$$\\widehat{H}=\\left(\\begin{array}{cc}v{k}_{x} & J{e}^{i\\phi}\\\\ J{e}^{-i\\phi} & -v{k}_{x}\\end{array}\\right)-i\\gamma \\left(\\begin{array}{cc}1 & 1\\\\ 1 & 1\\end{array}\\right)$$<\/p>\n
\n (1)\n <\/p>\n
Here, we set \u03c9K\u2009=\u20090 without loss of generality. The first term contains the Hermitian diffractive coupling parameter Jei\u03d5 between counter-propagating modes resulting in a gap of size 2\u2009J, as in Fig. 1c<\/a>. The second anti-Hermitian term describes coupling \u03b3 of the modes to the lossy radiative continuum through a Friedrich-Wintgen type process35<\/a> (see Supplementary Note 1<\/a> for more details). Thus, in Fig. 1c<\/a> the linewidths of the modes at k\u2009=\u20090 are 2\u03b3 and 0 for the lossy and BIC modes, respectively.<\/p>\n The mirror symmetry of the grating, taken to be at x\u2009=\u20090 so that u(x) = u(-x), implies that \u03d5 \u2208 {0, \u03c0}, and guarantees the presence of a symmetry protected photonic BIC at the \u0393-point in the antisymmetric energy branch36<\/a>. It is worth noting that the BIC is also topologically tied to a polarization vortex in the far-field37<\/a>. The phase \u03d5 of the diffractive coupling mechanism can be flipped from 0 \u2192 \u03c0 by adjusting the pitch and filling factor of the grating11<\/a>,33<\/a>,34<\/a>. Physically, this phase flip corresponds to inverting the energy hierarchy between the symmetric and antisymmetric standing wave Bloch states.<\/p>\n The eigenvalues of Eq. (1<\/a>) describe the dispersion of the guided photons at low momenta, corresponding to the angle-resolved reflectance simulation (see \u201cMethods\u201d) shown in Fig. 1c<\/a> using the Rigorous Coupled-Wave Analysis (RCWA) technique,<\/p>\n $${\\omega }_{\\pm }\\left({k}_{x}\\right)=-i\\gamma \\pm \\sqrt{{\\left(v{k}_{x}\\right)}^{2}+{J}^{2}-{\\gamma }^{2}-2{iJ}\\gamma \\cos (\\phi )}$$<\/p>\n \n (2)\n <\/p>\n The eigenvectors are derived in Supplementary Note 1<\/a>. Here, \u03c9\u00b1 refer to the upper (+) and lower (-) energy branches (bands) of the grating standing waves. The diffractive coupling J\u2009\u226b\u2009\u03b3 is the main parameter responsible for the bandgap opening at kx\u2009=\u20090. When \u03d5\u2009=\u20090, a BIC mode of an infinite lifetime appears in the center of the lower \u03c9- antisymmetric branch while a lossy 2\u03b3 mode appears in the upper symmetric \u03c9+ branch (see Fig. 1c<\/a>). If \u03d5\u2009=\u2009\u03c0, equivalent to J\u2009\u2192\u2009-J, band inversion takes place and the antisymmetric state containing the BIC is now higher in energy while the symmetric state is lower. The presence of the BIC at kx\u2009=\u20090 can be appreciated from,<\/p>\n $${\\rm{Im}}\\left[{\\omega }_{\\pm }(0)\\right]\\approx -\\gamma [1\\pm \\cos (\\phi )]$$<\/p>\n \n (3)\n <\/p>\n Theoretical model for photonic Jackiw-Rebbi interface state<\/p>\n Our photonic guided mode Hamiltonian Eq. (1<\/a>) can be transformed into a spinless one-dimensional non-Hermitian Dirac equation with a mass term m through the unitary transformation \\(\\widehat{U}=\\left({\\widehat{\\sigma }}_{x}+{\\widehat{\\sigma }}_{z}\\right)\/\\sqrt{2}\\),<\/p>\n $${\\widehat{U}}^{\\dagger }\\widehat{H}\\widehat{U}={\\widehat{H}}_{{\\rm{Dir}}}=c{\\widehat{\\sigma }}_{x}{\\widehat{p}}_{x}-{\\widehat{\\sigma }}_{z}m{c}^{2}-i\\gamma$$<\/p>\n \n (4)\n <\/p>\n Here, px = \u210fkx, c\u2009=\u2009v and m\u2009=\u2009\u210f(J cos(\u03d5) – i\u03b3)\/v2. It has been known for some time in relativistic quantum field theory that the 1D Dirac equation hosts a topologically protected localized state with fractional particle numbers known as a Jackiw-Rebbi mid-gap state9<\/a>,38<\/a>. Formally, the JR state appears spatially at the interface of two Dirac systems with opposite mass signs m(x) = m0(2H(x)-1) where H(x) is the Heaviside function, and the state wavefunction is written<\/p>\n $$\\left|{\\psi }_{{\\rm{JR}}}\\right\\rangle =\\sqrt{\\frac{c{m}_{0}}{2{\\rm{\\hbar }}}}{e}^{-{c|m}(x)\\cdot {x|}\/\\hbar }\\left({1}\\atop{i}\\right)$$<\/p>\n \n (5)\n <\/p>\n The JR state is energetically positioned exactly in the center of the gap of the Dirac Hamiltonian. Its topological origin can be appreciated from the fact that no energy mismatch is imposed on the structure. That is, flipping the sign of m does not alter the eigenvalues in Eq. (4<\/a>), precluding the presence of trivial bound states. A photonic JR state can be constructed if