# A short introduction to aberration correction with focus on Scherzer's and Rose's contributions

The aim to achieve ideal focusing of glass lenses or the problem to compensate for their aberrations, respectively, dates back about 800 years. Towards the end of the 19th century, Abbe developed the theory of image formation in the microscope and provided a scientific basis for the fabrication of microscopes. In 1873, he published his fundamental work on the resolution limit of optical microscopes after he had realized that diffraction determined the achievable resolution [ABB73]. Assuming a perfect lens, he showed that the resolution limit is given by

Here λ is wavelength, α is the aperture angle and n is the refraction index. This formula is quite universal because it also holds for electron microscopes where the usable aperture angle depends on the aberrations^{1}. One major achievement in the design of optimum compound lenses was the replacement of the extensive trial and error process by precise calculations of the objective lenses. The new wave theory of optics provided a formal and sufficient basis for the engineering of lenses. As a result, apochromats [ABB86] have been designed which are corrected for three colors. The performance of these lenses was significantly better than that of any previous objective lens. The optical community owes Ernst Abbe not less than 50 important inventions and discoveries. Due to the development of advanced fabrication techniques by Abbe and Zeiss at the beginning of the 20th century, microscope fabrication was concentrated in Germany at that time.

In 1926, the focusing property of rotationally symmetric magnetic fields was discovered and analyzed by H. Busch. He showed theoretically that a coil carrying a current deflects electrons in the same way as a glass lens refracts the light rays. The lens equation of the light optics can thus also be applied to electrons making it the basis of electron optics. The equation for a short lens is

Here f, b and g are the focal length, the distance between lens and image and the distance between lens and object, respectively. Unlike a glass lens, whose refracting surfaces can be shaped as desired, the electromagnetic field of an electron lens is determined by Maxwell’s equations and cannot be formed arbitrarily. This constraint results in lens aberrations (Fig. 1) which prevent perfect imaging and thus limit the performance of the microscope. The spherical aberration arises in light optics when lenses or mirrors with spherical surfaces are used. This aberration broadens the image of each object point. Based on the pioneering work of H. Busch, it became possible to calculate the paths of electrons in magnetic fields, in particular the focusing properties of coils encapsulated in iron.

These short magnetic lenses are the basic elements of all electron microscopes. After studying the Busch formula for the focal length of magnetic lenses, Knoll and Ruska developed the first single-stage magnetic electron microscope. A short time later, they built a two-stage instrument and named it transmission electron microscope (TEM)^{2}. Already in 1934, the resolution of the TEM surpassed that of the light microscope [BOR38].

The development of the electron microscope was accompanied by theoretical studies of the properties of electron lenses, multipoles and other electron optical components. In 1936, Scherzer found the so-called "Scherzer theorem" [SCH36] which states that the spherical and the chromatic aberrations of static rotationally symmetric and space-charge-free electron lenses are unavoidable if the object and image are real. Spherical aberration of round lenses is always positive because the marginal rays, which intersect the outer zones of the lens, are always more deflected than the paraxial rays. Scherzer himself showed theoretically, in 1947, that the aberrations can be corrected by lifting any one of the constraints of his theorem. Scherzer has made many ingenious fundamental contributions to the correction of spherical and chromatic aberration of electron lenses. Therefore, he is quite often called "the father of aberration correction" [MSA11]. Nevertheless, he had to defend the validity of his theorem against critics from his competitor W. Glaser who asserted that the theorem was erroneous by proposing a magnetic electron lens free of spherical aberration [GLA40]. In 1941, Recknagel showed that the magnetic field of this lens does not provide a real image and thus does not fulfill the conditions required for the validity of the Scherzer theorem [REC41]. Scherzer proposed several correction procedures (Fig. 2), in particular the C_{S}-corrected system shown in Fig. 3 [SCH47]. At Scherzer’s time computers were not yet available and numerical calculations very time consuming.

From 1948 onwards, theoretical studies and experiments on aberration correction were conducted at Scherzer’s institute in Darmstadt over a period of many years [ROS08]. In 1951, Seeliger showed experimentally that the Scherzer corrector (Fig. 3) compensated for the spherical aberration of the objective lens. However, he could not improve the resolution limit of the basic electron microscope due to electrical and mechanical instabilities of the corrector elements [SEE51].

In 1953, Scherzer’s corrector design was used as a basis for correction efforts in England. Archard proposed to substitute four quadrupoles for the two cylinder lenses and the inner round lens of the Scherzer corrector in order to achieve a higher mechanical stability [ARC54]. However, also this corrector could not improve the resolution of the electron microscope.

In 1956, Moellenstadt made further experiments with the Scherzer-Seeliger corrector in Tübingen [MOE56]. By employing critical illumination with a large cone angle of 20 mrad, he increased the spherical aberration to such an extent that it became by far the most dominant aberration blurring severely the image. After compensating for the spherical aberration by the octopoles, the resolution improved by a factor of about 7 accompanied by a remarkable improvement in contrast.

