Z<sub>2</sub> topological edge state in honeycomb lattice of coupled resonant optical waveguides with a flat band

Xue-Yi Zhu; Samit Kumar Gupta; Xiao-Chen Sun; Cheng He; Gui-Xin Li; Jian-Hua Jiang; Xiao-Ping Liu; Ming-Hui Lu; Yan-Feng Chen

doi:10.1364/OE.26.024307

1. Introduction

The exploration of various topological states in quantum systems mainly focuses on condensed-matter and artificial photonic systems [1–3]. Optical topological states may have important applications in optical devices due to its characteristic robustness. By breaking time-reversal symmetry using external magnetic fields, gapless topological edge states can emerge in gyromagnetic photonic crystals [4–6]. Alternatively, topological edge states can be realized by introducing effective magnetic gauge fields originating from periodic time-dependent [7] or spatial z-direction [8] modulations. However, the integration of time-dependent and z-direction modulations into semiconductor platforms is still quite challenging, while magnetic fields are incompatible with integrated miniature optical chips. Recently, 2D coupled resonant optical waveguide (CROW) which exhibits topologically protected edge state has been theoretically proposed [9, 10] and experimentally demonstrated [11, 12]. 2D dielectric CROWs, which can be fabricated using standard nanofabrication processes and are compatible with conventional photonic integrated circuits, pave promising ways toward manipulation of light through topological edge states.

A flat band is a dispersionless band that extents in the whole Brillouin zone. The flat band dispersion leads to a zero group velocity, which has important applications on the dynamic and localization properties, such as slow light devices [13, 14], nonlinear enhancement [15–17], zero index metamaterials [18, 19] and so on. And the nontrivial topological structure with a flat band combines dispersionless and robust edge state in an optical system to make it more tunable and functional. Flat bands or near flat bands has been present in some kinds of photonic structures, for example in a Lieb lattice [20–22], polaritonic systems [23], or waveguide networks [24]. However, the CROW mode is easier to couple with on-chip devices and does not require the magneto-optical materials, which means more compatibility and integration.

In this Letter, we design a 2D honeycomb CROW photonic structure which supports both two-fold and three-fold Dirac points at different high symmetry points in the Brillouin zone. The designed lattice geometry gives rise to destructive wave interference which results in a near-flat photonic band in the telecommunication frequency regime. The effective gauge fields implemented by the pseudo-spin-orbit interaction in the CROW opens up bandgaps with nontrivial topology through the Dirac points. The collective localization in the near-flat band depends on the design of lattice geometry, impurities or structural disorders [20–22, 25, 26]. These modes have a large modal area, similar to the bulk propagating ones, but with a much lower group velocity. Gapless edge states between the near-flat band and the adjacent energy bands are revealed. The field distributions of CROW structure confirm the robustness of topological edge states. A 2D tight-binding model is then set up to analyze both the bulk and edge spectra, which are in conformity with the numerical simulations. Based on these intriguing properties, the proposed honeycomb CROW structure may play a crucial role in the optical devices such as slow light devices [14, 27], topological cavity [28], large-mode-area laser [29, 30] and so on. Based on the honeycomb CROW lattice, we design and simulate the shape-independent topological cavity and the beam splitter, which demonstrate the potential application of honeycomb CROW structure in practical optical devices and new possibilities for photonic engineering. In the viewpoint of general wave theory, the underlying physics and exotic phenomena can be extended to other regimes, such as acoustics [31–34], plasmonics [35] and other systems.

