Bound states in the continuum and strong phase resonances in integrated Gires-Tournois interferometer

Abstract Photonic bound states in the continuum (BICs) are eigenmodes with an infinite lifetime, which coexist with a continuous spectrum of radiating waves. BICs are not only of great theoretical interest but also have a wide range of practical applications, e.g. in the design of optical resonators. Here, we study this phenomenon in a new integrated nanophotonic element consisting of a single dielectric ridge terminating an abruptly ended slab waveguide. This structure can be considered as an on-chip analog of the Gires-Tournois interferometer (GTI). We demonstrate that the proposed integrated structure supports high-Q phase resonances and robust BICs. We develop a simple but extremely accurate coupled-wave model that clarifies the physics of BIC formation and enables predicting BIC locations. The developed model shows that the studied BICs are topologically protected and describes the strong phase resonance effect that occurs when two BICs with opposite topological charges annihilate.


Introduction
Resonances are a key concept in photonics since they lie behind many intriguing optical effects arising in various photonic structures. Resonances occur when an eigenmode of the structure is excited, which is manifested in pronounced peaks in the transmittance and reflectance spectra and leads to local field enhancement, giving rise to extraordinary magneto-optical, nonlinear, and other optical effects. This makes resonant structures an indispensable building block for a wide range of photonic devices. High-Q resonators are particularly important in the design of lasers, filters, and sensors.
Concerning the number of open scattering channels, most of the existing studies on BICs consider structures with two channels (reflected and transmitted); however, in Refs. [16,21], the authors investigate BICs supported by structures with only one (reflected) channel open.
It was recently shown that BICs can be endowed with a topological invariant -topological charge, which takes integer values and is conserved upon the variation of the parameters of the structure [32]. BICs having nonzero topological charge are robust to these variations. Besides, the topological charge conservation law determines the possible interaction scenarios of several BICs [20,24,32,33].
In this paper, we propose a new on-chip photonic structure -the integrated Gires-Tournois interferometer (GTI), which consists of a single dielectric ridge terminating an abruptly ended slab waveguide and operates in the total internal reflection geometry (i.e. it has a single open scattering channel). We demonstrate that, in contrast to its conventional (nonplanar) counterpart [34], the proposed integrated GTI supports high-Q phase-only resonances associated with the excitation of leaky eigenmodes of the ridge. Moreover, we show that these modes can accidentally decouple from the radiation continuum of the guided modes supported by the slab waveguide, which results in the formation of BICs. To the best of our knowledge, integrated structures supporting BICs associated with phaseonly resonances have not yet been investigated despite their important potential applications in dispersion engineering, pulse compression, and phase encoding [34][35][36]. We present a simple and accurate analytical model that describes the optical properties of the studied integrated structure and reveals the mechanism behind BIC formation. By introducing the topological charge of the BICs, we prove that they are robust in the considered parameter space. Moreover, we show that the considered structure supports BICs only when the the coupling between the TE and TM guided modes at the second edge of the ridge is relatively weak. We formulate this "weak polarization coupling" condition analytically and show that when the structure is altered so that this condition becomes violated, BICs having opposite topological charges group in pairs, coalesce, and annihilate each other, leading to the strong phase resonance effect.

Integrated Gires-Tournois interferometer
First, let us recall the conventional GTI, which dates back to 1964 [34]. This interferometer (etalon) consists of a dielectric slab with the first interface being partially reflective and the second one having unit reflectivity (see the inset to Figure 1). Similar to the Fabry-Pérot interferometer, GTI exhibits resonant optical properties as a result of multiple reflections between the interfaces of the slab. If there is no absorption inside the GTI, the intensity of the reflected light is always unity, whereas its phase changes in a resonant manner. This makes GTI extremely important for various applications including dispersion compensation and compression of frequency-modulated light pulses [34][35][36]. Let us consider an integrated analog of the GTI, which is shown in Figure 1. The proposed structure consists of a single dielectric ridge with thickness h r and width w terminating an abruptly ended slab waveguide with thickness h a (h a < h r ). We restrict our consideration to the case where both the slab waveguides with thicknesses h a and h r support only the fundamental TE-and TM-polarized guided modes. However, the effects discussed below can also occur in thicker multimode waveguides.