two gratings that differ by \u03c0 in their diffractive coupling phase \u03d5 are put together11<\/a>. The interface between the two gratings corresponds then to a jump J\u2009\u2192\u2009-J which can be expressed in terms of the complex mass,<\/p>\n $$m(x)=\\frac{\\hbar }{{v}^{2}}[J(2H(x)-1)-i\\gamma ]$$<\/p>\n \n (6)\n <\/p>\n The topological origin of the JR state can be traced back to its condensed matter analog in the electron bands of 1D chains of conjugated polymers, the SSH model10<\/a>,38<\/a>. Therein, the topological invariant is known as the quantized Zak phase \\({\\mathcal{Z}}=i{\\int }_{\\mathrm{BZ}}\\,\\left\\langle {\\psi }_{\\pm }\\right|{\\partial }_{k}\\left|{\\psi }_{\\pm }\\right\\rangle {dk}\\in \\{0,\\pi \\}\\) 8<\/a> where |\u03c8\u00b1\u3009 are the symmetric and antisymmetric Bloch modes. The integral is over the Brillouin zone and \u3008\u03c8\u00b1|\u2202k\u2009|\u2009\u03c8\u00b1\u3009 is integrated over the real space unit cell. In the photonic grating, \u03d5\u2009=\u20090 \u2192 \u03c0 corresponds to \\({\\mathcal{Z}}\\)\u2009=\u20090 \u2192 \u03c0 implying that the band inversion represents a topological phase transition which invokes the bulk-boundary correspondence with consequent presence of a midgap JR state. The existence of the JR state and the associated Zak phase mismatch can also be linked to the surface impedance condition ZL\u2009+\u2009ZR\u2009=\u20090, where ZL,R are the surface impedances of the left and right grating at the interface39<\/a>. Moreover, the band inversion can also be linked to a different type of a topological invariant in the far-field, given by the half-integer charge of an optical Skyrmion number40<\/a>.<\/p>\n Design and fabrication of inverted double-grating structure<\/p>\n Here we explain the rationale behind the unique design of our grating structures and detail the subsequent fabrication process. As per our analytical model presented above, to obtain photonic band inversion and thus a Jackiw-Rebbi interface state, one needs to tune the grating filling factor to close the photonic gap and reopen it. However, the high refractive index of a WS2 grating leads to a band gap so large, that no amount of tuning of the filling factor can close the gap (see Supplementary Note 2<\/a>). For this reason, we employ an additional WS2 flake on top of our gratings to controllably reduce the effective refractive index contrast between the etched and unetched sections of the gratings, thus reducing the photonic band gap. This design also leverages on the intrinsic vdW adhesive forces of such materials, allowing simple transfer of the top bulk WS2 flake (i.e. slab) onto any prefabricated nanostructures and devices as illustrated schematically in Fig. 2a<\/a>. Here, a gold substrate was chosen owing to its strong reflectivity, thus yielding high contrast with the optical grating modes in reflectance measurements. In addition, gold further illustrates the range of substrates compatible with vdW materials, whilst also acting as a natural etch stop to produce as-designed structures. We note that plasmonic effects are not expected owing to the s-polarized incident light used to excite only TE grating modes.<\/p>\n Fig. 2: Simulated angle-resolved reflectance contrast of WS2 single and double inverted grating structures on gold.<\/b> a<\/b> Schematic of the single inverted grating structure. a, w, tgr, and tslab correspond to the grating period, width of WS2 beams, and thicknesses of the grating and slab respectively. \u03f5WS2 and \u03f5vac correspond to the WS2 and vacuum permittivities respectively. Blue double arrow denotes incident polarization direction E<\/b>i parallel to the grooves of the grating, corresponding to TE excitation. b<\/b> Schematic of the double inverted grating structure with respective periods and widths a1,2 and w1,2. Interface between the two gratings denoted by the black dashed line. c<\/b> Simulated reflectance contrast of single WS2 grating on gold with increasing thicknesses of tslab from left to right leading to reduction of the band gap. Total thickness of the structure is kept constant for each. d<\/b> Simulated tuning of the grating filling factor to achieve photonic band inversion. The period is also slightly adjusted to shift each mid-gap position to the same energy. The three panels show reflectance contrast for single inverted gratings. Left panel shows BIC on the lower energy branch for a