RiemannSilberstein Vortices For Paraxial Waves
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SPATIOTEMPORALLY LOCALIZEDNULL A. M. Shaarawi
waves due originally to Robinson and Troutman is motivated by which,in turn,is the paraxial approximation of the scalarwave Then,the Riemann-Silberstein vector associated with the Robinson-
arXiv:quant-ph/0511011v3 12 Jan 2006
The Riemann-Silberstein vectorand theclassical-quantum correspondence Iwo Bialynicki-Birula∗ Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotniko´w 32/46, 02-668 Warsaw, Poland and Institute of Theoretical Physics, Warsaw University Zoﬁa Bialynicka-Birula Institute of Physics, Polish Academy of Sciences, Al.
Beams of electromagnetic radiation carrying angular momentum
 where the Riemann Silberstein vector plays the cen-tral role. 2. Succinct description of the electromagnetic ﬁeld A natural tool in the analysis of the solutions of the Maxwell equations, in both classical and quantum theories, is the Riemann Silberstein (RS) vector F F ¼ ﬃﬃﬃﬃ 0 2 r ðE þ icBÞ: ð1Þ
Riemann Silberstein vortices for paraxial waves  Hajnal J V 1987 Singularities of the transverseﬁelds of electromagneticwaves. I. TheoryProc. R. Soc.A 414 433 46  Hajnal J V 1987 Singularities in the transverseﬁelds of electromagneticwaves. II. Observations on the electric ﬁeld Proc. R. Soc.A 414 447 68