We study the diffraction of the TE-polarized guided mode obliquely incident from region A at the angle of incidence θ (Figure 1). We consider the monochromatic case with the time dependence exp(−iωt), where ω = 2πc/λ is the angular frequency and λ is the free-space wavelength of the incident light. The incident guided mode has a plane wavefront with the x-y dependence exp(ik x x + ik y y), where the incident TE-polarized guided mode ( Figure 1). In some works, such waves are referred to as "semiguided planar waves" [37,38].
At small angles of incidence, the outgoing field includes not only the reflected TE-and TM-polarized guided modes but also a continuum of propagating waves scattered out of the waveguide core layer. However, at relatively large angles of incidence, all these "parasitic" nonguided waves become evanescent and do not carry energy. We present a brief discussion of this scattering cancellation mechanism in the Supplementary Material section; several integrated nanophotonic elements exploiting this mechanism have been proposed recently [13,14,[37][38][39].
Since there is no waveguide in region B (see Figure 1) and no out-of-plane scattering, all the energy is reflected at the second edge of the ridge and eventually returns to region A. Therefore, the structure works in the total internal reflection geometry. Moreover, if n TE,inc sin θ > n TM,inc , where n TM,inc is the effective refractive index of the TMpolarized guided mode in region A, the reflected field contains only the TE-polarized mode, which has the same intensity as the incident wave. Therefore, the proposed integrated structure, similar to [16,21], has only one open scattering channel.
In this regime, the structure indeed turns out to be an integrated analog of the GTI: the first interface of the ridge acts as a "weak" front mirror, whereas the second interface corresponds to a back mirror with unit reflectivity. Thus, the ridge itself can be considered as an interferometer cavity. There is, however, an important difference between the investigated on-chip structure and the conventional GTI. Indeed, at angles of incidence θ satisfying the inequality where n TM is the effective refractive index of the TMpolarized mode in the ridge region, the outgoing field, as discussed above, consists of a single wave (the reflected TE mode), whereas in the ridge region, two different waves (the TE-and TM-polarized modes) propagate. This makes the proposed integrated structure somewhat similar to the recently investigated anisotropic layered structure [16].
In the rest of the article, we focus on the angle of incidence range defined by Eq. (2) and show that in this angular range the structure possesses remarkable resonant optical properties. To illustrate these properties, we consider an example with the following parameters: free-space wavelength λ = 630 nm; refractive indices of the superstrate, waveguide core layer, and substrate n sup = 1, n inc = 3.32 (GaP), and n sub = 1.45, respectively; and the waveguide thicknesses in region A and in the ridge region h a = 90 nm and h r = 110 nm, respectively. We emphasize that the used parameter values are not unique (no optimization of the structure was performed) and that the effects described below can be observed for a wide range of materials and waveguide thicknesses.
For the example under study, the slab waveguide in region A supports a TE-polarized mode with effective refractive index n TE,inc = 2.681 and a TM-polarized mode with effective refractive index n TM,inc = 1.796. The slab waveguide with thickness h r corresponding to the ridge region supports TE-and TM-polarized modes with effective refractive indices n TE = 2.819 and n TM = 2.187, respectively. At these parameters, the inequality (2) is satisfied in the following angle of incidence range: 42.05° < θ < 54.64°.
