2.1.

Realization of Pseudomagnetic Fields in Photonic Crystals

Figure 1(a) shows a schematic of the PhC with a honeycomb lattice implemented on the silicon-on-insulator (SOI) platform. The unit cell is illustrated in the red-dashed rhombic boxes and contains two inverted equilateral triangular holes with side lengths $d_1$ and $d_2$. The lattice constant is $a_0$. When the spatial inversion symmetry is unbroken ($d_1=d_2$), the PhC lattice exhibits a $C_{6v}$ symmetry, featuring a Dirac cone at the $K$ and $K^{\prime }$ points in the momentum space, as shown by the band diagram of the transverse electric (TE) modes of the PhC in the middle of Fig. 1(a). Using the plane wave expansion method and applying the $\overrightarrow{k}·\overrightarrow{p}$ approximation,^{49 –56} the two lowest bands can be described analytically by an effective Hamiltonian near the $K/K^{\prime }$ points: $H_{K/K^{\prime }}=\pm (v{k}_x{\sigma }_x+v{k}_y{\sigma }_y)$, where $\pm$ represents the $K$ ($K^{\prime }$) valley pseudospin, ${\sigma }_{x,y,z}$ denotes the Pauli matrices, $v$ is the group velocity, and $(k_x,k_y)$ is the reciprocal vector with respect to the degeneracy points $K$ and $K^{\prime }$ (see Sec. 1 in the Supplementary Material for more details). By introducing asymmetry between the sizes of the two holes within the unit cell ($d_1\ne d_2$), the spatial inversion symmetry is broken, and the lattice symmetry is reduced from $C_{6v}$ to $C_{3v}$. An effective mass term $m$ is introduced to the Dirac cone, which lifts the Dirac degeneracy and opens a bandgap of $\mathrm{\Delta }\omega =2|m|$, as shown on the right of Fig. 1(a). More details about the effective mass and the size difference of the triangular holes can be found in Sec. 2 in the Supplementary Material. If the mass term varies with position, the effective Hamiltonian becomes

The state shows the behavior of a propagating wave along the $y$ direction and the characteristics of a Gaussian wave along the $x$ direction, indicating the existence of a propagative bulk state near $x=0$. Moreover, the field distribution of the state is directly related to the effective mass term, i.e., $\psi (x,y)\propto e^{-\frac{1}{v}\int m(x)\mathrm{d}x}$, which means the confinement of light near $x=0$ is guaranteed as long as $m(x)$ is an odd function such as $m(x)=a{x}^{1/3}$ and $m(x)=a{x}^3$, and can be tuned by choosing different functions for $m(x)$ (see Sec. 5 in the Supplementary Material for details). In Fig. 1(c), we show another example with a linearly varying mass term $m(y)=by$, which corresponds to a PMF $B_x=b$. The side length of the triangular holes $d_1$ ($d_2$) decreases (increases) linearly in the $y$ direction, as shown in the supercell in the red-dashed boxes. The red dotted line in the band diagram represents the propagative bulk states of the zeroth-order Landau level. The profile of the mode corresponding to the blue star point is shown on the bottom right of Fig. 1(c). As one can see, the field is well confined to the region near $y=0$ [purple-shaded area in Fig. 1(c)]. More details about the propagative bulk states under such a PMF can be found in Sec. 1 in the Supplementary Material.

