Advances in Imaging and Electron Physics by Peter W. Hawkes

By Peter W. Hawkes

The sequence bridges the distance among educational researchers and R&D designers by means of addressing and fixing day-by-day matters, which makes it crucial reading.This quantity seems to be at concept and it really is software in a realistic feel, with an entire account of the tools used and sensible distinctive program. The authors do that through reading the newest advancements, ancient illustrations and mathematical basics of the intriguing advancements in imaging and electron physics and observe them to life like sensible events. * Emphasizes wide and intensive article collaborations among world-renowned scientists within the box of photograph and electron physics* provides idea and it is software in a realistic experience, offering lengthy awaited ideas and new findings* offers the stairs find solutions for the hugely debated questions

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In the xy-plane we have in leading order Ma ðqÞev ¼ cosr¯ % Mb ðqÞev r¯ ð125Þ Mf ðqÞev % sinr¯ r¯ ð126Þ pffiffiffi for r¯ large (the Bessel functions are O(1/ r¯ ). This shows that evanescent waves along the xy-plane also have an O(1/q) part that survives in the far field. The evanescent part of Green’s tensor is in leading order for r¯ large $ $ w ðqÞev % ð I À er er Þ cosr¯ r¯ ð127Þ and Green’s vector is hðqÞev % er sinr¯ : r¯ ð128Þ It follows from Eqs. (24) and (28) that the unsplit Green’s tensor and vector for large q are $ $ w ðqÞ % ð I À qˆ qˆ Þ hðqÞ % Àiqˆ eiq q eiq : q In the xy-plane we have q ¼ r¯ and qˆ ¼ er, and therefore we find ð129Þ ð130Þ EVANESCENT WAVES IN THE NEAR AND THE FAR FIELD $ $ 25 Re w ðqÞ % w ðqÞev ð131Þ RehðqÞ % hðqÞev : ð132Þ Hence, in the xy-plane the real parts of Green’s tensor and vector consist purely of evanescent waves in the far field and the imaginary parts are pure traveling.

These series are Taylor series in j¯zj for a given r, ¯ and the Taylor coeYcients became functions of r, ¯ involving Bessel functions. We now seek series expansions in q, around q ¼ 0, for a given y. In this way, the Taylor coeYcients become functions of y. 46 ARNOLDUS To this end, we start from the result from Section XV, Eqs. (138)–(143). For the Bessel functions we substitute their series expansion as given by Eq. (136). For ReMa ðqÞtr we then find  2kþ1 1 X 1 r¯ j¯zj X ‘! ðk þ ‘ þ 1Þ! This double series can be written as a single series by using the Cauchy product, which yields  2nÀ2‘ 1 X n r¯ 1X ‘!

For j¯zj ¼ 0 only the first term, ‘ ¼ 0, contributes, and we get exactly the result from Eqs. (115)–(117) in Section XIII. For j¯zj 6¼ 0 we need to keep more terms. Then, if we calculate ReMk ðqÞtr with the series expansions above, we can also find the evanescent parts near the z-axis with Eq. (96), where ReMk ðq) are the real parts of the right-hand sides of Eqs. (68)–(73). For instance Ma ðqÞev ¼ cosq j¯zj 1 j¯zj3 þ J1 ðrÞ J2 ðrÞ ¯ À ¯ þ ... : q r¯ 3 r¯ 2 ð144Þ Figure 4 shows Ma ðqÞev for r¯ ¼ 5, computed this way, and with the series summed up to ‘ ¼ 20.

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