Abstract
The era of Big Data requires nanophotonic chips to have large information processing capacity. Multiple frequency onchip nanophotonic devices are highly desirable for density integration, but such devices are more susceptible to structural imperfection because of their nanoscale. Topological photonics provides a robust platform for nextgeneration nanophotonic chips. Here we give an experimental report of an onchip nanophotonic topological rainbow realized by employing a translational deformation freedom as a synthetic dimension. The topological rainbow can separate, slow, and trap topological photonic states of different frequencies into different positions. A homemade scattering scanning nearfield optical microscope with high resolution is introduced to directly measure the topological rainbow effect of the siliconbased photonic chip. The topological rainbow based on synthetic dimension have no restrictions for optical lattice types, symmetries, materials, wavelength band, and is easy for onchip integration. This work builds a bridge between silicon chip technologies and topological photonics.
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Introduction
In the past few years, topological photonics have made great progress from physical concept to new applications^{1,2,3,4,5,6}. Photonic devices become more robust against disorder and immune to scattering due to topological protection^{7,8,9}. However, topological nanophotonic devices^{9,10,11,12,13,14,15,16} are difficult to realize because of complexity of fabrication, challenges of measurement in the nanoscale, and the fact that magnetic response is inherently weak for natural materials in the visible and near infrared range. The introduction of synthetic dimensions has brought opportunities and insight into topological photonics where the topology goes beyond the geometric dimensions^{17,18,19,20,21,22,23,24,25,26}. Synthetic dimension provides a new degree of freedom which enables us to construct all dielectric onchip topological nanophotonic components, breaking the restrictions of magnetic materials^{27}. In topological photonics research, much attention has been paid to engineer the bulk band dispersion to realize topological states in order to achieve unidirectional propagation, topological laser, highorder topology, etc.^{7,28,29,30}. Multifrequency (or multiwavelength) devices are essential components of nanophotonic chips for large information capacity applications, among which, topological rainbow, as a basic multiwavelength topological photonic device, can separate and distribute different wavelengths of topological photonic states into different positions; but related topological devices have not yet been fully explored. Moreover, no effective method has been found to give direct measurement of topological photonic devices with multiple wavelengths at the nanoscale to date. These challenges have restricted the development and applications of topological rainbow and various topological nanophotonic devices, including topological router, topological temporary storage, and many other onchip integrated topological nanophotonic devices.
In this work, we have constructed a topological rainbow device based on a synthetic dimension, which provides a method that is universally applicable for all optical lattice types, symmetries, materials, dimensions, and wavelength range. Topological rainbow can separate and distribute different wavelengths of topological photonic states into different positions, and the light can be slowed and trapped through controlling the group velocities of the topological photonic states^{31,32,33}. Here, the topological photonic states are realized through constructing topological “Chern insulators” without the need of a magnetic field. The onchip siliconbased topological photonic crystal (PC) was successfully fabricated with a small size of only 4.5 μm × 22 μm. A homemade scattering scanning nearfield optical microscope (sSNOM) system with high enough resolution was set up to directly measure the electric field of the topological rainbow. Topological states of different wavelengths from 1540 to 1630 nm around the optical communication range have been separated, slowed, and trapped into different positions along the interface of PCs, which forms a nanoscale topological rainbow on the silicon chip. The group velocities can be decreased to zero in different interface positions of PCs, where the slowlight effect of the topological states is realized. The topological nanophotonic device based on a synthetic dimension is robust against disorder and can be implemented with purely dielectric materials and is hence compatible with photonic chip integration.
