Experimental Observation of Dissipatively Coupled Bound States in the Continuum on an Integrated Photonic Platform

Bound states in the continuum (BICs) are a special type of waves that coexist with continuous waves without any radiation loss. Owing to eliminated dissipative coupling to the environment, the coupling between BICs is naturally considered to be dispersive. Here, dissipative coupling between BICs in coupled photonic waveguides is exploited on an etchless lithium niobate integrated platform, and for the first time, supermode BICs are experimentally observed on a chip. It is discovered that, under the variation of the waveguide gap, the two supermodes have complementarily varied loss rates and the supermode BIC appears periodically, which verifies respectively the dissipative nature of coupling and periodic variation of the coupling coefficient between the two coupled photonic waveguides. The observed dissipatively coupled BICs can enable many on‐chip applications such as high‐sensitivity sensing, optical communication, and topological physics.

It was previously demonstrated that waveguide-based photonic BICs require a certain combination of structural parameters, and deviation from the BIC-required structural parameters introduces a propagation loss of the waveguide mode to the substrate continuum. This property has inspired us to take the substrate continuum as a common reservoir for realizing dissipative coupling on a planar photonic chip. We further exploited dissipative coupling between photonic waveguides on an etchless lithium niobate integrated platform, and for the first time, experimentally observed supermode BICs on a chip. We verified the dissipative nature of coupling between the photonic waveguides by demonstrating the complementarily varied loss rates of the two supermodes. The periodic appearance of the supermode BIC under the variation of the waveguide gap indicates periodic variation of the coupling coefficient. The observed supermode BICs open the way for a new category of dissipatively coupled BICs that may enable new applications in signal processing, sensing, chiral transmission, and topological physics.

Figure 1a
illustrates the coupled-waveguide structure fabricated on a lithium-niobate-on-insulator (LNOI) substrate, which can support dissipatively coupled BICs. Figure 1b shows the cross section of the coupled-waveguide structure with dimension labels. The thicknesses of the lithium niobate layer and polymer waveguides are h = 400 nm and t = 350 nm, respectively. The thickness of the silicon oxide layer is 2 µm. w 1 and w 2 denote the widths of the two waveguides and g denotes the gap between the two waveguides. Such a coupled-waveguide structure can support propagation of light in both the transverse-electric (TE) and transverse-magnetic (TM) polarizations along the longitudinal (y) direction. At the wavelength of 1.55 µm, the refractive indices of lithium niobate are n o = 2.21 and n e = 2.13, and the refractive indices of the polymer and silicon oxide are 1.55 and 1.44, respectively. With these data, Figure 1c plots the effective refractive index distributions for both the TE and TM polarizations, where n I,TE (n I,TM ) and n II,TE (n II,TM ) are the effective refractive indices of the TE (TM)-polarized light of the slab waveguide in the regions without and with polymer, respectively. It is clear that all the TM-polarized light, due to the lower effective refractive index, is located inside the continuous spectrum of the TE-polarized light. Figure 1d shows the cross-sectional E x profiles of the TEpolarized continuous modes and the cross-sectional E z profiles of the TM-polarized bound modes. The coupling between the two waveguides leads to the formation of supermodes with even or odd symmetry with respect to the center of the two waveguides. Similar to the case for a single waveguide, the TM-polarized supermodes for the coupled-waveguide structure can interact with the TE-polarized continuous modes, causing the former to dissipate energy into the substrate continuum. However, such energy dissipation can be eliminated by carefully designing the structural parameters, where the TM-polarized supermodes turn into supermode BICs.
