Abstract
Nonlinear transmission lines (NLTLs) are nonlinear electronic circuits used for parametric amplification and pulse generation, and it is known that lefthanded NLTLs support enhanced harmonic generation while suppressing shock wave formation. We show experimentally that in a lefthanded NLTL analogue of the SuSchriefferHeeger (SSH) lattice, harmonic generation is greatly increased by the presence of a topological edge state. Previous studies of nonlinear SSH circuits focused on solitonic behaviours at the fundamental harmonic. Here, we show that a topological edge mode at the first harmonic can produce strong propagating higherharmonic signals, acting as a nonlocal crossphase nonlinearity. We find maximum thirdharmonic signal intensities five times that of a comparable conventional lefthanded NLTL, and a 250fold intensity contrast between topologically nontrivial and trivial configurations. This work advances the fundamental understanding of nonlinear topological states, and may have applications for compact electronic frequency generators.
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Introduction
Topological edge states—robust bound states guaranteed to exist at the boundary between media with topologically incompatible band structures—were first discovered in condensed matter physics^{1}. Recently, electronic LC circuits have emerged as a highly promising method of realizing these remarkable phenomena^{2,3,4,5,6,7,8,9}. Compared to other classical platforms like photonics^{10,11,12,13}, acoustics^{14,15,16}, and mechanical lattices^{17,18,19}, which have also been used to realize topologically nontrivial band structures and topological edge states, electronic circuits have several compelling advantages: extreme ease of experimental analysis; the ability to fabricate complicated structures via printed circuit board (PCB) technology; and the intriguing prospect of introducing nonlinear and/or amplifying circuit elements to easily study how topological edge states behave in novel physical regimes. Notably, circuits have been used to study the Su–Schrieffer–Heeger (SSH) chain (the simplest onedimensional topologicallynontrivial lattice)^{4,20}, nonlinear SSH chains supporting solitonic edge states^{8}, twodimensional topological insulator lattices^{2}, and the corner states of highorder topological insulators^{5,9}.
One of the most interesting questions raised by the emergence of topologically nontrivial classical lattices is how topological edge states interact with nonlinear media. Previous studies have focused on nonlinearityinduced local selfinteractions in the fundamental harmonic, which can give rise to solitons with anomalous plateaulike decay profiles in nonlinear SSH chains^{8,21}, or chiral solitons in twodimensional lattices^{22,23,24,25,26,27}. It has also been suggested that topological edge states in nonlinear lattices could be used for robust travelingwave parametric amplification^{28}, optical isolation^{29}, and other applications^{30,31,32,33}.
In this paper, we report on the implementation of a nonlinear SSH chain based on a lefthanded nonlinear transmission line (NLTL)^{34,35,36,37,38,39,40,41}, in which the topological edge state induces highly efficient harmonic generation. Although previous studies have emphasized the role of local selfinteractions, including in a previous demonstration of a nonlinear SSH circuit based on weaklycoupled LC resonators^{8}, an important feature of our circuit is the decisive role of higherharmonic signals in modulating the firstharmonic modes: they can drive the entire lattice, not just the edge, deeper into the nontrivial regime at the firstharmonic frequencies. This behavior is aided by the fact that the lefthanded NLTL has an unbounded dispersion curve supporting travelingwave higherharmonic modes^{38,39,40,41,42}.
Our measurements on the nonlinear circuit reveal a firstharmonic mode that is localized to the lattice edge, similar to a linear topological edge state, as well as higherharmonic waves that propagate into the lattice bulk and have voltage amplitudes reaching over an order of magnitude larger than the firstharmonic signal. The intensity of the generated thirdharmonic signal has a maximum of ≈2.5 times that of the input firstharmonic signal, compared to <0.5 for a comparable conventional lefthanded NLTL without a topological edge state. The important role played by the topological edge state is further demonstrated by the fact that the thirdharmonic intensity is 250 times larger than in a trivial circuit, which has equivalent parameters but lacks a topological edge state in the linear limit, using the same input parameters.
Results
Circuit design
The transmission line circuit is shown schematically in Fig. 1a. It contains inductors of inductance L and capacitors of alternating (dimerized) capacitances C_{a} and C_{b}. We will shortly treat the case where the C_{b} capacitors are nonlinear (the L and C_{a} elements are always linear). First, consider the linear limit where C_{b} is a constant. We define the characteristic angular frequency ω_{a} = (LC_{a})^{−1/2}, and the capacitance ratio
The case of α = 1 corresponds to a standard (nondimerized) lefthanded transmission line. This type of transmission line is characterized by having sites separated by capacitors, and connected to ground by inductors, rather than vice versa. Lefthanded NLTLs have been shown to be useful for parametric amplification and pulse generation^{38,39,40,41}.
