Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension

Weyl points, as a signature of 3D topological states, have been extensively studied in condensed matter systems. Recently, the physics of Weyl points has also been explored in electromagnetic structures such as photonic crystals and metamaterials. These structures typically have complex three-dimensional geometries, which limits the potential for exploring Weyl point physics in on-chip integrated systems. Here we show that Weyl point physics emerges in a system of two-dimensional arrays of resonators undergoing dynamic modulation of refractive index. In addition, the phase of modulation can be controlled to explore Weyl points under different symmetries. Furthermore, unlike static structures, in this system the non-trivial topology of the Weyl point manifests in terms of surface state arcs in the synthetic space that exhibit one-way frequency conversion. Our system therefore provides a versatile platform to explore and exploit Weyl point physics on chip.

(c-e) Time-reversal symmetry breaking with modulation phases φ A = −φ B = π/3. The Weyl points are located at k y = 0, (k x , k f ) ∈ {(0.69 π a , 0), (1.46 π a , 0), (0.85 π a , π Ω ), (1.62 π a , π Ω )} with δ = [1.33, 0.78, −0.78, −1.33]γ, respectively. This differs from the tight binding model, in which the first pair of Weyl points has frequency δ = t and the second pair has δ = −t. (c) The band structure in the k x − k f plane at k y = 0. (d) The band structure in the k y − k f plane at k x = 0.69 π a . (e) The band structure in the k y − k f plane at k x = 0.85 π a . The Bloch wavevectors for the Weyl points are different from the tight binding model in which both Weyl points at k f = 0 and π Ω occurs at k x = 4π 3 √ 3a ≈ 0.77 π a .   3 √ 3a , where the Weyl points occur. These differ from the Weyl points in the bulk band structure due to Brillouin Zone folding. The waveguide amplitude model and the tight binding model both predict surface state arcs connecting a pair of Weyl points at k f = ± π 2Ω . They also predict the same charge and position of the Weyl points in the wavevector space.
The two models also have a few differences. In the waveguide amplitude model, the Weyl point near k x = 2π 3 √ 3a has lower frequency than the ones near k x = 4π 3 √ 3a . As a result, the surface states now have positive group velocity along x, which is evident by the tilt of the surface arcs in (a). On the contrary, the tight binding model predicts zero group velocity along x since the two Weyl points have the same frequency, as shown in (c). The frequency difference of the Weyl points in the waveguide amplitude model also means that if one considers a constant frequency cut between the frequencies of the Weyl points, the surface state arcs connecting the two regions of bulk states cross in the wavevector space, as shown in (b). Here the surface state arcs connect the bulk band above the Weyl point near k x = 2π 3 √ 3a to that below the Weyl point near k x = 4π 3 √ 3a . If we take a constant frequency cut above both Weyl points as shown in Fig. 4c in the main text, the region of bulk states around k x = 2π 3 √ 3a would be larger than that around k On the contrary, in the tight binding model, since the Weyl points have the same frequency, any constant frequency cut intersects either the bulk band above the Weyl points or below. As shown in (d), the surface state arcs remain parallel, and the two regions of bulk states have the same size.
(e,f) Surface state excitation on a stripe with 5 rows using the waveguide amplitude model. Calculated with δ = 0, k x = π √ 3a and 60 sidebands. Only 20 frequencies closest to the excitation are shown. A and B sites are plotted separately along y axis. Input frequencies are marked by pink arrows, with a phase difference of π/2 between squareended and circular-ended arrows. (e) Excited with S M0+1 /S M0 = −i to couple to k f = − π 2Ω modes, indicated by the green dot in (b). (f) Excited with S M0+1 /S M0 = i to couple to k f = π 2Ω modes, indicated by the red dot in (b). Both are excited from the bottom edge B rings. Other parameters are: modulation strength α = 0.65, inter-ring coupling γ = J 1 (α) = 0.3, round-trip transmission r = 0.9 and waveguide-ring coupling η = 0.1.   ]. In addition, the pair of Weyl points at k f = 0 is shifted to smaller k x , while that at k f = π Ω is shifted to larger k x , as shown in (b,e). This means that the k x for the surface state with optimal confinement is different near k f = 0 and k f = π Ω . (l,m) Surface state excitation on a stripe with 5 rows using the waveguide amplitude model. Calculated with k x = π √ 3a and 60 sidebands. Only 20 frequencies closest to the excitation are shown. A and B sites are plotted separately along y axis. Input frequencies as marked by pink arrows, with a phase difference of π between the squareended and the circular-ended arrows. (l) Excited with S M0 /S M0+1 = 1, δ = γ to couple to k f ≈ 0 modes, indicated by the green dot in (e). (m) Excited with S M0 /S M0+1 = −1, δ = −γ = to couple to k f ≈ π Ω modes, indicated by the red dot in (f). Both are excited from the top edge A rings. Other parameters are: modulation strength α = 0.65, inter-ring coupling γ = J 1 (α) = 0.3, round-trip transmission r = 0.9 and waveguide-ring coupling η = 0.1.    Supplementary Table 1. Two designs for silicon ring modulator based system. Given a FSR determined by the modulation bandwidth, we choose a coupling constant t f that is significantly larger than loss but still preserves the validity of the rotating wave approximation. The inter-ring coupling t xy should match t f by adjusting the waveguide distance and coupling region length. FSR determines the ring size. t f and the achievable modulation strength determine the modulation region length. The modulation length ratio shows that in both cases, less then 1/3 of the ring needs to be dynamically modulated to achieve the desired coupling constant. The ring size and modulation region length determine the round-trip transmission. The round-trip transmission and the operating wavelength, assumed to be 1550 nm, determine the quality factor Q. The FSR, the modulation strength and the loss value are taken from state-of-the-art published results on depletion-mode silicon waveguide modulators [4][5][6]. In this section we will show a ring-resonator-based system that implements the Hamiltonian in Eq. 1 of the main text. We also prove the validity of the tight binding model in Eq. 2 of the main text in describing such a ring resonator lattice.

Zig-zag edge
Consider the infinite honeycomb lattice of rings shown in Supplementary Figure 1a. The red and blue dots represent the centre of rings for the A and B sublattice, respectively. The dashed lines represent evanescent coupling between adjacent rings.
Supplementary Figure 1a introduces a coordinate system (x, y) for labelling the rings. Each horizontal zig-zag chain of rings, which we will called a row, is lablled with the same y. Within each row, the resonators are labelled with increasing x. As such, each integer pair (x, y) labels a resonator. All resonators in the A sublattice have indices x, y summing to an even integer, while those in the B sublattice have indices x, y summing to an odd integer. A general form of tight binding Hamiltonian describing such a honeycomb lattice is: where i, j ∈ {(x, y)} labels the position of each lattice site, c and c † are the standard ladder operators, and c.c. denotes complex conjugate. The second terms sums over all neighbouring pairs of i, j. In Supplementary Equation 1, each site supports a single resonant mode with a frequency ω i , as described by the first term. The second term represents coupling between all adjacent sites, i.e. pairs of sites connected by the dashed line in Supplementary Figure 1a.
Rewriting Supplementary Equation 1 in the explicit (x, y) label, we obtain: The second and third terms represent coupling in horizontal and vertical links, respectively. The horizontal coupling terms between x and x + 1 column sum over all rows. The vertical coupling between y and y + 1 rows, however, occurs only over every other column. We now use the Hamiltonian of Supplementary Equation 1 to describe the static coupling among the m-th resonant modes of the rings in the lattice shown in Supplementary Figure 1b. For such a ring resonator lattice, the coupling is between the counter-clockwise rotating states on the A sublattice and the clockwise rotating states on the B sublattice. We label the m-th resonant modes on the A and B sublattices as |a x,y,m and |b x,y,m , where x + y is odd or even, respectively. Denote |a(b) x,y,m = c † x,y,m |0 . Under rotation of an angle θ around the centre of a resonator, its resonant modes transform as: where the subscript of the rotation operator R represents the angle of rotation, and the superscript represents the centre of rotation.
Using the basis of the m-th resonant modes, the static coupling among these modes is described by: Below we will constrain the coupling constants by the rotational symmetry of the structure. Consider the terms in H m related to the coupling terms involving one particular resonator in the A sublattice labelled as (x, y), as schematically shown in Supplementary Figure 1c: H m = .... + (t 1 |b x,y−1,m a x,y,m | + t 2 |b x+1,y,m a x,y,m | + t 3 |b x−1,y,m a x,y,m | + c.c.) + ...
