Figure 1

Zeeman splitting of photonic states when a static magnetic field is applied along the $z$ direction. The top figure shows a gyromagnetic cylinder surrounded by air. Formula in the dashed box displays the relation between frequency shift and angular momentum density of light ($j_z$). The first column (black lines) shows the original degenerate states in the gyromagnetic cylinder in the absence of magnetic field $H_0=0$ (i.e., with isotropic permeability ${\mu }_r$). The second column (green lines) shows the shifts due to the change of effective permeability from ${\mu }_r$ to ${\mu }_r^{\prime }$ (indicated as “index shift”) when a static magnetic field of $H_0=800$ Oe is considered. The third column (red and blue lines) shows the final Zeeman splitting with the adjacent signs indicating the sign of $m$. Only the states with $n=1,2$ and $\left|m\right|=4,5$ are shown.Reuse & Permissions

Figure 2

Electric field patterns excited by an out-of-plane line source in the presence of external magnetic field ($H_0=800$ Oe). The line source lies on the surface of the gyromagnetic cylinder. (a) The $(n=1,m=-4)$ mode is excited at its resonant frequency $f=8.72$ GHz. (b) Same as (a), but for $(1,4)$ at its resonant frequency $f=10.40$ GHz. (c), (d) Same as (a), but for modes $(2,-4)$ and $(2,4)$ at their respective resonant frequencies $f=13.14$ GHz and $f=13.58$ GHz.Reuse & Permissions

Figure 3

Resonant frequency vs azimuthal momentum number $m$. The solid lines with open circles show exact results obtained from Mie resonant conditions [Eq. (2)]. The dashed lines with solid circles indicate the approximated results obtained from the perturbation theory [Eq. (14)]. Blue lines and red lines indicate the negative $m$ modes and the positive $m$ modes, respectively. Low-quality states with $\left|m\right|<4$ are not shown.Reuse & Permissions

Figure 4

Proposed experimental measurement for the Zeeman-like splitting of photonic angular momentum states. The upper panel shows a schematic of the experimental setup. The gyromagnetic cylinder in Fig. 1 is placed between two straight waveguides with refractive index $2.2$ and width $5.6$ mm. The gap between the cylinder and each waveguide is $1.8$ mm. (a) and (b) show the electric field patterns at the resonant frequency of the degenerate $(n,m)=(1,\pm 4)$ states in the absence of a static magnetic field when an incident wave is launched from left (port 1) and right (port 2), respectively. (c) and (d) are, respectively, the same as (a) and (b) except the waves are launched at the resonant frequency of the nondegenerate $(n,m)=(1,-4)$ state in the presence of the static magnetic field.Reuse & Permissions

Figure 5

(a) Transmittance spectra of the one-way coupled waveguides in a wide frequency range. The black (red) line corresponds to the transmittance $T_{1\to 3}$ from port 1 to 3 ($T_{2\to 4}$ from port 2 to 4, which is identical to $T_{3\to 1}$ in our case). The excited negative modes and positive modes are marked. (b) The corresponding differential transmission. The parameters are identical to that described in Fig. 4.Reuse & Permissions

Figure 6

Decoupling among photonic angular momentum states in an array of closely packed identical gyromagnetic cylinders. The yellow arrows indicate the location of the point source. (a) Strong resonant coupling when a static magnetic field is absent. Clockwise and counterclockwise rotational states $(1,\pm 4)$ are both supported. (b) Weak resonant coupling when a static magnetic field is present. Only the clockwise rotational state $(1,-4)$ is supported.Reuse & Permissions