Table 2 The dynamical and topological properties of generic EM wave, linear polarized surface EM wave, deep-water gravity wave, and acoustic wave fields.

From: Dynamical and topological properties of the spin angular momenta in general electromagnetic fields

 

Generic EM wave

Linear polarized surface EM wave

Gravity water wave

Acoustic wave

Field components

Electric field E;

Magnetic field H;

Electric or magnetic Hertz potential Ψ;

In-plane velocity V;

Normal velocity W;

Velocity v;

Pressure p;

Kinetic momentum

\({{{{{\mathbf{\Pi }}}}}}=\frac{1}{2{c}^{2}}{{{{\mathrm{Re}}}}}\{{{{{{{\bf{E}}}}}}}^{\ast }\times {{{{{\bf{H}}}}}}\}\)

\({{{{{\mathbf{\Pi }}}}}}=\frac{\varepsilon {k}^{2}{k}_{p}^{2}}{2\omega }{{\mbox{Im}}}\{{\varPsi }^{\ast }\nabla \varPsi \}\)

\({{{{{{\mathbf{\Pi }}}}}}}_{G}=\frac{{\rho }_{G}{k}_{G}}{{\omega }_{G}}{{\mbox{Im}}}\{{W}^{\ast }{{{{{\bf{V}}}}}}\}\)

\({{{{{{\mathbf{\Pi }}}}}}}_{A}=\frac{1}{2{c}_{A}^{2}}{{{{\mathrm{Re}}}}}\{{p}^{\ast }{{{{{\bf{v}}}}}}\}\)

Spin angular momenum

\({{{{{\bf{S}}}}}}=\frac{1}{4\omega }{{\mbox{Im}}}\{\begin{array}{c}\varepsilon {{{{{{\bf{E}}}}}}}^{\ast }\times {{{{{\bf{E}}}}}}\\ +\mu {{{{{{\bf{H}}}}}}}^{\ast }\times {{{{{\bf{H}}}}}}\end{array}\}\)

\({{{{{\bf{S}}}}}}=\frac{\varepsilon {k}_{p}^{2}}{4\omega }{{\mbox{Im}}}\{\nabla {\varPsi }^{\ast }\times \nabla \varPsi \}\)

\({{{{{{\bf{S}}}}}}}_{G}=\frac{{\rho }_{G}}{2{\omega }_{G}}{{\mbox{Im}}}\{{{{{{{\bf{V}}}}}}}^{\ast }\times {{{{{\bf{V}}}}}}\}\)

\({{{{{{\bf{S}}}}}}}_{A}=\frac{{\rho }_{A}}{2{\omega }_{A}}{{\mbox{Im}}}\{{{{{{{\bf{v}}}}}}}^{\ast }\times {{{{{\bf{v}}}}}}\}\)

Helicity

Spin-1 photon

σ = ±1

Spin-1 photon

σ = ±1

Spin-0 phonon

σG = 0

Spin-0 phonon

σA = 0

Spin-momentum locking

\(\begin{array}{c}{{{{{{\bf{S}}}}}}}_{t}=\frac{1}{2{k}^{2}}\nabla \times {{{{{\mathbf{\Pi }}}}}}\\ {{{{{{\bf{S}}}}}}}_{l}=\mathop{\sum}\limits_{i}\hslash {\sigma }_{i}{\hat{{{{{{\bf{k}}}}}}}}_{i}+\mathop{\sum}\limits_{i\ne j}\hslash {\sigma }_{ij}{\hat{{{{{{\bf{k}}}}}}}}_{ij}\end{array}\)

\(\begin{array}{c}{{{{{{\bf{S}}}}}}}_{t}=\frac{1}{2{k}^{2}}\nabla \times {{{{{\mathbf{\Pi }}}}}}\\ {{{{{{\bf{S}}}}}}}_{l}=0\end{array}\)

\({{{{{{\bf{S}}}}}}}_{G}=\frac{1}{2{k}_{G}^{2}}{\nabla }_{2}\times {{{{{{\mathbf{\Pi }}}}}}}_{G}\)

\({{{{{{\bf{S}}}}}}}_{A}=\frac{1}{{k}_{A}^{2}}\nabla \times {{{{{{\mathbf{\Pi }}}}}}}_{A}\)

  1. The field, kinetic momentum, SAM, helicity and spin-momentum locking properties of general EM field, linear polarized surface EM field, gravity water waves and acoustic waves. Therein, the field, kinetic momentum and SAM properties can be found in Ref. 6 and the Supplementary Table. S1 of Ref. 15, respectively. For the longitudinal acoustic wave, \({c}_{A}^{2}=1/{\beta }_{A}{\rho }_{A}\) is the speed of the acoustic wave, where βA is the compressibility of the acoustic medium; ρA is the mass density of the acoustic medium; ωA and kA = ωA/cA are the angular frequency and wavenumber, respectively; σW = 0 for the phonons corresponding to the longitudinal acoustic waves; and ρG is the mass density of the fluid.
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