2. Structure design

The designed honeycomb CROW structure is shown in Fig. 1(a). The ring waveguides with refractive index $n = 3$ are periodically arranged in the x-y plane. The topological edge states appear at the boundary of the structure, as shown by the red curves in the Fig. 1(a).The part of the white dashed box in Fig. 1(a) represents the unit cell of the honeycomb lattice, while the detailed structure is shown in Fig. 1(b). In this structure, two identical optical ring waveguides occupy the “site” positions of the graphene-like configuration (denoted as “site” rings with an inner diameter $R_{s} = 1.653 u m$ ). The optical ring waveguide supports both clockwise and counterclockwise modes, which constitute the two pseudo-spins (pseudo-spin up (down) denote the clockwise (anticlockwise) mode) in optical quantum spin Hall effect [10, 11, 36, 37]. The coupling between the clockwise and counterclockwise modes in an isolated ring is forbidden by time-reversal and rotation symmetries. Each site ring waveguide is connected with its nearest-neighbor site-ring waveguides through three coupling waveguides which are denoted as “link” rings with the same inner diameter $R_{l} = 1.653 u m$ . Five ring waveguides in unit cell are numbered by $τ = 1, ..., 5$ , as shown in Fig. 1(b). By designing directional coupling between two adjacent rings, the optical mode in the link ring always undergoes opposite rotation as compared to when it is in the site ring. As shown in the left inset of Fig. 1(a), the widths (outer radius minus inner radius) of both the site and link rings are the same $w = 0.2 u m$ . A small space gap is assigned between adjacent rings $g = 0.1 u m$ , allowing efficient coupling between them. In the frequency range of 190.0 THz to 196.0 THz, the transmission of the ring resonator is over 20% (the right inset of Fig. 1(a)). Such efficient coupling gives rise to the photonic energy bands in the CROW lattice.

Fig. 1 (a) a schematic diagram of honeycomb CROW structures. The topological edge state are shown as the red line. The insets represent the size parameters and the transmission/reflection spectra of ring resonators, respectively. (b) The detailed unit cell diagram of honeycomb CROW structures. The black dashed line represents a single repeating unit, and the arrows denote three nearest neighbor vectors $a_{1} = (0, 1), a_{2} = (\frac{\sqrt{3}}{2}, - \frac{1}{2}), a_{3} = (- \frac{\sqrt{3}}{2}, - \frac{1}{2})$ .

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A tight-binding model is employed to describe the honeycomb CROW lattice, which results in the following Hamiltonian [11, 12, 26]:

H_{0} = \sum_{< i j >} t_{i j} \cdot c_{i}^{†} c_{j}^{}

where

c_{i}^{†}

and

c_{i}^{}

are the photon creation and annihilation operators at lattice site

i

, respectively,

t

is the tunnel coupling between two adjacent rings. The two pseudospin states are simple duplication [10, 38], so we only consider one of them in the Hamiltonian. The photonic band structure of the CROW system can be obtained from the Hamiltonian in wavevector space

H = \sum_{k} Ψ_{k}^{†} H_{k} Ψ_{k}

, where

Ψ_{k} = {(c_{1 k}, c_{2 k}, c_{3 k}, c_{4 k}, c_{5 k})}^{T}

and

H_{k} = (\begin{matrix} 0 & e^{- i k \cdot a_{2} / 2} & 0 & e^{i k \cdot a_{2} / 2} & 0 \\ e^{i k \cdot a_{2} / 2} & 0 & e^{i k \cdot a_{1} / 2} & 0 & e^{i k \cdot a_{3} / 2} \\ 0 & e^{- i k \cdot a_{1} / 2} & 0 & e^{i k \cdot a_{1} / 2} & 0 \\ e^{- i k \cdot a_{2} / 2} & 0 & e^{- i k \cdot a_{1} / 2} & 0 & e^{- i k \cdot a_{3} / 2} \\ 0 & e^{- i k \cdot a_{3} / 2} & 0 & e^{i k \cdot a_{3} / 2} & 0 \end{matrix})

The three nearest neighbor vectors of the honeycomb structure (site rings) are given by

a_{1} = (0, 1), a_{2} = (\frac{\sqrt{3}}{2}, - \frac{1}{2}), a_{3} = (- \frac{\sqrt{3}}{2}, - \frac{1}{2})

