A decomposition for transverse spins in structured vector fields

TL;DR

This work addresses how the transverse spin angular momentum (SAM) of classical vector waves can be decomposed beyond familiar evanescent contexts. Using a Hertz-vector formalism, the authors derive a universal decomposition $\mathbf{s}=\mathbf{s}_{\rm p}+\mathbf{s}_{\rm e}$ with $\mathbf{s}_{\rm p}=\frac{1}{k^{2}}\nabla\times\mathbf{p}$ and an explicit higher-derivative form for $\mathbf{s}_{\rm e}$, then validate it on four vector beams (cosine evanescent, Bessel, Gaussian, Airy) in both propagating and evanescent regimes. The results show that $\mathbf{s}_{\rm p}$ reflects the inhomogeneity of momentum, while $\mathbf{s}_{\rm e}$ encodes polarization, near-field effects, and non-planar wave fronts, occasionally altering the classic double relation between the two components. The framework is experimentally testable via SAM mapping, torque-based measurements, or spin-resolved techniques, and it extends to scalar waves as a consistency check. Overall, the paper provides a unified, generalizable picture of SAM decomposition with broad implications for structured light and spin–momentum control.

Abstract

Classical vector waves can possess intricate spin angular momenta (SAM), which are \emph{perpendicular} to the propagation direction, as revealed by the recent recognition of surprisingly transverse SAM in electromagnetic (EM) fields. In this paper, we employ the Hertz potential method to define structured vector fields and analytically decompose the SAM of the wave fields in two parts. Our novel approach of decomposition not only confirms that transverse SAM may originate from the first-order spatial inhomogeneity of the Poynting momentum, but also points out that for \emph{non-planar vector waves with near fields}, an extraordinary spin appears as a distinct part out of transverse spin. By four examples of vector beams, we further demonstrate that the proposed transverse spins prevail universally in both propagating and evanescent waves. This work renews our fundamental understanding of the decomposition of SAM for classical vector waves.

Figures (9)

• Figure 1: (a) The normalized energy density $\tilde{w}$ and (b) normalized momentum $\tilde{p}$ distributions of the two different fields $\hat{n}=\hat{z}$ and $\hat{n}=\hat{x}$ when $z=0$. Parameters: $k_x=\beta\sin(\pi/3)$, $k_y=\beta\cos(\pi/3)$, $\kappa=\beta\sin(\pi/16)$, $\beta=k/\cos(\pi/16)$, and $k=2\pi/\lambda$.
• Figure 2: The cosine evanescent wave field defined by $\bm{\Pi}_m=\Pi_m\hat{x}$. (a) Normalized momentum $\tilde{p}$ and (b) Normalized SAM $\tilde{S}$ vector distribution. (c)-(d) Normalized Poynting spin $\tilde{S}_{\rm{p}}$ and extraordinary spin $\tilde{S}_{\rm{e}}$ vector distributions respectively. The orange background image represents the normalized energy density $\tilde{w}$ distribution in (a)-(d) and this applies for Figs. \ref{['figs-fig3']}, \ref{['figs-fig6']}, \ref{['figs-fig8']} and \ref{['figs-fig9']}. The red dashed line in (b)-(d) represents $z=0.1\lambda$ and $y=0$. (e)-(f) shows the curves of $x$ component and $z$ component densities of the three spins versus the lateral position $x$ when $z = 0.1\lambda$ respectively. (g) The vector distribution of three spins on plane $z=0.1\lambda$. Parameters: $k_x=\beta\sin(\pi/3)$, $k_y=\beta\cos(\pi/3)$, $\kappa=\beta\sin(\pi/16)$, $\beta=k/\cos(\pi/16)$, $k=2\pi/\lambda$.
• Figure 3: The Bessel beam defined by $\Pi_m=\Pi_m\hat{z}$. (a) Normalized momentum $\tilde{p}$ and (b) Normalized SAM $\tilde{S}$ vector distribution; (c)-(d) Normalized Poynting spin $\tilde{S}_{\rm{p}}$ and extraordinary $\tilde{S}_{\rm{e}}$ vector distributions. (e) the density distribution of three spins in $r$ direction and $\delta \tilde{s}_\phi=\tilde{S}_{\rm{p}}/2+\tilde{S}_{\rm{e}}$; (f) the spin vectors on the red dash line ($y=0.5\lambda$, $z=0$) in $x\text{-}y$ plane. Parameters: $k_r=k\cos(\pi/3)$, $k_z=k\sin(\pi/3)$, $k=2\pi/\lambda$.
• Figure 4: The Bessel beam defined by $\bm{\Pi}=\Pi\hat{x}$. (a) Vector density distribution of SAM $\textbf{s}_E=s_{E_x}\hat{x}$ provided by the electric field; (b)-(c) distribution for two components of magnetic field $\textbf{s}_H=s_{H_x}\hat{x}+s_{H_y}\hat{y}$; (d) distribution of the total spin $\textbf{s}=\textbf{s}_E+\textbf{s}_H$. The arrow represents the size and direction of the spin vector, the background represents the spin density. Parameters: $k_r=k\cos(\pi/3)$, $k_z=k\sin(\pi/3)$, $k=2\pi/\lambda$.
• Figure 5: (a) The phase difference $\delta_{\phi_E}=\phi_{E_y}-\phi_{Ez}$ distribution between the two components of the Gaussian electric field defined by Eq. \ref{['58']}. From Eq. \ref{['59a']}, $E_z=0$ for plane $y=0$, so we remove the factor $y$ here. The orange and white dashed line represent the wave front of $E_y$ and $E_z$ respectively, which can also characterize the phase difference between the two components; (b) The electric field vector $\text{Re} (\mathbf{E} )$ and the energy density $\tilde{w}$ of such a Gaussian beam distribution. The white arrow represents the electric vector and the background represents the normalized energy distribution. Here, we choose the position $z=0.1\lambda$ near focus, with $z_R=\lambda$ and $k=2\pi/\lambda$.