Comparing Supplementary Equations 5-8 we obtain: Without loss of generality we pick the phase of t 1 to be zero, i.e., t 1 = t xy . Supplementary Equation 5 becomes Reverting to the second quantization form, we therefore have: We note that the coupling along x has a directional, m-dependent phase as shown in Supplementary Figure 2a. Now consider applying an index modulation with frequency Ω = ω m+1 − ω m to each ring to couple the different orders of resonant modes. We assume that the modulation at a ring (x, y) has a modulation phase φ x,y . The term in the Hamiltonian describing the effect of modulation is: Overall, the Hamiltonian describing the honeycomb lattice of resonator under dynamic modulation is: which describes a three-dimensional lattice lablled by (x, y, m). Performing a gauge transformationc x,y,m ≡ e imx 2π 3 c x,y,m results in: x,y,m+1c x,y,m + c.c.) As a result of the gauge transformation, the static (spatial) coupling is homogeneous and m-independent. On the other hand, the coupling along the synthetic frequency dimension now has an additional x-dependent, directional phase, as shown in Supplementary Figure 2b. Further simplifying the above Hamiltonian by changing the (x, y) labelling to the i, j ∈ {(x, y)} labelling for the static coupling terms, we obtain: 3c † x,y,m+1c x,y,m + c.c.) This Hamiltonian is very similar to Eq. 1 in the main text once we replacec i,m with a i,m . Defining c i,m =c i,m e −iωmt which performs a transformation to a rotating frame, and applying the rotating wave approximation to remove the time-dependency of Eq. 15, we obtain: With a redefinition of φ i = φ x,y + 2πx 3 , Eq. 16 matches Eq. 2 in the main text. Therefore, the Hamiltonian of Eq. 2 in the main text can be implemented in a lattice of rings provided that the modulation phases are appropriately chosen.

Supplementary Note 2. Waveguide amplitude model
In this section, we provide a more detailed description of our system using coupled mode theory, where the dynamic variables are the position-dependent waveguide modal amplitudes in each ring. We refer to such a model as the waveguide amplitude model. Compared to the tight binding model, the waveguide amplitude model provides more microscopic details of the effect of inter-ring coupling and the dynamic index modulation. Here we first set up the general waveguide amplitude equations describing the system. In the next three sections we apply this model to calculate band structures and surface states.
In general, the electric field E inside a ring can be written as [1,2] where r ⊥ denotes the direction perpendicular to the waveguide that forms the ring, and E(r ⊥ ) is the modal profile of the waveguide mode. z denotes the longitudinal direction of the waveguide. ω m and E m (t, z) are the resonant frequency and the modal amplitude of the m-th mode. Here we assume that the waveguide is single-moded. In addition, since the modulation frequency is typically much smaller than the optical frequency, we assume that the model profile does not change over the frequency range of interest. E m is assumed to be slow-varying, which implies where v g is the group velocity in the waveguide. For each ring, static coupling to its neighbouring rings occur at position z 1 , z 2 , z 4 , as shown in Supplementary  Figure 3a. The equations at each coupler are: where describes the coupling between two rings. γ is the coupling strength. Dynamic coupling between modes of different order m in the same resonator is achieved by an EOM at z 3 = (z 2 + z 4 )/2, which is modeled by [1]: Using the Jacobi-Anger expansion: where J q (α) is the Bessel function of the first kind, we obtain a relation of the electric field before and after modulation: where m ≡ m − q. Switching the labels m and m , we obtain: where δω is a small detuning from resonant frequency, z 0 is a position in the section. Let a(b) k,m be the timeindependent modal amplitudes E m (z) defined at the points in the k-th section as shown in Supplementary Figure 3a.
For k = 1, 4, a(b) k,m are at the centre of the respective sections. For k = 2, 3, a(b) k.m are immediately before and after the modulator, which is located at the midpoint between the two couplers. E m (z) can be expressed using a(b) j,m section-wise as: Note that for a resonant mode β(ω m )L = 2mπ, where L is the circumference of the ring, and z 2 − z 1 = z 4 − z 2 = z 1 − z 4 = L/3, as shown in Supplementary Figure 3a. Furthermore, define δ = (δω/v g )L as the round-trip phase detuning. For convenience, in the following we will refer to δ as detuning, bearing in mind its aforementioned relation to the actual frequency detuning δω. Now we can re-write Supplementary Equations 19-27 using a(b) j,m to obtain the set of equations for the steady state modes: Supplementary Equations 30-33 thus define a discrete linear system with each ring represented by the amplitudes a(b) k,m . In the following, we will solve this discrete system to obtain band structures for the bulk and the surfaces.