In 1957, Scherzer advised his PhD student Meyer to examine the obstacles that limit the actual resolution of the spherical-aberration-corrected electron microscope. In his thesis Meyer investigated both theoretically and experimentally the requirements for achieving an improvement in resolution by correction of aberrations [MEY61]. He realized that two kinds of parasitic aberrations prevented an improvement of resolution: (1) adjustment aberrations resulting from poor alignment, lens imperfections and insufficient shielding of static external fields and (2) time-dependent aberrations caused by electrical and mechanical instabilities and by insufficient shielding of external alternating fields [MEY61].

In another thesis performed at Scherzer‘s institute, Tretner determined the lower limit of the coefficients C_{S} and C_{C} of spherical and chromatic aberration for several constraints [TRE59]. His results have subsequently been utilized by Riecke for designing the so-called condenser-objective lens which minimizes C_{S} and C_{C} [RIE66]. This lens is still used in most high-resolution electron microscopes and enables a resolution of about 100λ for non-corrected microscopes [ROS08].

Although Scherzer could show that the departure from rotational symmetry is sufficient to eliminate chromatic aberration, no practical corrector could be found up to 1961 when Kelman showed that the combined electrostatic-magnetic quadrupole can have a chromatic aberration coefficient with negative sign opposite to that of a round lens [KEL61]. In 1967, Hardy first obtained experimentally a negative chromatic aberration with a system of four electric-magnetic quadrupoles [HAR67]. However, it remained a proof of principle because the system could not be used as a corrector for an electron microscope. The successful correction of chromatic aberration in an actual electron microscope was first demonstrated within the frame of the so-called Darmstadt Project by means of a novel corrector. It enabled compensation of chromatic and spherical aberration [KOO77] in 1976. This corrector was composed of three magnetic quadrupoles, two electric-magnetic quadrupoles and three octopoles. During their experiments Scherzer, Rose and co-workers had systematically eliminated the parasitic aberrations which had prevented the successful aberration correction.

In 1963, Dymnikov showed that a symmetric quadrupole quadruplet with anti-symmetric excitation of the quadrupoles can be used as a substitute for a round lens [DYM63]. Later improved versions of this system served as correctors [ROS71, BEC74, ZAC95, KRI96]. The so-called Chicago-corrector consisting of four magnetic quadrupoles and octopoles was built and tested in the scanning transmission electron microscope (STEM) from 1972 to 1978 [BEC74]. Also this corrector failed to improve the resolution of the STEM.

In 1965, Hawkes discovered that hexapoles produce rotationally symmetric third-order combination aberrations apart from other aberrations [HAW65]. His findings remained largely unnoticed until 1979 when Beck showed that a combination of a single round lens and two hexapoles is able to produce a negative spherical aberration [BEC79]. Unfortunately, his systems introduced a large fourth-order aperture aberration which prevents an appreciable improvement in resolution [BEC79, CRE80]. Subsequent studies showed that hexapoles are suitable elements for nullifying successfully C_{S} because the correctors require fewer elements and can be adjusted more easily than quadruple-octopole correctors [ROS81, CRE82, SHA88, CHE91]. In 1990, H. Rose proposed the semi-aplanat depicted in Fig. 4 which is composed of the objective lens, two round-lens transfer doublets and two hexapoles [ROS90].

In following two small companies - CEOS in Heidelberg, Germany and Nion near Seattle in the US - are market leaders for aberration correctors. In 1996, CEOS was founded by Maximilian Haider and Joachim Zach, students of Harald Rose at the Technical University Darmstadt. In 1997, Nion was founded by Ondrej Krivanek and Niklas Dellby, researchers from Cambridge University, UK. CEOS has started with supplying hexapole-based correctors based on the design originally developed by Harald Rose [ROS90, HAI98] and Nion has started supplying quadrupole/octopole correctors originally developed by Crewe and co-workers [THO71, BEC74, KRI99]. The correctors developed by CEOS and Nion have increased the resolution of electron microscopes from about 0.2 nm to better than 0.1 nm over the past decade [BAT02, FRE04].

The successful correction of spherical (Cs) and chromatic (Cc) aberration by Rose [ROS05] and the companies CEOS and FEI Electron Optics together with Lawrence Berkeley National Laboratory (LBNL) [HAI08, KIS08] (Fig. 5, 6) opened the door for improvements of resolution at 80 kV in the so-called TEAM project [KAB09]. After reduction of accelerating voltage to 20 kV [KAI11] and decreasing of Johnson-Noise [UHL13] the final electron microscope SALVE III, equipped with the optimized Rose-Kuhn Cs/Cc-corrector (Fig. 7) [ROS92], was developed and completed by the companies CEOS and FEI Electron Optics together with Ulm University. With this microscope the resolution of electron microscopes was increased up to a new world record of 15 times the wavelength (90 pm at 40 kV) [LIN16].

More details about the SALVE III corrector can be found here.

Footnotes:H. Boersch (1909 - 1986) showed that Abbe’s theory of image formation in the light microscope also holds for the electron microscope.

M. Knoll and E. Ruska built the first two-stage microscope in 1931 after they realized the focusing properties of solenoids.

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