Results
Construction and design of the topological rainbow
In this section, the basic ideas about the construction of the topological rainbow are given in detail. The geometric structure of the topological rainbow is shown in Fig. 1a, which consists of three regions. The blue region (Region II) is the dispersing region, which separates and distributes different wavelengths of topological photonic states into different positions (and hence called “rainbow”) due to the nontrivial topology in synthetic dimension. The red regions (Regions I and III) are barrier regions, which provide a bandgap to prevent the leakage of light. The purpose of adding blue or red color is to display different regions visually. The barrier regions consist of ordinary PCs with a complete bandgap, and the lattice vectors are denoted by a_{1}, a_{2}. The dispersing region is constructed by an undeformed PC on one side of the yaxis and a graded translationally deformed PC on the other, where the vector of the displacement is along lattice vector a_{2}. The ratio between the displacement of the ith layer and the lattice vector is defined as the translational parameter ξ_{i}, as is shown in the inset of Fig. 1a. The parameters N_{a}, N_{ξ}, N_{b} denote the number of layers of regions I, II, III along a_{1} direction, and N_{u}, N_{d} denotes the number of layers of undeformed region and the deformed region in region II, respectively. External light signal is incident from a waveguide, and the frequency range lies within the bulk bandgap of the PC.
The functionality of the dispersing region results from the nontrivial topology in the lattice translational deformation dimension. We find two ways to achieve nontrivial topology using a synthetic dimension in PCs, one is based directly translational PC, and the other is based on twisted PC (see Supplementary Fig. 1). The details are illustrated in the Supplementary Materials Sec. 1. We take the first way of constructing synthetic dimension for the following experimental verification. Here we consider a uniformly deformed PC along a_{2} direction, and the ratio between the displacement and the lattice constant is denoted by ξ. The Bloch wavevector is decomposed by using the basis of reciprocal lattice vectors k = k_{1}b_{1} + k_{2}b_{2}, where b_{1} and b_{2} satisfy \({{\mathbf{a}}}_{i}\cdot{{\mathbf{b}}}_{j}=2\pi\delta_{ij}\) . Here, the Bloch states and the wavevectors are defined either in the deformed or in the undeformed domains alone, where the periodicity is preserved so that the wavevectors are well defined. The parameters (k_{1}, k_{2}, ξ) form a threedimensional synthetic space. With the value of k_{1} fixed, the Zak phase^{34} of the k_{2} dimension is defined as Eq. 1,
where \({\theta }_{n}^{({{{{{\rm{Zak}}}}}})}({k}_{1},\xi )\) denotes the Zak phase corresponding to parameter (k_{1}, ξ) and band n. As the displacement parameter ξ goes from −1/2 to 1/2, the PC goes back to itself and the Zak phase evolution along the ξ path can be used to define a Chern number. In the threedimensional synthetic space, when k_{1} is fixed, and the twodimensional subspace (k_{2}, ξ) chosen, the Chern number is related to the evolution of Zak phase when ξ changes by a period, as is shown by Eq. 2^{35},
where C_{n}(k_{1}) is the Chern number of the band n and parameter k_{1}. Equation 2 means that the Chern number of the subspace (k_{2}, ξ) specified by a fixed value of k_{1} equals the winding number of the Zak phase when ξ changes by a period. The Wannier center will change when ξ changes by a period, so that the Zak phase changes by 2π, as is shown in Fig. 1b, and the Chern number of the 2D synthetic subspace (k_{2}, ξ) is 1. For all values of k_{1}, the results are the same. More detailed discussion of the nontrivial topology of synthetic dimension is shown in the Supplementary Materials Sec. 1.
According to the bulkedge correspondence^{36,37}, the net number of interface states going in one direction equals the difference of the Chern numbers of systems on either side of the interface. For synthetic dimensions, the “direction” of the interface dispersion is also defined in synthetic space, that is, the direction along which eigenfrequencies changes when the synthetic parameter changes by a period. Because the nontrivial topology utilized here lies in the synthetic space (k_{2}, ξ), the deformed and the undeformed PC act as the topological and trivial structures, and are joined together in the direction of a_{2}. The nontrivial “topology” ensures that the eigenfrequencies of the interface states change with ξ and cross the bulk bandgap when ξ changes by a period. Therefore, the interface states with different frequencies are separated when ξ is modulated spatially, which realizes the functionality of a rainbow whose dispersion relation is protected by the topology in the synthetic dimension^{27,36}.