Supplementary Note 3. Band Structure of an Infinite Lattice
First we solve for the band structure of a lossless, infinite system in both spatial dimensions as well as in the synthetic frequency dimension. For such a system we observe certain symmetries that will simplify the form of our solution. Supplementary Equation 33 is invariant under a transformation of m → m + 1, while Supplementary Equations 30-32 are invariant under a transformation of m → m + 3. Below, we will prove that a steady state solution of Supplementary Equations 30-33 satisfies: where a particular resonance with resonant frequency ω 0 is defined as the zeroth order mode, and the m-th order mode is related to the zeroth order mode by ω m = ω 0 + mΩ. Supplementary Equation 34 is similar to the standard form of a Bloch wave function, but with the θ j term introduced to account for the different translational symmetry in m between implies that θ 3 = θ 2 . Without loss of generality we set θ 2 = θ 3 = 0, then The system also has translational symmetry in the spatial dimensions. Therefore b, b , b are related by: as required by the Bloch theorem applied to the spatial dimensions. a is the nearest neighbour distance. We note that the θ and θ are j-independent since the spatial translational symmetry applies to the ring as a whole. Here we separate the Bloch phase into two parts to provide a convenient definition of the wavevectors k x and k y .
and similarly from Supplementary Equations 31 and 32 we derive: Combining Supplementary Equations 33, 36, 44 and 45 we obtain: By applying the choice of modulation phases as shown in Supplementary Figure 2c, we have φ B = φ B + 2π 3 and φ B = φ B − 2π 3 . b, b , b will then have the same phase α cos(φ B + k f Ω) resulting from the dynamic coupling.
In summary, the system of equations derived from Supplementary Equations 30-33 for the infinite 3D lattice is: Supplementary Equations 48-50 can be formulated as an eigenvalue problem from which we can determine the band structure that relates δ to (k x , k y , k f ). Here: where T 1,2,3 are defined in Supplementary Equations 48-50. k x , k y , k f thus play the role of Bloch wavevectors in characterizing the band structure. The solution to Supplementary Equation 51 defines the band structure for the three-dimensional lattice. Below we will refer to such a band structure as the "bulk" band structure.
Using the model developed above, we solve the band structure for the case of inversion-symmetry-breaking case and time-reversal breaking, as shown in Supplementary Figure 4.

Supplementary Note 4. Band Structure of a Stripe
In this section, we study the band structure and surface states of a lossless stripe of a honeycomb lattice of rings infinite in the spatial dimension x and the frequency dimension, and have N rows along y, as shown in Fig. 4a in the main text. The eigenstates are thus Bloch states with well-defined k x and k f . The wave function in a unit cell is characterized by {a j,n,0 , b j,n,0 |j = 1, 2, 3, 4; n = 1...N }'s, where the first subscript j labels the section in each ring, and the second subscript n labels the n-th row along y. The last subscript 0 labels a particular resonance with resonant frequency ω 0 chosen as the zeroth order mode.
Supplementary Equations 48-50 hold for all n, with k y = 0. The system of equations is: These equations can again be written in the form of an eigenvalue problem: where u = [a 1/2/4,1...N,0 , b 1/2/4,1...N,0 ] T and T = T (k x , k f ; γ, α, φ A , φ B ) captures the static and dynamic couplings and the boundary conditions. This eigenvalue problem defines a band structure with δ as a function of k x and k f . Below we refer to such a band structure as a projected band structure. The projected band structure contains the bulk states as defined by Supplementary Equation 51 projected onto the k x and k f plane, as well as additional surface states associated with the surface only. Comparisons between the band structure calculated using the waveguide amplitude mode and the tight binding model are shown in Supplementary Figures 5 and 6.

Supplementary Note 5. Surface State Transport
Now we consider a stripe of honeycomb lattice of lossy rings infinite in x, as shown in Fig. 4a in the main text. To excite the system, a set of continuous wave signals is sent into the rings on the edge of the stripe through waveguides coupled to the rings. The input signals have a spatial phase distribution in the x direction to probe states with Bloch wavevector k x . Input signal into each waveguide consists of two frequencies separated by Ω with tunable phase delay, to allow probing of surface states associated with different Weyl points.
We solve for the steady state modal amplitudes of the driven lossy system. The modal amplitudes characterizing the system are {a j,n,m , b j,n,m |j = 1...4, n = 1...N, m = 1...M }'s, where j labels the section on a ring, n labels the row along y and m labels the frequency modes. Only M frequencies closest to the excitation frequencies are considered, since the loss in the rings results in an exponential decay in the amplitudes of modes as a function of their frequency difference from the excitation frequencies.
Under the excitation with a constant k x and a small detuning δ from the resonant frequencies, Supplementary Equations 30-32 hold for all n and m: for all the modes except the ones that directly couple to inputs.