We employed the coupled-mode theory to analyze the modal coupling behaviors. The two waveguides support TM-polarized propagating optical modes with effective refractive indices of n 1 and n 2 . An overlap between the two waveguide modes results in direct modal coupling with a real coupling coefficient 1 , which decreases exponentially with the waveguide gap g (see the Supporting Information). Meanwhile, the two TM-polarized waveguide modes have dissipation to the TE-polarized substrate continuum, with the dissipation rates of 1 and 2 . As a result, the two TM-polarized waveguide modes are also coupled with each other indirectly via their dissipation into the TE-polarized continuous modes. As all the continuous modes are distributed periodically along the x axis, the coupling coefficient can be expressed as j 2 exp(j + j2 g/g 0 ) with 2 = ( 1 2 ) 1/2 . denotes the phase shift of a continuous mode propagating across a single waveguide. 2 g/g 0 denotes the phase shift of a continuous mode propagating across the waveguide gap, where g 0 denotes the wavelength of the continuous mode. The system's Hamiltonian can be expressed as where k 0 is the wave number of the propagating light. Without loss of generality, here we focused on the coupling of identical waveguides with w 1 = w 2 , n 1 = n 2 , and 1 = 2 (= ). When the waveguide gap is wide ( 1 << 2 ), the effective Hamiltonian can be expressed as When + 2 g/g 0 = N (N is an arbitrary integer), the system's eigenvalues are D ± = ± (− 2 ) 1/2 -j . Therefore, this system has two eigenmodes (or termed the supermodes). One supermode has the corresponding eigenvalue D + = 0, which is the BIC. The other supermode has the corresponding eigenvalue D − = −2j , which is highly lossy. It is clear that the coupling coefficient j exp(j + j2 g/g 0 ) varies periodically under the variation of the waveguide gap g. At periodic values of waveguide gap g, the coupling becomes purely dissipative, causing the supermode BIC to appear. It should be noted that the dissipative coupling here works effectively for the TM-polarized modes but not for the TE-polarized modes. We simulated the properties of the supermode BICs with a finite-element method in COMSOL. Figure 2a,b plots the simulated propagation loss rates of the even and odd supermodes as a function of the waveguide gap and wavelength. It is clear that the conditions for obtaining the BIC are quite different for the even and odd supermodes. Furthermore, due to the small structural and material dispersion in this hybrid coupled-waveguide structure, the supermode BICs can maintain ultralow loss in a large wavelength range. Figure 2c,d plots the simulated propagation loss rates of the even and odd supermodes as a function of the waveguide gap and width (under the condition of w 1 = w 2 ).
It is clear that for any waveguide width, one can always choose an appropriate gap size that supports a supermode BIC. This is fundamentally determined by the topological properties because the even and odd supermode BIC has a topological charge of −1 and 1, respectively (see the Supporting Information for detailed discussion). The required gap for a specific supermode BIC decreases as the waveguide width increases. Note that at the width of w 1 = w 2 = 1.80 µm, which is the condition for supporting the BIC in the two individual waveguides with a topological charge of 0, the supermode BICs always exist regardless of the gap size. Figure 2e,f plots the propagation loss rate and effective refractive index as a function of the waveguide gap for propagating light at the wavelength of 1.55 µm. It is clear that both the propagation loss rate and the effective refractive index oscillate as the waveguide gap varies. More interestingly, the even and odd supermodes have complementary loss rates during their periodic oscillations, which is different from the behavior of the modes in an individual waveguide on the same platform (see the Supporting Information).
Experimentally, we employed auxiliary waveguides supporting both the fundamental (TM 0 ) and 1st-order (TM 1 ) modes to demonstrate the supermode BICs (see the Supporting Information for detailed device fabrication processes). More specifically, we used the TM 0 (TM 1 ) mode to excite the even (odd) supermode because both of them have symmetric (anti-symmetric) distributions. Figure 3a,b plots the |E| field distribution as the TM 0 (TM 1 ) mode is converted into the even (odd) supermode through a taper with a length of L t . Figure 3c,d plots the simulated conversion loss between the TM 0 (TM 1 ) mode and the even (odd) supermode   The main waveguide (left part) has a width w 0 = 6.7 µm. The split waveguides (right part) have the same widths w 1 = w 2 = 1.1 µm with a waveguide gap g = 4.5 µm. c,d) Coupling losses from the TM 0 mode to the even supermode (c) and from the TM 1 mode to the odd supermode (d) as a function of the waveguide gap g and wavelength . e) Optical microscope image of the fabricated device structure for measuring the propagation loss rates of the even and odd supermodes. The propagating TM 0 (TM 1 ) mode is converted into the even (odd) supermode and then back to the TM 0 (TM 1 ) mode. f,g) Simulated (f) and measured (g) normalized optical transmission spectra for a device with L d = 80 µm. The simulated results were obtained from a structure with 2-nm roughness on waveguide surfaces.
