In-plane excitation of a topological nanophotonic corner state at telecom wavelengths in a cross-coupled cavity

Xin-Tao He; Meng-Yu Li; Hao-Yang Qiu; Wen-Sheng Ruan; Li-Dan Zhou; Lin Liu; Xiao-Dong Chen; Wen-Jie Chen; Fu-Li Zhao; Jian-Wen Dong

doi:10.1364/PRJ.419569

Abstract

In silicon photonics, the cavity mode is a fundamental mechanism to design integrated passive devices for on-chip optical information processing. Recently, the corner state in a second-order topological photonic crystal (PC) rendered a global method to achieve an intrinsic cavity mode. It is crucial to explore such a topological corner state in silicon photonic integrated circuits (PICs) under in-plane excitation. Here, we study both theoretically and experimentally the topological nanophotonic corner state in a silicon-on-insulator PC cavity at a telecommunications wavelength. In theory, the expectation values of a mirror-flip operation for the Bloch modes of a PC slab are used to characterize the topological phase. Derived from topologically distinct bulk polarizations of two types of dielectric-vein PCs, the corner state is induced in a 90-deg-bend interface, localizing at the corner point of real space and the Brillouin zone boundary of reciprocal space. To implement in-plane excitation in an experiment, we fabricate a cross-coupled PC cavity based on the bend interface and directly image the corner state near 1383 nm using a far-field microscope. Finally, by means of the temporal coupled-mode theory, the intrinsic

Q

factor of a cross-coupled cavity (about 8000) is retrieved from the measured transmission spectra. This work gives deterministic guidance and potential applications for cavity-mode-based passive devices in silicon PICs, such as optical filters, routers, and multiplexers.

1. INTRODUCTION

The optical cavity mode is a fundamental mechanism for light manipulation and light–matter interaction at nanoscale [1]. In silicon photonics, the implementation of passive devices, such as on-chip optical filters, routers, and multiplexers, is highly relevant to the control capability of cavity modes. A photonic crystal (PC) slab provides a useful tool to manipulate cavity modes in silicon photonic integrated circuits (PICs), through finely engineering point defects in the precise location of the PC slab [2,3]. This local method enables us to flexibly control the cavity modes in an extrinsic way, along with increasing the design complexity. Due to recent advances of topological photonics [4 –6], a 2D PC slab provides a standard and advanced platform to experimentally explore a variety of topological crystalline phases that are difficult to implement in atomic scale [7 –13], and gives a new paradigm to design novel devices in PICs [14 –19]. One can apply the simple topology language to achieve an intrinsic cavity mode with a global method [17,20] (i.e., the cavity mode will be induced by the global feature of topological bulk states). In other words, such an intrinsic cavity mode is totally predictable by investigating the bulk topology, regardless of the local region of PC slab.

In 2D topological photonics [21,22], there are two strategies to design an optical cavity. One is the first-order topological phase that a 2D PC with 2D insulating bulk states supports 1D topological edge states. Due to the robust propagation of edge states along the sharp-bending interface, it is possible to achieve a whispering-gallery-mode ring cavity with an arbitrary profile [20,23,24]. Another is the second-order topological phase, in which a 2D PC with a 2D insulating bulk states supports 0D topological corner states [25 –31]. Such 0D corner states with in-plane localization will induce a small-mode-volume cavity mode, and thus the cavity size can be reduced for miniaturization. Nowadays, based on III–V semiconductor materials, the second-order topological photonic crystal (SOTPC) slab has been extensively explored for light–matter interaction [17,32 –35], such as light emission of quantum dots, lasing, and quantum electrodynamics. On the other hand, exploration of second-order corner states in silicon PICs is important for the design of a cavity and its passive devices. To do this, two main issues must be overcome: (i) how to quantitatively determine the second-order topological phase (i.e., bulk polarization) in a PC slab, and (ii) how to experimentally couple the corner state by propagating waves along waveguides (i.e., in-plane excitation), so it can be compatible with other integrated devices.