By synthesizing uniform magnetic fields, one can realize the Landau levels and the straight waveguiding of light in the $x$ and $y$ directions. However, more complicated manipulation of light waves is required for constructing photonic integrated circuits for real applications. We go beyond the Landau levels and design complex nonuniform PMFs to flexibly control the flow of light on a chip, as shown in Fig. 1(d). By introducing a nonuniform PMF varying in both the $x$ and $y$ directions, the deflection of light can be realized and exploited to build an S-bend in a PhC platform, as indicated by the purple-shaded stripe on the left of Fig. 1(d). Different from the sharp bends that exploit the spin- and valley-Hall effects,^{45 ,49} the S-bend is solely determined by the predefined PMF and therefore provides more flexibility in designing the optical path. Furthermore, the field distribution of the propagative bulk state can be engineered by judiciously designing the PMF through the relationship $\psi (x,y)\propto e^{-\frac{1}{v}\int m(x,y)\mathrm{d}x}$. In other words, once the light path or the field distribution function $\psi (x,y)$ is given, one can derive the corresponding effective mass distribution $m(x,y)$ by taking the first-order derivative of $\psi (x,y)$ with respect to the spatial coordinate $x$. This method can be employed to build a power splitter with a splitting ratio of 50:50 in a PhC platform, as schematically illustrated on the right side of Fig. 1(d). The light paths in the PhC are shaded in purple as a guide to the eye. The design and the implementation of the PMFs for these two devices will be detailed in Sec. 2.2. Most importantly, we argue that the methodology presented here is universal and systematic. A new family of nanophotonic devices can be developed by designing and synthesizing nonuniform PMFs with this method.

2.2.

Device Design and Experimental Demonstration

We start with the straightforward-propagating states of the zeroth-order Landau levels under the uniform PMFs $m(x)=ax$ and $m(y)=by$. The design details, simulation results, and experimental results are given in Sec. 6 in the Supplementary Material. The ILs of the two devices are lower than 1.32 and 0.8 dB in the wavelength range of 1520 to 1580 nm, manifesting the capability of on-chip waveguiding for these two kinds of PMFs. Then, we move on to the more general cases where the more sophisticated nonuniform PMFs are designed and implemented for various basic functional devices such as an S-bend and a power splitter. Figure 2(a) shows a schematic of a PhC-based S-bend operating at telecommunication wavelengths, which is built on an SOI platform with a 220-nm-thick top silicon layer, a $3\text{-}\mu \mathrm{m}$-thick buried oxide layer, and a $1\text{-}\mu \mathrm{m}$-thick silica upper cladding layer. The PMF is designed to be $m(x,y)=cx+d$ in region I, $m(x,y)=ax+by$ ($b/a=-10$) in region II, and $m(x,y)=cx-d$ in region III. It leads to a zero magnetic vector potential $A_z=0$ on the line $x=-d/c$ in region I, $y=-ax/b$ in region II, and $x=d/c$ in region III, which defines the propagation path of light in the PhC as indicated by the purple shaded stripe in Fig. 2(a). We choose $b/a=-10$ for region II to achieve a small bending radius and a relatively low IL at the same time. The trade-off between the bending radius (the partition abruptness) and the IL is discussed in Sec. 7 in the Supplementary Material. The lattice constant of the PhC is $a_0=450\mathrm{nm}$. The side lengths of the triangular holes $d_1$ and $d_2$ are 225 nm on the propagation path line and change linearly with a variation step of 14.7 nm ($\mathrm{\Delta }{d}_1=-\mathrm{\Delta }{d}_2=14.7\mathrm{nm}$) along the direction perpendicular to the line. The light is coupled into and out of the chip by grating couplers, as illustrated in Fig. 2(a). The PhC structure is connected to the grating couplers via two $3\mu \mathrm{m}$-wide silicon stripe waveguides whose fundamental TE mode excites the zeroth-order propagative bulk state in the PhC. To prove that the bending of light based on the nonuniform PMF works as well as the straight waveguiding of the Landau level state under the uniform PMF, we used the three-dimensional finite-difference time-domain (3D FDTD) methods to simulate the propagation of light in both the straight waveguide and the S-bend. Figure 2(b) presents the simulated propagation profiles for the two devices at the wavelength of 1550 nm. It is evident that the light propagates along an S-shaped trajectory predefined by $A_z=0$ in a nonuniform PMF just as in the case of a straight PhC waveguide, which is in good agreement with our theoretical prediction. These devices were also fabricated on the SOI platform using complementary metal-oxide-semiconductor (CMOS)-compatible process (the fabrication details can be found in Sec. 8 in the Supplementary Material).^{58 –60} The scanning electron microscopy (SEM) images of the fabricated devices are displayed in Fig. 2(c), with the light propagation paths shaded in purple as a guide to the eye. The zoomed-in images of the interface between the silicon stripe waveguide and the PhC, the area around $x=0$ of the straight PhC waveguide, and the area in the proximity of the propagation path in region II of the S-bend are shown in the red, blue, and orange dashed boxes on the bottom of Fig. 2(c). By employing a scanning near-field optical microscope (SNOM) system with atomic force microscope tips which offers a high resolution of $\sim 20\mathrm{nm}$, the light propagation routes in the PhCs can be directly monitored.⁶¹ However, due to the lack of such an equipment in our lab, we did not perform the SNOM experiment here. Figure 2(d) shows the simulated and measured transmission spectra of both devices. The measured transmission spectra are all normalized to that of a pair of reference grating couplers fabricated on the same chip (see Sec. 9 in the Supplementary Material for the measurement details). The measured ILs of the straight waveguide and the S-bend are lower than 1.32 and 1.83 dB, respectively, in the wavelength range of 1520 to 1580 nm, which are in reasonable agreement with the simulated ILs of $<2\mathrm{dB}$ (straight waveguide) and $<2.27\mathrm{dB}$ (S-bend). Lower ILs can be reached by optimizing the variation step $\mathrm{\Delta }{d}_1$ and the width of the silicon stripe waveguide $W_{\mathrm{Si}}$ to maximize the mode overlap between the silicon stripe waveguide and the PhC (see Sec. 10 in the Supplementary Material for more details). We argue that the ILs mainly originate from the coupling losses at the interfaces between the silicon stripe waveguide and the PhC structure while the propagation losses inside the PhC are negligible. It is noteworthy that the S-bend structure introduces almost no extra losses as compared with the straight waveguide, confirming the feasibility of achieving low-loss nanophotonic devices with as-designed nonuniform PMFs. To further demonstrate the universality of this PMF design method, we also simulate a straight waveguide and an S-bend based on PhCs with circular holes (see Sec. 11 in the Supplementary Material for more details).