Because the topology discussed above can exist as long as the bulk PC has a bandgap, the geometric structure and the material of the rainbow can be designed to satisfy the target frequency. In order to realize rainbow separation of light in the range from 1540 to 1630 nm, which is around the optical communication range and is also our tunable laser source range, we design a PC structure using all dielectric materials of silicononinsulator (SOI) whose bandgap covers the target range. The geometric structure and the band dispersions are shown in Fig. 1c. The lattice constant is a = 485 nm, the thickness of the silicon slab is h = 220 nm, and the side length of the triangle is l = 0.75a. Figure 1c shows the bulk bands of transverse electric (TE)like modes, where the orange stripe marks the photonic bandgap from 172.4 to 201.1 THz (from 1491 to 1739 nm in wavelength), and the grey region denotes the light cone. Under the bandgap marked by the orange stripe, there is one isolated band with Chern number 1 in the synthetic dimension, and therefore only one interface band is expected at the interface of spliced structure. Figure 2a shows the calculated results of the dispersion of interface states for different values of the synthetic dimension ξ. The inset shows the geometric structure of the supercell, and the translational deformation is marked by the red vector and the wave vector is marked by the blue vector. The grey and greyishgreen regions denote the light cone and the projected bulk bands, respectively. The colored curves are the bands of interface states corresponding to different ξ. As is shown in Fig. 2a, when ξ changes by a period, the frequency of the interface states for given wavevector k varies across the bandgap. It should be pointed out that the topological rainbow shows advantage to conventional designs, because as long as a bandgapped photonic crystal is constructed, the topological rainbow will definitely exist due to the nontrivial topology.
The group velocities of interface states can also be controlled by modulating of dispersion. In Fig. 2b, the calculated results of group velocities are shown in the space of parameter ξ and wavelength (see Supplementary Materials Sec. 2 for the detail of the calculation method). Because of the timereversal symmetry, only the group velocities of states with the positive wavevectors are shown. The zerogroupvelocity plane is marked in grey, and the area bounded by the states with zero group velocity is marked in cyan. Because the zerogroupvelocity points corresponds to wavelength extrema, the cyan region is also the projection of the interface dispersion bands on the ξ dimension. We call the region “projected interface bands” in the following text. Because of the topology in ξ dimension, when ξ increases, the wavelengths of the states with zero group velocity increase with ξ, which can be employed to stop or separate light with different frequencies. It demonstrates that the notion of a synthetic dimension provides an effective way to tune group velocity of topological photonic states. Therefore, topological photonic states with different frequencies can be slowed and used for information temporary storage. Furthermore, by coupling the interface modes to output waveguides, a topological router can be constructed, as is shown in Supplementary Fig. 2. A detailed discussion is available in the Sec. 3 of Supplementary Materials.
We calculate the light intensity distributions of the topological rainbow by using the finite difference time domain (FDTD) method. The intensity distributions at the interface of undeformed PC and deformed PC are shown in Fig. 2c, and the boundaries of the projected interface bands are marked by cyan dashed lines, where the parameter ξ is substituted by the corresponding coordinate y. The maximum intensity of different wavelengths is trapped in different positions of the interface. As the propagating interface modes are only supported in the projected interface bands, light mainly concentrates in the region of projected interface bands, and the intensity decreases exponentially outside that domain. A timedomain simulation is shown in the Supplementary Videos S1 and S2 of the Supplementary Materials to demonstrate that light is trapped at the interface.
It should be pointed out that the intensity distributions of modes with different wavelengths can be well separated by using other materials or structures, and an example is given in Supplementary Fig. 3 in the Supplementary Materials. A topological rainbow with highly localized interface states can be constructed for at least 7 distinguishable wavelengths (1200, 1240, 1277, 1323, 1459, 1606, and 1772 nm) within the range of bulk bandgap (from 1185 to 1811 nm). The corresponding y coordinates of the point with maximal intensity are −1953, −1619, −1279, −953, 400, 1699, and 2606 nm, respectively. The displacement rate is Δy/Δλ = 7.97, which is the ratio of the total displacement of the positions with maximal intensity over the working bandwidth, and different modes are separated completely in space because the modes are highly localized.