Let us treat the points adjacent to the capacitors as lattice sites, indexed by an integer k, and close the circuit by grounding the edges [the left edge is the site labeled A in Fig. 1a]. Using Kirchhoff’s laws, we can show that a mode with angular frequency ω satisfies (see Supplementary Note 1):
where v_{k} denotes the complex voltage on site k. The matrix \({\cal H}\) has the form of the SSH Hamiltonian:
Thus, the eigenfrequency modes of the circuit have a onetoone correspondence with the SSH eigenstates.
The band diagram for the linear closed circuit is shown in Fig. 1d. The lack of an upper cutoff frequency is a characteristic of lefthanded transmission lines^{40}. There is a bandgap in the range \(\omega _a{\mathrm{/}}\sqrt 2 < \omega < \sqrt {\alpha /2} {\kern 1pt} \omega _a\). For α > 1, the bandgap contains edge states, which are zeroeigenvalue eigenstates of \({\cal H}\) that can be characterized via a topological invariant derived from the Zak phase^{1}. The edge state’s angular frequency is
Note that the edge state are not at zero frequency, nor do they lie at precisely the middle of the bandgap; this is due to the aforementioned mapping from the circuit equations to the SSH model—specifically, the fact that ω is not the eigenvalue in Eq. (2).
For α < 1, there is a finite bandgap below \(\omega _a{\mathrm{/}}\sqrt 2\), which is topologically trivial and contains no edge states. If we swap the two types of capacitors, so that the C_{b}type capacitors are the ones at the edge, then the α > 1 bandgap is trivial and the α < 1 bandgap nontrivial, as shown in Fig. 1e.
Next, consider a nonlinear circuit with each C_{b} capacitor consisting of a pair of backtoback varactors. The nonlinear capacitance C_{b} decreases with the magnitude of the bias voltage (the voltage between the endpoints of the capacitor), as shown in Fig. 1c. For theoretical analyses, it is convenient to model this nonlinearity by
where α_{nl}(t) ≡ C_{a}/C_{b}(t), and ΔV(t) is the bias voltage. The key feature of the nonlinearity is that at higher voltages, the effective value of α increases. Depending on the chosen boundary conditions, this drives the circuit deeper into the topologically trivial or nontrivial regime.
Experimental results
The implemented NLTL, shown Fig. 1b, contains a total of 40 sites, or 20 unit cells. The linear circuit elements have L = 1.5 μH and C_{a} = 47 pF, so that ω_{a}/2π ≈ 19 MHz. By fitting Eq. (5) to manufacturer data for the varactors at low bias voltages (see Supplementary Note 3), we obtain A = 1.32 and B = 0.51 V^{−2} (thus, in the linear limit, α ≈ 1.3 > 1). The fitted capacitance–voltage relation is shown in Fig. 1c.
We supply a continuouswave sinusoidal input voltage signal, with tunable frequency f_{in} and amplitude V_{in}, to either of the points labeled A and B in Fig. 1a. This allows us to study the cases corresponding to Fig. 1d, e, which we refer to as the “nontrivial” and “trivial” lattices, respectively (see Methods). In both cases, the input site is denoted as k = 0.
A typical set of measurement results is shown in Fig. 2a–c, for f_{in} = 16 MHz and V_{in} = 2.5 V. On each site k, the spectrum of the voltage signal is shown in Fig. 2c, with prominent peaks at odd harmonics (f_{in}, 3f_{in}, 5f_{in}, etc.); even harmonics are suppressed due to the symmetry of the capacitance–voltage relation^{39}. Focusing on the first and third harmonics, we define the respective peak values as \(\left {v_{k}^{\, f}} \right\) and \(\left {v_k^{3f}} \right\), and use these to plot Fig. 2a, b. We verified that these experimental data agree well with results from the SPICE circuit simulator (see Supplementary Note 5).