high FF. Center panel exhibits two exceptional points (EPs)41<\/a> where the modes cross for FF\u2009=\u20090.5. Right panel shows BIC flipped to the upper energy branch for low FF. e<\/b> Simulated reflectance contrast of a double inverted grating structure combining the high and low FF gratings as in (b<\/b>). A topologically protected JR interface state is observed at the mid-gap energy as a result<\/p>\n The effects of adding a top WS2 slab to the gratings were simulated using the RCWA technique (see \u201cMethods\u201d) as shown in Fig. 2c<\/a>. The electric field component of the incident light was fixed along the direction parallel to the grooves (i.e. along y in Fig. 2a<\/a>), corresponding to TE polarization. Here, the angle-resolved reflectance contrast with respect to the wavevector component kx from the inverted grating structure is plotted while changing top slab thickness tslab and keeping the total structure thickness constant. A clear reduction of the band gap is observed with increasing tslab, corresponding to a decrease in the effective refractive index contrast between the repeating high and low index sections of the structure.<\/p>\n To initiate photonic band inversion, we then simulated a grating with a top slab with parameters tgr\u2009=\u200935\u2009nm and tslab\u2009=\u200945 and varied the FF as in Fig. 2d<\/a>. The FFs of the gratings were chosen to obtain two different topological regimes through a band inversion mechanism. The left panel shows a grating structure with a BIC on the lower energy branch. By continuously decreasing the filling factor, the gap is reduced until it closes and forms two exceptional points (EPs)41<\/a>, non-Hermitian equivalent of a Dirac point, as seen in the middle panel. In the context of our 2\u2009\u00d7\u20092 model given in Eq. (1<\/a>), these points appear when J\u2009=\u20090. By further decreasing the filling factor, the gap reopens, as seen in the right panel with the BIC on the upper energy branch, leading to the change of the topological nature of the structure. Here, the grating period was also tuned to compensate for the redshift induced by the change of the filling factor, thus ensuring that the mid-gap energy was the same for each structure. Using this inverted grating design, we show that photonic band inversion via closing and re-opening of the band gap is possible. To obtain the JR interface state, we then placed the high and low filling factor gratings from the left and right panels of Fig. 2d<\/a> adjacent to one another, as illustrated by the schematic in Fig. 2b<\/a>. Via further reflectance contrast simulations, this time using a Finite-Difference Time-Domain (FDTD) solver (see \u201cMethods\u201d), a clear state within the photonic band gap was realized as in Fig. 2e<\/a>, which we attribute to a topologically protected JR interface state.<\/p>\n To realize such inverted double-grating structures in experiment, we began by mechanically exfoliating WS2 flakes of a range of thickness (\u223c10\u2013500\u2009nm) onto electron-beam evaporated gold on silicon wafers (see \u201cMethods\u201d). Subsequent electron-beam lithography of the substrates coated with a positive resist produced a pattern of alternating high and low filling factor grating pairs. The grating period and electron beam dosage were varied across multiple pairs to account for different flake thicknesses and etching rates. Reactive ion etching using a partially chemical (SF6) and physical (CHF3) etch recipe produced grating structures etched to the gold with parallel beams of WS2. The interface between high and low filling factor gratings were imaged via scanning electron microscopy (SEM) and atomic force microscopy (AFM) as in Fig. 3a, b<\/a>, respectively, for characterization of the fabricated period, filling factor, and thickness.<\/p>\n Fig. 3: WS2 double grating structure fabrication and experimental Fourier setup.