Moreover, higherdimensional spatial topological rainbow trapping can also be constructed beyond 2D planar nanostructures based on synthetic dimensions. We give an example of a 3D topological rainbow based on a dielectric PC in a diamond lattice. Similar to the 2D planar cases, the 3D topological rainbow also shows rainbow effect, and the difference lies in that the 3D topological rainbow supports dispersed conductive channels rather than dispersed localized states. A detailed discussion is in Supplementary Materials Sec. 5 (Supplementary Figs. 4, 5, and 6). This also demonstrates the notion of synthetic dimension is applicable for the study of higherdimensional and more complex physical systems.
Experimental realization of topological rainbow
The topological rainbow proposed here can be verified experimentally in the optical frequency range by siliconbased technologies. The samples are fabricated on a common SOI chip comprising a 220 nmthick silicon layer and a 2 μmthick SiO_{2} layer. The fabrication methods and scanning electron microscope (SEM) images of samples in a large scale are shown in Supplementary Fig. 7 in the Supplementary Materials. The geometric parameters for the structure are N_{a} = 6, N_{b} = 9, N_{ξ} = 30, N_{u} = 3, N_{d} = 6, matching the geometry of the input waveguide. The parameter N_{ξ} = 30 determines the variation rate of the synthetic parameter ξ along the interface. A detailed discussion is in the Sec. 7 of the Supplementary Materials, and the calculated results of different N_{ξ} are shown in Supplementary Fig. 8. The central axis of the whole structure is aligned to the waveguide. The SEM images are shown in Fig. 3a, where the barrier regions and dispersing region are colored with pseudo red and green color respectively to display the pattern clearly.
To accurately characterize the performance of such an onchip topological rainbow device, it requires a SNOM system that has the simultaneous capability of substructure spatial resolution, high optical collection efficiency, and scanning repeatability. There are two typical ways for measurement based on SNOM. One is to employ tapered fibers with an aperture, which is also called collected SNOM with a resolution of about 100 nm, depending on the diameter of the aperture. The other is to employ atomic force microscope (AFM) tips that offer a higher resolution (less than 20 nm) depending on the diameter of the tip, which is also called scattering SNOM (sSNOM)^{38}. Nearly all previous nanophotonic topological functionalities can also be conveniently identified in the far field, but for the onchip ultracompact topological rainbow device, the sSNOM system may be the unique technique to give a direct measurement with high enough resolution for both topography and optical field. In our work, different from aperturefiber SNOM system used in recent works^{16,39}, we adopt a homemade reflectionbased sSNOM system with a higher spatial resolution of ~20 nm, which can directly measure each lattice topography and its corresponding optical intensity simultaneously with high enough resolution to demonstrate each nanophotonic lattice.
Figure 3b shows the homemade reflectionbased scattering scanning nearfield optical microscope (sSNOM), which consists of an AFM module (NTMDT Spectrum Instruments, NTEGRA Solaris) and a fiber Mach–Zehnder interferometer (MZI) with heterodyne detection. The nearfield microscopy is based on a cantilevered AFM probe as a nearfield probe. In this work, the silicon AFM probe (Scansens GmbH, HA_NC) has a typical tip radius of ~10 nm and cone full angle of 30°, which ensures that the topography and the optical field can be acquired simultaneously with a spatial resolution of ~20 nm. Operating in the tapping mode, the AFM probe vibrates vertically at a fixed frequency of Ω ≈ 140 kHz with the amplitude of ~100 nm. The interaction of light with the AFM probe results in the conversion of near field to backward guided modes in the input waveguide, forming the modulated signal light. This reflectionbased sSNOM has high light collection efficiency because the waveguide can naturally play the role of a highnumericalaperture lens, as is shown in Fig. 3c.
