}\,{\mathrm{meV}}\)) made the use of monochromatization to improve the CTEM spatial resolution fairly routine from 2004 onward, before fourth- and fifth-order aberrations were eliminated [13.208, 13.209, 13.339, 13.340, 13.341]. With correction half-angles presently increasing to \({\mathrm{50}}\,{\mathrm{mrad}}\) and above thanks to improved geometrical aberration correction, monochromatization will need to decrease the energy spread of the illuminating beam in the CTEM to below \({\mathrm{30}}\,{\mathrm{meV}}\) [13.244, 13.245] and even smaller. This is not readily achievable, for two reasons: the medium- and long-term stabilities of the microscope high voltage are typically worse than \({\mathrm{30}}\,{\mathrm{mV}}\), and decreasing the energy width of the illumination from \({\mathrm{700}}\,{\mathrm{meV}}\) (typical of Schottky sources) to \({\mathrm{30}}\,{\mathrm{meV}}\) would reduce the illumination intensity more than 20\(\times\). Correction of chromatic aberration offers a way around this difficulty.

When the chromatic limit is overcome, seventh-order principal aberrations and sixth-order parasitic aberrations are expected to become dominant. This limit, shown by the green lines in Fig. 13.20, indicates that when aberrations including fifth-order and chromatic ones are eliminated, \({\mathrm{0.3}}\,{\mathrm{\AA{}}}\) STEM probe sizes should become possible at \({\mathrm{200}}\,{\mathrm{kV}}\), \({\mathrm{0.45}}\,{\mathrm{\AA{}}}\) at \({\mathrm{100}}\,{\mathrm{kV}}\), and \({\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{20}}\,{\mathrm{kV}}\). These three primary energies correspond to electron wavelengths \(\lambda\) of \(\mathrm{2.51}\), \(\mathrm{3.7}\) and \({\mathrm{8.59}}\,{\mathrm{pm}}\), and this means that the probe size, which is closely related to resolution, divided by the wavelength (\(d/\lambda\)) will be approaching about 12 when such performance is reached. This is a far cry from optical microscopies, for which resolutions \(d<\lambda\) or even \(d\ll\lambda\) have been available for some time. But it is a major improvement on the pre-aberration-correction days, in which the best-performing electron microscopes were reaching \(d/\lambda\) values of only \(60{-}100\).

Figures 13.21–13.23a-d demonstrate how this kind of performance is approached in practice. Figure 13.21 shows graphene and monolayer \(\mathrm{MoS_{2}}\) imaged with the \(C_{\mathrm{s}}/C_{\mathrm{c}}\) SALVE corrector in the bright-field phase-contrast mode at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.244]. Transfer to spatial frequencies close to \({\mathrm{1}}\,{\mathrm{\AA{}^{-1}}}\) has been reached in both cases, giving \(d/\lambda={\mathrm{15}}\). The contrast modulation is about \({\mathrm{6}}\%\) in the graphene image, and about \({\mathrm{40}}\%\) in the \(\mathrm{MoS_{2}}\) image, owing to the presence of heavier atoms and to the fact that two \(\mathrm{S}\) atoms are projected on top of each other.

Figure 13.22a,b shows a pair of out-of-focus \((\Updelta z\approx{\mathrm{500}}\,{\mathrm{nm}})\) Ronchigrams extending to \({\mathrm{50}}\,{\mathrm{mrad}}\) half-angle, recorded at \({\mathrm{30}}\,{\mathrm{keV}}\) in a monochromatized STEM using an amorphous sample [13.342]. The Ronchigram on the left was recorded with the energy-selecting slit retracted and thus admitting the full beam produced by the microscope’s cold-field emission gun ( ), with a measured energy spread of \({\mathrm{320}}\,{\mathrm{meV}}\) (full width at half-maximum \(=\) FWHM; this value is typical for a CFEG). The Ronchigram on the right was obtained with the energy-selecting slit inserted, restricting the energy spread \(\updelta E\) to \({\mathrm{110}}\,{\mathrm{meV}}\). The unmonochromatized Ronchigram shows blurring in the radial direction for angles larger than about \({\mathrm{20}}\,{\mathrm{mrad}}\). The blurring is due to the defocus spread that occurs when the full range of energies of the CFEG are used: the Ronchigram magnifications are different for the different defocus values, which causes blurring near the outer edges of the Ronchigram. The blur-free center is the voltage axis of the microscope, and the virtual objective aperture ( ) used to limit the range of angles contributing to the probe was centered on it. Had the Ronchigram been recorded close to zero defocus, it would have been largely featureless, and the blurring due to the energy spread would not have been visible. The blurring might have even *improved* the appearance of the Ronchigram by fuzzing out any features intruding into the *flat phase* (*sweet spot*) part of the Ronchigram, thereby giving a misleading impression regarding the *perfection* of the implemented tuning.