. The energy bands of the Hamiltonian are obtained by diagonalizing the above matrix. At the middle of the photonic spectrum, a near-flat band appears across the entire Brillouin zone and the wave functions associated with the flat band have zero amplitude at link rings, demonstrating collective localization phenomenon. Indeed, the Hamiltonian matrix can be block-diagonalized into two subsets: one subset constitutes only site-rings, while the other consists of only link-rings. The decoupling of our system arises from the fact that the honeycomb CROW lattice has a bipartite structure which allows for an effective decoupling between the site rings (

τ = 2, 4

) and the link rings (

τ = 1, 3, 5

). And it is worth noting that while the two site rings form the background honeycomb lattice, the three link ring waveguides constitute a kagome lattice [39]. It turns out that the band structure can be derived from the energy spectrum describing these two sublattices [40]. The single-particle Schrödinger equation describing noninteracting particles on the honeycomb CROW structure can be directly derived from the second-quantized Hamiltonian. Here we consider the first-quantized Hamiltonian and the

ϕ_{τ k}

is introduced to represent the wave function corresponding to the

c_{τ k}

, one can find two separate eigensystems given by:

ε_{Γ} (k) (\begin{matrix} ϕ_{1} \\ ϕ_{3} \\ ϕ_{5} \end{matrix}) = 2 (\begin{matrix} 0 & \cos k \cdot (a_{2} - a_{1}) / 2 & \cos k \cdot (a_{2} - a_{3}) / 2 \\ \cos k \cdot (a_{2} - a_{1}) / 2 & 0 & \cos k \cdot (a_{3} - a_{1}) / 2 \\ \cos k \cdot (a_{2} - a_{3}) / 2 & \cos k \cdot (a_{3} - a_{1}) / 2 & 0 \end{matrix}) (\begin{matrix} ϕ_{1} \\ ϕ_{3} \\ ϕ_{5} \end{matrix}) = H_{Γ} (k) (\begin{matrix} ϕ_{1} \\ ϕ_{3} \\ ϕ_{5} \end{matrix})

and

ε_{K} (k) (\begin{matrix} ϕ_{2} \\ ϕ_{4} \end{matrix}) = 2 (\begin{matrix} 0 & \sum_{v = 1}^{3} \exp (- i k \cdot a_{v}) \\ \sum_{v = 1}^{3} \exp (- i k \cdot a_{v}) & 0 \end{matrix}) (\begin{matrix} ϕ_{2} \\ ϕ_{4} \end{matrix}) = H_{K} (k) (\begin{matrix} ϕ_{2} \\ ϕ_{4} \end{matrix})

The two decoupled systems, described by the Hamiltonians

H_{Γ} (k)

and

H_{K} (k)

, correspond to the link and site rings respectively, with the energy bands given by:

\begin{array}{l} ε_{K} (k) = (1 \pm \sqrt{4 A_{k} - 3}), ε_{K} (k) = - 2, \\ ε_{Γ} (k) = \pm | e x p (i k \cdot a_{1}) + e x p (i k \cdot a_{2}) + e x p (i k \cdot a_{3}) | \end{array}

where

A_{k} = \cos^{2} | k \cdot (a_{2} - a_{3}) / 2 | + \cos^{2} | k \cdot (a_{3} - a_{1}) / 2 | + \cos^{2} | k \cdot (a_{1} - a_{2}) / 2 | .

The theoretical result from tight-binding model is shown in Fig. 2(a). As we can see in formula (5), an eigenvalue with near-flat dispersion appears. The bipartite CROW lattice is similar to the edge-centered square (Lieb) lattice [20, 21]. The flat band arises from the destructive wave interference and can be described by Lieb theorem.

Fig. 2 The energy bands $E (k)$ corresponding to the Hamiltonian (a) without the Spin-Orbit term, and (b) with the Spin-Orbit coupling term.