Discussion
Our reflectionbased sSNOM system has several advantages. Firstly, the AFM tip can penetrate the nanostructures and couple with the light confined in the air holes, which can significantly increase the collection efficiency. Secondly, thanks to the high resolution, the topography, and the optical field can be acquired simultaneously, and the deviation of the scanning area in different scanning process can be corrected according to the topography image, providing an effective method to measure the surface intensity of topological nanophotonic device directly. Thirdly, the allfiber setup with a single coupling point configuration can provide a convenient, high light collection efficiency, and low background noise platform^{40} for the characterization of onchip structure. Based on the above characteristics, the sSNOM makes the direct observation of onchip nanophotonic topological rainbow with high resolution.
Next, we show the experimental and calculated results of the topological nanophotonic rainbow. In order to characterize the light separation ability of the rainbow, the y coordinates of the points with maximal intensity along the interface are measured, and displacement rate Δy/Δλ, is defined to characterize the light separation ability of the rainbow device. For the calculated results, the top view of the calculated FDTD model is shown in Fig. 4a, where the blue part denotes the silicon wafer and the white part denotes the hole, and the light intensity distributions (E^{2}) for four wavelengths are shown in Fig. 4b. For wavelengths 1540, 1570, 1600, and 1630 nm, the ycoordinates of the points with maximal intensity are −635, −154, 776, and 1257 nm, respectively, and the displacement rate is 21.02. For the experimental results, the topographic image of the sample and the surface intensity distributions are measured simultaneously by the sSNOM system. Figure 4c is the topographic picture of the surface of sample. The color denotes the height of the sample surface. Figure 4d shows the measured intensity distributions for the four wavelengths. The positions with maximal electric intensity are marked by the cyan dashed rhombuses. The position of maximal intensity moves towards the positive direction of yaxis when wavelength increases. The y coordinates of the maximal electric intensity are −1699, −81, 848, and 1849 nm for the four wavelengths, respectively, and the displacement rate is 39.42. This relation is consistent with the topology in the lattice translational dimension. As is predicted by the dispersion bands, when wavelength increases (frequency decreases), the position of the interface state moves to the position with the increase of ξ.
Figure 4e, f shows the comparison between the interface intensity of the topological rainbow and the projected interface bands. Similar to Fig. 2d, the ξ dimension of the projected interface bands is substituted by y, and the interface intensity is averaged in x direction within a distance of \(\sqrt{3}a/4\) from the interface. For the calculated results, the patterns of intensity distributions are the same as discussed previously, that the light concentrates in the projected interface bands region and decreases exponentially out of the region. For the experimental results, the light is also mainly concentrated in the projected interface bands region, and the states in the interface of rainbow moves when the wavelength changes. The positions of different rainbow states are related to the dispersion of topological edge states, though the intensity distributions of calculated and experimental results have slight differences. There are three main reasons that cause the differences. First, the AFM probe dose not only reflect the light in the rainbow device to the waveguide, but also scatters the light in the free space into the waveguide, which adds noise and causes the signal of stray light. Second, the waveguide is wide compared to the wavelength of incident light, which will support multimode propagation. The interference of different modes will influence the spatial distribution of rainbow states. Third, some evanescent modes within the light cone may be excited under continuous excitation during the measurement, and they may be collected by the AFM probe. Apart from the above triangular hole structure, the topological rainbow can also be constructed by other structures. In the Supplementary Materials, we also present the calculated and experimental topological rainbow for square air holes nanostructures in Supplementary Figs. 9 and 10.