Figure 13.23a-d shows a medium-angle annular dark-field ( ) image recorded at \({\mathrm{30}}\,{\mathrm{kV}}\) in the monochromatized condition illustrated by Fig. 13.23a-db [13.342]. Single atoms of carbon are well resolved and the image noise is not excessive, despite the beam current being only about \({\mathrm{10}}\,{\mathrm{pA}}\). The highest spatial frequency transferred is \(({\mathrm{107}}\,{\mathrm{pm}})^{-1}\), similar to the phase-contrast image shown in Fig. 13.21, but the modulation for monolayer graphene is \({\mathrm{50}}\%\), as shown by a line profile taken along the dashed line in the as-acquired image. The modulation increase is mostly due to different contrast mechanisms: phase-contrast imaging for the earlier figure and medium-angle annular dark-field (MAADF) imaging for the present one. The MAADF image contrast modulation is similar to the modulation achieved in graphene and monolayer boron nitride (BN ) in an unmonochromatized \({\mathrm{60}}\,{\mathrm{keV}}\) STEM in 2010 [13.337, 13.343]. The energy width was \({\mathrm{110}}\,{\mathrm{meV}}\) and the system \(C_{\mathrm{c}}\) \({\mathrm{0.96}}\,{\mathrm{mm}}\), of which \({\mathrm{0.6}}\,{\mathrm{mm}}\) was due to the objective lens, and the rest was due to the electron gun, condenser lenses, monochromator and the aberration corrector. The \(C_{\mathrm{c}}\)-limited resolution, computed using (13.27b) above (with \(b={\mathrm{0.5}}\)), was therefore only \({\mathrm{84}}\,{\mathrm{pm}}\), and allowed strong transfer of \({\mathrm{107}}\,{\mathrm{pm}}\) spatial frequency even in the presence of other probe-broadening effects such as finite source size.

Defining reproducible resolution criteria is an important subject (see Chap. 12), with many prior contributions, e. g., *Sato* [13.344]. It is useful to remember that the MAADF image contributions arising from different energy electrons add up mostly incoherently, and that, as discussed for example by *Nellist* [13.345] and *Krivanek* et al [13.346], the resultant probe shape for such an addition can be a small narrow maximum surrounded by an intense broad *tail*. This may be why the \({\mathrm{107}}\,{\mathrm{pm}}\) spacing was observed in unmonochromatized MAADF STEM images obtained at \({\mathrm{30}}\,{\mathrm{keV}}\) with an energy spread of \({\mathrm{0.4}}\,{\mathrm{eV}}\) and a \(C_{\mathrm{c}}\) coefficient of \({\mathrm{0.61}}\,{\mathrm{mm}}\) [13.328], for which (13.27b) predicts \(d_{\mathrm{c}}={\mathrm{120}}\,{\mathrm{pm}}\). The intensity of the tail can be evaluated from the intensity distribution of the experimental image and it can be subtracted from the images; this is necessary for quantitative comparisons of the image intensities of different atoms [13.337]. The main effect of the tail is therefore to increase the image background level and also its statistical noise.

Because the structure and thickness of every single-layer graphene sample that is free of defects is exactly the same as that of any other graphene sample, using graphene as a test object to obtain a quantitative evaluation of microscope performance can be highly reproducible. This makes graphene an ideal test object for evaluating the relative performance of different microscopes. Comparing the quality of the images produced by the three techniques that have achieved atomic resolution in graphene at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.244, 13.328, 13.342] gives the edge to monochromatized STEM (Fig. 13.23a-d), but this is of course likely to change in the future with further progress in aberration correction.