Since the reflected TE-polarized mode has a constant (unit) amplitude in the angular range of interest, it is the phase of the reflected radiation that has to be investigated. The rigorously calculated dependence of the phase arg R on the ridge width w and the angle of incidence θ is shown in Figure 2A. The plot was arrived at using an efficient in-house implementation of the aperiodic Fourier modal method (AFMM), an established numerical technique for solving Maxwell's equations in integrated optics problems [40][41][42]. Figure 2A demonstrates that along with the smooth fringes caused by Fabry-Pérot resonances of TE-polarized guided modes in the ridge region, sharp resonant features are also present in the phase spectrum. The calculation of arg R in a wider angular range (not presented here) shows that the high-Q resonances occur only at the angles of incidence considered in Figure 2. As discussed above, at these angles the reflected radiation contains only the TE-polarized guided mode, whereas in the ridge region both the TE-and TM-polarized modes exist. The fact that the high-Q resonance region is located between the cut-off angles of the TM modes outside and inside the ridge suggests that these sharp features arise as a result of the excitation of cross-polarized (quasi-TM) eigenmodes of the ridge.
However, the most interesting feature of Figure 2A is that the width and, consequently, the quality factor of the phase resonances strongly vary along the dispersion curves. At the points marked with magenta circles, the resonances vanish, i.e. their widths turn to zero and their quality factors diverge. This is demonstrated by the rigorously calculated [13,43,44] Q-factor plot shown with a blue line in the inset to Figure 2A, which indicates the presence of BICs in the considered structure. Let us note that absorption, which is inevitable in realistic structures, turns infinite-Q BICs into high-Q resonances, referred to as quasi-BICs (see the red line in the inset to Figure 2A).
Simulation results presented in the Supplementary Material section suggest that quality factor of the order k −1 can be achieved at quasi-BICs, where k is the extinction coefficient of the waveguide material. In addition, in the presence of absorption, the resonances are manifected not only in the phase of the reflected light but also as narrow dips in the reflection coefficient. As we demonstrate in the Supplementary Material section, the reflectance can almost vanish near quasi-BICs in lossy structures, which makes the considered structure promising for sensing applications [45]. Figure 3A shows the cross-sections of Figure 2A in the vicinity of the BIC marked with "3". The resonances in this plot are manifested in an arctangent-shaped rapid change of the phase. The angular widths of the resonances decrease when approaching the BIC condition. Similarly, the upper panel of Figure 3B, which shows the wavelength dependence of the phase arg R(λ), also contains resonant features narrowing to zero when approaching the BIC. The presented plots suggest that the proposed integrated GTI can be used as a narrowband all-pass filter for transforming optical pulses and beams. The lower panel of Figure 3B shows the group delay −d arg R/dω introduced by the ridge structure (note the log scale used for the vertical axis). The group delay plots exhibit sharp resonant peaks, which are characteristic of GTIs [34]. Note that the peaks provided by conventional GTIs are evenly spaced with respect to the frequency, whereas in the considered integrated structure, we can obtain a single group delay peak on a wide, constant background. Moreover, when approaching the BIC, the maximum group delay increases by orders of magnitude (see the lower panel of Figure 3B).
The maximum group delay ranges from ~1 ps at θ = 42.5° to ~60 ps at θ = 46.5°. The width of the peaks is inversely proportional to the quality factor of the mode. For the considered parameters, the quality factor ranges from ~600 to ~47,000. In order to obtain such high-Q resonances using a conventional GTI with comparable thickness and materials, the reflectance at the first interface (see the inset to   = 47,000). At the same time, the rest of the incident wave energy has to be transmitted (and not absorbed), which would normally require the use of multilayer dielectric structures. In the proposed integrated GTI, high-Q phase resonances are achieved using a different mechanism based on BICs.
In order to explain the BIC formation mechanism, in the following section we derive a simple but very accurate coupled-wave model, which proves the existence of BICs and enables predicting their locations in the parameter space.

Bound states in the continuum
In this section, we develop a coupled-wave model describing resonant optical properties of the considered structure and, in particular, BICs. The model is based on the fact that once we neglect the near-field effects, we can represent the field inside region A and inside the ridge region as a superposition of a small number of slab waveguide modes, which are coupled at the ridge edges.
