By Giulio Pozzi (Eds.)
Advances in Imaging and Electron Physics merges long-running serials, Advances in Electronics and Electron Physics and Advances in Optical and 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, photograph technological know-how, electronic picture processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in these kind of domains.
- Contains contributions from best experts at the topic matter
- Informs and updates the entire most modern advancements within the box of imaging and electron physics
- Provides practitioners attracted to microscopy, optics, photo processing, mathematical morphology, electromagnetic fields, electron, and ion emission with a precious resource
- Features prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photograph technological know-how, and electronic photo processing
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Additional resources for Particles and Waves in Electron Optics and Microscopy
Example text
In order to manage this fact, we follow the analysis of Goursat (1933) (but see also Kaufmann, 1962 and Barozzi, 2004) and divide the integral into two parts: Z π=8 ReÀR Z 2 cos2θ 0 dθ + π=4 π=8 ReÀR 2 cos2θ dθ (15) pffiffiffi When θ varies between 0 and π/8, cos2θ is larger than 1= 2 and therefore p ffiffi 2 2 eÀR cos 2θ < eÀR = 2 . pffiffi 2 The first integral is less than π8 ReÀR = 2 and therefore approaches 0 when R increases. With regards to the second integral, we note that in the interval π/8 pffiffiffi θ π/4, we have that 2 sin 2θ > 1, so that the second integral is less than Z π=4 π=8 pffiffiffi 2 R 2 sin 2θeÀR cos2θ dθ (16) But now the integrand has an antiderivative, so that the value of the definite integral is given by !
The magnetic vector of this wave can be evaluated by using the Maxwell equation r^E ¼ iωμ0H (where ω ¼ kc is the angular frequency of the monochromatic field). In particular, denoting by Hor the radial component of the magnetic vector, we find Hor ðρ, φÞ ¼ À μ0 @Eo μ exp½ikρ φ ¼ À 0 pffiffiffiffiffi cos iωρ @φ 2iωρ 2 kρ (86) It is easy to see that Hor vanishes for φ ¼ π or, which is the same, for x < 0. By comparing the values of Hor for φ ¼ 0 and for φ ¼ 2π, we further see that there is a discontinuous change of the tangential component of the magnetic field across the z ¼ 0,x > 0 half-plane.
47 Complements of Wave Optics Therefore, the field distribution produced by the incident wave across the plane z ¼ 0 equals exp ½ikxcos α, and the problem is to find a scattered field E(s), which is generated by currents flowing in the half-plane x > 0, z ¼ 0, and which for x > 0, z ¼ 0, equals Àexp ½ikxcos α, such that the boundary condition of having field zero on the plane is satisfied. 2 The Elementary Solution We now consider the function exp ½ikρ φ Eo ðρ,φÞ ¼ pffiffiffiffiffi sin ¼ Eo ðx,zÞ, 2 kρ where ρ and φ are the polar coordinates given by & x ¼ ρcos φ z ¼ ρ sinφ (84) (85) It can easily be verified that Eq.