<\/b> a<\/b> SEM image of example double grating interface before top slab transfer. Dashed light blue line corresponds to the interface between high and low FF gratings at the top and bottom of the image respectively. b<\/b> AFM image of double grating interface before top slab transfer. Measured grating thickness 47\u2009nm. c<\/b> Dark field microscope image of WS2 inverted double grating structure with parameters tgr\u2009=\u200947\u2009nm, tslab\u2009=\u200941\u2009nm, a1,2\u2009=\u2009279, 319\u2009nm, F1,2\u2009=\u20090.81, 0.41. High and low filling factor inverted gratings denoted by green and orange dotted boxes respectively. d<\/b> Schematic of Fourier spectroscopy setup with angle-resolved reflectance and PL capability. All optics on flip mounts denoted by curved arrows. Sample sits on the x, y stage beneath the beam splitter (BS) and objective (denoted by black dashed circle) with numerical aperture (NA) 0.7. For reflectance measurements the multi-mode (MM) fiber is coupled to a white light source with long-pass filter removed. Lens L1 ensures uniform K\u00f6hler illumination of the sample, whilst L2 images the back focal plane (i.e. Fourier plane) of the objective. Variable aperture (VA) allows selection of signal region from the real-space plane. L3 and L4 act as a beam expander for the Fourier spot. L5 and L6 focus the Fourier plane onto the sample camera and spectrometer slit respectively. For PL the single-mode (SM) fiber is coupled to a 637\u2009nm laser with both short- and long-pass filters in the optical path and L1 removed<\/p>\n We note that with such thin layers of material used (\u2009<\u200950\u2009nm), partial etching down to nanometer resolution required to achieve band inversion at the same mid-gap energy would be challenging to realize experimentally. A way around this issue is through our inverted grating design, with both the grating and unetched top flake thicknesses chosen with nanometer precision, thus allowing reliable tuning of the photonic modes.<\/p>\n The transfer step involved first a calibration via simulating the reflectance contrast spectra of the gratings with the measured parameters upon varying top slab thickness. This process enabled accurate determination of the slab thickness required to achieve two topologically distinct adjacent gratings, and thus a JR interface state. WS2 was then exfoliated onto PMMA spin-coated on silicon covered with a PVA sacrificial layer. A flake of the required thickness was selected with the help of AFM and transferred onto the gratings, covering the interface (see details in \u201cMethods\u201d). The resulting inverted double-grating structures on gold were imaged via dark field microscopy as shown in Fig. 3c<\/a>. Here, the top slab was of thickness tslab\u2009=\u200941\u2009nm and the grating had tgr\u2009=\u200947\u2009nm. The darker blue rectangles of the etched bottom flake correspond to the high filling factor gratings with a period a1\u2009=\u2009279\u2009nm and a filling factor F1\u2009=\u20090.81, and the lighter blue rectangles correspond to the lower filling factors gratings with a2\u2009=\u2009319\u2009nm and F2\u2009=\u20090.41. The top WS2 slab was positioned to cover at least one of each type of grating, as well as the interface between them for measurement of all three topological regimes. The covered high and low filling factor grating regions (i.e. inverted gratings) of interest are denoted by the green and orange dotted boxes in Fig. 3c<\/a>, respectively.<\/p>\n Far-field measurements of Jackiw-Rebbi interface state<\/p>\n Angle-resolved reflectivity contrast measurements of the structure were obtained using a Fourier spectroscopy setup as depicted schematically in Fig. 3d<\/a>. A white light illumination source was used with TE configuration to compare to the simulated spectra (more details in \u201cMethods\u201d). The reflectivity contrast maps in the kx direction (for ky\u2009=\u20090) of the low FF grating region, interface, and high FF grating region are presented in Fig. 4a<\/a>. In agreement with our previous simulations from Fig. 2d<\/a>, one can observe a BIC on the high energy branch of the spectra from the left panel of Fig. 4a<\/a> corresponding to the low FF grating, and a (quasi-) BIC flipped to the low energy branch for the high FF grating in the central panel of Fig. 4a<\/a>. Here we observe a quasi-BIC at kx\u2009=\u20090 as the top WS2 slab did not fully cover the entire high FF grating (green dotted box in Fig. 3c<\/a>), thus leading to a finite linewidth and obscured reflectance at negative kx. Additional experimental data from a separate structure are presented in Supplementary Note 3<\/a> exhibiting true BICs for both the high and low FF gratings, whilst also highlighting the repeatability of our transfer fabrication method, and tunability of the bands in energy. By measuring at the interface region shown in the right panel of Fig. 4a<\/a>, there is a clear feature at the mid-gap energy, not present in the two gratings alone, corresponding to a JR state. These experimental results therefore not only confirm the successful photonic band inversion via tuning of the filling factor, but also the different topological phases of such inverted band structures, resulting in the formation of an interface-localized Jackiw-Rebbi state within the band gap.