The bends on a quantum waveguide and cross-products of Bessel functions

Abstract

A detailed analysis of the wave-mode structure in a bend and its incorporation into a stable algorithm for calculation of the scattering matrix of the bend is presented. The calculations are based on the modal approach. The stability and precision of the algorithm is numerically and analytically analysed. The algorithm enables precise numerical calculations of scattering across the bend. The reflection is a purely quantum phenomenon and is discussed in more detail over a larger energy interval. The behaviour of the reflection is explained partially by a one-dimensional scattering model and heuristic calculations of the scattering matrix for narrow bends. In the same spirit we explain the numerical results for the Wigner-Smith delay time in the bend.

Steady fields of coherent synchrotron radiation in a rectangular pipe

“…• In introducing a symbol which is used in another paper in a different meaning from the present paper, we use the symbol with double quotation marks in order to avoid confusion. For example, "k" denotes the radial wavenumber of the field in [21], which differs from k in the present paper. iR + = {ix|x ∈ R + }: All purely imaginary numbers on the upper half-plane excluding 0.…”

“…Eqs. (20)(21)(22) are the general wave equations in the frequency domain, which hold regardless of the assumptions listed in section 2.3. That is, Eqs.…”

“…That is, Eqs. (20)(21)(22) hold anywhere in the beamline regardless of the straight or bending section including their boundary area. ∇ 2 ⊢ and ∇ 2 v are symbols which denote the differential operators of the wave equations [excluding αx ∂s and ±(2κ ∂s + α)/g] for the (x, s) and y components of the fields in the frequency domain (∇ ⊢ and ∇ v do not exist unlike ∇),…”

“…On the other hand, in the present study we do not neglect the α-terms in Eqs. (20)(21)(22) since our purpose is to find the exact expression of the fields. Since this is a problem which requires a somewhat intricate consideration due to a singular behavior of the field at the edge of the bend, we will discuss it in section 3.2 in detail.…”

“…Although Ẽx,s and Bx,s couple in Eqs. (20)(21), if they do not have the α-terms, we can solve Eqs. (20)(21) exactly as shown in appendix H. There is another way to find the expressions of Ẽx,s and Bx,s , because the electromagnetic field, Ẽ and B, has only two degrees of freedom.…”