To excite the surface states on the top edge of the stripe, consider inputs to the row n = 1 for the A sublattice, using two frequencies ω M0 + δ and ω M0+1 + δ, where ω M0 and ω M0+1 are the frequencies of the M 0 -th and (M 0 + 1)-th resonant modes with ω M0+1 − ω M0 = Ω, and δ is a small detuning. The amplitudes of the excitation at these two frequencies are S M0 and S M0+1 , respectively. The boundary conditions are modified for the M 0 -th and (M 0 + 1)-th modes as: where η is the coupling between the input waveguides and the rings. By choosing the relative phase of the inputs S M0 and S M0+1 , we can excite certain k f modes selectively.
Similarly, to excite surface states on the bottom edge of the stripe, we can instead impose the following boundary conditions at the row n = N for the B sublattice for the M 0 -th and (M 0 + 1)-th modes:  For the case of inversion symmetry breaking, the Weyl points are near δ = 0, k x = 2π/3 √ 3a or 4π/3 √ 3a, and k f = ±π/2Ω. Input with δ = 0, k x = π/ √ 3a and a relative phase of S M0+1 /S M0 = ±i thus selectively couples to the surface states at the midpoint of the surface state arcs near k f = ±π/2Ω, respectively. In the presence of input to the top row of rings, steady state modal amplitudes are plotted in Figs. 4f,g of the main text. Supplementary Figures 5e,f show the steady states with input to the bottom edge. The resulting modal amplitudes are confined to the rings on the respective edges, demonstrating the existence of surface states. With a choice of input that selectively couples to k f = −π/2Ω, the system exhibits one-way frequency down conversion on the bottom edge, and one-way frequency up conversion on the top edge [ Fig. 4f in the main text]. Frequency conversion in the opposite direction is observed with a choice of input that selectively couples to k f = −π/2Ω. This proves that the surface states are one-way, and have different chirality near k f = ±π/2Ω. Supplementary Figures 6l,m show the case of time-reversal symmetry breaking with φ A = −φ B = π/3 and input to the top edge. In this case the Weyl points near k f = 0 and k f = π/Ω have frequencies close to δ/γ = ±1, respectively. To excite the modes at the midpoint of the surface state arcs near k f = 0, we choose δ = γ, k x = π/ √ 3a and S M0 /S M0+1 = 1. This leads to one-way frequency up-conversion, as shown in Supplementary Figure 6l. To excite the modes at the midpoint of the surface state arcs near k f = π/Ω, we choose δ = −γ, k x = π/ √ 3a and S M0 /S M0+1 = −1. This leads to one-way frequency down-conversion, as shown in Supplementary Figure 6m. These results again confirm the existence and one-way frequency conversion of surface states.
For the time-reversal symmetry breaking case, the excitation is not as well confined to the surface as in the case of inversion symmetry breaking. For the case of time-reversal symmetry breaking shown in Supplementary Figure 6, the third row of y(i), corresponding to the second row of A resonators, has non-zero intensity. The reason is the existence of bulk states at the same frequencies and k x as the surface arcs. For example in Supplementary Figure 6l, k x = π/ √ 3a, δ = γ and S M0+1 /S M0 = 1 is chosen to maximize excitation of the surface states near the midpoint of the pair of Weyl points at k f = 0. However, there are bulk states near k f = π Ω at the same frequency and k x . Although the excitation phase eliminates coupling to states with exactly k f = π Ω , coupling to other k f states are not completely suppressed. Some of the excitation leaks through these bulk states. In comparison, for the case of inversion symmetry breaking, the Weyl points are nearly degenerate. For δ ≈ 0 the bulk state density is very low, and is absent for most of the range of 2π/3 √ 3a < k x < 4π/3 √ 3a that supports surface states. This results in better edge confinement shown in Supplementary Figures 5e,f  To facilitate comparison between different systems, we normalize the frequency units with respect to the free spectral range (FSR) of the resonators. With this normalization, the inter-ring coupling constant t xy = γ×FSR, nearest-neighbour coupling is t f = J 1 (α)×FSR, and the internal loss rate is κ in = − ln(r)×FSR, where r is round-trip amplitude transmission in the ring. To preserve the validity of rotating wave approximation, the coupling strength and the modulation frequency has to satisfy [3] t xy , t f ≤ Ω/6 ≈FSR, where Ω = 2π×FSR is the modulation frequency. In order to observe one-way frequency conversion in edge state before loss become dominant, the system also need to satisfy t xy , t f κ in .