<\/p>\n Fig. 4: Experimental reflectance contrast characterization of the JR interface state.<\/b> a<\/b> Angle-resolved reflectance contrast measurements of the low FF grating (a2\u2009=\u2009330\u2009nm, FF2\u2009=\u20090.3), high FF grating (a1\u2009=\u2009261\u2009nm, FF1\u2009=\u20090.7), and whole structure including the interface respectively, in the kx direction at ky\u2009=\u20090\u2009\u00b5m\u20131. Wavevector normalized to the respective grating periods a, where the average between the high and low FF gratings is taken for the interface region (right panel). b<\/b> Angle-resolved reflectance contrast of the grating interface region with respect to the ky direction for three different values of kx, as denoted by the blue dashed lines in the right panel of (a<\/b>). c<\/b> Three-dimensional tomographic reconstruction of the grating and interface state modes in momentum space<\/p>\n We further considered the reflectance contrast along the ky direction by rotating the sample whilst keeping the TE excitation configuration as in Fig. 4b<\/a>. Three measurements were taken at the interface region for three different values of kx as denoted by the dashed blue lines in the right panel of Fig. 4a<\/a>. For all three sets of reflectance contrast spectra, we observe that the dispersion of the two lossy grating modes is parabolic with positive curvature. As kx is reduced to zero, the JR state emerges and follows the parabolic dispersions of the two lossy modes, remaining at the mid-gap energy with positive curvature in the ky direction. Via fitting to a Lorentzian function, we extract a linewidth of 10\u2009meV, and further calculate that the JR state yields an angular emission bandwidth in the direction perpendicular to the grooves of just \u2206\u03b8x\u2009=\u20098\u00b0, yet can be in- and out-coupled throughout the full measurement region in the direction parallel to the grooves with \u2206\u03b8y\u2009=\u200988\u00b0 (\u00b144\u00b0, the maximum angle range for an 0.7NA objective). These results therefore highlight the highly confined nature of the JR interface state in TMD-based inverted double-gratings, with strong localization in both energy and momentum space.<\/p>\n We subsequently plot the full 3D reconstruction of the experimental grating mode structure in kx,y in Fig. 4c<\/a>. One can observe that the lower energy mode has a saddle shape, with negative curvature in kx but positive curvature in ky. In contrast, the higher energy mode displays a 3D parabolic shape with positive curvature in both directions, but greater curvature in kx. In addition, the JR state is a parabolic stripe along ky, with a width of 1.1\u2009\u00b5m\u20131 in kx. As well as being able to visualize the full dispersion of all modes in momentum space, this three-dimensional reconstruction also highlights the relative linewidths of each mode in energy. We expect the BICs at the band edges and JR state to be most strongly confined to the structure for kx\u2009=\u20090, owing to the symmetry conditions required to host such states42<\/a>,43<\/a>. This can be observed experimentally via the narrower linewidths of the modes along the line kx\u2009=\u20090, but also shows that even at large angles in the direction parallel to the grooves up to \u00b144\u00b0 from the normal, the linewidths of the resulting modes change negligibly. Such 1D double grating structures therefore possess highly selective directivity profiles depending on the plane of incidence.<\/p>\n Near-field measurements of Jackiw-Rebbi interface state<\/p>\n From our RCWA simulations and experimental angle-resolved reflectivity measurements, the existence of a topological photonic JR interface state in WS2 double inverted grating structures is clearly demonstrated in the far-field. We now investigate the near-field localization of this state through both simulation, and experimental probing of the local electric field distribution via s-SNOM.