From Fig. 2a, b, we see that the nontrivial and trivial lattices exhibit very different behaviors for both the first and thirdharmonic signals. First, consider the firstharmonic signal. In both lattices, there is an exponential decay away from the edge, but the decay is sharper in the nontrivial lattice, which may be attributed to the enhanced intensity arising from the coupling of the input signal to the topological edge state. As a quantitative measure of the localization of the firstharmonic signal, Fig. 2d, e shows the inverse participation ratio (IPR) \(\mathop {\sum}\nolimits_k \left {v_k^f} \right^4{\mathrm{/}}\left( {\mathop {\sum}\nolimits_k \left {v_k^f} \right^2} \right)^2\); a larger IPR corresponds to a more localized profile^{43}. We see that the IPR is substantially larger in the nontrivial lattice than in the trivial lattice, over a broad range of f_{in} and V_{in}. The strong difference in localization is a key signature of nonlinearity: in the linear regime, a driving voltage on the edges of the nontrivial and trivial lattices would produce different overall amplitudes, but the same exponential decay profile (see Supplementary Note 1). It is interesting to note that the region of enhanced IPR, shown in Fig. 2d, closely resembles the nontrivial bandgap in Fig. 1d.
We can also see from Fig. 2a, c that strong higherharmonic signals are present in the nontrivial lattice. Moreover, Fig. 2a indicates that the thirdharmonic signal is extended, not localized to the edge. To understand this in more detail, we define
which quantifies the intensity of the thirdharmonic signal relative to the input intensity at the first harmonic. Here, 〈⋯〉 denotes an average over the first ten lattice sites. Figure 3a, b plots the variation of χ with f_{in} and V_{in}. In the nontrivial circuit, the maximum value of the normalized intensity is χ ≈ 2.5 for f_{in} ~ 16 MHz and 1 V ≲ V_{in} ≲ 4 V. The fact that χ peaks over a relatively narrow frequency range, as shown in Fig. 3a, may be a finitesize effect: the highfrequency modes of the lattice form discrete subbands due to the finite lattice size [see Fig. 1d, e]. In computer simulations, we obtained a similar maximum value of χ ≈ 2.4 for the nontrivial lattice, whereas a comparable lefthanded NLTL of the usual design (containing only identical nonlinear capacitances) has maximum χ ≈ 0.47 (see Supplementary Note 7).
The trivial lattice exhibits a much weaker thirdharmonic signal. As indicated in Fig. 3c, for certain choices of f_{in} and V_{in}, the value of χ in the nontrivial lattice is 200 times that in the trivial lattice. Figure 3d plots the normalized thirdharmonic signal intensities versus the site index k, showing that they do not decay exponentially away from the edge. In the nontrivial lattice, the normalized thirdharmonic signal increases with V_{in} (i.e., stronger nonlinearity).
Discussion
Our results point to a complex interplay between the topological edge state and higherharmonic modes in the SSHlike NLTL. When a topological edge state exists in the linear lattice, it can be excited by an input signal at frequencies matching the bandgap of the linear lattice. The importance of the edge state is evident from the comparisons between the topologically trivial and nontrivial lattices (Figs. 2, 3). Note also that when the excitation frequency lies outside the linear bandgap, the two lattices behave similarly and the harmonic generation is relatively weak.
In the topologically nontrivial lattice, the resonant excitation generates third and higherharmonic signals that penetrate deep into the lattice, unlike the firstharmonic mode which is localized to the edge. Away from the edge, the higherharmonic signals become stronger than the first harmonic, and hence dominate the effective value of the nonlinear α parameter. In the linear lattice, α is the parameter that drives the topological transition, and increasing α leads to a larger bandgap and hence a more confined edge state. In the nonlinear regime, Fig. 2 shows an orderofmagnitude increase in the thirdharmonic signal amplitude in the nontrivial lattice, relative to the trivial lattice; this implies an effective increase in α, and indeed we see that the firstharmonic mode profile is more strongly localized. A more localized edge state, in turn, produces a stronger response to an input signal.
The above interpretation is supported by a more detailed analysis of the coupled equations governing the different circuit mode harmonics (see Supplementary Notes 2–4). These equations involve an effective α parameter whose approximate value, in the nth unit cell, is \(\left\langle {\alpha _n} \right\rangle \approx A + 2B\mathop {\sum}\nolimits_m \left {W_n^m} \right^2\), where \(\left {W_n^m} \right\) is the mth harmonic of the bias voltage on the nonlinear capacitor in the nth unit cell, and m = 1, 3, 5, … We are able to show that propagating waves can be selfconsistently realized for higher (m ≥ 3) harmonics in the presence of nonlinearity, even if the fundamental (m = 1) mode only has decaying solutions. The firstharmonic mode is localized to the edge, with localization length decreasing with 〈α_{n}〉 in a manner similar to the linear SSHlike lattice. The generation of the higherharmonic signals occurs mainly near the edge of the lattice, where the firstharmonic mode is largest. The nonlinearityinduced harmonic generation is aided by the wellknown fact that the SSH edge state changes sign in each unit cell, corresponding to the fact that the gap closing in the SSH model takes place at the corner of the Brillouin zone^{20}. This feature increases the bias voltages across the nonlinear capacitors, which can thus exceed the values of the voltages at individual sites.