Next, we present another example of a 50:50 power splitter to show how to design the nonuniform PMF according to the light field distribution $\psi (x,y)$. Figure 3(a) shows a schematic of the consecutive supercells of the power splitter along the propagation direction. The field of the propagative bulk state is initially localized around $x=0$ and gradually becomes concentrated on two well-separated positions in a supercell, as indicated by the red Gaussian-like wave packets in Fig. 3(a). The separation between the two maxima of the field grows linearly with propagation distance, implementing the splitting of light during propagation. The relationship between the light field and the PMF can be described by $\psi (x,y)\propto e^{-\frac{1}{v}\int m(x,y)\mathrm{d}x}$ (see Sec. 1 in the Supplementary Material for details). By taking the first-order derivatives of $\psi (x,y)$ with respect to the spatial coordinate $x$, one can obtain the effective mass distribution $m(x,y)$ in every supercell, as shown on the bottom of Fig. 3(a). The two zeros of the mass term separate slowly along the propagation direction. The whole structure of the power splitter is schematically illustrated in Fig. 3(b) where the propagation paths of light are shaded in purple. The lattice constant of the PhC is $a_0=450\mathrm{nm}$. The side lengths of the triangular holes $d_1$ and $d_2$ are 225 nm on the propagation paths where $A_z=0$ and change gradually with a variation step of 6.9 nm away from the paths to mimic the nonuniform PMF described above (the relationship between the effective mass term and the sizes of the triangular holes can be found in Sec. 2 in the Supplementary Material). Full-wave numerical simulations were carried out using the 3D FDTD methods to investigate the propagation of light in the power splitter. Figure 3(c) shows the propagation profile for the device at the wavelength of 1550 nm. The zeroth-order propagative bulk state is excited by the fundamental TE mode in a silicon stripe waveguide, split equally into two beams in the PhC, and output from two silicon stripe waveguides. Again, the proposed device was fabricated with a CMOS-compatible process. Figure 3(d) presents the SEM images of the fabricated device, with the purple-shaded area indicating the light propagation paths. The zoomed-in images of the input port, splitting area, and output ports are displayed in the red, blue, and orange dashed boxes, respectively, at the bottom of Fig. 3(d). The simulated and measured transmission spectra of the device are plotted in Fig. 3(e). In the simulations, the imbalance of the power splitter is close to 0 dB, and the EL is below 2.06 dB in the wavelength range of 1527 to 1573 nm. In the experiments, the measured imbalance and EL are lower than $\pm 0.5$ and 2.11 dB, respectively, in the same wavelength range, which is in good agreement with the simulation results. Moreover, the EL of the power splitter is comparable to the ILs of the straight waveguide and the S-bend shown in Fig. 2, proving that the EL mainly comes from the coupling loss between the silicon stripe waveguide and the PhC, and the beam splitting process causes rare extra losses. More importantly, the arbitrary control of the flow of light with PMFs can be envisioned as both the light field distribution and the corresponding PMF can be designed at will using our method. For example, we also demonstrate power splitters with arbitrary splitting ratios by adjusting the distribution of the effective mass term $m(x,y)$. The simulation details are provided in Sec. 12 in the Supplementary Material. In addition, to further verify that other functional devices can also be constructed through this mechanism, we propose and experimentally demonstrate a $2<span\; class="naked\_sign">\times </span><span\; class="naked\_aural">ばつ</span>2$ directional coupler using PMFs in silicon PhCs at telecommunication wavelengths (see Sec. 13 in the Supplementary Material for details).