<\/p>\n Figure 5a<\/a> depicts the simulated electric field confinement in a cross-sectional slice through the middle of the double gratings as calculated via FDTD. To emulate an s-SNOM measurement, the incident excitation was a p-polarized plane wave traveling at 60\u00b0 to the substrate normal, in the direction parallel to the grooves (see \u201cMethods\u201d). It is clear that the JR state is strongly localized at the interface between the two topologically distinct gratings, with an up to 50 times enhancement of the incident wave intensity. In addition, we observe that confinement is highest within the top WS2 slab rather than the grating itself, which we attribute to the higher overall effective refractive index of the slab portion. The field also protrudes significantly out of the top of the structure, enabling direct probing via a nanoscale tip.<\/p>\n Fig. 5: Near-field study of the JR interface state.<\/b> a<\/b> FDTD simulation of the near-field distribution of the JR state for 60\u00b0 incident illumination angle. b<\/b> Profiles of the simulated electric field strength along the top surface of the structure for changing incident wavelength. c<\/b> Schematic of the s-SNOM setup from side (top panel) and top-down (bottom panel) views. Incident light depicted by the pink arrow propagating at 60\u00b0 to the tip axis. Blue double arrow denotes polarization direction (p-polarized) with component along the tip axis. The grating is oriented parallel to the propagation direction of the incident light allowing to excite the edge mode along the ky direction. d<\/b> Experimental normalized near-field scattering amplitude image at 736\u2009nm excitation of the WS2 double grating structure. Light blue arrow highlights interface between low and high FF gratings where near-field enhancement is observed. Dashed white light corresponds to edge of the top WS2 slab. e<\/b> Averaged near-field scattering amplitude profiles over multiple scanning paths displaced in the y direction for varying incident wavelength. Grating interface centered at x\u2009=\u20090 showing enhancement only around 736\u2009nm illumination<\/p>\n We then simulated a range of illumination wavelengths, and plot profiles of the electric field strength along the top surface of the structure, i.e. where we are able to probe with an s-SNOM tip, as in Fig. 5b<\/a>. These simulations show that the JR state peaks in electric field confinement at around 737\u2009nm (1.68\u2009eV). We also note that at lower wavelengths, the overall field strength is higher to the right of the grating interface (x\u2009=\u20090) which corresponds to the lossy mode of the high FF grating, whereas the fields are weaker to the left of the interface corresponding to the BIC. This behavior flips as we tune through the JR state to higher wavelengths owing to the band inversion between the high and low filling factor gratings.<\/p>\n To successfully probe the JR interface state via experimental s-SNOM measurements, the sample had to be precisely orientated owing to its strong directivity. With the high incident illumination angle to the substrate normal (see s-SNOM \u201cMethods\u201d), excitation of the JR state was only possible via light propagating in the plane of incidence parallel to the grooves of the grating, as depicted by the pink arrow in Fig. 5c<\/a>. This illumination angle corresponds to zero wavevector in the kx direction, with mode dispersion along ky given by the central panel of Fig. 4b<\/a>. We further extrapolated the JR state dispersion in ky up to the incident angle of 60\u00b0 (see Supplementary Note 4<\/a>), yielding an estimated excitation energy of 1.68\u2009eV (735\u2013738\u2009nm), which agrees perfectly with our simulations from Fig. 5b<\/a>. Importantly, this excitation regime means a component of the incident electric field lies parallel to the grooves as shown by the blue double arrow in Fig. 5c<\/a>, which is required to excite the TE-polarized grating modes and thus the JR interface state.<\/p>\n Following careful sample alignment, we performed s-SNOM measurement at 736\u2009nm excitation as in Fig. 5d<\/a> and observed enhanced scattering from the interface between the two gratings as marked by the light blue arrow. The data shown have been median line levelled separately either side of the grating interface to minimize the intrinsic material response, which differs depending on the effective refractive index experienced by the tip. We thus focus on scattering arising mostly from probing the localized electric fields associated with the grating modes, as described in more detail in Supplementary Note 5<\/a>.