Cavity-induced backscattering in a two-dimensional photonic topological system

The discovery of robust transport via topological states in electronic, photonic and phononic materials has deepened our understanding of wave propagation in condensed matter with prospects for critical applications of engineered metamaterials in communications, sensing, and controlling the environment. Topological protection of transmission has been demonstrated in the face of bent paths and on-site randomness in the structure. Here we measure the propagation of microwave radiation in a topological medium possessing time reversal symmetry with a cavity adjacent to the edge channel. A coupled-resonance model analysis shows that the cavity is not a spin-conserving defect and gives rise to negative time delay in transmission.


Introduction
Recent progress in the study of photonic topological structures has opened new possibilities for robust wave transport from microwave to optical frequencies [1,2].Topologically protected edge states, whose progress is undeterred by point defects or bent paths propagate along the boundary between topological and trivial domains with different topological invariants [3].
Several schemes have been proposed to emulate the quantum spin Hall (QSH) effect and test the robustness of edge modes. Hafezi et al. [4] propose a network of coupled resonator optical waveguides in which the pseudo-spin is emulated by clockwise and counterclockwise circulation in the ring. They showed in numerical simulations that transmission through the edge mode remains near unity in the face of random fluctuations in resonator frequencies as the system size increases. QSH-like topological insulators (TIs) were also realized in bianisotropic metamaterials [5,6]. The measured time delay of edge modes averaged over disorder is consistent with ballistic transport time [7].
In addition to perfect transmission, the ability to control the time delay in transmission and in paths within the TI could be important in many applications. The time delay can be modified by either creating localized modes inside the bulk region by introducing disorder or by changing the length of the boundary line between domains. The impact of defects is also of interest because no system is perfect and there may be defects in a topological structure that support discrete modes in the vicinity of the edge. The random disorder that will be considered is not strong enough that the band gap in the nontrivial system begins to close. Several different types of disordered have been considered, including highly-correlated disordered crystals [8] and random fluctuations in onsite energy [9]. An immediate question is whether unidirectional propagation of edge mode persists in a large-scale of disorder in which localized mode are created within the bulk bandgap. This is of particular interest because it is possible to engineer the type of disorder and its proximity to the edge in order to slow down an edge mode by coupling to such states.
The magnitude and phase of the field transmitted through a continuum channel coupled to a discrete mode is a Fano resonance. This approach to scattering was first proposed to explain the asymmetric spectrum of the electron-molecule scattering cross section [10]. The interference between the discrete autoionized state and the freely propagating continuum gives rise to a characteristic asymmetric shape in the scattering spectrum. This phenomenon can be extended to classical interference between a discrete state and a slowly changing background. The sharp Fano resonance in transmission has many applications in the photonic crystal waveguide structures, such as an optical switch from complete transmission to complete reflection [11]. Here we observe a Fano resonance in a non-trivial photonic crystal. A cavity introduced inside the bulk region is found to support several discrete modes within the band gap. We will analyze the observed spectrum by generalizing the standard Fano formula to the case of multiple-modes using coupled-mode theory. Understanding the coupling between the edge mode and modes of a cavity is a first step in tailoring transport in TIs via cavity modes.
We study a TI with TR symmetry. This system does not require real or auxiliary magnetic fields to realize topological protection in bent paths and weak onsite distributed disorder. The sample has a topological domain with the triangular lattice shown in the upper half of Fig. 1a adjoining a trivial domain with the triangular lattice shown in the lower half of the figure. The topological lattice is composed of rods with a concentric collar. The position of the collar can be changed by pushing the rod, which protrudes through the holes drilled in the two plates. The waveguide supports both transverse-electric-like (TE) and transverse-magnetic-like (TM) modes. When the collar is midway between the plates, the TE and TM modes are degenerate at the Dirac point. Pseudo-spin-up and spin-down states are the bonding and antibonding combinations of these modes. TE and TM modes are coupled when the collar is displaced from a midpoint between the plates, effectively emulating spin orbit coupling in electronic systems and leading to topological order. This system can be described by the Kane-Mele , where the Pauli matrices  , s and  act on the subspace of valley, spin, and double states respectively. is the group velocity, and m is the mass term due to bi-anisotropy [5]. This Hamiltonian is a good approximation near the Dirac point. Pushing the rod to the opposite plate changes the sign of m. The electromagnetic wave is confined to the boundary between the 2D lattice of rods and collars and a 2D array of triangular prisms with a gap between them, which form a trivial insulator with bandgap coinciding with that of the adjacent TI [12,13]. Two pseudo-spin-polarized edge modes propagate in opposite directions along the interface to form a Kramers pair. To create a cavity, seven rod-collar units are pushed up so that the collars are in contact with the upper plate.
In an electronic TI system with TR symmetry, the Kramers theorem ensures the decoupling of the single pair of helical edge states [14]. In a photonic system, previous theoretical, numerical, and experimental studies show spin flipping was inhibited and reflection was suppressed. However, the only disorder discussed so far has been introduced via point defects or bent paths. The effect of a large cavity near the edge channel on the reflection rate has not been studied previously in a real structure.
In this work, we carry out microwave measurements of wave transport through a QSH TI in which a single defect cavity is introduced adjacent to the edge. We extract the reflection through a fit of an analytical model of the Fano resonance formed by the cavity and continuum edge mode to the measurements of field spectra inside the cavity.

