Asymmetric topological pumping in nonparaxial photonics

Topological photonics was initially inspired by the quantum-optical analogy between the Schr\"odinger equation for an electron wavefunction and the paraxial equation for a light beam. Here, we reveal an unexpected phenomenon in topological pumping observed in arrays of nonparaxial optical waveguides where the quantum-optical analogy becomes invalid. We predict theoretically and demonstrate experimentally an asymmetric topological pumping when the injected field transfers from one side of the waveguide array to the other side whereas the reverse process is unexpectedly forbidden. Our finding could open an avenue for exploring topological photonics that enables nontrivial topological phenomena and designs in photonics driven by nonparaxiality.

Of particular interest is the topological pumping of light beams in periodically-modulated waveguide arrays 9,16,[33][34][35] , which may find applications in quantum information processing, such as preparing entangled states of photons 36,37 and transferring quantum states [38][39][40][41] . The first topological pumping scheme is known as Thouless pumping [42][43][44][45][46][47] , and it was proposed and realized in photonic systems [48][49][50][51] , in which the light field fills uniformly a bulk band and transfers by integer unit cells. An alternative topological pumping is to transfer a localized light field from one side to another, realized in optical frequency regime 16 . Up to now, the topological pumping of light was studied in the framework of quantum simulations, where specific optical properties do not play any role 38,[52][53][54][55] . In the lowfrequency regime, the nonparaxial light propagation cannot be simply mapped to a Schrödinger-type wave equation. For example, a recent study 56 revealed that a tightly focused nonparaxial light beam can enhance nonlinear optical interactions. However, no attempt has been done to explore the topological properties of nonparaxial light.
Here, we consider the case when the quantum-optical analogy is no longer valid, and when the paraxiality requirement is removed. We demonstrate experimentally a remarkable effect of asymmetric topological pumping when the injected field transfers from one side of the waveguide array to the other side, whereas the reverse process becomes forbidden. Intriguingly, the pump setup of the light field is designed and fabricated by varying the waveguide structural configurations according to the celebrated Rice-Mele model 57 , which predicts the symmetric topological pumping. The key to understanding this effect of asymmetric topological pumping is that an effective negative next-nearestneighboring (NNN) coupling is induced by nonparaxiality, which levitates the symmetric spectral spacing of the energy spectrum of the paraxial light beam. This spectral levitation allows to maintain one pump channel but block the other channel by obeying and violating adiabatic conditions, respectively, in an asymmetric fashion. Notably, the negative NNN coupling cannot arise from the direct coupling between waveguides 58 , but instead it stems from the nonparaxial beam propagation of microwaves. Indeed, our experimental results are nicely consistent with both simulations and analysis from the exact Helmholtz wave equation. We believe these findings may open an avenue to topological photonics, and thus may provide further insights for the study of topological phases and dynamics triggered by nonparaxiality of light.