In this work, a physics quantity of Bloch-mode symmetry with respect to a mirror-flip operation is proposed to evaluate the bulk polarization of SOTPC in theory, and to experimentally observe the topological nanophotonic corner state in a cross-coupled cavity at a telecommunications wavelength. Two types of dielectric-vein PCs, with topologically distinct bulk polarizations, are designed on a standard silicon-on-insulator (SOI) platform that allows integration with other optoelectronic devices in a silicon PIC. In simulation, the nanophotonic corner state in a 90-deg-bend interface constructed by such two topologically distinct PCs is characterized by the local density of state spectra, mode pattern, and 2D spatial Fourier transformation. To experimentally observe the corner state, we fabricate a cross-coupled PC cavity based on the bend interface of SOTPC and image the corner state via a far-field microscope. In addition, we measure the transmission spectra of the interface to characterize the cross-coupled cavity property via a temporal coupled-mode theory.

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2. RESULTS

A. Second-Order Topological Photonic Crystal in a Silicon Membrane

Here, the second-order topological structure is designed by a dielectric-vein PC, whose unit cell consists of four square-hole clusters in a silicon background, as shown in Fig. 1. In theory, such an SOTPC can be preliminarily described by the 2D Su–Schrieffer–Heeger (SSH) model [36,37], which gives a clear picture to analyze the intrinsic mechanism. The topology of SOTPC is related to the mirror symmetry of its eigenfields at high-symmetry $k$ points (e.g., $Γ$ , $X$ , $Y$ ), which is dominant for the nearest-neighboring hoppings. To simplify in the following discussion, we only focus on two cases of nearest-neighboring hopping (i.e., intra-neighboring hopping $t_{a}$ dependent on intra-cluster distance $d_{a}$ and inter-neighboring hopping $t_{b}$ dependent on inter-cluster distance $d_{b}$ ). For $d_{a} = d_{b} = a / 2$ [as yellow points in Fig. 1(a)], there are equivalent hopping ( $t_{a} = t_{b}$ ) between intra-neighboring and inter-neighboring clusters that the band structure shows as a four-fold degenerate point at $X$ , as well as gives a double degenerated line along the Brillouin zone boundary (i.e., $X − M$ ). Appendix B has more details about band structures. To open a band gap, one can shrink/expand the clusters to the center/corner of unit cell, so that the structure will generate inequivalent hoppings between inter- and intra-clusters. The shrunken structure (blue) is dominant for intra-cluster hopping ( $t_{a} > t_{b}$ ), while the expanded case (red) is for inter-cluster hopping ( $t_{a} < t_{b}$ ). In terms of the specific condition that $d_{a} + s = a$ (or $d_{b} + s = a$ ), the intra-clusters (or inter-clusters) can be reconsidered as a new square hole with side length double to $2 s$ . This type of specific case can help precise nanofabrication avoid approximate separation between two nearest-neighboring clusters. Such a model gives a brief guidance for the practical design of SOTPCs.

Figure 1.2D Su–Schrieffer–Heeger (SSH) model to describe SOTPCs with four square-holes clusters. To shrink/expand the clusters to the center/corner of unit cell, the structure will generate inequivalent inter-intra hopping ( $t_{a} \neq t_{b}$ ). The shrunken structure (blue) is dominant for intra-cluster hopping ( $t_{a} > t_{b}$ ) with trivial topology, while the expanded case (red) is for inter-cluster hopping ( $t_{a} < t_{b}$ ) with nontrivial topology.

In SOTPC design, we consider a dielectric-vein PC slab based on a free-standing silicon membrane (the refractive index of silicon is $n_{Si} = 3.464$ ) with 220 nm thickness, as shown in Fig. 2(a). Blue and red boxes indicate shrunken and expanded unit cells, in correspondence with the depiction of Fig. 1. The square holes (side length $2 s = 266 nm$ ) are arranged in square lattice with the periodicity of $a = 430 nm$ . For practical PC, the details of bands and eigenmodes are beyond the 2D SSH model with nearest-neighboring hopping. In our calculation, we perform the 3D plane wave expansion method to accurately simulate the band structures [38]. Figure 2(b) shows the TE-like bulk band structure, possessing a 13.2% TE-like gap from 1274 to 1454 nm [as shown in the yellow box in Fig. 2(b)]. The gray region in the band structure represents a light cone of air.

Figure 2.Bulk states of SOTPC in a silicon membrane. (a) 3D schematic of the designed SOTPC in free-standing silicon membrane with 220 nm thickness. It can be viewed as a dielectric-vein PC in a square lattice with the periodicity $a = 430 nm$ and the side length of square holes $2 s = 266 nm$ . (b) TE-like band structure of SOTPC, showing a 13.2% TE-like gap from 1274 to 1454 nm (yellow). (c)–(d) Expectation values of mirror-flip operation for shrunken PC and expanded PC eigenmodes along $Γ X$ (circle) and $Γ Y$ (asterisk) directions. $η_{x}$ ( $η_{y}$ ) represents the expectation values of mirror-flip operation with respect to $x = 0$ ( $y = 0$ ) plane.

