Theory of intrinsic propagation losses in topological edge states of planar photonic crystals

Using a semi-analytic guided-mode expansion technique, we present the theory and analysis of intrinsic propagation losses for three topological photonic crystal slab waveguide structures with honeycomb lattices of circular or triangular holes. Although conventional photonic crystal waveguide structures, such as the W1 waveguide, have been designed to have lossless propagation modes, they are prone to disorder-induced backscattering. Topological structures have been proposed to help mitigate this effect as their photonic edge states may allow for topological protection of backscattering. However, the intrinsic propagation losses of these structures are not well understood and the concept of the light line becomes blurred. Traditional numerical methods, such as the finite-difference time-domain method, are not very efficient for computing such losses. Therefore, the semi-analytical guided-mode expansion method is a natural method of choice to analyze these structures. For the three example topological edge-state structures, photonic band diagrams, loss parameters, and electromagnetic fields of the guided modes are computed. Results show that these topological structures have significant intrinsic propagation losses, more than 100 dB/cm, which is comparable to or larger than typical disorder-induced losses using slow-light modes in conventional photonic crystal waveguides.


I. INTRODUCTION
Semiconductor photonic crystals (PCs) are dielectric structures that allow the manipulation of light on the nanoscale. Such manipulation of light can be achieved by tailoring the periodicity of the dielectric constant 1-4 . In particular, planar photonic crystals slabs (PCSs) have a two-dimensional in-plane periodicity in their lattice structure, which can be used to realize slow light modes on semiconductor chips 5 . These PCSs are also often introduced with defects within their lattice structures to create waveguides 3,6-12 , which allow the propagation of light in a particular direction, or cavities 13-24 , thus allowing the confinement of light. The fabrication of these types of structures is usually done through semiconductor growth techniques 2 , such as etching 25 or lithography 26 .
The PCSs combine the features of two-dimensional photonic crystals, which control the in-plane propagation of light with Bragg reflection, and slab waveguides, which control the vertical propagation of light with total internal reflection. Slabs are generally used when the confinement of light in the vertical direction is desired 27 . By combining slabs with two-dimensional waveguides, light is able to be manipulated both in-plane and vertically. Light modes leaking out from the slab lie above the light line and are "quasiguided modes" that are subject to intrinsic losses 28 .
It has been recently proposed that topological structures can help mitigate the problem of disorder-induced losses, thanks to the special properties of their photonic edge states. These edge states of topological waveguides may allow scatter-free propagation for nanoscale PCs and are known to have applications in quantum computing due to their strong interactions with quantum emitters [43][44][45][46][47][48][49][50] . Electromagnetic modes for these topological edge states have been experimentally measured by Barik et al. 51 , indicating that these topological edge states can function as waveguides, which local spin control. However, for a PCS geometry, the role of out-ofplane losses on the propagating modes is not well understood as the concept of the light line might be ill-defined for these mixed lattice structures. Therefore, quantifying such radiative losses is fundamental to fully characterize the topological edge states in PCSs.
To properly understand the behaviour of light within PCSs, solving Maxwell's equations in the full threedimensional geometry is required. Well-known numerical approaches, such as the finite-difference time-domain (FDTD) method 52 or the plane wave expansion (PWE) method 53,54 , have been commonly employed during the last two decades. The need for using numerical methods is because Maxwell's equations cannot be solved analytically for complex structures [55][56][57][58] . For example, using FDTD techniques directly solves Maxwell's equations by iterating through time. Its solutions are numerically exact, however it is a brute-force method which can be computationally inefficient 55,59 . This computational inefficiency is especially clear when computing modes above the light line in 3D, as the computational space becomes quite large, thus significantly increasing the runtime, and lossy modes can be hard to resolve with a time-dependent solution. The PWE method, on the other hand, works in the frequency domain rather than in the time domain. This numerical method solves Maxwell's equations as an eigenvalue problem and is significantly more efficient than FDTD. However, a major limitation with PWE is that it assumes periodicity in all spatial directions. As a result, PWE can only be accurately used for lossless systems and modes, such as standard PCs below the light line. This limitation becomes evident for PCSs, since the PWE does not (and cannot) include leaky modes in the basis expansion and therefore out-of-plane losses cannot be directly estimated 2,27 .
An alternative method to the brute-force solvers is the semi-analytical method originally proposed by Andreani and Gerace known as the guided-mode expansion (GME) 28 method. In the GME, the magnetic field of the PCS is expanded in the basis of the guided mode of the slab's effective waveguide, and the resulting eigenvalue equation is solved numerically. The benefits of the GME method are two-fold: (i) it is much more computationally efficient than other numerical methods such as FDTD, because the matrix elements of the Maxwell operator become analytical in the guided mode basis; and (ii) the imaginary part of the eigenvalue, which accounts for the the out-of-plane losses, can be easily estimated by means of time-dependent perturbation theory in the low loss regime. This makes GME an ideal theoretical tool for solving PCSs when the imaginary part of the mode frequency is much smaller than its real part. The GME method is therefore an excellent method of choice to analyze the intrinsic propagation losses in the edge states of topological PCS structures above the light line.
In this paper, we apply the GME method to study three different topological PCS structures, which are taken after recent designs and experiments in the literature: two separate structures proposed by Anderson and Subramania 43 , and one proposed by Barik et al. 44 . Anderson and Subramania earlier presented two topological structures for a modified honeycomb lattice of circular holes, each with a different interface between the topological and trivial lattices: one with an armchair interface and one with a zigzag interface. Anderson and Subramania show theoretical photonic band structure calculations, as well as power flow diagrams along the interfaces. However, losses were not considered for these two structures. Barik et al. introduced a similar topological structure with a modified honeycomb lattice 44 , but with triangular holes separated by an armchair interface. The interactions between this structure and quantum emitters have been investigated experimentally, and electromagnetic modes have been found 51 . Some partial loss calculations for this structure are available in the supplementary material of Ref. 51, which are presented in the form of minimum propagation length using brute-force numerical solutions. Experimentally, a loss length of 22 μm has been shown for this structure, and they have predicted that a loss length of up to 40 μm can be achieved with appropriate parameter adjustments. With such a brute-force FDTD approach, the origin of such losses is not so clear; alternative techniques are needed not only to highlight the underlying physics, but also to explore parameter space for lower loss designs. Figure 1. Schematic 3D models of the PCS structures of interest that are studied in this work, with: (a) an armchair interface, (b) a zigzag interface, and (c) a triangular hole armchair interface. The interfaces for these structures separate a topologically trivial lattice structure with shrunken honeycomb clusters (blue, left) and a topologically non-trivial lattice structure with expanded honeycomb clusters (brown, right). Figure 1 shows 3D models of the three structures studied in this work. For all three models, the interface separates a topologically trivial lattice structure with shrunken honeycomb clusters and a topologically non-trivial lattice structure with expanded honeycomb clusters. These structures will be analyzed using the GME approach in order to compute the radiative losses above the light line. Our goal is to identify the regions where out-of-plane losses are minimized and characterize the corresponding intensity profiles of these quasi-guided modes. All structural parameters used from now on are taken from Refs. 43 and 44.