Results
Model. To illustrate the topological photonic pumping in our microwave waveguide platform, we take advantage of the celebrated Rice-Mele model 57,59 . Two essential factors of the Rice-Mele pump setup are time-periodic staggered on-site potentials (β 1,2 ) and dimerized nearest-neighbor (NN) hoppings (κ 1,2 ), as shown schematically in Fig. 1a, b. By regulating the periodically curved waveguides into an array 60 , we can construct the timedependent NN hopping along the propagation direction z, where the z-direction acts as the synthetic time [see Fig. 1a]. The coupling constants between the jth and (j + 1)th waveguides can be approximated as κ j ðzÞ % κ 0 þ ðÀ1Þ j δκ cosð2πz=ΛÞ (see Fig. S1 for details). As to the periodically staggered on-site potential energy, the structural parameters of the waveguide vary along the zdirection to make the time-dependence of the propagation constant follow a sinusoidal function. The Rice-Mele setting can thus be performed by the flexible designs 61,62 of the periodic "Hshaped" structural unit imprinted on the microwave waveguides. Here, we alter the height parameter of the "H-shaped" unit as a sinusoidal function of the propagation direction z to obtain the periodically-modulated on-site potential in the Rice-Mele setting (see Fig. S2 for more details), leading to the desired on-site potentials β j ðzÞ % β 0 þ ðÀ1Þ j δβ sinð2πz=ΛÞ. Via the above grooves-corrugated waveguide design, our coupled waveguide array can offer us a photonic analog of the Rice-Mele model (as shown in Fig. S3), in which the corresponding tight-binding Hamiltonian of guiding mode amplitudesĉ j can be described as: For our setup, the experimental parameters are N = 10, Λ = 42 cm, κ 0 = 0.034 mm −1 , δκ = 0.021 cm −1 , β 0 = 0.47 mm −1 , and δβ = 0.03 mm −1 , and the distributions of the functions β 1,2 (z) and κ 1,2 (z) are explicitly presented in Fig. 1c, d, respectively. More specifically, κ 1,2 (z) depends not only on the spacings between waveguides, but also on the geometry of spoof surface plasmon polariton (SPP) structures. Therefore, the precise coupling function in Fig. 1d is a complicated one, as shown in Fig. S4. In such periodically-modulated waveguide arrays, the pump dynamics is governed by the function ψðzÞ with the time-ordering operator T . Microwaves injected from one side should be transported to the other side after a pump cycle, that is to say, the initial state ψ(0) is an instantaneous zero mode at one edge, and the final state ψ(T) becomes the other zero mode at the other edge after one cyclic evolution. Three points need to be emphasized for this prediction before proceeding to our experimental results. First, the paraxial approximation is taken into account to derive the above Hamiltonian within the coupled-mode theory. Second, the adiabatic pumping requires the adiabatic condition that the modulation frequency is much smaller than the minimal spectral spacing between the pump channel and bulk states. Third, in the pump process from the left boundary ψ L!R ðzÞ , and the reverse process from the right boundary ψ R!L ðzÞ , the spatial probability distributions obey the relation |ψ L→R (z, j)| 2 = |ψ R→L (z, N + 1 − j)| 2 , indicating that the pumping from the left to right edges should be symmetric to that from the right to left.
Experimental results. In our photonic Rice-Mele setting consisting of ten spoof SPP waveguides, microwaves are injected from the left (the 1st) and the right (the 10th) boundary waveguides to excite two pump channels, respectively, i.e., one from left to right and the counterpart one from right to left. We then carry out experimental near-field measurements of the field distribution of the electric component E y (see Fig. S5). We find that when injecting 17 GHz microwaves from the 1st waveguide, the near-field distribution E y shows that the microwave gradually transfers to the 10th waveguide, as like the microwave spreads into the bulk and converges again on the right boundary of the array [see Fig. 1e], which indeed confirms with the left-toright topological pump process. However, as shown in Fig. 1f, when this microwave is injected from the 10th waveguide, the propagation dynamics exhibits some scattering into the bulk waveguides and resides mainly near the right boundary waveguide without further transferring to the left boundary, reflecting the breakdown of the right-to-left pump channel. This failed pumping obviously contradicts the symmetric pumping prediction in our well-designed Rice-Mele setup as mentioned above. To this end, the observation of this dramatic asymmetric pump behavior is one of the main achievements of our work.
Energy spectrum and beam propagation. To explain the unexpected experimental observation of asymmetric pumping, an intuitive understanding suggests that the Rice-Mele Hamiltonian (1) is insufficient to fully describe the pump process. The Hamiltonian can be amended by adding the most natural term from the inevitable direct NNN coupling (κ NNN ) between the same sub-lattices [see Fig. 1(b)]. We notice that the direct NNN coupling should be always positive, as similar to the nearestneighboring coupling profile [see Fig. 1d]. However, as we demonstrate below, such a positive NNN coupling still insufficient to explain the experimental data.