The input signal can also be applied to the middle of the lattice. In this context, it is interesting to note that when we choose to excite a single site in the bulk of an SSHlike lattice, the sections to either side of the excitation have different topological phases: either trivial on the left and nontrivial on the right, or vice versa, depending on the two possible choices of excitation site. If the source impedance is sufficiently low, the effect is similar to exciting independent chains to the left and right; thus, the enhanced higherharmonic signal is preferentially emitted toward the topologically nontrivial side (see Supplementary Note 6).
The presence of higherharmonic signals distinguishes our system from previous studies of nonlinear topological edge states, which were based on nonlinear selfmodulation at a single harmonic. For instance, in a nonlinear SSH lattice where the coupling depends on the local intensity of a single mode, solitonlike edge states with anomalous mode profiles were predicted^{21}, and subsequently verified using a NLTLlike circuit^{8}. That circuit, unlike ours, had narrow frequency bands and thus did not support propagating higherharmonic modes. Topological solitons based on nonlinear selfmodulation are also predicted to exist in higherdimensional lattices^{22,23,24,25,26,27}. In our case, the effective value of α away from the edge is dominated by the higherharmonic signals; from the point of view of the firstharmonic mode, these act as a nonlocal nonlinearity, driving the entire lattice deeper into the topologically nontrivial regime, not just the sites with large firstharmonic intensity.
Our work opens the door to the application of topological edge states for enhancing harmonic generation, not just in transmission line circuits, but also a variety of other interesting systems. These include twodimensional electronic lattices, where topological edge states have already been observed in the linear regime^{2}, and the unidirectional nature of the edge states may be even more beneficial for frequencymixing^{28}. Higher dimensional circuit lattices may possess different thresholds for bulk propagation in different directions, with an extreme generalization being that of a corner mode circuit constructed in ref. ^{4}. Electronic circuits incorporating amplifiers and resistances may also be able to explore behaviors analogous to topological lasers^{44,45,46,47,48}, combining topological states with both nonlinearity and nonHermiticity. Finally, circuits containing varactors that are explicitly timemodulated may be suitable for generating synthetic dimensions to realize topological features in higher dimensions^{49,50,51,52,53,54}.
Methods
Sample fabrication and experimental procedure
The NLTL was implemented on a PCB (Seeed Tech. Co.), with each nonlinear capacitor consisting of a pair of backtoback varactors (Skyworks Solutions, SMV1253004LF). The transmission line, as fabricated, is topologically nontrivial, as shown in Fig. 1a. To probe the trivial circuit, we use a switch to add one sublattice unit cell at the rightmost end of the transmission line, and disconnect the leftmost C_{a} and L in Fig. 1a. This yields a nontrivial circuit of same length, with the C_{a} and C_{b} capacitors swapped.
A function generator (Tektronix AFG3102C) supplies the continuouswave sinusoidal input voltage, and the voltages on successive lattice sites, k ≥ 1, are measured by an oscilloscope (Rohde & Schwarz RTE1024) in highimpedance mode. Numerical results were obtained using the SPICE circuit simulator.
Data availability
Raw experimental data and Python code used to generate all plots can be found at https://doi.org/10.21979/N9/I74ZP1. All other data are available from the authors upon reasonable request.
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Acknowledgements
The authors are grateful to H. Wang, D. Leykam, and Z. Gao for helpful discussions. Y.W., L.J.L., B.Z., and Y.D.C. were supported by the Singapore MOE Academic Research Fund Tier 2 Grant MOE2015T22008, the Singapore MOE Academic Research Fund Tier 3 Grant MOE2016T31006, and the Singapore MOE Academic Research Fund MOE2018T21022(S).
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Y.W. designed the printed circuit board and performed the measurements. Y.W. and L.J.L. analyzed the data. B.Z. and Y.D.C. supervised the project. Y.W., L.J.L., C.H.L., B.Z., and Y.D.C. participated in the design and interpretation of the experiment, and all contributed substantially to the work.
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Wang, Y., Lang, LJ., Lee, C.H. et al. Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial. Nat Commun 10, 1102 (2019). https://doi.org/10.1038/s41467019089669
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DOI: https://doi.org/10.1038/s41467019089669
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