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An intrinsic pseudo-spin-orbit coupling term $H_{S O} = i λ_{s o} \sum_{< i j > α β} {[(d_{_{i j}}^{1} \times d_{i j}^{2}) \cdot σ]}_{α β} c_{i α}^{†} c_{j β}$ is introduced to account for next-nearest-neighbor hopping in the honeycomb lattice. $α$ and $β$ represent the pseudo spin indexes. Here the $λ_{S O}$ corresponds to the next-nearest-neighbor (NNN) coupling strength, $d_{_{i j}}^{1}$ and $d_{i j}^{2}$ are two vectors connecting the NNN site ring, and $σ$ stands for the Pauli matrices for the pseudo-spins. In this system, the time reversal symmetry and the pseudo time reversal symmetry are always maintained. Here the time reversal operator exchanges the wave functions of the two basis without changing the phase, and the pseudo time reversal symmetry operator has the same form as the fermionic time reversal symmetry operator, under which the clockwise and counterclockwise modes becomes a photonic Kramers doublet [3]. And time reversal is a special case for the reciprocity. If there is no absorption and time reversal is kept, then the reciprocity of our CROW system holds [41]. It is worth noting that each spin sector time-reversal symmetry is effectively broken by the pseudo-spin-orbit coupling term, enabling nontrivial energy gaps and topological edge states that are backscattering-immune as long as the spins are decoupled . In CROW lattice, the pseudo spin-orbit coupling term results from the direction-dependent hopping phase, which arises from the difference in the path lengths traveled in the linking rings [38]. In the vicinity of the high symmetry points, the effective Hamiltonians become:

\begin{array}{l} h_{p}^{K} = v_{K} (p_{x} σ_{1} + p_{y} σ_{2}) - \frac{\sqrt{3}}{2} α λ_{s o} σ_{3} \\ h_{p}^{Γ} = v_{Γ} (p_{x} S_{1} + p_{y} S_{2}) - 2 \sqrt{3} λ_{s o} S_{3} \end{array}

where

p = k - K_{\pm} (Γ_{0})

and

α

is the spin index. This pseudo-spin-orbit coupling term opens up gaps in the bulk bands associated with the nontrivial Z₂ index [26], and the theoretical results from tight-binding model are shown in Fig. 2(b). We also simulate and obtain the photonic bands numerically by full wave simulation, which are shown in Fig. 3. The existence of the flat band coincides with the picture illustrated in the tight-binding model. It is worth noting that topological phase of honeycomb CROW was discussed in [42]. But it was described by the network model and the link sites were simplified as a coupling term in the theoretical model. However, the flat band appears only when the link rings are considered as lattice sites and the Hamiltonian matrix is expanded to

5 \times 5

.

Fig. 3 (a) Schematic diagram of the energy band. (b) Schematic diagram of the projected band. The gapless edge states are shown in the red and blue dots, which corresponds to the upper and lower borders, respectively. The calculation interval of $k_{x}$ is $- Γ \to K \to Γ$ ( $Γ$ and $K$ are high symmetric points in the first Brillouin zone), and $k_{y}$ is always zero in the calculation.

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The bulk and edge photonic spectrum of a strip of the honeycomb CROW structure is presented in Fig. 3(b) for zigzag edges. The gapless edge states in the band diagram are for the upper and lower boundaries for the pseudo-spin up mode (shown in Fig. 3(b) as the red and blue dots), respectively.

In full-wave simulation, the pseudo spin-orbit coupling term results from the direction-dependent hopping phase, which arises from the difference in the path lengths traveled in the linking rings. And the effective coupling can be derived by considering two site rings connected by a link ring through full wave calculation, as shown in the inset of Fig. 1(a). The hopping phase can be estimated by the formula: $ϕ = 2 π Δ v / F S R$ [11, 38], where the $F S R \approx 9 e 3 G H z$ represents the free spectral range of the ring. The resonance frequency of the site rings is detuned from the link rings by $Δ v \approx 600 G H z$ . $J \approx 15 G H z$ represents the intersite couping strength for antiresonant sites and links. As we can see, $J ≪ (Δ v, F S R)$ , the rings are weakly coupled high quality factor resonators. When we consider the next nearest neighbor hopping between two link rings, the hopping phase can be estimated by $J \exp (i ϕ / 3) c s c (ϕ / 2)$ , and $J \exp (i ϕ / 2) c s c (ϕ / 2)$ for the next nearest neighbor hopping between two site rings in full wave simulation. The pseudo spin orbit coupling term is related to the imaginary part of the hopping phase.