Figure 13.24 illustrates another important aspect of aberration-corrected electron microscopy. It shows a zero-loss peak with an FWHM of \({\mathrm{5}}\,{\mathrm{meV}}\), acquired at \({\mathrm{30}}\,{\mathrm{keV}}\) primary energy with a monochromator that is corrected to the third order and an EELS that is corrected up to the fifth order. The peak was acquired with an acquisition time of \({\mathrm{100}}\,{\mathrm{ms}}\), (i. e., spanning six periods of North American mains). The excellent energy resolution was possible only because the STEM–EELS system used stabilization schemes that made the EELS system insensitive to fluctuations in the high voltage and in the prism current [13.288, 13.307]. The monochromator was briefly described in Sect. 13.2.4 above. The energy-loss spectrometer is a recently developed one, with aberration correction up to the fifth order and several other features that optimize the attainable energy resolution [13.289]. At \({\mathrm{60}}\,{\mathrm{keV}}\), the zero-loss peak FWHM in this system broadens to \({\mathrm{6}}\,{\mathrm{meV}}\), but even so, EELS in the electron microscope, which began exploring vibrational signals only in 2014 [13.347, 13.348], is now able to resolve vibrational peaks due to C–C, C–N and C–O bonds in proteins and other biological materials [13.349]. When carried out using the aloof EELS technique [13.350], which avoids radiation damage even in highly sensitive materials such as water [13.351] and ice, this development promises to open new possibilities for sensitive and damage-free analysis of biological and other organic materials at a spatial resolution of a few tens of nanometers.

Vibrational EELS in the EM has recently also demonstrated isotopic sensitivity, whereby deuterium is distinguished from hydrogen and \(\mathrm{{}^{13}C}\) from \(\mathrm{{}^{12}C}\), by energy shifts in vibrational spectra [13.349, 13.351]. Labeling biological compounds with different isotopes and analyzing cells that have taken them up may allow metabolic pathways to be traced in an electron microscope, at nanometer-level resolutions.

## 13.4 Concluding Remarks

Aberrations correction has progressed considerably since *Science of Microscopy* was published. Correction of geometrical aberrations up to the third order allows the spatial resolution to be improved some \(2{-}3\times\) relative to uncorrected electron microscopes, and several hundred such instruments have now been installed in laboratories around the world. The next major resolution-limiting influence, chromatic aberration, can be overcome by \(C_{\mathrm{s}}/C_{\mathrm{c}}\) correctors or by monochromatizing the illuminating beam. Combined with correction of geometrical aberrations up to the fifth order, these strategies have enabled aberration-corrected imaging to achieve a resolution of \(15\lambda\), a very substantial improvement on the resolution of \(60\lambda{-}100\lambda\) achievable with uncorrected instruments.

With a suitable test sample, the resolution achieved by aberration correction is typically similar to the STEM probe size, and the two terms are sometimes used interchangeably. However, there is an important distinction: the probe size defines the best spatial resolution for *incoherent* signals such as high angle dark field (HAADF) imaging and EELS and EDXS maps. For *coherent* signals such as bright field imaging, the resolution can be improved beyond the probe size limit by *ptychographic reconstruction techniques* (Chap. 17) [13.352, 13.353]. Using these techniques, probe-limited resolution of about \({\mathrm{1}}\,{\mathrm{\AA}}\) has recently been improved to a resolution of less than \({\mathrm{0.4}}\,{\mathrm{\AA}}\) at \({\mathrm{80}}\,{\mathrm{keV}}\), i.e. about \(10\lambda\) [13.354].

Aberration-corrected monochromators and electron energy-loss spectrometers have improved the energy resolution in electron microscopes to better than \({\mathrm{10}}\,{\mathrm{meV}}\). This has improved the quality of EELS results substantially, and has opened up a new research field: vibrational spectroscopy combined with high spatial resolution. The spatial resolution can be greatly enhanced by using *dark-field EELS*, in which the forward-scattered electrons (which can undergo delocalized scattering events that result in blurred image information) are excluded [13.355]. This approach has now allowed *Hage* et al [13.356] to reach \({\mathrm{2}}\,{\mathrm{\AA}}\) resolution in vibrational images of h-BN using dark field EELS. The forward-scattered electrons arise by dipole scattering, which is strong in polar materials, but only a weak second-order effect in non-polar materials such as Si. Taking advantage of the absence of dipole scattering in non-polar materials, *Venkatraman* et al [13.357] reached similar (\(\approx{\mathrm{2}}\,{\mathrm{\AA}}\)) resolution in vibrational images of Si without employing dark field EELS—by simply making sure that large angle scattering events, which carry high spatial resolution information, contributed to the images.

