Time reversal and the spin angular momentum of transverse-electric and transverse-magnetic surface modes

The polarization state of a surface mode is a hybrid between linear and elliptical states, meaning that although one of its electric- and magnetic-field components is linearly polarized, the other rotates on the transverse-longitudinal plane, effectively in its elliptical polarization state. We show that the rotation of the electric-field component can induce transverse spin angular momentum of the surface mode. The rotation of the magnetic-field component, however, cannot generate the spin. We argue that this results from the fact that the rotation direction of the magnetic field is invariant under time reversal.

Figure 1

(a)–(d) Schematics showing the propagation of the electric-field component of a TM-polarized surface mode at $t=0$, $\pi /2\omega$, $\pi /\omega$, and $3\pi /2\omega$, respectively. We assumed an interface between materials whose permittivities have opposite signs, which supports a forward mode. Peak positions of the induced (and accumulated) polarization charge density are denoted as $+$ and $-$. (e) Polarization states of the $\mathbf{E}$ field near the dotted area in (a)–(d).

Figure 2

As Fig. 1 for the magnetic flux density ($\mathbf{B}$) of a TE-polarized surface mode. We assumed similarly an interface between materials whose permeabilities have opposite signs, which also guides a forward mode. Peak positions of the induced (and accumulated) magnetization current density are denoted as $\odot$ and $\otimes$. Straight arrows near the dotted area indicate the local directions of the $\mathbf{B}$ field there.

Figure 3

(a) As Fig. 1e under time reversal. We took Fig. 1d as a new initial field configuration ($t^{\prime }=0$). The rotation direction of the $\mathbf{E}$ field is reversed or is odd under time reversal. (b),(c) The case of TE polarization. We begin with (b) [which is equal to Fig. 2d]. Under time reversal, the surface mode propagates along the $-z$ direction, resulting in (c${}^{\prime }$). However, the direction of the magnetization current should be reversed as well, which produces the field pattern shown in (c). It is actually equal to Fig. 2a. That is, the rotation direction of the $\mathbf{B}$ field is even under time reversal.

Figure 4

(a) Rotation directions of the $\mathbf{E}$ field at the left and right sides of the interface which supports a forward TM mode ($\beta >0)$. (b)–(d) Directions of the Minkowski and the Abraham forms of the spin AM density at various types of interface composed of (b) PIM-NIM, (c) PIM-ENG, and (d) MNG-ENG layers. One interface we neglected here is that between MNG and NIM layers. This is in order to avoid any confusion because it usually supports a backward mode, and in such a backward case, the directions of the spin AM (since $\beta <0$) as well as those of the $\mathbf{E}$-field rotation are all reversed [3].