Here we consider a concrete example using silicon ring resonator on silicon-on-insulator platform. Supplementary Table 1 lists parameters for two possible designs using carrier-depletion modulator in silicon rib waveguide. In this system, the RF modulation frequency is limited to a few tens of GHz [6], which coincides with the free spectral range of the ring. Consequently, the length of the waveguide forming the ring needs to be on the order of one millimetre.
For a modulation frequency of 26 GHz, a simple ring with 445 µm radius is required, as experimentally demonstrated in [4]. As shown in Supplementary Figure 8, a 4 × 3 array of ring resonators are already sufficient for demonstrating the Fermi arc surface states, thus the overall dimension of the ring array is 3 × 4 mm 2 , which fits well on a single chip. On this length scale, however, fabrication-related variation may still lead to large resonant mismatch between rings [7][8][9]. Since the ring size is relatively large, and there are only 12 resonators, it is possible to use heaters to thermally aligned the resonances. In addition, one can also reduce the size of the resonator by folding the waveguide [10], e.g. as shown in Supplementary Figure 9, so that the resonator can be enclosed in a 150 µm radius area, 1/3 of its original size. With this, the whole array can be shrunk down to 1 mm in size. The size can be further reduced by increasing the RF modulation frequency to 50 GHz, which has been achieved in silicon carrier-depletion modulator [6] with careful optimization.
Index modulation in silicon is based on the plasma dispersion effect, which is intrinsically lossy. Carrier-depletion based modulation, which changes the width of the depletion region in a pn junction formed inside the waveguide, results in lower optical losses as compared to modulation schemes based on carrier injection. Loss due to carrier absorption is proportional to the carrier density, while coupling strength between adjacent modes separated in frequency by one FSR is proportional to both the carrier density and the modulation voltage. Coupling strength of 0.5 mm −1 accompanied by absorption loss of less than 1 dB mm −1 with 3 V modulation voltage has been demonstrated in silicon carrier-depletion modulator based Mach-Zehnder interferometer [5,6]. With such coupling and absorption loss, the requirements t xy , t f ≤FSR and t xy , t f κ in can be satisfied, with specific examples shown in Supplementary  Table 1. For a FSR=26 GHz ring with 1/3 of its circumference modulated, one may obtain a coupling constant t f = 8 GHz ≈ FSR/3, and a loss rate κ in = 2.7 GHz ≈ t f /3. An intrinsic quality factor (including waveguide loss and carrier absorption) of 2 × 10 5 is required, which is well within the reach of high Q passive silicon ring resonators [11,12], but still a few times higher than that demonstrated for active rings [4]. However, carrier-depletion based silicon ring modulator has not been well-optimized for the modulation frequency matching to the free-spectral range, thus careful design of doping concentration and waveguide cross-section [5] could in principle achieve the desirable quality factor in active ring modulators.
Using the coupling constant t f = 8 GHz and loss κ in = 2.7 GHz from the 26 GHz FSR example, we simulate the excitation of one-way frequency conversion in edge states on a 4 × 3 array of ring resonators with inversion-symmetrybreaking modulation phase. The result is shown in Supplementary Figure 8. We observe a good confinement to the edge, and one-way frequency conversion propagating over 6 cavity modes before the intensity is reduced below 1/10.
In our simulation shown in Supplementary Figure 8, the loss is large enough that the fields do not propagate to the boundary along the frequency axis, which may be caused by a sudden increase of refractive index dispersion or absorption. On the other hand, if the loss is low enough, and there is a sharp boundary along the frequency axis, then upon reaching the boundary, there is always scattering into the bulk states since the bulk is not gapped. In practice, such scattering into the bulk state may be small due to the small bulk density of states near the frequency range of the Weyl points. For system without inversion symmetry symmetry (Supplementary Figure 5), such a boundary also creates back-scattering since the same edge supports counter-propagating edge states at the same frequency. For system without time-reversal symmetry (Supplementary Figure 6), back-scattering does not occur since there is no counter-propagating edge state with the same frequency at the same edge as the forward propagating state.
While we use carrier-depletion modulation based silicon ring resonator as an example due to their compactness for on-chip integration and popularity as an integrated photonics platform, LiNbO 3 [13][14][15] and polymer [16] modulators may have high optical Q, stronger and intrinsically lossless electro-optic effect and higher modulation bandwidth, making them attractive platforms for demonstrating our proposed scheme for achieving Weyl point in synthetic frequency dimension as well.