As was shown in the previous section, the structure exhibits resonant properties when it has only one scattering channel, namely the reflected TE-polarized mode. We denote the complex amplitude of the wave corresponding to this channel by R and the amplitude of the incident wave by I. Inside the ridge region, there are both TE and TM slab waveguide modes with amplitudes U 1,2 and V 1,2 , respectively. Here, the subscript "1" is used for the modes propagating from the first (left) edge of the ridge towards the second (right) one, and the subscript "2" denotes the modes that are reflected from the second edge and propagate backwards. The propagation directions of these six waves are schematically shown in Figure 1.
The considered modes are coupled at the edges of the ridge. The coupling at the first edge can be described by a 3 × 3 scattering matrix, which relates the complex amplitudes of the "scattered" modes (R, U 1 , and V 1 ) with the amplitudes of the modes that are incident on the first edge of the ridge (I, U 2 , and V 2 ): Here, r and t denote reflection and transmission coefficients, respectively. The subscripts denote the scattering channels being coupled: "1" and "2" correspond to the TE-and TM-polarized waves inside the ridge, whereas "c" corresponds to the cross-polarization coupling of the modes. Note that these coefficients depend on the angle of incidence θ and can be easily calculated using AFMM [40]. Similarly, the coupling at the second edge of the ridge is described by a 2 × 2 scattering matrix S 2 : where the subscripts have the same meaning as in Eq. (3). Since U 1,2 and V 1,2 denote the amplitudes of the modes at the first edge of the ridge, the scattering matrix S 2 couples the amplitudes of the waves multiplied by the exponents describing the phase change acquired upon propagation of the modes between the ridge edges. The phases φ and ψ can be obtained by multiplying the wave vector components of the modes by the ridge width w: x y x y k k n k k n (6) and k y is defined in Eq. (1).
Equations (3) and (4) give a system of five coupledwave equations describing the optical properties of the considered structure. In the Supplementary Material section, we solve these coupled-wave equations and obtain a closed-form expression for the reflection coefficient R. The obtained expression is quite complicated; however, since the considered structure is lossless, the scattering matrices S 1 and S 2 are unitary and it is possible to present the reflection coefficient in the following compact form: Here, D is the determinant of the matrix of the system of linear equations (3) and (4); D * is the complex conjugate of D; and d i = det S i , i = 1,2, denote the scattering matrix determinants. Since the matrices S 1 and S 2 are unitary, d 1,2 have unit magnitudes, and, hence, the presented form of the reflection coefficient makes it obvious that |R | = 1. The phase of the reflected light arg R, however, is not constant but changes resonantly when the eigenmodes of the structure are excited. Figure 2B shows the phase of the reflected light (arg R) calculated using Eq. (7). By comparing Figure 2A and B, we see that the developed coupled-wave model is in excellent agreement with the full-wave simulation results. In particular, the presented model reproduces the sharp phase resonances with varying width, which are present in the rigorously calculated spectrum of Figure 2A.
The obtained Eq. (7) also makes it possible to prove BIC existence and allows one to calculate their positions in the parameter space. Since BICs are infinite-Q eigenmodes, the denominator D in Eq. (7) vanishes at a real frequency ω in the case of a BIC. This, however, does not lead to the divergence of R since the numerator in the case of a BIC vanishes as well [8,13,19]. In the Supplementary Material section, we present a rigorous proof of the fact that both the numerator and the denominator do indeed vanish at some real frequencies when the following constraint on the cross-polarization reflection coefficient q c holds: According to this inequality, the considered structure supports BICs when q c is relatively small. In particular, BICs always exist when no cross-polarization coupling occurs at the second edge of the ridge (at q c = 0). Therefore, BICs arise only in a certain part of the parameter space where the inequality (8), which can be called "weak polarization coupling" condition, is fulfilled. Let us note that inequality (8) is the cut-off condition for the BICs, which, however, is not related to the cut-off of any guided or scattered waves of the structure. In the Supplementary Material section, we provide closed-form expressions for the phases φ and ψ satisfying the BIC condition. These phases, which are expressed in terms of the scattering matrices S 1 and S 2 , are real when the inequality (8) holds. In this case, from Eqs. (5) and (6), we can find expressions for the ridge width w and the angle of incidence θ at which the structure supports a BIC: 2  2  2  2  2  2  TM  TE  2  2  2  2  2  TE  TM  TE, The BIC positions calculated using Eq. (9) and shown in Figure 2A are in perfect agreement with the features in the rigorously calculated phase of the reflected wave. Note that the positions of different BICs in Figure 2A were calculated by adding different integer multiples of 2π to the phases φ and ψ in Eq. (9). Similar equations can be obtained for the positions of the BICs in the ω-k y parameter space.