3. Nontrivial Z₂ index

In this part, we compute the four Z₂ indices associated with the four bulk gaps opened by the Pseudo-spin-orbit coupling term in the honeycomb CROW structure. Since the honeycomb structure possesses inversion symmetry, the Z₂ indices $v_{N}$ associated with the N-th bulk gap can be easily computed through the following formula [43, 44]:

\prod_{i = 0}^{3} \prod_{m = 1}^{N} ξ_{2 m} (Γ_{i}) = {(- 1)}^{v_{N}}

In this formula, $ξ_{2 m} (Γ_{i}) = \pm 1$ represents the parity eigenvalue associated with the 2m-th energy bands. All the eigenvalues combine two degenerated pseudo-spin states, corresponding to the clockwise and anti-clockwise states of the site ring. So we only consider the even numbered energy bands. As shown in Fig. 4, we calculate the field distributions of the honeycomb CROW unit cell through full wave simulation at the four high symmetric points: $Γ_{0} = (0, 0),$ $Γ_{1} = (0, \frac{2 π}{3 a}),$ $Γ_{2} = (\frac{\sqrt{3} π}{3 a}, - \frac{π}{3 a}),$ $Γ_{3} = (\frac{\sqrt{3} π}{3 a}, \frac{π}{3 a}),$ where ‘ $a$ ’ represents the distance between two adjacent site ring center.

Fig. 4 The field distributions at the four high symmetric points for the energy bands.

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Choosing the center site ring inside the unit cell as the center of inversion, the parity eigenvalues of the even numbered energy bands are determined. For example, the field distribution of band 1 calculated by full wave simulation at high symmetric point $Γ_{0} = (0, 0)$ is shown at the left lower corner of Fig. 4. As we can see, the phase in the ring waveguides coincides after inversion transformation, so the parity eigenvalue is + 1. On the contrary, the phase of band 1 at $Γ_{1} = (0, \frac{2 π}{3 a})$ becomes opposite after inversion transformation, which means the parity eigenvalue is −1. All the parity eigenvalues of four bands at different high symmetric point can be obtained, as shown in Table 1. We can see that the formula (7) gives ${(- 1)}^{v} = - 1$ for every bulk gap, which essentially means that the Z₂ indices generated by the Pseudo-spin-orbit term are all nontrivial.

Table 1. Parity-eigenvalue patterns at the four high symmetric points for the four different energy bands. All the energy gaps are associated with a nontrivial Z₂ index .

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4. Robust edge state and flat band state

Through full-wave simulation, we obtain the field distribution of the topologically protected edge states in a finite-size lattice of 7 × 4 unit-cells. The excitation port is on the right-lower corner of the structure, as shown in the inset of Fig. 5(a). The photonic edge states are shown to be robust against site defects [Fig. 5(b)], where the inset represents the specific location of the site defect. Robust unidirectional light propagation around the site defect is observed. Besides, the edge state can also be robust against an link ring defect as well as an additional site ring.

Fig. 5 The field distributions of topologically protected edge state and the bulk localized mode. (a) The field distribution of topologically protected edge state, white dashed box represents the input port of CROW structure. (b)Robust edge state propagation with the site ring defect.

