By A.L. Septier
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Extra info for Applied Charged Particle Optics
Well-known manifestations of shadow radiation are Fresnel diffraction and forward scattering. The PO drawbacks are the following. It is not self-consistent. When the observation point approaches the scattering surface, the PO integrals do not reproduce the initial Geometrical Acoustics (GA) values for the surface ﬁeld. Also, the PO ﬁeld does not satisfy rigorously the boundary conditions and the reciprocity principle. The reason for these shortcomings is the Geometrical Optics (GO) approximation for the surface ﬁeld, which does not include its diffraction components.
8), π l = 1, 2, 3, . . 8), π vl = l , l = 0, 1, 2, 3, . . 7). The Bessel functions Jvl (kr) can be used in the region r ≤ r0 , because they are ﬁnite at the edge r = 0, and the Hankel functions are appropriate in the region r ≥ r0 , because they satisfy Sommerfeld’s radiation condition at inﬁnity: lim √ r du − iku = 0, dr with r → ∞. 15). These expressions satisfy the boundary conditions, as well as the reciprocity principles; that is, they do not change after interchanging r and r0 , ϕ and ϕ0 .
21), multiply both sides by sin vt ϕ, where vt = tπ/α, and integrate them over ϕ from 0 to α. 22) t = 1, 2, 3, . . , and obtain d d α kr0 al Jvl (kr0 ) Hv(1) (kr0 ) − Hv(1) (kr0 ) Jv (kr0 ) = I0 . 23) The expression inside the brackets is the Wronskian W [Jv (x), Hv(1) (x)] = Jv (x) d (1) d 2i H (x) − Hv(1) (x) Jv (x) = . 24) it follows that al = π I0 . 25) In the case of the hard boundary conditions, we carry out similar manipulations. 21), multiply both sides by cos vt ϕ, and integrate over ϕ from 0 to α.