Topological charge of the BICs
In this section, we demonstrate that BICs in the considered structure are topologically protected. We do this by showing that a nonzero integer topological charge can be assigned to each BIC. Usually, topological charge is defined as a curvilinear integral of some quantity calculated along a curve encircling a BIC in a certain parameter space [32]. Topological charge is introduced differently depending on the geometry of the structure. Approaches exploiting far-field polarization vector [32,33] and argument of a quasimodal expansion coefficient [24] were recently proposed. For the considered structure, we propose the following quantity to define the topological charge: which can be interpreted as the coupling coefficient between the incident light and the eigenmode of the integrated GTI (see Supplementary Material). To define the topological charge, we take the contour integral of the gradient of the argument of P: where θ θ θ We note here that the phases φ and ψ in P depend on both θ and w, whereas the coupling coefficients q 1,c and t 1,2 depend solely on θ.
When the integration path γ encircles a BIC, Eq. (11) defines its topological charge. Moreover, when the topological charge is nonzero, there should be a point inside the integration contour where the phase of P is undefined. Therefore, at this point P = 0 and, as we prove in the Supplementary Material section, at this very point of the parameter space the considered structure supports a BIC. Figure 4 shows arg P calculated in the considered w-θ parameter space. BICs are marked with white circles. One can see the singularities of arg P appearing at these points. Hence, if we move along a contour encircling a BIC, the phase will not return to its initial position but will differ by an integer multiple of 2π. For example, in the case of the BIC marked with "1" in Figure 4, the phase increases by 2π when encircling the BIC counterclockwise, and therefore its topological charge is +1. The phase around the BIC marked with "2", however, changes in the opposite direction, resulting in topological charge −1.
Small perturbations of the parameters of the structure result in small changes in the quantity P. The integer value of the topological charge C, however, can only change in a discrete manner. Therefore, small perturbations will not change the topological charge and the BIC will not disappear but move in the parameter space. Such BICs are called "robust" or "topologically protected". Calculations based on Eq. (11) show that all the BICs investigated in this paper have nonzero topological charge and are, hence, robust.
Another important application of the topological charge is the investigation of the interaction of several BICs. To do this, let us choose the integration contour γ in such a way that it encircles several BICs. In this case, the value of C in Eq. (11) will be the sum of the topological charges of the encircled BIC. If the BICs interact in some manner, after the interaction the sum of the topological charges is conserved. One scenario for such an interaction is the annihilation of the BICs having opposite charges, which is discussed in the following section.

Annihilation of the BICs and strong phase resonances
As we have shown in the previous section, adjacent BICs in the considered structure have opposite topological charges (−1 and +1). The total topological charge of the two BICs is zero. Therefore, according to the charge conservation law, the coalescence of these BICs may result in their annihilation, after which no BICs remain, resulting in the same total charge equal to zero. In this section we demonstrate this effect.
To "move" the BICs in the parameter space, we introduce a nonzero thickness h b of the waveguide layer in region B (see the inset to Figure 5B). We will assume that h b is small enough so that the scattering channels corresponding to the transmitted modes in region B remain closed. In this case, the scattering matrix S 2 changes but the coupled-wave model of Section 3 remains applicable.
