Ultra-thin grating coupler for guided exciton-polaritons in WS₂ multilayers

2.1 Theoretical calculation for a WS₂ grating structure

Figure 1A shows a schematic of a 1D photonic crystal structure (i.e., grating structure) made of layered WS₂. Among TMDC materials, WS₂ is chosen because it shows a large oscillator strength of A exciton and high exciton resonance energy (∼1.98 eV). This makes a WS₂ layer promising polariton waveguide in a wide spectral range at visible wavelengths. The thickness of the WS₂ grating structure was designed to support guided exciton-polariton modes in the WS₂ layer. Owing to the periodic modulation of the grating structure, far-field light can couple to the guided exciton-polariton mode. WS₂ material has strong excitonic resonances and a large background permittivity, not only for monolayers but also for multilayers. Therefore, compared with conventional dielectric and semiconductor waveguides such as silicon (Si) and silicon nitride (Si₃N₄), TMDC waveguides can provide very strong mode confinement, as shown in Figure 1B. The thickness limit of the WS₂ layer for the guided exciton-polariton modes is ∼7 nm when the WS₂ layer is situated on a SiO₂ substrate, which is comparable to the thickness of the plasmonic waveguides.

Figure 1:

Characteristics of the one-dimensional WS₂ grating. (A) Schematic structure of the grating. The design parameters are the thickness t of WS₂, period Λ, and the width of WS₂ in one period. (B) Normalized intensity of slab waveguides with different materials (thickness of 100 nm). (C) An example of reflectance spectra with variation of period Λ at θ = 12° (t = 20 nm). (D) Calculated optical dispersion relations for TE-polarized light (t = 20 nm, Λ = 500 nm, F = 0.4). The anti-crossing behavior near exciton resonance (1.97 eV) indicates strong light–matter interaction, and the spatial periodicity in the y-direction contributes to the dispersion folding. (E) Cross-section of simulated TE-polarized electric field profiles of the grating in y–z plane corresponding to different energies. (F) An SEM image of the grating on SiO₂/Si substrate. Inset shows height profile of the WS₂ layer measured by AFM. Scale bar = 500 nm.

To design a grating structure with layered WS₂, we characterized the following parameters: thickness t, period Λ, unit width w, and filling factor F = w/Λ. The guided mode resonance energy and angle are determined by the design parameters t, Λ, and F according to the phase-matching condition. Figure 1C shows the reflectance spectrum for a sample at t = 20 nm with an incident angle of θ _inc = 12°. We calculated the angle-resolved reflection data of grating structures by sweeping the period Λ from 100 nm to 1000 nm. We can see that guided mode resonances exist when Λ is approximately 400–650 nm.

Figure 1D shows an example of the optical dispersion relation of the WS₂ sample where t = 12 nm, Λ = 500 nm, and F = 0.4. A typical feature of guided mode resonance, that is, the folded dispersion curve of the guided mode, was observed. One can also notice anti-crossing behavior near the exciton energy, indicating a strong coupling regime between excitons and photons. The vacuum Rabi splitting energy is approximately 100 meV. The large Rabi splitting energy is due to the monolithic structure of the WS₂ grating, which leads to a large wavefunction overlap between the guided mode resonance and the exciton [23]. Figure 1E shows the TE-polarized electric field distribution in the y–z cross-section. The electric-field profiles confirm that guided mode resonance is formed in a very thin WS₂ grating structure.

Based on the calculation results, we fabricated a grating pattern on multilayer WS₂ using an e-beam lithography technique, followed by reactive ion etching. Patterned WS₂ structures were placed on a SiO₂/Si substrate with a SiO₂ thickness of 280 nm. Figure 1F shows a scanning electron microscope (SEM) image of the fabricated 1D grating structure. The inset in Figure 1F shows the atomic force microscope (AFM) scanning profile of the WS₂ grating. The minimum thickness of the fabricated WS₂ grating structure was 10 nm, which is approximately 50 times smaller than the wavelength of light.