To resolve the contradiction between the observed asymmetric pumping and the modified Rice-Mele model, we start by inspecting the particle-hole symmetric case with κ NNN = 0, as shown in Fig. 2a by the tight-binding band structure of the 10waveguide system as a function of the phase parameter ϕ ≡ 2πz/Λ. Pump channels from the left to right (red solid line) and from the right to left (blue dashed line) can be seen clearly in the bulk-band gap, which is symmetric mutually with the same magnitude and opposite sign of the energy dispersion. In Fig. 2d, by explicitly calculating the instantaneous wavefunction distribution at each values of ϕ, we simulate the propagating behaviors of microwaves injected from the left and right boundary waveguides, respectively. Obviously, both pump channels are unblocked with symmetric propagating patterns, and microwaves can thus be pumped from left to right and vice versa.
However, in the presence of a positive direct NNN coupling κ NNN > 0, which can be approximated by a constant (~0.007 mm −1 ) with small fluctuations depending on ϕ, the band structure is no longer symmetric, as shown in Fig. 2b. The bandwidth and overall level spacing of the valence (conduction) band at each ϕ get significantly compressed (enlarged). Notably, the spectral spacing between the left-to-right pump channel (red solid line) and the valence band is severely compressed, while the spectral spacing between the right-to-left pump channel (blue dashed line) and the conduction band is enlarged. According to the Landau-Zener-Stückelberg transition process 63 , the adiabatic condition would presumably be violated for the left-to-right pumping because of the severely-concentrated spectral spacing, but it remains satisfied for the right-to-left pumping. The asymmetric breakdown of the adiabatic condition further leads to the scattering of the light field in the left-to-right pump channel to the bulk, and thus blocks this channel. As illustrated in Fig. 2e, the numerical simulations indicate that the left-to-right pumping fails whereas the right-to-left pumping holds. Although the asymmetric pumping could emerge with the positive direct NNN coupling in the lower panel of Fig. 2e, it runs counter to the experimental observation in Fig. 1f, in which the right-to-left pump channel is blocked.
Therefore, in light of the above discussion, if the NNN coupling is negativeκ NNN < 0, in contrast to the positive case, we find that the spectral spacing between left-to-right (right-to-left) pump channel and valence (conduction) bands becomes enlarged (compressed), as shown in Fig. 2c. As a result, this leads to the compelling asymmetric pumping [ Fig. 2f], in consistent with the experimental observations.

Discussion
Now, a natural question arises that how the negative NNN coupling appears persuasively in the nonparaxial waveguide system? We recall that the paraxial approximation was adopted in the effective photonic tight-binding Hamiltonian (1) through the coupled-mode theory, which requires the stringent condition ∂ 2 ψ=∂z 2 ( 2k ∂ψ=∂z . Here, ψ is the field distribution of the electric component (e.g., E y ), k is the wavevector. However, for the microwave system with much smaller wavevectors, this condition is not satisfied, and we need to go beyond the coupledmode theory and start from the original exact two dimensional Helmholtz equation of the waveguide systems, where H ¼ 1 n 0 is the defined Helmholtz-Hamiltonian operator, n 0 is reference refractive index, and for monochromatic electromagnetic wave Eðx; z; tÞ ¼ ψðxÞe iωtÀikn 0 z , the wavevector is k = ω/c. Indeed, solving the exact Helmholtz equation can provide almost the same results as those observed experimentally (see Fig. S6 for details).
