As shown in Fig. 1a, WS2 thin flakes are exfoliated onto standard SiO2/Si wafers. We use the oblique illumination of the He-Ne laser with circular polarization (*λ*0 = 632.8 nm), which is close to the low energy branch of EP in WS2 waveguides we used in the following experiments (32). With SOC, we anticipate the unidirectional propagation of EPs, originating from the high-spatial-frequency radiations asymmetrically coupled into the waveguide modes.

For better understanding of SOC here, we examine the waveguide modes first. The dielectric WS2 nanoflakes is uniaxial with the optic axis along z direction. The Air/WS2/SiO2 three-layer structure exhibits ordinary (TE) and extraordinary (transverse magnetic, TM) waveguide modes (28). The associated in-plane modal wavevectors are shown in Fig. 1b where the thickness of WS2 nanoflake varies. For thin flakes (*d* < 30 nm), only the fundamental TE0 mode is supported, while for thicker WS2 flakes (*d* > 50 nm), more waveguide modes, particularly the extraordinary TM modes, can be observed with the increasing in-plane momentum as the thickness increases.

In our experiments to map the SOC, distinguishing those TE and TM modes from the viewpoint of the near-field modal symmetry is important, for both the calibration and the SOC feature extraction. For this purpose, we consider the elliptical dipolar excitations. As for a dipole (*p**x*,*0,p**z*) polarized perpendicularly to the dielectric slab, TM modes can be induced and the electric field distribution shows significant asymmetric spatial frequency, which drives the TM modes of the dielectric slab to propagate unidirectionally. By decomposing the electric field of dipole into TM modes with in-plane momentum (*k**x*,*k**y*) as *E**z*(*x,y,z*) = *∫*\({\stackrel{\sim}{E}}_{z}\)(*k**x*,*k**y*,*z*)*e**i*(*k**x**x*+*k**y**y*)d*k**x*d*k**y*, the unidirectional property of the TM modes propagating along *x* axis (*k**y* = 0) is shown as:

$${\tilde {E}_z}({k_x},z)= - \frac{{ic}}{{8{\pi ^2}\omega \varepsilon }}\frac{{ \mp {k_x}{k_z}{p_x}+k_{x}^{2}{p_z}}}{{{k_z}}}{e^{i{k_z}|z - {z_{{\text{dipole}}}}|}}$$

1

where *k**z* = (*k**0*2 - *k**x*2)1/2 is the wavevector along *z* axis, *ω* is the angular frequency and *k*0 = *ω/c*. Due to the vertical circularly polarized dipole is composed by two orthogonal dipole orientations *p**x* and *p**z*, the linear superposition of these two components results in an asymmetric distribution while the *p**z* component has even parity and the *p**x* component shows odd parity about *k**x* (see Supplementary Information, Section S1). For example, as illustrated in Fig. 1c and 1d, numerical simulations demonstrate that the TM0 mode is allowed in a WS2 slab with the thickness of 62 nm and it exhibits unidirectional propagation, which depends on the chirality of illumination.

Importantly, we should note that the WS2 flake also support TE modes. Owing to the oblique illumination of the circularly polarized laser, the modulation from the apex of AFM deviates the circularly polarized dipole elliptically for both the horizontal and the vertical components. The horizontal components could generate an in-plane circular polarized dipole (*p**x*, *p**y*,*0*). When such an in-plane circular polarized dipole placed parallel to the dielectric nanoflake, TE modes can be excited and their magnitudes are independent to the circularly polarized state of the dipole. After similar process above, TE modes expansion can be expressed as *E**y*(*x,y,z*) = *∫*\({\stackrel{\sim}{E}}_{y}\)(*k**x*,*k**y*,*z*)*e**i*(*kxx*+*kyy*)d*k**x*d*k**y*, where

$${\tilde {E}_y}({k_x},z)= - \frac{{ic}}{{8{\pi ^2}\omega \varepsilon }}\frac{{k_{x}^{2} \pm k_{z}^{2}}}{{{k_z}}}{p_y}{e^{i{k_z}|z - {z_{{\text{dipole}}}}|}}$$

2

The electric field shows symmetric spatial frequency distribution, and it results in the TE modes propagating in the dielectric flake only have a *π/*2 phase difference between opposite chirality. Corresponding numerical simulations are performed in a 10 nm-thick WS2 slab, which only supports TE0 mode and the symmetric field distributions of TE modes for opposite horizontal chirality component (*p**x*, *p**y*,*0*) are demonstrated in Fig. 1e and 1f.

To quantitively characterize the directionality of the modes, based on Eq. (1) and Eq. (2), the ratio of unidirectionality is defined as *R* = |\({\stackrel{\sim}{E}}_{{{\sigma }}^{-}}\)(*k**x*)*/*\({\stackrel{\sim}{E}}_{{{\sigma }}^{+}}\)(*k**x*)|, where \({{\sigma }}^{-}\) denotes the left circular polarized (LCP) state and \({{\sigma }}^{+}\) represents the right circular polarized (RCP) state. As illustrated in Fig. 1g, unidirectionality exhibits negative correlation (*k**x*>*k**0*) with the in-plane momentum *k**x*, where an elliptical polarized dipole with (*p*x, *p*y, *p*z) = (1, 0, -0.3i) is chosen as the illumination based on the experimental condition. For instance, *R* = 2.23 for TM0 mode in a 62 nm-thick WS2 slab, whereas *R* = 1 for the TE0 mode in a 10 nm-thick WS2 slab, showing strong signatures of SOC in EPs. Besides, this distinguished feature can allow us to use the TE mode to calibrate our system, which will be discussed later.