2.2 Reflection measurement for WS₂ grating structure

First, we measured the optical dispersion relation of 1D photonic crystals made of WS₂ using an angle-resolved microscopy setup to analyze the guided mode resonance. A supercontinuum laser was used as the white light source for reflectance measurement. A homemade Fourier plane optical microscopy and spectroscopy setup was used to measure the angle-resolved spectrum [23]. The polarization of incident light along the grating bars is defined as transverse electric (TE) and across the grating bars as transverse magnetic (TM), according to the coordinates described in Figure 1A.

For TE-polarized incident light (Figure 2A, B and E, F), the guided mode resonant feature was clearly seen as an intensity peak in the reflectance spectra, which matches well with the simulation results. The guided mode resonance observed in the reflectance spectra was the result of interference between the reflected light from two distinct pathways: one is directly reflected from the WS₂ layer, and the other is reflected after the guided mode resonance takes place [8, 9]. Owing to the spatial anisotropy of the 1D photonic crystal structure, the optical dispersion relations exhibit propagation direction dependency (i.e., k _x vs. k _y or q _x vs. q _y ). In y-direction, where the translational symmetry is broken, there is a steep folded dispersion of the guided mode in the WS₂ waveguide. In contrast, a parabolic dispersion is visible in x-direction, where translational symmetry holds. Importantly, for both x and y directions, we can observe anti-crossing behavior near the exciton energy of WS₂. However, for TM-polarized incident light (Figure 2C, D and G, H), there is no guided mode resonance because the layer is too thin to support TM-guided modes. The parabolic shape appeared as the intensity deepened in the reflection spectra owing to the Fabry–Perot modes in a finite thickness of the SiO₂ layer (280 nm).

Figure 2:

Angle-resolved reflectance spectra of the one-dimensional WS₂ grating (t = 15 nm, Λ = 500 nm, F = 0.4). The top/bottom row shows the measured (A–D)/simulated results (E–H), respectively. (A, B) and (E, F) The angle-resolved reflectance spectra of the grating for TE-polarized incident light. The spectra are sensitive to direction of measurement due to the spatial anisotropy of the grating. (C, D) and (G, H) The angle resolved reflectance spectra of the grating for TM-polarized incident light.

We investigated the thickness dependence of the optical dispersion relations. WS₂ grating structures were fabricated with thicknesses of 10, 15, and 20 nm, with the period Λ and filling factor F fixed. Figure 3 shows the measured optical dispersion relations for various thicknesses of the WS₂ grating structures. As the sample thickness increased, the guided-mode resonance peaks shifted to lower energies and the widths of the peaks broadened. This was attributed to the stronger confinement of the guided modes in thicker layers. Figure 3D shows the calculated electric field distributions of the guided mode resonance. For thicker samples, the electric fields were highly confined in the WS₂ material, which appeared as a higher effective refractive index of the guided modes. The increased real part of the effective mode index is shown as a redshift of the resonance peak, and the increased imaginary part is shown as broadening of the dispersion (Figure 3A–C). Figure 3E presents the calculated linewidth of the guided mode resonance as a function of thickness at a fixed angle of 10°. The experimental results agree well with the calculation results, which showed a broadening of the linewidth with increasing thickness. We also emphasize that an anti-crossing behavior near the exciton energy (∼1.97 eV) was observed for all thicknesses. The thickness-dependent optical dispersion results suggest broad controllability of the guided mode resonance in the WS₂ grating.

Figure 3:

Angle-resolved reflectance spectra of the one-dimensional WS₂ grating for different thicknesses (Λ = 500 nm, F = 0.4). The top/bottom row of (A–F) shows the measured (A–C)/simulated (D–F) results, respectively. (A–C) Angle-resolved reflectance spectra of the grating for t = 10 nm, 15 nm, and 20 nm, respectively. The redshift of resonance peaks and the broadened linewidth of peaks in the spectra are attributed to the increased effective index of the grating modes with increasing thickness. (G) Cross-section of simulated TE-polarized electric field profiles of the grating in y–z plane corresponding to different thicknesses with an energy of 1.61 eV. (H) Simulated and measured linewidth of resonance peaks for different WS₂ grating thicknesses at θ = 10°.