Aiming in the opposite direction, vibrational spectroscopy that admits the forward-scattered electrons and positions a narrow electron beam just outside the sample ensures that only long-distance interactions take place between the fast electrons and the sample. The technique is called *aloof spectroscopy* and, in a first for electron microscopy, it avoids radiation damage by limiting the amount of energy that can be deposited in the sample by the fast electrons [13.350].

Aloof spectroscopy is not the only way of avoiding radiation damage. Following the example of *diffracting then destroy* with ultrashort pulses from x-ray lasers [13.358, 13.359], the new field of ultrafast electron microscopy (Chap. 8) may allow the microscopist to *outrun* radiation damage by recording image or diffraction data before damage has had time to occur, as has been suggested by *Egerton* et al and *Spence* et al [13.360, 13.361, 13.362, 13.363]. But outrunning radiation damage in the electron microscope remains to be demonstrated in practice, whereas minimizing the damage by aloof positioning of the electron beam is now a practical technique.

- 1.
The resolution is higher in a STEM than in a CTEM with equivalent optical parameters.

- 2.
The interpretation of incoherent dark-field STEM images is more intuitive than the interpretation of coherent bright-field CTEM images.

- 3.
The STEM can analyze the sample composition by spectroscopic techniques, often at the same time as recording image data.

- 4.
STEM is compatible with ptychographical reconstruction techniques, which can significantly enhance the spatial resolution and improve the signal-to-noise ratio in images recorded with limited electron doses.

*only*\(2{-}3\,{\mathrm{\AA{}}}\), uncorrected CTEM used for bright-field imaging is the primary tool (Chap. 4).

Second, as the STEM resolution improves as a result of advances in aberration correction and the bright-field disk becomes larger and larger, an increasing fraction of the scattered electrons contributes to the signal within the bright-field disk rather than outside it, making dark-field techniques less efficient. The electrons scattered into the bright-field disk owe their origin to a wide variety of contrast mechanisms, and the corresponding signals can be used for phase-contrast imaging, magnetic imaging and several other types of imaging experiments. An electron Ronchigram is of course a diffraction pattern recorded with a highly convergent probe, typically out of focus. When the illumination is made more parallel, the Ronchigram becomes a regular diffraction pattern with Bragg disks, and recording such patterns from every image point is another way of capturing a large amount of information about the sample. See Chap. 17 and the earlier reviews of these techniques by *Rodenburg* [13.352] and *Zuo* and *Tao* [13.364]. Their complete blossoming is facilitated by detectors able to record the 2-D diffraction pattern at every image pixel, with high DQE for single electron detection, high dynamic range and high speed. Recently introduced detectors record diffraction patterns \({\mathrm{128}}\times{\mathrm{128}}\) to \({\mathrm{256}}\times{\mathrm{256}}\) in size at a rate of about 1000 patterns per second [13.365, 13.366, 13.367], and faster detectors with a greater number of pixels are being planned. The recent improvement in the attainable spatial resolution [13.354] is an important outcome of these developments. Moreover, the efficiency of STEM phase-contrast imaging (the amount of useful information extracted per each incident electron) is likely to rival that of CTEM phase-contrast imaging.

In summary, aberration correction has progressed to a new development phase: with the utility of probe and image correctors firmly established, it has moved on to the exploration of new contrast mechanisms and even signals that are new to electron microscopy. Electron microscopists have also realized that aberration correction may be useful in areas other than simple improvement of the spatial or energy resolution. An example is in situ microscopy, where ample space is needed between the polepieces of the objective lens; the coefficients of spherical and chromatic aberration are then inevitably large. By correcting these aberrations, in situ microscopy should become a high-resolution technique. In the life sciences, chromatic correction will make it possible to work with thicker specimens. The overall development of aberration correction—from a technique once dismissed as too complicated to become practical, to a useful tool focused on improving spatial resolution and subsequently to a multifaceted tool making many different kinds of contributions—has exceeded expectations more than once. The development is by no means over, and many fundamental surprises are likely waiting in the wings.

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