In conclusion, we have experimentally realized onchip topological rainbow nanophotonic devices by introducing a synthetic dimension to a dielectric photonic crystal. The siliconbased PC sample was successfully fabricated with an ultracompact size of 4.5 μm × 22 μm. Topological states of different wavelengths from 1540 to 1630 nm around the optical communication range have been separated and trapped at different positions along the interface of the photonic crystals. The rainbow effect is directly demonstrated using a homemade sSNOM system for different optical lattice nanostructures. It offers an effective way to implement onchip nanophotonic rainbow devices enabled by topological ideas, using the notion of synthetic dimensions. This work provides a method for constructing topological states using dielectric materials without using magnetic field or magnetic materials in the optical frequency range. The topological rainbow effect can be used for topological slow light, topological router, and topological temporary storage, enabled by the fact that different frequencies of topological photonic states can have different group velocities with suitably chosen sample structural parameters. The work provides an avenue to the implementation of onchip nanophotonic topological devices beyond the structure’s band topology, such as realizing nonzero Chern numbers in dielectric systems, and can stimulate the development of topological nanophotonic chips.
Methods
Theoretical calculations
The detailed derivation of the nontrivial topology in synthetic dimensions for arbitrary lattice type is shown in Supplementary Materials Sec. 1. The bands of the bulk states and the edge states are calculated by the finite element method, and the intensity distribution of the topological rainbow is calculated by the FDTD method. The numerical calculation method of the group velocity is shown in Supplementary Materials Sec. 2.
Experimental measurements
The samples are mainly fabricated by a focusedionbeam system. The details of the sample fabrication method are shown in Supplementary Materials Sec. 6. The sample is illuminated from a tunable laser (Agilent, 81940A), with the wavelength ranging from 1520 to 1630 nm and the linewidth of <100 kHz. The continuouswave laser light is separated to signal and reference arm through a fiber directional coupler. In the signal arm of the MZI, a lensed fiber (CXFIBER) launches the light (ω_{0}) into the waveguide under tested and collects the modulated reflection light (ω_{0} ± Ω) produced by the AFM probe from the same facet. The threepaddle polarization controller ensures that the guided modes in the waveguide are quasiTE polarization. In the reference arm of the MZI, the light is frequencyshifted by Δω = 30 kHz via an inline LiNbO_{3} phase modulator (SWT Sci. & Tech.) linked with a sawtooth waveform generator. The allfiber characteristic brings about convenience (alignment free with single coupling point), compactness, and low background noise, which is suitable for the nearfield imaging of on chip photonic circuits. Then, both the signal (ω_{0} ± Ω) and the reference (ω_{0} + Δω) light are combined and sent to an InGaAs amplified photodetector (Thorlabs, PDA10CS2). The photocurrent can accurately yield the amplitude and phase of the signal light through a lockin amplifier (Zurich Instruments, HF2LI) at the demodulation frequency of Ω + Δω.
Data availability
All data and materials in the manuscript or the supplementary materials are available from Cuicui Lu.
Code availability
All codes in the main text or the supplementary materials are available from Cuicui Lu.
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Acknowledgements
We gratefully thank Prof. Ling Lu for kind help to use some related optical measurement equipment in Chinese Academy of Sciences and thank Prof. Z. Q. Zhang for the helpful discussions. We thank the support of the National Natural Science Foundation of China (91850117, 11654003, 92150302, 11734001, 91950204, 92050110, 11904264, and 62075083), National Key Research and Development Program of China under Grant Nos. 2018YFB2200403, Guangdong Basic and Applied Basic Research Foundation 2021B1515020030, the Fundamental Research Funds for the Central Universities (21620316), and Hong Kong Research Grants Council through grant 16307420 and AoE/P02/12, and the Croucher Foundation (CAS20SC01). The authors also acknowledge the Analysis & Testing Center of Beijing Institute of Technology for the use of FIB.
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C.L. conceived the program. C.L., X.H., W.D., and C.C supervised the projects. C.W., C.L., M.X., W.Z., and C.C. contributed to the theory and calculations. C.L. fabricated the samples. Y.S., C.L., C.W., H.Z, Y.L., X.H., and W.D. performed the measurements and analyzed the results. All authors discussed the results and contributed to the manuscript.
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Lu, C., Sun, YZ., Wang, C. et al. Onchip nanophotonic topological rainbow. Nat Commun 13, 2586 (2022). https://doi.org/10.1038/s4146702230276w
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DOI: https://doi.org/10.1038/s4146702230276w
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