Coupled-mode theory
We treat multiple modes and internal loss in the cavity using coupled mode theory [15,16].
Complex field of the nth mode an with central frequency n

Experimental results and spectral analysis
To determine the linewidth and central frequency of the cavity modes in this system, we measure the spectrum of the field inside the cavity. The probe is placed at the upper-left corner of the cavity, which is indicated by the blue point of Fig. 1a. The position of probe relative to the cavity is fixed in the following measurements. The results for separation between the cavity and boundary of 1, 3, 5, and 7 layers are shown in Fig. 2 Here n V describes the coupling between the cavity and edge modes, which depends on the detector position and field distributions of the cavity modes.
The parameters n  and tot n  are retrieved from the modal decomposition of the spectrum using the method of harmonic inversion [18].    Fig. 2e. The 3rd/4th mode are not detected in the experiment. Some measured modes, such as the mode at 20.707GHz in column a, only appear once in the measurement. When the cavity is moved to another position, this mode disappear. Such kind of modes related to the local imperfections inside the cavity. An abrupt phase change of -0.6  rad is observed at 21.06GHz, which is a signature of a Fano resonance. The intensity is also near zero at this point. The mode analysis described above shows that the transmission spectrum is the result of the interference between the edge mode and the cavity mode at 20.922-0.028i GHz. The neighboring small peaks are due to other nearby modes.
We first consider the case of a single localized mode and simplify Eq. (2) to The subscript n is omitted. Since we focus on a narrow frequency range, t can be taken to be a constant real number.
Scattering matrix is not unitary in a dissipative system, we need another constraint to determine φ. Yoon et al. point out the quasi-reversibility [19] of the scattering matrix as  Figs. 3c,d. The edge mode and resonance mode interfere destructively at the minimum in intensity. The analytical results ignore the nearby modes, so that the phase at two ends does not align with the measurement.
We consider the quasi-normal mode at 20.922 GHz. It should be noticed that the parameters obtained in the fit are not uniquely determined, however, 1  cannot be zero. Comparing Eq.

 
indicates that cavity is coupled to both the forward and backward edge channel and pseudo-spin polarization is not conserved for light trapped inside the cavity. Based on the Hamiltonian approximation mentioned previously, the disorder introduced here is proportional to z s , so that this kind of disorder should not mix the spin up and spin down states. Additionally, the spin flipping should be inhibited as long as the disorder has TR symmetry. Our measurement shows this Hamiltonian approximation does not works well for the whole bandgap in the presence of a large resonant cavity. Spin coupling is not negligible.
The delay time in transmission at a given frequency, which is the delay of a pulse in the limit of vanishing bandwidth, is equal to the derivative of the phase of the transmitted field with angular frequency / dd  [20]. The negative phase derivative at the zero-intensity point indicates the pulse is reshaped at this frequency.

Generalization to system broken TR symmetry
The analysis of the transmission spectrum shows that reflection is present in this TI system. However, transmission can be robust in a nonreciprocal system, for instance, chiral edge states in gyromagnetic photonic crystals [21][22][23]. We generalize this argument to a case in which only the forward channel can be supported.
In the case of a single discrete mode, D becomes the outgoing coupling coefficient d while K becomes the incoming coupling coefficient k. Mann et al. [24] demonstrate that * td k =− for a nonreciprocal system. Equation (2) can be rewritten to give [25] 0 tot 0 tot using an open-source package Kwant [27]. The transmission is unity since there is no backward propagating channel. We therefore focus on the phase change. The phase variation of the transmitted field is shown in Fig. 4d. The phase increases by 2 twice, indicating that the cavity supports two modes within this frequency range. The speckle patterns at two central frequencies of these two modes are plotted in the inset. In this case, the phase derivative equals the intensity integral of the system divided by 2 .

Conclusion
We have observed a Fano resonance between a continuum edge state and an extended defect in a time-reversal invariant TI structure. Our results contrast with previous work, in which the edge state resisted spin flipping in the presence of point defects [6,12]. The time delay near resonance increases with increasing energy inside the system. In the present work, we find backscattering induced by an extended cavity, which accounts for 30% of the linewidth of the cavity mode. Thus it is not possible to increase the transport time while maintaining perfect transmission in a bianisotropic structure with TR symmetry by introducing a cavity. However, this cavity design can be utilized in systems without reciprocity. Based on the coupled-mode theory, as long as decay rate due to internal dissipation is smaller than the rate of coupling, the phase change is always positive. Transmitted pulses can be modified and the time delay lengthened by introducing a cavity structure in nonreciprocal systems. This work points the way to a broader class of disordered TI system in which the edge mode is coupled to extended defects along the length of the edge. The defects may be extended cavities arranged either periodically or randomly, or the systems with random disorder in which spatially localized modes are introduced into the band gap as a result of Anderson localization.