We should emphasize that the topology of an expanded PC is quite different from that of a shrunken PC, although they share the same band structure. Such second-order topological phase can be characterized by bulk polarization $P$ . $P = (P_{x}, P_{y})$ is a vector quantity defined by the integration of a Berry connection over the first Brillouin zone. Due to the inversion symmetry constraints on $P$ , we can have a simple form to determine bulk polarization with respect to the parities at high-symmetry wavevector points (i.e., $Γ$ , $X$ , $Y$ ) [39]: $P_{x} = \frac{- i}{2 π} \ln [\prod_{n \in occ} \frac{η_{x}^{n} (X)}{η_{x}^{n} (Γ)}], P_{y} = \frac{- i}{2 π} \ln [\prod_{n \in occ} \frac{η_{y}^{n} (X)}{η_{y}^{n} (Γ)}],$ (1)where $η_{i}^{n} (k) = ⟨ u_{n} (k) | {\hat{σ}}_{i = 0} | u_{n} (k) ⟩ / ⟨ u_{n} (k) | u_{n} (k) ⟩$ is the expectation value of mirror-flip operation ${\hat{σ}}_{i = 0}$ with respect to the $i = 0$ plane. $i$ represents $x$ or $y$ axis, $u_{n} (k)$ is the Bloch function of $n$ -th-band eigenfields at $k$ point and the subscript ‘occ’ in Eq. (1) implies the summation over the bands below the bandgap based on bulk-edge correspondence. In regard to our designed PCs, there is only one TE-like passing band below the topological bandgap. At $Γ$ point, the bulk state of the first TE band is a zero-frequency mode, which is always homogeneous in real space, such that $η_{i}^{1} (Γ) \equiv 1$ . Thus, the expression of bulk polarization can be simplified as $P = (P_{x}, P_{y}) = (\frac{- i}{2 π} \ln [η_{x} (X)], \frac{- i}{2 π} \ln [η_{y} (Y)]),$ (2)where $η_{i}$ represents $η_{i}^{n} (k)$ of the first band and the superscript 1 is ignored here. In other words, one can estimate the bulk polarization based on the mode symmetry at $X$ and $Y$ points.

To determine the second-order topological phase in PCs, a general method is to “see” the eigenmode distributions and qualitatively judge the parity at high-symmetry $k$ points according to their symmetry. In this work, the expectation value of a mirror-flip operator provides a global parameter to quantitatively evaluate the parity and topological phase. To prove the prediction above, we plot the expectation values of mirror-flip operation of a shrunken PC [as in Fig. 2(c)] and an expanded PC [as in Fig. 2(d)], calculated from their eigenmodes. Circle points represent $η_{x}$ along $Γ X$ direction, while asterisk points are for $η_{y}$ along $Γ Y$ direction. For shrunken PC with $t_{a} > t_{b}$ , the mirror-flip operation eigenvalues $η_{i}$ are always positive. It is a lack of mode-symmetry inversion from $Γ$ to $X (Y)$ points so that the shrunken PC has trivial phase with zero bulk polarization $P = (0, 0)$ . On the other hand, due to $t_{a} < t_{b}$ in expanded PC, the inversion symmetry at $X (Y)$ will change to be “odd” compared to the “even” mode at $Γ$ , and thus the expanded PC has a nontrivial phase with nonzero bulk polarization $P = (1 / 2, 1 / 2)$ . Therefore, the expectation value of mirror-flip operation is a deterministic parameter to quantitatively evaluate the second-order topological phase in PC, particular for the 3D slab whose eigenmode distributions are more complex than the 2D effective model.

B. Localized Cavity Mode Based on the Topological Corner State

For the second-order topological phase, one essential feature is that a 2D PC with nontrivial bulk polarization supports a 0D topological corner state. As schematically shown in Fig. 3(a), when considering a 90-deg-bend interface constructed by shrunken (blue) and expanded (red) PCs, the corner state will be induced by the differences in bulk polarizations.