as a function of the waveguide gap g and wavelength with w 1 = w 2 = 1.1 µm and L t = 25 µm. It is clear that the loss varies with the gap but is essentially independent of the wavelength. We fabricated a series of devices on a z-cut LNOI substrate with an etchless fabrication process. Figure 3e shows an optical microscope image of a fabricated device, where the TM 0 (TM 1 ) mode of the input single waveguide is converted into the even (odd) supermode of the coupled waveguides (length L d ) and then back to the TM 0 (TM 1 ) mode of the output single waveguide. Figure 3f,g plots the simulated and measured normalized spectra of optical transmission from the input to the output port for the four combinations of the TM 0 and TM 1 modes with L d = 80 µm and g = 2.0 µm. It is clear that most of the output light maintains in the same waveguide mode as the input light. The transmission loss of the same mode (i.e., TM 0 -TM 0 , TM 1 -TM 1 ) is below 2.2 dB and the crosstalk between different modes (i.e., TM 0 -TM 1 , TM 1 -TM 0 ) is below −12.3 dB. It can be further concluded that the light propagating in the coupled waveguides is predominantly in the even (odd) supermode when the light input into the single waveguide is in the TM 0 (TM 1 ) mode. The TM 0 -TM 0 (TM 1 -TM 1 ) transmission loss consists of the conversion loss between the TM 0 (TM 1 ) mode and the even (odd) supermode and the propagation loss of the even (odd) supermode. With a fixed L t , the conversion loss between the single waveguide and coupled waveguides is fixed. Then, we further obtained the propagation loss rates for both the supermodes by varying L d . Figure 4a,b plots the measured results for the even and odd supermodes as a function of the waveguide gap g and wave- length (under the condition of w 1 = w 2 = 1.1 µm). Figure 4c,d plots the corresponding numerical results simulated in Lumerical. Figure 4e,f plots both the measured and simulated propagation loss rates for the even and odd supermodes as a function of the waveguide gap g at the wavelength of = 1.57 µm. It is clear that the measured and simulated propagation loss rates agree well with each other, while the slight deviation may be attributed to the unavoidable fabrication imperfections and measurement errors. The demonstrated architecture of coupled low-refractive-index waveguides on high-refractive-index substrate provides a new paradigm for manipulating light propagation. With such a structure, we can choose a proper waveguide gap in addition to waveguide width to laterally confine the propagating light. This enables structural designs with ultranarrow waveguide width, where light resides predominantly in the high-refractive-index substrate for maximal light-matter interaction. On the other hand, the demonstrated architecture can also be used as a testbed for exploring interesting physical phenomena. The amplitude and phase of the coupling coefficient are determined by the waveguide width and waveguide gap, respectively. This property enables designs with arbitrary coupling coefficient, which has clear advantages in investigation of topological properties of non-Hermitian waveguide arrays. [24] With this on-chip integrated photonic platform, we expect to demonstrate many dissipative-coupling-related theoretical proposals, including nonreciprocal transmission, [33] quantum computation, [34] superscattering, [35] chiral dynamics, [36] and non-Hermitian physics. [37]

Discussion
In conclusion, we experimentally demonstrated dissipatively coupled BICs in photonic waveguides on an etchless lithium niobate integrated platform. These BICs are laterally confined to and longitudinally guided by low-refractive-index polymer waveguides patterned on the high-refractive-index lithium niobate substrate. We employed the coupled-mode theory and the topological theory for analyzing the physical properties of the supermode BICs. We found that a supermode BIC has a nonzero topological charge so that it can exist for any waveguide width. The theoretical results were validated by our experimental observation of dissipative coupling between the two coupled waveguides and periodic appearance of the supermode BIC under the variation of the waveguide gap.
Our work has a significant impact on the development of both practical applications and fundamental physics. For practical applications, the demonstrated on-chip supermode BICs break the limitations on both material refractive index and waveguide width in the conventional waveguiding mechanisms. These breakthroughs enable a new class of integrated photonic devices where photons are confined to waveguides with an arbitrary width. For example, the demonstrated supermode BIC can be used for obtaining enhanced light-matter interaction in applications of telecommunication, sensing, and signal processing because light is distributed mainly in the substrate or in the surrounding medium for structures with a narrow waveguide width. For fundamental physics, realizing dissipative coupling between Laser Photonics Rev. 2023, 17,2200961