2.3 Photoluminescence of WS₂ grating structures under non-resonant pumping

We further analyzed the optical dispersion curve of the grating structure under non-resonant pumping conditions. To excite excitons at the direct bandgap of the WS₂ layer, we used a 514 nm continuous-wave (CW) diode laser. The laser was focused on the middle of the grating structure with a pumping spot size of approximately 1 μm, as shown in Figure 4A. The photoluminescence (PL) process in a WS₂ grating structure is different from that of an emitter in an optical grating system. When an emitter emits light in a conventional grating structure, the radiation rate is affected by the presence of the grating structure, which is known as the Purcell effect. Consequently, the radiation pattern of the emitter was modified according to the modified optical density of state with the grating structure.

Figure 4:

Angle-resolved PL spectra with non-resonant pumping. (A) Schematic illustration of the angle-resolved PL measurement with pumping laser (514 nm). (B) Schematic process of the relaxation of exciton-polaritons. Excitons in the exciton reservoir pumped by non-resonant laser pumping subsequently relax to polariton states via scattering processes. (C and D) Measured and simulated angle-resolved PL spectra for θ _y with TE-polarized pumping laser. (E and F) Measured and simulated angle-resolved PL spectra for θ _x with the same pumping laser.

In the case of our WS₂ grating structure, excitons are directly relaxed from the excited states to the polaritonic guided mode resonance states, rather than having a modified radiation process with the Purcell effect. Figure 4B shows a schematic of the PL process of the exciton-polariton modes under non-resonant excitation pumping. Most carriers excited by the pumping laser accumulate at the exciton reservoir. The excitons in the exciton reservoir can then relax into the polaritonic dispersion branch. The polaritons at the guided mode resonance can couple to the far field, which can be observed using a far-field microscope.

Figure 4C shows the angle-resolved PL of the WS₂ grating with a thickness of 10 nm. Bright emission at the guided mode resonance was observed near the WS₂ exciton energy (1.97 eV). Owing to the thermal relaxation process, the emission of the lower polariton branch is brighter than that of the upper polariton branch. In addition, the intensity of the lower polariton branch decreases when the polariton energy is far from the exciton energy. This is attributed to both the large energy difference from the exciton reservoir and the limited density of states of the polariton modes that reduce the relaxation probability for the polariton occupation [21]. Moreover, unlike in the weak coupling regime, the background emission not coupled to the grating modes is not visible. There also exists slight emission in the 1.4–1.6 eV range, which originates from indirect bandgap transition [21]. For the PL dispersion of the TM-polarized mode, we could not see the momentum-dependent dispersion, which corresponds to the result in Figure 2C, D. The polaritonic dispersion measured with PL agrees well with the reflectance measurement. Note that the double parabolic modes observed in the reflection measurement (Figure 2B) were not observed in the PL measurement (Figure 4E). This implies that such double parabolic branches in the reflection data may results from interference between the guided mode resonance and background signals. We also verified that the polaritonic dispersion can be predicted by and agrees well with the simulation by calculating the absorption of the resonance mode of the WS₂ grating (Figure 4D–F).

2.4 Far-field coupling of guided exciton-polariton at a WS₂ grating

Finally, we demonstrated that the patterned WS₂ layer can act as an efficient grating coupler for guided exciton-polaritons. A CW laser with a wavelength of 514 nm was used for non-resonant excitation of the guided exciton-polariton modes in the non-patterned region of WS₂ (Figure 5A). The excitation spot was placed a few micrometers away from the 1D grating coupler. The distance between the grating coupler and pumping spot was gradually increased from 2.4 μm to 12.8 μm. All measurements were conducted under the same excitation power.

Figure 5:

Far-field coupling of guided exciton-polaritons at the WS₂ grating. (A) Schematic side view of the WS₂ grating and pumping laser. (B) An optical microscope image of the grating (t = 15 nm, Λ = 500 nm, F = 0.4). The scale bar is 5 μm. (C–E) Show images of the laser spot and coupled guided mode on CCD camera with different distances between the laser spot and edge of the grating. The distances are 4.8 μm, 7.2 μm, and 9.6 μm, respectively. The right panels show the cross-section of the intensity profile in the y-direction (along with laser centered on the grating). (F) Spectrum of the pumping laser and the guided mode. Top: spectrum at the center of the laser spot. Bottom: spectrum of the guided mode coupling at the edge of the grating for different distances. (G) Intensity of the coupled exciton-polariton (wavelength of 660, 670, and 680 nm) with different distances between the laser center and the edge of the grating. The dashed lines indicate fitting lines for each wavelength.

Figure 5C–E shows the charge-coupled device (CCD) images with pumping spots at 2.4 μm, 6 μm, and 12.8 μm apart from the grating coupler, respectively. Strong emission is visible at the pumping spot, indicating the direct radiation from the exciton. In addition, scattered light was observed at the position of the grating coupler. This scattered light suggests that the guided polariton modes were coupled out from near-field to far-field with assistance from the grating coupler. This far-field scattering of the guided polariton mode was observed only for TE polarization.

To further distinguish between the scattered light from the grating and the direct radiation of the exciton at the pumping spot, we compared the spectrum measured at different positions. The upper graph in Figure 5F shows the spectrum at the pumping spot, with an exciton peak at a wavelength of ∼630 nm. The lower graph shows the distance-dependent spectrum measured at the grating position. Because the polaritons occupied the lower polariton branch, the polariton spectrum was located at a longer wavelength than the exciton spectrum. Moreover, as the distance increased, the spectral intensity decreased with a continuous red-shift of the peak. This is attributed to the wavelength-dependent loss of polariton modes. As noticeable from the linewidth of the dispersion relations of the polariton modes, the polaritonic modes with higher energy show higher optical loss owing to exciton absorption. The red-shift of the spectrum as a function of propagation distance shows direct evidence of the far-field coupling of the guided polariton mode instead of direct excitation of the laser in the grating coupler.

Figure 5G shows the intensity of the spectrum as a function of propagation distance at various wavelengths. We fitted these experimental data with a single exponential function to estimate the propagation distance of the polaritons. The estimated propagation distances for the guided polaritons were 1.7, 2.5, and 3.3 μm at 660, 670, and 680 nm wavelengths, respectively. As demonstrated here, a grating coupler is useful for examining the fundamental properties of guided polariton modes in WS₂ layers. If we collect the scattered guided polariton modes at the end of a bare layer, the collection efficiency is very low because the scattered light loses the momentum information of the guided polariton modes. In contrast, the coupled light from the grating coupler shows the folded polariton dispersion curves. Therefore, a grating coupler is a useful element to investigate the non-radiative polariton modes in the far-field [24, 25].

4.1 Theoretical modeling

The finite-difference frequency domain method developed in MATLAB was used to calculate the electric field profile of the slab waveguides of Si₃N₄, Si, and WS₂. Electric field distributions around grating structures were calculated by Finite-difference time-domain method (FDTD, Lumerical). Simulations for the reflection and PL spectra of the grating was carried out using Rigorous Coupled-Wave Analysis (RCWA) developed in MATLAB. The refractive indices of SiO₂ and Si are set as 1.46 and 3.85, respectively. The in-plane permittivity of WS₂ was obtained from the Lorentz oscillator model as follows:

Here, we used four oscillators: exciton resonance energy E _j = 1.97, 2.4, 2.69, 2.95 eV, oscillator strength f _j = 0.53, 1.38, 3.18, 30 eV², linewidth of the oscillator γ _j = 0.04, 0.18, 0.16, 0.52 eV and background permittivity ε∞ = 12 [21].

4.2 Sample preparation

Multilayer WS₂ flakes were mechanically exfoliated from purchased bulk WS₂ crystals (2D Materials). The grating patterns was fabricated by standard electron-beam lithography (EBL) process followed by reactive ion etching (RIE).

4.3 Optical spectroscopy setup

Optical measurements were performed using a home-built microscope with 50× objective of NA = 0.75. To characterize the optical dispersion relation of patterned multilayer WS₂, angle-resolved spectroscopy was performed using a Fourier imaging setup. For reflection measurements, a white laser (supercontinuum laser, NKT Photonics) with a beam diameter of 15 μm was focused on the patterned multilayer WS₂. For PL measurements, a continuous-wave diode laser with a wavelength of 514 nm with a beam diameter of 2 μm was focused to excite carriers in the sample. The polarization of the excitation laser and the collected light was controlled by half-waveplates. To avoid errors arising from polarization-dependent collection efficiency of the optical setup, a fixed linear polarizer was placed in front of the detection system.