Figure 3.Localized cavity mode based on topological corner state in a 90 deg bend interface. (a) Schematic illustration of a bend interface constructed by shrunken (blue) and expanded (red) PCs. (b) Photonic LDOS spectrum at the corner point of the bend interface, which is normalized by the LDOS in vacuum. (c) $H_{z}$ field patterns of corner states for the $z$ center ( $z = 0$ ) and $y$ center ( $y = 0$ ) planes, excited by a dipole source at the corner point at resonant wavelength ( $λ = 1400.515 nm$ ). (d) 2D spatial Fourier transformation (FT) of $H_{z}$ at the $z$ center plane. The green solid box and dashed circle are the first Brillouin zone and light cone of air, respectively.

To confirm the corner state in simulation, a general method is used to distinguish the eigenmode of corner state from other eigenmodes within the bandgap. This method is applicable for the 2D model, whereas it is not as straightforward to be generalized into a 3D PC slab because the corner state in a PC slab is buried in lots of additional air modes. On the other hand, the corner state is a high- $Q$ cavity mode, so we can investigate its Purcell enhancement phenomena, which can be demonstrated by the local density of states (LDOS) feature. By exciting a dipole source in the FDTD simulation, one can accumulate the time domain Fourier transforms of the field at a given real-space point to obtain the entire LDOS spectrum in a single calculation [40,41]. Figure 3(b) gives the LDOS spectra at the corner point of bend interface, which is normalized by the LDOS in vacuum. The spectrum is excited by a dipole source at the corner point. There is a resonant mode at $λ = 1400.515 nm$ within the bandgap (yellow region), which is in correspondence with topological corner state. The peak value that represents the LDOS at the corner can be enhanced 40,000 times more than in vacuum.

Next, we will focus on the resonant mode distributions in both the real space and momentum space. Figure 3(c) gives the $H_{z}$ field patterns at a resonant wavelength ( $λ = 1400.515 nm$ ) for $z = 0$ and $y = 0$ planes, respectively. The mode energy is mainly confined around the corner of bend interface in 3D. At this mode, the calculated $Q$ factor is about 9000, while the mode volume is $V_{m} = 0.366 {(λ / n_{si})}^{3}$ . Note that the effect of the 0D corner state works on in-plane confinement, while the vertical confinement should be induced by the total internal reflection (TIR) of the PC slab. To further investigate the strength of the vertical confinement by TIR, we extract the excitation $H_{z}$ fields with various wavevectors by a 2D spatial Fourier transformation (FT) [3,42]. Figure 3(d) gives the 2D spatial FT spectrum of $H_{z}$ at the $z$ center plane. The green solid box and dashed circle are the first Brillouin zone and light cone of air, respectively. In the FT spectrum, the wavevector components of the excitation fields are mainly distributed along Brillouin zone boundary, in particular, for the up (right) and down (left) $M$ points. It is because the photonic bandgap opens from the $X − M$ degenerated bands (see Appendix B). According to a little component in the out-of-plane leaky region (inside the green dashed circle), it is accessible to engineer a 3D-confined cavity mode from the topological corner state. In the next section, we will experimentally explore the corner state.

C. Far-Field Microscope Images of the Corner State in a Cross-Coupled Cavity

In the experiment, one of the key issues is how to excite the 3D-confined corner state. This work aims to discuss in-plane excitation of the topological corner state in an SOI platform, which is promising for integrated passive devices. As shown in Fig. 4(a), we design a cross-coupled cavity based on the SOTPC bend interface, of which some edge rectangular holes are filled with silicon to form a line-defect photonic crystal waveguide (PCW). The incident waves propagate along the strip waveguide (waveguide width $w_{g} = 430 nm$ ) and line-defect PCW, and then excite the corner state when the coupling length $l_{cp}$ is proper. Here, $l_{cp} = 5 a$ . By employing an advanced nanofabrication technique, Fig. 4(b) shows the scanning electron microscope (SEM) images of fabricated samples containing $40 \times 40 PC$ unit cells. The sample was patterned on a standard 220 nm thick silicon layer based on an SOI platform, but the ${SiO}_{2}$ layer below the PC pattern has been removed to avoid TM–TE coupling of localized fields. Insets give more details about the sample. The lattice constant is almost the same as the design (i.e., $a = 430 nm$ ) and the side length of the square holes is $2 s = 270 nm$ . More details about the nanofabrication process can be seen in Appendix A.2.