III. THEORY
For linear and non-magnetic media, one can rewrite Maxwell's equations in the frequency domain, such that a second-order eigenvalue equation in terms of the magnetic field H(r) is obtained: where ǫ(r) is the dielectric constant of the slab. To solve this eigenvalue problem using the GME method, the magnetic field is expanded in an orthonormal set of basis states: with the orthonormality condition, Then, Eq.
(1) is rewritten as a linear eigenvalue problem: where the matrix elements H µν are defined as To solve for H µν , the GME method involves solving for the magnetic field for each Bloch wave vector k as a sum of the guided modes over the reciprocal lattice vectors and the mode index α. Therefore, the GME for the magnetic field can be rewritten as where G is a reciprocal lattice vector for the PCS's lattice structure. The analytical definition for the guided mode H guided k+G,α (r) varies depending on the slab's layer, and whether the mode is transverse electric (TE) or transverse magnetic (TM) 28 . Notice that the matrix elements H µν in Eq. (4) depend on the Fourier transform of the inverse dielectric function in each slab layer j = {1, 2, 3}, through where ρ = (x, y). However, from a numerical perspective, it is much more convenient to calculate the matrix elements of the dielectric function directly as 28 and use numerical matrix inversion to find η j (G, . This is the approach that we take. The guided mode basis is computed in an effective homogeneous slab whose dielectric constant is usually taken as the spatial average of ǫ j (ρ): where A is the unit cell area and j represents one of the slab's three layers: the lower cladding, the core and the upper cladding. Once the magnetic field is obtained from Eq. (6), the electric field is obtained by where unit cell Although performing the GME in this way is accurate for photonic modes below the light line, it does not take possible out-of-plane losses into account. However, since such losses are small, one can can estimate these losses perturbatively. When a photonic mode escapes the slab's core into the claddings, it couples to lossy radiation modes and falls above the light line. The mode becomes quasiguided and is now subject to intrinsic losses, which can be accurately computed from the imaginary part of the eigenfrequency, Im(ω). Similarly to the Fermi's golden rule from quantum mechanics, these losses can be computed by second-order time-dependent perturbation theory 28 , where λ represents either a TE or TM mode, and the matrix element between a guided and lossy radiation mode is given by and ρ j is the one-dimensional photonic density of states for a given wave vector g = k + G in layer j: with θ representing the Heaviside step function. Similarly to those for the guided modes, the analytical definitions for the radiation modes H rad k+G ′ ,λ,j (r) depend on the slab's layer and polarization 28 . Finally, the imaginary part of the frequency is obtained from:

IV. NUMERICAL RESULTS: COMPLEX BAND STRUCTURES AND PROPAGATION LOSSES
In this section, we apply the GME to the three topological PCS structures shown in Fig. 1. For each of the three designs, photonic band diagrams in the k x direction are computed, along with the nominal light line. Since these band diagrams are symmetric about k x = 0, only the results for k x ≥ 0 are shown. The topological edge states are represented by two of the bands and are shown below in a zoomed-in region of interest. Propagation losses for the two topological states within the regions of interest are also presented in terms of loss length L α and group index n g = |c/v g |, where v g is the group velocity. Using the fact that D = ǫǫ 0 E, normalized electric displacement fields of the guided modes at the points of minimum loss for these two states are also shown, which gives a visual representation of how well the modes remain confined along the waveguides for their respective topological structure. If these modes were truly bound, then such intrinsic losses would be zero, as is usually the case for W1-like modes.
The computational implementation of the GME for the structures below was done via MATLAB. To obtain all necessary results, a choice in the number of k points and the number of basis states were chosen for the dispersion calculations. These numbers are dependent on PCS structure. For the structures we analyze below, both the armchair and zigzag interface structures of circular holes use a total of 81 basis states and 1002 k points. For the armchair interface structure of triangular holes, a total of 144 basis states were computed with a total of 3002 k points. For all three structures, the cut-off in reciprocal lattice vectors G was set to 30.

A. Armchair Interface of Circular Holes
We first show results for the PCS structure of the armchair interface of circular holes, proposed by Ander- son and Subramania 43 . The GME computations use a slab dielectric constant of ǫ s = 11.5, a slab thickness of d = 0.25a, a hole radius of r = 0.13a and a lattice constant of a = 870 nm. Assuming the radius of each honeycomb cluster is R and the lattice constant is a, the topologically non-trivial side has expanded honeycomb clusters with R exp = a/2.9 and the topologically trivial side has shrunken honeycomb clusters with R shr = a/3.1. Figure 2 shows the photonic band diagram for this topological structure, assuming propagation in the x direction. One might expect the topological edge states in this case to be below the light line, however the GME identifies them to be entirely above the light line due to having non-zero losses. It is important to note that the propagation losses here do not arise from backscattering, but rather from radiation leaking from the plane while the mode is propagating along the waveguide. The zoomed-in region of interest indicates the point of minimum loss at k x a = 0.09102. Fig. 3 displays the propagation losses for this structure, indicating that the loss   Fig. 2 and (b) the lower guided band labelled as State 2 in Fig. 2. length at minimum propagation loss is L α = 97a, and finite throughout all of k space. Assuming the lattice constant of a = 870 nm, the minimum losses in this structure were found to be equal to 510 dB/cm, which is significantly larger than typical disorder-induced losses of conventional PC modes 2,29-31,33,36 , which are around 5-30 dB/cm for the fast light regimes, and around 100-1000 dB/cm for the slow light regime (n g ≈ 100). For thin samples, the disorder-induced losses scale inversely with the group index squared 2, 35 . The above light-line intrinsic losses of W1 waveguides have also been measured to be around 400 dB/cm 30 , which is close to the values of the topological edge states.
The x and y components of the Bloch-mode displacement fields D at z = 0 (i.e., in the vertical centre of the slab) are shown in Fig. 4. These modes, shown for State 1 and State 2 from Fig. 2, are taken at the points of minimum loss. As expected, the modes remain mostly along the interface, however they are still quite lossy and confinement seems to be rather poor for these edge states. Next, we study the results for the PCS structure of the zigzag interface of circular holes, also proposed by Anderson and Subramania 43 . Similar to the PCS structure of the armchair interface of circular holes, the GME computation used a slab dielectric constant of ǫ s = 11.5, a slab thickness of d = 0.25a, a hole radius of r = 0.13a and a lattice constant of a = 870 nm. Additionally, the size of the expanded and shrunken honeycomb clusters remain consistent with the armchair interface, with R exp = a/2.9 and R shr = a/3.1.