Let us focus on the two BICs bounded by the white rectangle in Figure 2A. In the upper panels of Figure 5, we show the phase of the reflected wave calculated at different values of h b ranging from 0 to 25 nm. The lower panels of Figure 5 show the corresponding rigorously calculated [43] quality factors of the modes. Note that the upper panel of Figure 5A is simply a magnified 450 500 550 600 650  Figure 2A, where two BICs are present. With an increase in h b , these BICs approach each other ( Figure  5B). Then, at a critical thickness h b = 22.5 nm, the BICs coalesce, as shown in Figure 5C. At this thickness, the topological protection of the BIC is lifted, i.e. the considered "second-order" BIC in Figure 5C gets topological charge zero. If we further increase h b , the BIC will disappear; indeed, instead of a BIC, only a finite-Q resonance is seen in Figure 5D corresponding to h b = 25 nm. This demonstrates the annihilation of the BICs governed by their topological properties. Now, let us focus on the lower panels of Figure 5 and investigate the w-dependence of the quality factor near the ridge width w 0 satisfying the BIC condition. To do this, we write the complex frequency of the mode ω p in the big O notation: Re ω p = ω + O(w − w 0 ), Im ω p = O((w − w 0 ) 2 ), which follows from the causality condition [19]. This expansion leads to a quadratic decay rate of the quality factor when moving away from the BIC: Q = Re ω p /(−2Im ω p ) ~ (w − w 0 ) −2 . Indeed, in Figure 5A and B, the quadratic decay law (w − w 0 ) −2 shown with dashed lines provides a good fit of the quality factor. However, this is not the case in Figure 5C, where the two BICs coalesce and the decay rate becomes quartic: Q ~ (w − w 0 ) −4 (see the dotted lines in Figure 5). Note that a similar phenomenon was predicted for different periodic structures (photonic crystal slabs and arrays of cylinders or spheres) [20,24,33,46,47] and referred to as a strong resonance [24,33,46,47]. Therefore, in the case of a "second-order" BIC, the proposed integrated GTI exhibits a strong phase resonance, which provides a slower decay of the quality factor when moving away from the BIC. This effect is of practical importance for designing high-Q resonators with relaxed fabrication tolerances.
Therefore, in this section we have shown that if the structure is altered in such a way that the weak polarization coupling condition (8) becomes violated, BICs having opposite topological charges group in pairs, coalesce, and, finally, annihilate each other.

Conclusion
In this work, we demonstrated that an integrated analog of the GTI, which consists of a single dielectric ridge terminating an abruptly ended slab waveguide, supports topologically protected BICs. We developed a coupledwave model that accurately describes the resonant optical properties of the proposed structure. In particular, simple closed-form expressions were obtained for the condition of the BIC existence and for the positions of the BICs in the considered parameter space. The coupledwave model also allowed us to introduce the topological charge of the BICs, proving their robustness and limiting their possible interaction scenarios by the charge conservation law.
The existence of BICs in the proposed structure makes it possible to use it as an integrated optical waveguide with an unconventional guiding mechanism [29][30][31]. Detuning from the BIC condition allows one to obtain an all-pass filter exhibiting sharp phase resonances. This makes the proposed structure promising for on-chip dispersion engineering, in particular for phase equalization and pulse compression. The existence of the so-called strong resonances in the considered structure relaxes the fabrication tolerances and makes the proposed integrated GTI an excellent candidate for creating high-Q optical resonators, which are widely used in filtering, sensing, and lasing. The latter application may fruitfully exploit the fact that the proposed structure has only one open scattering channel.
We believe that the proposed concept of an integrated GTI supporting BICs and strong phase resonances can be extended to other on-chip platforms, in particular Bloch surface waves propagating along the interfaces of photonic crystals [48]. We also expect that the demonstrated charge-conserving interaction of the BICs including their coalescence and annihilation can be implemented not only by changing the geometry of the structure but also through an external stimulus of magneto-optical, electrooptical, or nonlinear nature.