Figure 4.Experimental observation of nanophotonic corner states through far-field images. (a) Schematic view and (b) SEM images of cross-coupled cavity based on the bend interface of SOTPC. Note that some edge rectangular holes are filled with silicon to form a line-defect photonic crystal waveguide. (c) Far-field images of bulk, edge, and corner states in a cross-coupled cavity sample via an optical microscope system. (d) Simulated field patterns of electromagnetic energy of bulk, edge, and corner states at the $z$ center plane.

For optical testing, a fiber-to-chip alignment system with a far-field microscope is applied to characterize the topological nanophotonic corner state (see Appendix A.3). A near-IR (NIR) continuous wave was firstly launched into a fiber polarizer to select the TE mode, and then coupled to the device with the aid of a lensed fiber. After passing through device, the out-of-plane radiation signals were collected by a $100 \times$ microscope objective and then imaged using an InGaAs CCD. Figure 4(c) gives the far-field microscope images of a cross-coupled cavity excited by three typical wavelengths, in correspondence with bulk, edge, and corner states of bend interface. For the far-field radiation pattern at $λ = 1383.033 nm$ , we do indeed observe the in-plane confined mode localized around the corner of the bend interface (i.e., the topological corner state). On the contrary, the excitation of the edge state at $λ = 1440.000 nm$ would smoothly propagate along the interface without out-of-plane scattering near the corner, while the bulk state at $λ = 1269.000 nm$ coupled to the extended slab mode and thus produced a wide range of vertical radiation inside the bulk crystals. Appendix C provides more details about the far-field images. It also confirms that all of our experimental results are in good agreement with the simulated near-field images at $z = 0$ plane [as in Fig. 4(d)], regardless of the blue shift of the operation wavelength. This cross-coupled cavity design enables us to experimentally “see” the topological corner states on the PC slab.

D. Q-Factor Measurement of the Corner State in a Cross-Coupled Cavity

Apart from the far-field observation of the corner state, it is necessary to deeply study the optical properties and coupling mechanism of the cross-coupled cavity. In this section, we will use the temporal coupled-mode theory (CMT) [2,43,44] to further study the optical properties of the above cross-coupled cavity, which can be modeled as shown in the schematic diagram of Fig. 5(a). In CMT, the cavity mode with field amplitude $A$ has three attenuation mechanisms that are characterized by their lifetimes (i.e., $τ_{r}$ radiated to free space and $τ_{w 1} / τ_{w 2}$ coupled to input/output waveguides 1/2). $s_{1 +} / s_{1 -}$ and $s_{2 +} / s_{2 -}$ are input/output field amplitudes for input waveguide 1 and output waveguide 2, respectively. As detailed in Appendix D, we can solve the temporal coupled-mode equations [see Eqs. (D1 )–(D3) in Appendix D] and retrieve the cross-coupled cavity transmittance with Lorentz features described by $T (λ) = \frac{Q_{r}^{2}}{4 Q_{r}^{2} Q_{w}^{2} {(λ_{C} / λ - 1)}^{2} + {(Q_{r} + Q_{w})}^{2}},$ (3)where $Q_{r} = π c τ_{r} / λ_{C}$ is the out-of-plane radiative quality factor and $Q_{w} = π c τ_{w 1} / (2 λ_{C}) = π c τ_{w 2} / (2 λ_{C})$ is the in-plane coupling quality factor of the cross-coupled cavity, respectively. $λ_{C}$ is the central wavelength at a Lorentz peak for the resonant cavity mode. In Fig. 5(b), we give the full-wave simulated transmittance (red dots) of the cross-coupled cavity. By the Lorentz curve fitting [as a green line in Fig. 5(b)] of Eq. (3), we have $λ_{C} = 1400.459 nm$ , $Q_{r} = 8994$ , and $Q_{w} = 11, 069$ for our designed cavity. It validates that CMT is available to describe the light confinement of topological corner state, because the value of $Q_{r}$ is very close to the intrinsic quality factor of bend interface in Fig. 3(c). Note that the fitting $λ_{C}$ has a slight deviation from that of bend interface due to the effect on the coupled waveguide.

Figure 5.Measurement of transmission spectra and $Q$ factor in a cross-coupled cavity. (a) Description of cross-coupled cavity mode by temporal CMT with Lorentz feature. (b) Simulated transmittance (red dots) and Lorentz fitting (green line) of cross-coupled cavity, that retrieve the resonant wavelength $λ_{C} = 1400.459 nm$ , the out-of-plane radiative quality factor $Q_{r} = 8994$ , and the in-plane coupling quality factor $Q_{w} = 11, 069$ for the designed cavity. (c) SEM images of fabricated samples and (d) measured transmission spectra for bulk crystal, flat interface, and cross-coupled cavity, respectively. The resonant peak near 1383 nm is in correspondence with topological corner state. (e) Measured transmission spectrum (red) of cross-coupled cavity around the resonant wavelength, normalized by the transmission spectrum of a line-defect PC waveguide.