To understand qualitatively the origin of the negative NNN coupling, we replace the Helmholtz-Hamiltonian operator via the geometric modeling of the Rice-Mele Hamiltonian H RM in our system, and rewrite Eq. (2) in an effective Schrödinger-type form In Eq. (2) the nonparaxial term, contributes to this iteration form, which can be formally expressed as an effective H eff by using many perturbation methods. Full details of Eq.  Table S1). Intriguingly, even though H RM contains positive NN coupling terms, it is found analytically that negative NNN couplings term can already emerge in the (1,1)-order Padé approximated Hamiltonian in which the term À 1 2kn 0 H 2 RM contributes a dominant negative NNN coupling, so that,κ NNN $ Àκ 2 0 =ð2kn 0 Þ. We also obtain the NNN (κ NNN ) term directly from the exact Helmholtz equation. The logical and pedagogical steps withκ NNN are described in Supplementary Materials. Therefore, the effective negative NNN couplings are qualitatively related to the intrinsic breakdown of the paraxial approximation in the microwave regime, as shown in Fig. S7 for details. Besides, when increasing the wavevector k (decreasing the NN coupling κ NN ) to some value, for instance, entering the optical infrared regime (enlarging the waveguide spacing), the effective negative NNN couplings can be neglected, and the problem is reduced to the paraxial approximation. To further confirm the above explanation, as shown in Figs. S8 and S9, we present the simulation results of the optical analog of the pump process with a much higher frequency at near-infrared regime (λ = 1.55 μm) and different waveguide spacing of the propagating electromagnetic waves. In this setup, the optical waveguide configurations are also deployed based on the Rice-Mele model, and the paraxial approximation is more strictly satisfied and the negative NNN coupling is negligible, so that the pump behaviors of both two channels appear to be symmetric as expected. Additionally, to exclude the nonadiabatic driving as the reason for asymmetry, we extend the propagation distances from 42 cm to 60 cm and 80 cm, as shown in Fig. S10, which can further confirm the nonparaxial mechanism of asymmetric pumping. We reveal that the asymmetric pump dynamics is ubiquitous in various waveguide designs and it should also be expected in other photonic waveguide systems where the paraxial approximation is violated. Figure 3a, b, c demonstrate three types of modulations of height, period and girder width of the "H-shaped" unit microstructure, to achieve the periodic on-site potentials in the photonic Rice-Mele model (Figs. S3 and S6 show the structural designs and experimental results measured at 17 GHz). For all the cases, the microwave field injected from the left boundary waveguide can almost be transferred to the right boundary, see Fig. 3d, e, f. However, the opposite right-to-left pump channels are blocked, in which the input microwave signal is scattered into a bulk, see Fig. 3g, h, i. For a better illustration of the difference between two pump channels of the three array designs, we plot the instantaneous intensity distributions in the arrays at the beginning (ϕ = 0), middle (ϕ = π), and end (ϕ = 2π) of the pump cycle, as presented in Fig. 3j, k, l.
In summary, we have predicted the asymmetric topological pumping of light extending substantially the conventional framework of topological photonics beyond the paraxial approximation. We have verified this concept experimentally for microwave waveguide arrays, and have demonstrated that the effect of nonparaxiality can block or open the topological pump channels by altering the adiabatic passage. Our results suggest that the existing quantum-optical analogy widely used to characterize topological phases in photonics should be extended to the nonparaxial regime described by the exact Helmholtz equation. This would bring unexplored richness to the well-studied quantum simulations of topological phases in microwave photonic lattices.

Data availability
The data supporting the findings of this study are available from the open Digital Repository at https://doi.org/10.17605/OSF.IO/58FGY.

Code availability
All numerical codes are available from the corresponding authors on reasonable request. Re(E y ) -1 1 Fig. 3 Experimental observation of asymmetric topological pumping. Three types of photonic-lattice configurations are considered, including engineering of height, period, and girder width profiles of the "H-shaped" structures in ultra-thin corrugated metallic coupled waveguide arrays. Schematic of altering a height, b period, and c girder width. d-f Near-field evolution of the measured electric field E y at the excited frequency 11 GHz from the 1st site shows the openness of the pump channel. g-i Near-field evolution of the measured E y from the 10th site show the disability of the pumping. Correspondingly, the two upper and lower panels in j, k, and l show the instantaneous field distributions at the propagation distances z = 0, 21, 42 cm, as shown by the black dotted lines in d-i, respectively, which are regarded as the chosen phase ϕ in one driven period equals to 0, π, and 2π, respectively.