Electromagnetic Vortices. Группа авторов

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far‐field expression from Eq. (1.A.9):

      (1.A.17)StartLayout 1st Row ModifyingAbove upper E With right-arrow Subscript italic f f Superscript italic upper L upper G Baseline left-parenthesis r comma theta comma phi right-parenthesis equals StartFraction italic j k 0 upper E 0 Superscript italic upper L upper G Baseline e Superscript minus italic j k 0 r Baseline Over 4 italic pi r EndFraction left-parenthesis ModifyingAbove theta With ampersand c period circ semicolon cosine phi minus ModifyingAbove phi With ampersand c period circ semicolon cosine theta sine phi right-parenthesis w Subscript g Baseline left-parenthesis negative 1 right-parenthesis Superscript p Baseline left-parenthesis negative j right-parenthesis Superscript l Baseline 2nd Row times StartRoot StartFraction 2 italic pi p factorial Over left-parenthesis p plus bar l bar right-parenthesis factorial EndFraction EndRoot left-parenthesis StartFraction sgn left-parenthesis l right-parenthesis normal upper Psi Over StartRoot 2 EndRoot EndFraction right-parenthesis Superscript bar l bar Baseline e Superscript minus StartFraction normal upper Psi squared Over 4 EndFraction Baseline upper L Subscript p Superscript bar l bar Baseline left-parenthesis StartFraction normal upper Psi squared Over 2 EndFraction right-parenthesis e Superscript negative italic j l phi Baseline comma EndLayout

      where Ψ = k0wg sin θ. For the definition of wg refer to the first paragraph of the appendix and Figure 1.A.1. The previous discussion refers to the far‐field where the radiation integral can be found in closed form. The near‐field calculation using the Fresnel–Kirchhoff diffraction integral [24] was carried out numerically in Section 1.2 and the results are shown in Figure 1.7.

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