In the experiment, Fig. 5(c) gives the SEM images of fabricated samples for bulk crystal, a flat interface, and cross-coupled cavity, respectively. The shrunken/expanded PCs are labeled as blue/red false color. The measured transmission spectra of bulk crystal (black), flat interface (blue), and cross-coupled interface (red) samples are shown in Fig. 5(d), which is normalized by the transmission spectra of a strip Si waveguide. In the low-transmission region of bulk crystal and flat interface, the spectrum of the cross-coupled cavity has a resonant peak near 1383 nm, in correspondence with the corner state. Around the resonant wavelength, we show another measured transmission spectrum of the cross-coupled cavity normalized by the transmittance of a line-defect PCW, as the red dots shown in Fig. 5(e). Based on CMT, we obtain a Lorentz fitting curve (green) to experimentally evaluate the resonant wavelength $λ_{C} = 1383.108 nm$ , the in-plane coupling quality factor $Q_{w} = 5688$ , and the out-of-plane radiative quality factor $Q_{r} = 8233$ (also can be considered as the intrinsic $Q$ of corner state). These results quantitatively confirm that we have successfully obtained a high- $Q$ cavity mode based on the topological corner state.

3. CONCLUSION

In summary, we have successfully implemented a cross-coupled PC cavity based on second-order topology and observed a topological corner state under in-plane excitation at a telecommunications wavelength. Both simulated results in the bend interface and measured results in the cross-coupled PC cavity show evidence of the topological corner state in silicon PIC. In addition, we measure the transmission spectra of the interface to characterize the optical property of the cross-coupled cavity, and retrieve the intrinsic $Q$ via the temporal coupled-mode theory. In future, we believe it is interesting to introduce the tight-binding model of SOTPC into the temporal coupled-mode equation, and thus develop a non-Hermitian Hamiltonian to more precisely describe the radiative topological corner states. In devices, combined with the coupled cavity-waveguide design [45,46], the topological corner state is promising for cavity-mode-based passive devices (e.g., optical filters, routers, and multiplexers) in silicon PICs, and its applications for on-chip optical information transfer.

APPENDIX A: METHODS

1.Details for Numerical Simulations

All of simulation results in this work are retrieved from a suspended slab in 3D model. The maximum scale of the discrete grid is smaller than 27?nm, making the resolution large enough to ensure the convergence.

H_{z} (k_{x}, k_{y})

Q

2.Process of Sample Fabrication

{SiO}_{2}

3.Schematic View of the Experimental Setup

z

Figure 6.Schematic diagram of the experimental setup.

APPENDIX B: OPENING THE PHOTONIC BANDGAP BY CLUSTER SHRINKAGE/EXPANSION

d_{x}

Figure 7.Evolution of photonic band structures with cluster shrinkage/expansion. (a) Schematic view of the photonic crystal unit cell, consisting of four air-hole clusters. (b)–(d) TE band structures with different intra-cluster distance.

n_{eff} = 2.88

X ? M

APPENDIX C: DETAILS FOR FAR-FIELD IMAGES OF THE CROSS-COUPLED CAVITY

λ = 1269 ?? nm

Figure 8.Far-field images of cross-coupled cavity, under in-plane excitation of (a) bulk state at $λ = 1269 nm$ , (b) corner state at $λ = 1383.033 nm$ , (c) edge state at $λ = 1440 nm$ and without eigenmode excitation of PC slab (d) at $λ = 1301 nm$ and (e) at the resonant wavelength, respectively.

APPENDIX D: TEMPORAL COUPLED-MODE THEORY

ω

e ?^{i ω t}

T (ω) = \frac{{| s_{2 ?} |}^{2}}{{| s_{1 +} |}^{2}} = \frac{2 {| A |}^{2} / τ_{w}}{{| s_{1 +} |}^{2}} = \frac{4 / τ_{w}^{2}}{{(ω ? ω_{C})}^{2} + {(1 / τ_{r} + 2 / τ_{w})}^{2}} .

ω

{| s_{1 +} |}^{2}

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