<\/p>\n We then repeated the scan for a range of wavelengths and plotted line profiles of the scattering intensity averaged over multiple rows in the y direction as in Fig. 5e<\/a>, where the grating interface and thus JR state is centered at x\u2009=\u20090. We observe that the peak in scattering at 736\u2009nm along the dashed line quickly decays when changing the incident wavelength, which correlates very well with our electric field profile simulations. The localization of the enhanced scattering exactly where we expect it spatially, also at the correct energy we predicted from our far-field measurements for light incident at 60\u00b0, strongly suggests that we have successfully probed the near-field response of this topological photonic JR interface state, confirming its highly localized nature in real space, k-space, and energy.<\/p>\n Photoluminescence enhancement at the Jackiw-Rebbi state<\/p>\n After demonstrating the near-field localization of the JR interface state in both simulation and experiment, we now study the PL enhancement and directivity of an emitter coupled to such a state. To do so, we consider a similar double grating structure shown schematically in Fig. 6a<\/a>, in which an hBN-encapsulated WSe2 monolayer is embedded between the grating layer and the top WS2 slab. The thin hBN layers prevent charge transfer between the WSe2 monolayer and bulk WS2 with negligible effect on the grating mode structure, thus enabling bright PL emission. A combination of PDMS and PMMA-based transfer techniques was used to build this structure as explained in more detail in \u201cMethods\u201d. Figure 6b<\/a> presents an optical microscopy image of the fabricated double-grating heterostructure, in which the top WS2 flake and encapsulated WSe2 monolayer are indicated by the blue and yellow dashed lines respectively. Here we used a top flake of thickness tslab\u2009=\u200932\u2009nm with a grating tgr\u2009=\u200969\u2009nm. The top and bottom hBN layers were of 3 and 6\u2009nm thicknesses, respectively. To confirm the presence of a JR interface state in this more complex heterostructure, we performed angle-resolved reflectance contrast measurements as with the previous sample. Figure 6c<\/a> shows results from the grating interface, taken from the white dashed rectangular region of Fig. 6b<\/a> using the variable aperture as detailed in the \u201cMethods\u201d section. A clear JR state is once again visible within the photonic band gap, shifted slightly to lower energy likely owing to coupling with the WSe2 exciton at 1.65\u2009eV.<\/p>\n Fig. 6: Photoluminescence enhancement via active grating heterostructure.<\/b> a<\/b> Schematic of the active double grating heterostructure composed of hBN-encapsulated WSe2 between a WS2 double grating and top slab on a gold substrate. b<\/b> Optical microscope image of the fabricated heterostructure. Top WS2 slab and monolayer WSe2 outlined by dashed blue and yellow lines respectively. White dashed box corresponds to Fourier RC measurement region. Colored circles correspond to the Fourier PL excitation spots. Light blue triangle denotes the Fourier PL reference measurement location on encapsulated monolayer WSe2 away from the gratings. c<\/b> Angle-resolved RC taken from the white dashed region in (b<\/b>) around the double grating interface showing JR state within the photonic band gap. d<\/b> Angle-resolved photoluminescence measurement of the low FF grating, high FF grating and the interface between the two corresponding to the orange, green, and white circles from (b<\/b>) respectively. e<\/b> Photoluminescence directional enhancement factor along the direction perpendicular to the grooves compared to uncoupled monolayer. Signal integrated over kx and energy as shown by the dashed white lines in (d<\/b>)<\/p>\n PL measurements were performed at room temperature using the Fourier setup from Fig. 3d<\/a>, with the introduction of a 637\u2009nm laser for excitation. The measurements were taken from three different locations indicated by the circles in Fig. 6b<\/a>, corresponding to the two separate low and high filling factor gratings, and the grating interface. Clear coupling of the WSe2 exciton PL to the lossy grating mode can be seen in the left panel of Fig. 6d<\/a> for the low FF grating, where enhanced signal follows the dispersion of the modes as measured via angle-resolved reflectance contrast in Fig. 6c<\/a>. In the center panel, a dip in PL can be observed around kx\u2009=\u20090 at \u223c1.6\u2009eV, corresponding to emission unable to outcouple to the far-field from the BIC within the high FF grating. Some enhancement is also observed around 1.64\u2009eV, however this corresponds to leaked signal from the JR state owing to the close spatial proximity of the measurement regions, and the finite extent of the JR state electric field profile. Finally, the right panel of Fig. 6d<\/a> shows the strongest PL enhancement from the grating interface region at the JR state energy. The PL is enhanced over the same range of kx as the JR state in reflectance contrast, therefore indicating coupling of the WSe2 excitonic emission to the state.<\/p>\n To quantify the directional PL enhancement from monolayer WSe2 coupled to the JR state, the PL intensity was integrated over the region depicted by the dashed white lines in the right panel of Fig. 6d<\/a>. The size of this integration region is equal to the fitted linewidth of the JR state-coupled PL peak in both kx and energy. This integration was repeated for PL dispersions measured at multiple x positions (vertical direction in Fig. 6b<\/a>) and then divided by the integrated counts from a PL measurement taken at a reference position as marked by the light blue triangle in Fig. 6b<\/a>. In this region, the bottom WS2 remains unetched, resulting in an hBN-encapsulated monolayer WSe2 embedded between two WS2 slabs (as detailed further in Supplementary Note 6<\/a>), which are transparent at the WSe2 emission energy of 1.65\u2009eV. The reference PL emission was therefore not coupled to any of the grating or interface modes, with no dispersion in k-space (see Supplementary Note 6<\/a>). As such, the directional enhancement factor corresponds to the ratio of PL from the grating heterostructure between 753-765\u2009nm wavelength emitted into a wavevector range of \u00b11.1\u2009\u00b5m-1 (angular bandwidth of \u00b17.5\u00b0), to that of the PL from the reference region over the same wavelength and wavevector range. Note that the wider angular bandwidth for the active grating heterostructure compared to the passive device presented in Fig. 4a<\/a> is likely owing to losses induced by the monolayer WSe2, and the presence of hBN which reduces the overall effective refractive index and thus broadens the modes. The directional enhancement results are plotted in Fig. 6e<\/a> and give an indication of how much additional emission is directed upwards of the sample plane via JR state coupling compared to the omnidirectional emission from uncoupled monolayer WSe2. We calculate a directional enhancement factor of between 11 and 13 for the monolayer coupled to the separate grating modes, and up to 22 for the JR state-coupled emission at the grating interface.<\/p>\n","protected":false},"excerpt":{"rendered":"Theoretical model for guided modes of photonic gratings Assuming continuous translational symmetry in the lateral y direction, the…\n","protected":false},"author":3,"featured_media":948501,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_share_on_mastodon":"0"},"categories":[25],"tags":[41753,34582,90806,16251,26984,39057,34154,37988,44421,45590,492,90807,159,67,132,68],"class_list":["post-948500","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics","tag-imaging-and-sensing","tag-lasers","tag-microwaves","tag-nanophotonics-and-plasmonics","tag-optical-and-electronic-materials","tag-optical-devices","tag-optics","tag-photonic-crystals","tag-photonic-devices","tag-photonics","tag-physics","tag-rf-and-optical-engineering","tag-science","tag-united-states","tag-unitedstates","tag-us"],"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/116955026222846080","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/948500","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=948500"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/948500\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/948501"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=948500"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=948500"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=948500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"Fig.](\"https:\/\/www.europesays.com\/us\/wp-content\/uploads\/2026\/07\/41377_2026_2392_Fig2_HTML.png\")
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