Introduction

The field of topological photonics1,2,3,4,5 emerged after the discovery of topological insulators for electrons6,7, and it utilizes topology and topological phases to tailor behaviors of light in an unusual way. Seminal studies in topological photonics include quantum simulations of topological edge states8,9,10,11,12,13,14 and topological pumping15,16,17,18,19,20,21. Similarities of those concepts are based primarily on a direct correspondence between the dynamics of the electron wavefunction governed by the Schrödinger equation and the propagation of a light beam described by the paraxial Helmholtz equation. Beyond quantum simulations, unusual topological effects in photonics are generated by intrinsic non-Hermiticity such as topological lasers22,23,24,25,26,27, nonlinear response including active tunability25,26,28, nonreciprocity29,30, and frequency conversion31,32.

Of particular interest is the topological pumping of light beams in periodically-modulated waveguide arrays9,16,33,34,35, which may find applications in quantum information processing, such as preparing entangled states of photons36,37 and transferring quantum states38,39,40,41. The first topological pumping scheme is known as Thouless pumping42,43,44,45,46,47, and it was proposed and realized in photonic systems48,49,50,51, 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 regime16. Up to now, the topological pumping of light was studied in the framework of quantum simulations, where specific optical properties do not play any role38,52,53,54,55. In the low-frequency regime, the nonparaxial light propagation cannot be simply mapped to a Schrödinger-type wave equation. For example, a recent study56 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Here, we consider the case when the quantum-optical analogy is no longer valid, and when the paraxiality requirement is removed.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 model57, which predicts the symmetric topological pumping. The key to understanding this effect of asymmetric topological pumping is that an effective negative next-nearest-neighboring (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 waveguides58, 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 model57,59. Two essential factors of the Rice-Mele pump setup are time-periodic staggered on-site potentials (β1,2) and dimerized nearest-neighbor (NN) hoppings (κ1,2), as shown schematically in Fig. 1a, b. By regulating the periodically curved waveguides into an array60, we can construct the time-dependent 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 \({\kappa }_{j}(z)\approx {\kappa }_{0}+{(-1)}^{j}\delta \kappa \cos (2\pi z/{{\Lambda }})\) (see Fig. S1 for details). As to the periodically staggered on-site potential energy, the structural parameters of the waveguide vary along the z-direction to make the time-dependence of the propagation constant follow a sinusoidal function. The Rice-Mele setting can thus be performed by the flexible designs61,62 of the periodic “H-shaped” 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 \({\beta }_{j}(z)\approx {\beta }_{0}+{(-1)}^{j}\delta \beta \sin (2\pi z/{{\Lambda }})\). 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 amplitudes \(\hat{c}_j\) can be described as:

$${H}_{{{{\rm RM}}}}= \mathop{\sum }\limits_{j=1}^{N}\left[{\beta }_{0}+{(-1)}^{j}\delta \beta \sin (2\pi z/{{\Lambda }})\right]{\hat{c}}_{j}^{{{{\dagger}}} }{\hat{c}}_{j}\\ +\mathop{\sum }\limits_{j=1}^{N-1}\left[{\kappa }_{0}+{(-1)}^{j}\delta \kappa \cos (2\pi z/{{\Lambda }})\right]{\hat{c}}_{j}^{{{{\dagger}}} }{\hat{c}}_{j+1}+{{{{{{{\rm{H.c.}}}}}}}}.$$
(1)

For our setup, the experimental parameters are N = 10, Λ = 42 cm, κ0 = 0.034 mm−1, δκ = 0.021 cm−1, β0 = 0.47 mm−1, and δβ = 0.03 mm−1, and the distributions of the functions β1,2(z) and κ1,2(z) are explicitly presented in Fig. 1c, d, respectively. More specifically, κ1,2(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.

Fig. 1: Asymmetric topological pumping in an array of SPPs waveguides.
Fig. 1: Asymmetric topological pumping in an array of SPPs waveguides.
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a Schematic illustration of the photonic Rice-Mele model consisting of periodically bending waveguides, with sinusoidal modulations of the nearest-neighbor (NN) waveguide spacing Gz G(z) and height parameters of the “H-shaped” unit along the propagation direction z. b Illustration of the extended Rice-Mele model with staggered on-site potentials (propagation constants β1,2), NN coupling strengths (κ1,2), and next-nearest-neighbor coupling strength (κNNN). The detailed values of the sinusoidally modulated (c) propagating constants and (d) coupling strengths κNN and κNNN along the propagation direction z. Experimental data for the near-field measurements of the electric field (Ey, perpendicular to the array surface) after injecting 17 GHz microwaves from e left boundary (the 1st waveguide) and f right boundary (the 10th waveguide), respectively.