As the inter-valley coupling is negligible among the counter-propagating waves affiliated with different valleys, the transport of the propagative bulk states under the synthetic PMFs is topologically protected.^{41 ,44} Here, we fabricated and measured different kinds of PhC devices (including the straight waveguide, S-bend, and power splitter) with intentionally introduced defects. The experimental details are provided in Sec. 14 in the Supplementary Material. No severe discrepancies were observed between the transmission spectra of the fabricated devices with and without defects. The experimental results show clear evidence that the performance of the PMF-based devices exhibits good robustness against small perturbations, which is very much desired in large-scale photonic integration and massive production. We also investigate the robustness of the PMF-based devices against large areas of defects. The simulation results are provided in Sec. 14 in the Supplementary Material.

2.3.

High-Speed Data Transmission Experiment

To verify the feasibility of using the aforementioned devices in realistic on-chip optical communication systems, we performed a high-speed data transmission experiment⁶² based on the proposed S-bend and power splitter, as shown in Fig. 4. The experimental setup and the digital signal processing (DSP) flow charts for the transceiver are illustrated in Figs. 4(a) and 4(b), respectively (for details, see Sec. 15 in the Supplementary Material). A Nyquist-shaped 70-GBaud PAM-4 signal is transmitted through each channel of the fabricated devices. The measured optical spectra of the signals before and after passing the devices are shown in Figs. 4(c) and 4(d). Figure 4(e) plots the bit error rates (BERs) for the channels of the S-bend and the splitter, which are all below the 7% hard-decision-forward error correction threshold of $3.8<span\; class="naked\_sign">\times </span><span\; class="naked\_aural">ばつ</span>{10}^{-3}$. The recovered eye diagrams of the PAM-4 signals for different channels are presented in Fig. 4(f). The demonstration provides unequivocal evidence that the basic functional devices built with PMFs in the telecom band can be used in real applications such as on-chip optical communications.

References