B. Zigzag Interface of Circular Holes
The photonic band diagram and its zoom-in on the region of interest for this topological structure are shown in Fig. 5, assuming, as before, propagation in the x direction. Once again, one may naively assume that these topological edge states reside above the light line, but that is not true for the mixed lattice structure; minimum propagation losses now occur at k x a = 0.02718, as indicated on Fig. 6, which yield a maximum loss length of L α = 166a. With a lattice constant of a = 870 nm, the minimum losses were found to be equal to 301 dB/cm. Although performing somewhat better than the armchair interface structure of circular holes, this PCS structure is still significantly more lossy than typical disorderinduced losses from regular PC modes.
The Bloch modes (D field) for z = 0 are shown in Fig. 7. Despite being less lossy than the previous topological PCS structure, these modes are significantly lossy as a waveguide structure. Note also that these calculations are for the perfect structure with no structural disorder. However, the physics of these topological structures is much richer than regular PC modes 43,44 , and these loss lengths are certainly large enough to probe many finite-size waveguide effects, exploiting topologydependent spin 51 .

C. Armchair Interface of Triangular Holes
We next consider the PCS structure of the armchair interface of triangular holes initially proposed by Barik  51 . The GME computation uses the following parameters: a slab dielectric constant of ǫ s = 12.11, a slab thickness of d = 160a/445, a length of one side of the equilateral triangular hole of L = 140a/445 and a lattice constant of a = 445 nm. In this case, the topologically non-trivial side has expanded honeycomb clusters with R exp = 1.05a/3 and the topologically trivial side has shrunken honeycomb clusters with R shr = 0.94a/3. We show in Fig. 8 the photonic band diagram for this topological structure, along with its zoom-in of the region of interest. Propagation is assumed to be in the x direction once again, however the topological edge states are not fully above the light line in this case. State 1 resides above the light line for |k x a| < 3.0495, whereas State 2 resides above the light line for |k x a| < 2.6286. Minimum losses occur at k x a = 0.017796, and the loss propagation parameters are shown in Fig. 9. The loss length achieved at the point of minimum loss is equal to L α = 79a, and the minimum propagation losses were found to be equal to 1242 dB/cm. This PCS structure seems to perform the worst among the three designs; it has much more significant losses on the order of 10 3 rather than 10 2 .
Barik et al. have performed experimental work on this structure and have acquired some values for loss length 51 . Knowing that a lattice constant of a = 445 nm was used, the loss length of L α = 79a for this structure is equivalent to L α = 35 μm. Comparing this loss length with their experimental value of 22 μm, it is clear that these two values are within the same order of magnitude. Figure 10 shows the components of the guided mode's electric displacement field for this structure at z = 0. Similarly to the two other topological structures, the modes remain mostly along the interface, however the edge state confinement is significantly worse in this case.

V. CONCLUSIONS
In this work, we have applied the guided-mode expansion method to three topological photonic crystal slab   44,51 . Photonic band diagrams were acquired for each structure, the propagation losses for their topological edge states were investigated, and electromagnetic field modes for the points of minimum loss were shown. Although these topological edge states are known to provide scatter-free light propagation, none of the structures seemed to perform particularly well in terms of minimizing propagation loss. Taking previously reported minimum losses of 15 db/cm and 5 dB/cm for the W1 waveguide as a comparison 30,31 , the three topological structures we have studied show minimum losses on the order of 10 2 and 10 3 dB/cm. The electromagnetic fields of the guided modes remain mostly along the structures' interfaces, however these edge states are not shown to be tightly confined.
From the studied structures, the zigzag interface of circular holes provides the lowest minimum loss (and therefore the greatest loss length), however it is only marginally better than the armchair interface of circular holes. The armchair interface, on the other hand, is shown to be much more lossy than the other two structures and has much more difficulty in confining the edge state modes. Nonetheless, the armchair interface structure of triangular holes was the only structure to have its edge states fall below the light line.
If these topological structures are expected to be worthwhile solutions to maximize the confinement of light, significant optimization methods to the structures' parameters or topologies must be applied. The guidedmode expansion is certainly an appropriate tool to do so, thanks to its ability to efficiently analyze complex photonic crystal slab structures. While finishing this work, we recently became aware of alternative topological edge-state PC structures, including those by Shalaev et al. 60 , and He et al. 61 , which may be more promising in terms of reducing intrinsic propagation losses. Future work will also examine these two structures. Vučković, "Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity