Experimental demonstration of the robust end state in a split-ring-resonator chain

One of the fascinating topological phenomena is the end state in one dimensional system. In this work, the topological photonics in the dimer chains composed by the split ring resonators are revealed based on the Su-Schrieffer-Heeger model. The topologically protected photonic end state is observed directly with the in situ measurements of the local density of states in the topological nontrivial chain. Moreover, we experimentally demonstrate that the end state localized at both ends is robust against a varied of perturbations, such as loss and disorder. Our results not only provide a versatile platform to study the topological physics in photonics but also may have potential applications in the robust communication and power transfer.


I. INTRODUCTION
The rise of topological photonics, 1,2 is along with the discovery of various topological phases in condensed matter physics. Topology, as a new concept in photonics, can be used to effectively control light-matter interaction and make use for robust one-way transmission. Recently, photonic topological one-way edge modes with broken time-reversal (T-reversal) symmetry have been widely studied in theory 3 and experiment 4-6 based on photonic crystals composed by gyro-magnetic materials and helical optical fibers, respectively. The quantum spin Hall effect (QSHE), 7,8 time-reversal symmetric electron systems with nontrivial topological properties, also attracted people's great attention for its novel spin-dependent topological phases.
Inspiring by the QSHE of electron, up to now, a variety of optical analogue have been proposed, by use of metamaterials, [9][10][11][12] coupled ring resonators array, [13][14][15] synthetic gauge field, 16,17 two-dimension (2-D) photonic crystal. 18 In addition to the edge state in the 2-D system, the topological states in the one dimensional (1-D) system also have been demonstrated by metamaterials, 19 resonant dielectric structure, [20][21][22] photonic/phononic crystals, [23][24][25] and 1-D waveguide array. 26,27 The topological interface state between two crystals with distinct topological gap has been demonstrated, which can be used for the field enhancement. 25 Very recently, this interface state has been used to determine the topological invariants of the polaritonic quasicrystals, 28 and the chiral edge states in 1-D double coupled Peierls chain have been observed for the first time. 29 Topological states are protected by the topological phase transition across the interface. Thus they are robust against the defects, the loss, and the disorder, which do not change the topological phase of the structure. 21,24,30 In 1-D system, the robustness of the topological interface state has been investigated in dielectric resonator chains and photonic crystals. 21,24 In this work, we experimentally demonstrate the robust end states of the Su-Schrieffer-Heeger (SSH) mode in a split-ring-resonator (SRR) chain.
SRRs with broken rotational symmetry provide a new degree of freedom, the azimuth in addition to the distance and the background between the resonators, to adjust the coupling in the chain. Based on the SRR chains with limited length, we systemically studied the robustness of end states which are insensitive to a variety of perturbations, such as loss and disorder in the structure. Our results provide a versatile platform to observe the robust topological states in photonics. In addition, the robust end states at the two ends of a chain may have some potential applications in the information transmission, power transfer, topological gap soliton and so on.
The paper is organized as follows. In Sec. II, using the near-field method, we experimentally measure the density of states (DOS) for two dimer chains with different topological property, and demonstrate the end states exist in the topological nontrivial chain. After that, in Sec. III, by adding a variety of perturbations into the structure, we experimentally investigate the robustness of the end states in the topological nontrivial chain. Finally, we conclude in Sec. IV.

II. PHOTONIC END STATES IN DIMER CHAINS COMPOSED OF SRRS
Our experimental setup is shown in the Below the cutoff frequency, there is only evanescence wave in bare waveguide.
Therefore, the bare waveguide can be regarded as an electromagnetic "topological trivial insulator". In meanwhile, the coupling between SRRs can only rely on the near-field interaction due to the suppression of their far-field radiation. So our system works well in the tight-binding regime.
Considering only the nearest-neighbor coupling, the motion equation for the infinite dimer chain can be described as: Where nor ω denotes the frequency, which is normalized with respect of the resonant  Fig. 2(a). One can find there are two isolated bands separated by a gap, which is indicated by the gray area. For this Type Ⅰ sample, the measured DOS in the band is relatively high, whereas in the gap it is almost zero, as shown in Fig. 2(a). It is consistent with the theoretical calculation (marked by the black dots). Here the DOS spectrum is obtained by averaging the local density of states (LDOS) spectral over all sites, and the LDOS spectrum of each site is obtained from the reflection by putting the probe to the center of the corresponding SRR. 34 Here all of the LDOS measurements have been normalized.
Next, we studied a 16-unit type Ⅱ chain (topological non-trivial) with ' 0.48 κ = , whose unit cell is shown in the inset of Fig. 2(b). Similarly, we calculate its Eigen frequencies and measure its DOS spectrum, which consists with each other well in Fig. 2 where k θ is the polarization vector angle. After calculation, we get the winding number for both upper and lower bands are 0 w = for the typeⅠchain, and ' 1 w = for the type Ⅱ chain (Details can be found in Appendix). According to the relationship between the band gap and the passband, the gaps of two chains considered above are completely different. 25 The band gap of type Ⅰ chain is trivial as the bare waveguide, while the band gap of type Ⅱ chain is nontrivial. The end states observed at the two ends of the type Ⅱ chain are topologically protected by the topological transition -from trivial bare waveguide to nontrivial dimer chain. The robustness of the end states is investigated experimentally in the following section.

AGAINST LOSS AND DISORDER
In this section, we will reveal that the end state in the typeⅡ chain is robust against certain loss and disorder. At first, we add the loss into the central 20 SRRs of the chain as shown in Fig. 3(a). The lossy SRRs marked by grey background are realized by adding absorbing materials into the interior of the rings. Measured DOS spectrum is shown in Fig. 3

3(c) and 3(d). The measured LDOS of the end state is still confined at the two ends
[the red triangles in Fig. 3(c)], just like that calculated in the lossless system [dashed line in Fig. 3(c)]. While for the bulk state, the LDOS is significantly affected, as shown in Fig. 3(d).
Secondly, we investigate the robustness of the end states against certain disorder perturbation. The structure disorder is realized by in-plane rotating the central 20 SRRs, as shown in Fig. 4(a). The detail of rotation is illustrated in the inset of can find that at different disorder levels the end state is still maintained, whereas the bulk state has been deteriorated seriously.
At last, the robustness of the end state is further examined by adding loss and structure disorder simultaneously. By comparing three structures with both loss and random rotate angles in the central 20 SRRs [just as the situation in Fig. 4(a)], we find the topological end state is still maintained, as shown in Fig. 5. Remarkably, our demonstration of topological robust effect using in-plane rotation of SRRs paves a new way to steer the random distribution, which does not resort to the structural disorder by randomly distributing the inter-site separations. 21 In discussion, our results have revealed that the topological end state cannot be affected when the loss and the random rotation are introduced into the center 20 SRRs of the structure. This robust end state may have some significant applications, such as the robust communication and power transfer. When the electromagnetic wave is fed into one end of a chain (called end A), the end state will be established quickly. Then the electromagnetic field will accumulate at another end of this chain (called end B) and the electromagnetic field at end B can be interpreted as information or gathered as energy. In some special occasions, this robust topological end state will show great superiority to the cable transmission. For example, when the wire is buried deeply in the ground, the cable might be broken due to the geological activities or other natural disasters. The broken conductor cannot be restored, resulting in the termination of communication and power transfer. This potential risk can be greatly reduced by using the robust topological end state existing in the SSH chain by means of near-field coupling.

IV. CONCLUSION
In summary, we experimentally demonstrate the end states of the SSH model by

Appendix: Calculation of winding number
For our system, the equation of motion in wave-vector space is governed by Here we rewrite Eq. (S.2) as follows:  α =° (green triangles). Detail of rotation is shown in the inset of Fig. 4(b). As a comparison, the calculated LDOS distribution of the end and the bulk states in the original chain are also presented (gray dashed line).