By Peter W. Hawkes

**Advances in Imaging & Electron Physics** merges long-running serials--**Advances in Electronics & Electron Physics** and **Advances in Optical & Electron Microscopy**. The sequence positive factors prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photo technology and electronic snapshot processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in a majority of these domain names.

**Read or Download Modern Map Methods in Particle Beam Physics, Volume 108 (Advances in Imaging and Electron Physics) PDF**

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**Example text**

For the remaining two functions we ﬁnd sﬃﬃﬃﬃﬃﬃ 2 2 ev Mc ðqÞ % sin ðq þ p=4Þ ð214Þ q pq 1 Me ðqÞ % q ev sﬃﬃﬃﬃﬃﬃ 2 cosðq þ p=4Þ pq ð215Þ and with Eq. (166) we verify that these solutions are asymptotically equivalent to the exact results from Eqs. (120) and (122). To conclude this section, let us summarize. For the evanescent waves in the far ﬁeld, Green’s tensor and vector are given by Eqs. (163) and (164), and with the three auxiliary functions given by Eqs. (205)–(207). This accounts for a far-ﬁeld contribution near the z-axis and near the xy-plane.

Evanescent Waves in the Far Field In the far ﬁeld, q is large and y is arbitary. The standard method to obtain an asymptotic solution for q large from an angular spectrum representation is by the method of stationary phase. Appendix B shows that it seems to follow from this method that only traveling waves contribute to the far ﬁeld and that this might not necessarily be true. In any case, we consider the contribution of the evanescent waves to the far ﬁeld by means of our asymptotic expansion from Section XVI.

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. It is seen that the series expansion perfectly reproduces the exact result, obtained with numerical integration, up to about |¯zj ¼ 12.