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In the honeycomb CROW structure, the near-flat band can be successfully excited in the frequency range of 193.07 THz to 193.10 THz by inputting the mode light into the lower right port of the structure [Fig. 6(a)]. It is worth noting that different field distributions can be excited by various excitation approaches or excitation positions, as the near-flat band is composed of several states. As an example, we notice that using the dipole light source (at the end of the white arrows in Fig. 6(b)) to stimulate the flat band, the excitation of the bulk mode field distribution is significantly different. The dispersionless collective localization modes have a large mode area, similar to the propagation mode, but a lower group velocity. Consequently, the propagation velocity of the light waves is reduced, which may be suggestive of potential applications to slow light devices and large mode area laser.

Fig. 6 The field distributions of the collective localization modes. (a) The flat band is activated by inputting the mode light into the lower right port of the structure. (b) The flat band is activated by using the dipole light source (at the end of the white arrows).

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5. Applications

The above topologically protected edge state for the photonic systems can be leveraged to design counter-intuitive devices that have immediate applications. An example of significance is a topological cavity of arbitrary geometry. Resonant cavities are essential derives governing many photonic phenomena, however their geometry limit the application of optical devices. The geometry-independent topological cavities provides the opportunity to develop complex topological circuitry and pave promising ways toward light manipulation through topological edge states. Our design and the full-wave simulation result are shown in Fig. 7(a). Here, we use the dipole light source to stimulate the cavity mode. In addition, we have also designed an optical beam splitter based on the honeycomb CROW structure. As shown in Fig. 7(b), the dipole light source is at the end of the white arrow. When the light source simultaneously excites the clockwise and counterclockwise modes of the input ring waveguide, the clockwise and counterclockwise edge states can be observed simultaneously in the structure, and the light beam is separated to the upper right and lower left, respectively.

Fig. 7 (a) The field distribution of edge state in an arbitrarily shaped topological cavity. The cavity mode is activated by using the dipole light source (at the end of the white arrows). (b) The field distribution of edge state in a topological beam splitter.

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6. Conclusions

In conclusion, we have designed a topological honeycomb CROW structure containing a near-flat band and topologically protected edge states. The system has been modeled by a 2D tight binding framework which efficiently corroborates the nontrivial energy band gap and the near-flat band. In the honeycomb CROW structure the excitations of topologically protected edge states and pertinent effects have been elucidated. The topologically protected edge states can be used in the asymmetric selective control of light intensity in different boundaries. The collective localized modes formed due to the near-flat band possess large mode area and lower group velocity which may be applied to large mode area laser and slow light devices. These characteristics may be suggestive of potential application prospects and research values in optical devices such as optical couplers, beam splitters and so on. Additionally, the photonic systems in analogy to the quantum or electronic systems can be used for controlling some exotic optical properties, which can envisage an excellent optical platform for quantum simulations.

Funding

National Key Research and Development Program of China (NKRDP) (2017YFA0303702 and 2017YFA0305100); National Natural Science Foundation of China (NNSFC) (Grant No.11625418, No. 11474158, No. 51732006, No. 51721001, No. 51472114 and No. 11675116); Natural Science Foundation of Jiangsu Province (BK20140019)

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Z₂ topological edge state in honeycomb lattice of coupled resonant optical waveguides with a flat band

Abstract

1. Introduction

2. Structure design

3. Nontrivial Z₂ index

4. Robust edge state and flat band state

5. Applications

6. Conclusions

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (7)

Optics Express

	$Γ_{0}$	$Γ_{1}$	$Γ_{2}$	$Γ_{3}$	$\prod_{i}$	$v$
Band 4	+ 1	+ 1	−1	−1	1	$v_{4} = 1$
Band 3	−1	−1	+ 1	+ 1	1	$v_{3} = 1$
Band 2	+ 1	+ 1	−1	−1	1	$v_{2} = 1$
Band 1	+ 1	−1	+ 1	+ 1	−1	$v_{1} = 1$

Abstract

1. Introduction

2. Structure design

3. Nontrivial Z2 index

4. Robust edge state and flat band state

5. Applications

6. Conclusions

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (7)

Optics Express

3. Nontrivial Z₂ index