By A. B. El-Kareh
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Extra info for Electron Beams, Lenses, and Optics. Volume 2
This formula can also be written as follows: Ci = / 16£(Φο)^1/2J ,J*. 229) m m and this is the form in which it is published by Scherzer. Let us once more remember that M is the magnification. Since we started with ry = 1 in the object plane, M = ry in the image plane; φ0 is the potential at the object, and raa is the radius of the aperture; and ra is the paraxial-ray solution with the boundary conditions ra' = 1 and ra = 0 at the object. 4 we know that coma is proportional to the first power of r at the object plane and to the second power of r at the aperture.
213) We shall now show that the j terms are equal to zero. We write d [ « V ) I ] = [lr. r? "ra2 | + r^r. 214) Substituting for ra" as in Eq. 209), and since the left-hand term is equal to zero upon integration between the limits z0 and zx, we obtain L " 2 J . 9 43 Spherical Aberration We now use Eq. 215) in the imaginary expression of Eq. 216) Now, let the coefficient of r3r' be A. 218) We have thus shown that the sum of they terms is equal to zero. 219) Since this form does not contain any terms with rar^3, we start by eliminating the term with rar'3 from Eq.
Therefore the radius of the disk of minimum confusion is one-fourth of the radius of the aberration circle at the Gaussian plane. The distance along the z axis from the Gaussian plane to where this disk occurs is BF. 75Q0! 2 . 4 The Spherical Aberration of the Combination of Two Thin Lenses Consider two thin lenses as shown in Fig. 4. We shall now calculate the coefficient of spherical aberration of the combined lens system. This arrangement is generally found when an intermediate image or cross-over is first 56 X THE SPHERICAL ABERRATION OF ELECTRIC AND MAGNETIC LENSES ' 1 1 i F «^ F 2 '^^\ \ 11 '• Lens L 2 Lens Li Fig.