By Peter W. Hawkes (Eds.)

ISBN-10: 0123858615

ISBN-13: 9780123858610

*Advances in Imaging and Electron Physics *merges long-running serials--*Advances in Electronics and Electron Physics* and *Advances in Optical and Electron Microscopy*.

This 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 photograph processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in a majority of these domains.

* Contributions from major foreign students and specialists

* Discusses sizzling subject components and offers present and destiny learn developments

* necessary reference and consultant for physicists, engineers and mathematicians

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**Extra resources for Imaging and Electron Physics**

**Example text**

Thus, in general, the electromagnetic field does not propagate at the speed of light. In fact, an instantaneous local velocity seems to be an altogether more appropriate concept. For a pulse, this local velocity approaches the speed of light from above as one moves farther from the source. , a hard-to-measure boundary between the field and the region of space that it has not reached yet) always travels 50 Neil V. Budko at the speed of light. We shall also consider the power flow from a small transient source.

Thus, the 2D convolution of two radially symmetric functions yields another radially symmetric function, as can be seen from Eq. (115). Moreover, by using the proper definition of a 2D convolution instead of using the tempting definition of a 1D convolution, the well-known relationship between convolutions in one domain leading to multiplication in the other domain is preserved—namely, that f (r) ∗∗ g(r) = f (r) ∗∗ g(r) ⇔ F(ρ)G(ρ). (117) The key point to this relationship is the proper definition of the convolution as a 2D convolution.

In polar coordinates, the 2D Laplacian takes the form ∇2 = ∂2 1 ∂ 1 ∂2 + + . r ∂r r2 ∂θ 2 ∂r2 (149) Consider a typical function written in standard 2D polar form: ∞ fn (r) ejnθ . f (r) = f (r, θ ) = (150) n=−∞ Taking the Laplacian of f (r) gives 1 ∂ 1 ∂2 ∂2 + + r ∂r r2 ∂θ 2 ∂r2 ∇ 2 f (r) = ∞ = d2 f n dr2 n=−∞ + ∞ fn (r) ejnθ n=−∞ n2 fn 1 dfn − 2 ejnθ . r dr r (151) Hence for a function written in the form of Eq. (150), the required form of the Laplacian is denoted with ∇n2 , where this operator is defined by ∇n2 = d2 1 d n2 − + .

### Imaging and Electron Physics by Peter W. Hawkes (Eds.)

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