In such periodically-modulated waveguide arrays, the pump dynamics is governed by the function \(\left|\psi (z)\right\rangle ={{{{{{{\mathcal{T}}}}}}}}\exp (i\int\nolimits_{0}^{z}{H}_{{{\rm RM}}}(z^{\prime} )dz^{\prime} )\left|\psi (0)\right\rangle\) with the time-ordering operator \({{{{{{{\mathcal{T}}}}}}}}\). 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 \(\left|{\psi }_{L\to R}(z)\right\rangle\), and the reverse process from the right boundary \(\left|{\psi }_{R\to L}(z)\right\rangle\), the spatial probability distributions obey the relation ∣ψL→R(z, j)∣2 = ∣ψR→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 Ey (see Fig. S5). We find that when injecting 17 GHz microwaves from the 1st waveguide, the near-field distribution Ey 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-to-right 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 (κNNN) between the same sub-lattices [see Fig. 1(b)]. We notice that the direct NNN coupling should be always positive, as similar to the nearest-neighboring 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 κNNN = 0, as shown in Fig. 2a by the tight-binding band structure of the 10-waveguide 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.

Fig. 2: Energy spectrum and beam propagation for topological pumping.
Fig. 2: Energy spectrum and beam propagation for topological pumping.
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The energy spectrum for a conventional Rice-Mele model without next-nearest-neighbor (NNN) hopping, and b, c extended Rice-Mele model with positive (κNNN = 0.007 mm−1) and negative (\({\tilde{\kappa }}_{{{{\rm NNN}}}}={{{{{{{\rm{-}}}}}}}}0.007  {{{\rm{mm}}}}^{-1}\)) NNN hopping terms, respectively. Two symmetric pump channels are shown in a but only one of the channels (b, c) remains when adding the NNN hopping. Correspondingly, the beam propagation on these photonic lattices demonstrate two symmetric pumping in d and asymmetric pumping in e, f. The beam propagation in the blocked channels [the upper panel in e, and the lower panel in f] show the topological edge state confinement along the array boundaries. The simulation parameters are taken from the experiment.

However, in the presence of a positive direct NNN coupling κNNN > 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 process63, 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 negative \({\tilde{\kappa }}_{{{{\rm NNN}}}} \, < \, 0\), 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 \(\left|{\partial }^{2}\psi /\partial {z}^{2}\right|\ll 2k\left|\partial \psi /\partial z\right|\). Here, ψ is the field distribution of the electric component (e.g., Ey), 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 coupled-mode theory and start from the original exact two dimensional Helmholtz equation of the waveguide systems,

$$i\frac{\partial \psi }{\partial z}-\frac{1}{2k{n}_{0}}\frac{{\partial }^{2}\psi }{\partial {z}^{2}}=H\psi ,$$
(2)

where \(H=\frac{1}{2k{n}_{0}}{\nabla }_{x}^{2}+\frac{k}{2}\left(\frac{{n}^{2}-{n}_{0}^{2}}{{n}_{0}}\right)\) is the defined Helmholtz-Hamiltonian operator, n0 is reference refractive index, and for monochromatic electromagnetic wave \(E(x,z;t)=\psi (x){e}^{i\omega t-ik{n}_{0}z}\), 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 HRM in our system, and rewrite Eq. (2) in an effective Schrödinger-type form

$$i\frac{\partial \psi }{\partial z}=\frac{{H}_{{{{\rm RM}}}}}{1+i\frac{1}{2k{n}_{0}}\frac{\partial }{\partial z}}\psi ={H}_{{{{\rm eff}}}}\psi .$$
(3)

In Eq. (2) the nonparaxial term, contributes to this iteration form, which can be formally expressed as an effective Heff by using many perturbation methods. Full details of Eq. (2) and Eq. (3) are provided in Supplementary Materials. It should be emphasized that the above iterative equation remains exact beyond the paraxial approximation, and Heff can be treated by the known Padé approximation58. The Padé approximation can provide an effective Hamiltonian \({H}_{\,{{{\rm eff}}}\,}^{(m,n)}=N(m)/D(n)\), where N(m), and D(n) are polynomials of HRM (The lower (m,n)-order Padé approximations are listed in Table S1). Intriguingly, even though HRM 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

$${H}_{{{{\rm eff}}}}^{(1,1)}\approx {H}_{{{{\rm RM}}}}-\frac{1}{2k{n}_{0}}{H}_{{{{\rm RM}}}\,}^{2},$$
(4)

in which the term \(-\frac{1}{2k{n}_{0}}{H}_{\,{{{\rm RM}}}\,}^{2}\) contributes a dominant negative NNN coupling, so that, \({\tilde{\kappa }}_{NNN} \sim -{\kappa }_{0}^{2}/(2k{n}_{0})\). We also obtain the NNN (\({\tilde{\kappa }}_{{{{\rm NNN}}}}\)) term directly from the exact Helmholtz equation. The logical and pedagogical steps with \({\tilde{\kappa }}_{{{{\rm NNN}}}}\) 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 κNN) 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.

Fig. 3: Experimental observation of asymmetric topological pumping.
Fig. 3: Experimental observation of asymmetric topological pumping.
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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 Ey 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 Ey 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.

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.