Aberrations Spherical

Haider and colleagues prove the concept of the TEM spherical aberration corrector... 

Figure Bl.17.5. Examples of CTFs for a typical TEM (spherical aberration = 2.7 mm, 120 keV electron energy). In (a) and (b) the idealistic case of no signal decreasing envelope fimctions [77] are shown, (a) Pure phase contrast object, i.e. no amplitude contrast two different defocus values are shown (Scherzer focus of 120 mn imderfocus (solid curve), 500 mn underfocus (dashed curve)) (b) pure amplitude object (Scherzer focus of 120 mn underfocus) (c) realistic case mcluding envelope fimctions and a mixed weak...

The construction of an aberration-corrected TEM proved to be teclmically more demanding the point resolution of a conventional TEM today is of the order of 1-2 A. Therefore, the aim of a corrected TEM must be to increase the resolution beyond the 1 A barrier. This unplies a great number of additional stability problems, which can only be solved by the most modem technologies. The first corrected TEM prototype was presented by Flaider and coworkers [M]- Eigure BE 17.9 shows the unprovement in image quality and interpretability gained from the correction of the spherical aberration in the case of a materials science sample. 

Figure Bl.17.9. A CoSi grain boundary as visualized in a spherical-aberration-corrected TEM (Haider et a/ 1998). (a) Individual images recorded at different defocus with and without correction of C(b) CTFs in the case of the uncorrected TEM at higher defocus (c) CTF for the corrected TEM at only 14 nm underfocus. Pictures by courtesy of M Haider and Elsevier.

The ratio F/d is the F number of the lens. For F numbers much less than unity, spherical aberration precludes reaching the ultimate diffraction-limited spot size. Therefore a practical limit for the minimum spot size obtainable is approximately the wavelength of the light. Commonly this is expressed as the statement that laser light may be focused to a spot with dimensions equal to its wavelength. 

The specimen is immersed in the next lens encountered along the column, the objective lens. The objective lens is a magnetic lens, the design of which is the most crucial of all lenses on the instrument. Instrumental resolution is limited primarily by the spherical aberration of the objective lens. 

No matter where is situated, its image will appear as indicated in Figure 4-12, but the quality of the image will vary owing to spherical aberration. For a spherical mirror, this aberration is least and virtually absent if... 

The use of a laser beam expander as a spatial filter has also been found to be satisfactory 42). The beam expander consists of an interchangeable negative input lens and a positive output lens. Both the input and output lenses are designed for minimum spherical aberration. The expansion power may be varied by using a different input lens (Fig. 23.) The laser beam... 

Figure 5. Spherical aberration rays corresponding to different aperture angles focus at different locations along the optical axis.

Furthermore, for any two-mirror telescope in which each mirror is individually corrected for spherical aberration (i,e. composed of two conics) one can solve for the third order aberrations in closed form. 

ASA angular diameter of spherical aberration blur circle. 

Examples of two-mirror designs in which each mirror is individually corrected for spherical aberration are ... 

With these closed form aberration equations it is possible to do something interesting Rather than use the mirrors at their individual conic conjugates which would correct for spherical aberration in each mirror independently, we can set the overall spherical and comatic aberration to zero and solve for a different set of conics. This is called the aplanatic condition, when 3 order... 

Spherical aberration at prime focus Financially important... 

Two-mirror telescopes are the most common optical design for ground based telescopes. These systems require a parabolic or hyperbolic primary mirror. As mentioned before, more complex optical systems can accommodate a spherical primary with its attendant simplifications, but several additional mirrors are needed to correct the spherical aberration, and the light loss and alignment complexity makes this configuration less commonly used. Here we will assume that a non spherical primary is needed and we will discuss the resulting surface shapes that segments will have. 

In the design presented here, the compensation of the enormous spherical aberration of the primary mirror inevitably falls on the quaternary one. With a FI. 42 spherical primary mirror, the aspheric departure to be figured into the quaternary is as large as 14 mm. 

Besides the increase of reflections implied by Owl optical design, a price to pay for the spherical primary mirror solution is the difficulty to compensate for its spherical aberration, and in particular the horrendous aspherization of the quaternary mirror (which is conjugated to the primary). A possible test setup has been identified and the state of current technology allows for cautious hope industrial studies are however still required to confirm feasibility and evaluate implied cost and schedule. 

The original bend and polish idea is due to Bernard Schmidt who built what has become known as a Schmidt telescope with a spherical primary and an aspheric corrector plate that introduces exactly the right amount of spherical aberration to cancel the spherical aberration that would be introduced by the spherical primary (Schmidt). The corrector has to be thicker at the edge in proportion to the radial distance in the aperture to the fourth power to provide the needed correction. 

The departure of these segments from spherical is given by the terms set out in the Tab. 1 in Ch. 8. Table 1 simply gives a similar Table of the peak-to-valley departure for the inner and outer rings of segments for the aberrations using those equations (and multiplying by 2 to get P-V except for spherical aberration where the multiplication factor is 1.5). 

For the high resolution case, the phase-contrast effects are automatically introduced owing to the combined effect of defocus and spherical aberration, which gives rise to an image of a structure complicated by the fact that also the amplitude term, resulting from the propagation process, interacts in a non-linear way with the phase term [16,89,90,96]. 

We have also not mentioned the last breakthrough in electron microscopy, where the spherical aberration problem has been finally solved by means of multipole lenses [117,118], and new aberration corrected electron microscopes are entering the market. 

The least expensive (and most common) objectives are the achromatic objectives that are designed to limit the effects of chromatic and spherical aberration. Achromatic objectives are corrected to bring two wavelengths of light (typically red and blue) into focus in the same plane. The limited correction of achromatic objectives leads to problems with color microscopy and photomicrography. 

Spherical aberration Inacccurate focusing of light due to curved surface of lense whereby light rays passing through the lens at different distances from its center are focused to different positions in the Z-axis. 

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