One-dimensional photonic bound states in the continuum

In 1985 Fridriech and Wintgen proposed a mechanism for bound states in the continuum based on full destructive interference of two resonances which can be easily applied to the two- and three-dimensional wave systems. Here we explicitly show that this mechanism can be realized in one-dimensional quantum potential well, owing to destructive interference of electron paths with different spin in tilted magnetic field. Due to one-by-one correspondence between the spin of the electron and the polarization state of light, we have found numerous bound states in the continuum in the one-dimensional photonic system and experimentally confirmed them. The experimental set-up consists of the one-dimensional photonic crystal conjugated with a liquid-crystalline anisotropic defect layer and covered by metal film. Bound states in the continuum have recently found application to sensing, lasing and optoelectronics, but have not been realised in 1D. Here, destructive interference of electron spin in a tilted magnetic field is shown to give rise to bound states in the continuum of a 1D layered photonic crystal.

I n 1929, von Neumann and Wigner 1 discovered that the longrange oscillating attractive one-dimensional (1D) potential can support localized solutions that correspond to isolated discrete eigenvalues embedded in the continuum of positive energy states. Extension and some correction of this work was done by Stillinger and Herrick 2 , who presented a few examples of attractive local potentials with bound states embedded in the continuum (BICs) of scattering states. The BIC is a classical paradox of a quantum particle with the energy enough to escape from the potential well and nevertheless remaining spatially confined. The BIC emerges due to precise destructive interference of waves scattered by a bound potential in such a way that, after enough distance, we obtain a localized state. The physics of localization is similar to Anderson localization in random potential 3 . However, the specially selected long-range bounded 1D potentials in refs. 1,2 have not been realized experimentally to consider the phenomenon of BICs as mathematical curiosity for a long time. In 1985, Friedrich and Wintgen 4 in the framework of generic two-state effective non-Hermitian Hamiltonian formulated the concept of the BIC as the result of complete destructive interference of two resonances undergoing an avoided crossing. When two resonant states approach each other as a function of a certain continuous parameter, the width of one of them vanishes. Since the energy remains above the threshold for decay into the continuum, this state becomes a BIC although each resonant state has a finite width.
This concept was applied to two-dimensional plane microwave open resonator 5 and two-dimensional photonic crystal (PhC) waveguide with two off-channel resonators 6 . In what follows, we define such a BIC as the Friedrich-Wintgen (FW) BIC. After the first experimental observation of the symmetry-protected (SP) BICs in two-dimensional PhC by Plotnik et al. 7 , the studies of BICs were intensively grown (see the review by Hsu et al. 8 ).
When Maxwell's equations are decoupled over transverse electric (TE) and transverse magnetic (TM) polarizations, they can be written in the form of the Helmholtz equations 25 to result in one-by-one equivalence with quantum mechanics 26 . Therefore, layered one-dimensional PhC is equivalent to the 1D quantum mechanical problem with a stepwise potential profile for each polarization. However, in the 1D quantum mechanics with an arbitrary bounded potential, there are no transmittance zeros 27 , and respectively there are no FW BICs because of the absence of degeneracy and thereby of avoided crossing of resonances, which could result in the complete destructive interference. Thus, in the 1D-layered PhC structure, FW BICs cannot occur if the polarizations can be separated. However, if the PhC structure holds a defect layer that mixes polarizations of light, the FW BICs can occur because of complete destructive interference of two channels with different polarizations in the defect layer [28][29][30] . Similar FW BICs were also reported in a dielectric slab on a surface 31 , 1D solid-fluid phononic crystal 32 , and elastic layer in liquid 33 .
In this paper, we show that BICs occur even in 1D quantum well potential owing to destructive interference of electron paths with different spin in a tilted magnetic field. Moreover, due to the one-by-one correspondence between the spin of the electron and the polarization of light, we report numerous BICs in the onedimensional PhC structure with anisotropic defect layer (ADL) that plays a role of the quantum well with a tilted magnetic field. In the experimental setup, the ADL is presented by liquidcrystalline anisotropic defect layer. We propose to replace one of the PhC arms with a metallic mirror in order to facilitate fabrication, decrease the structure size, and govern the BIC by external fields applied to liquid crystal. In the reflectance spectra, we show numerous events of the Fano resonance collapse 34 , which unambiguously witness the FW BICs. Thus, the 1D-layered structures pave the way to novel tunable high-quality devices both in spintronics and photonics.

Results
Friedrich-Wintgen BICs in a one-dimensional spin model. Due to the one-by-one correspondence between the spin of the electron and the polarization state of light (Table 1), we illustrate the one-dimensional FW BICs first in a toy quantum model of spin-polarized electron transmission. Let us consider three domains in which the external stationary magnetic field is applied as sketched in Fig. 1a. Assume that the external magnetic field B inside the central layer is tilted relative to the xoriented outer magnetic field. We also assume that the inner layer has the potential shifted relative to the outer layers by a value U 0 . Outside of the central layer, an electron has two split-energy spectra E ¼ k 2 σ Ç B, σ = ↑, ↓ which specify the continua by the wave vector k σ . In the central layer, the spin-dependent spectra have the following form E ¼ k 2 s þ U 0 Ç B; s ¼ 1; 2 which specifies spin-dependent channels by the vector k s . The energy band structure is given in Supplementary Figure 1b shows the electron reflectance in dependence on energy E. One can see numerous points of collapse of Fano resonances that are the unambiguous signatures of the BICs 5,34 . These points coincide with the analytic solutions for the BICs as the solution that has zero coupling with the spin-up continuum, but is coupled with the evanescent spin-down channel. These conditions and the corresponding equations for the BICs are presented in Supplementary Note 2, and two examples of the solutions are shown in Fig. 1c. Similarly, the numerous Fano resonance collapses appear for dependencies on such parameter as the angle of incidence θ (Fig. 1d).
Friedrich-Wintgen BICs in a one-dimensional photonic model. Although this spin model cannot be directly applied for electrons because of neglecting the orbital motion of electrons in the magnetic field, it has one-by-one correspondence to the polarized light reflection from an anisotropic defect layer (ADL). The aim of this paper is realization of the FW BICs in the optical analog of the spin model, in a 1D-layered PhC structure where the TE and TM polarizations of the light play the role of spin up and -down, and the defect layer with optical anisotropy plays the role of the layer with a tilted magnetic field. What is more remarkably predicted is that FW BICs are certified experimentally.
There are also some differences between the spin model and its optical analog. First of all, there is no counterpart of Zeeman interaction in optics, which could split the frequency of light with different polarizations. Instead, we use the optical materials in which propagation bands are split by the polarization of light. In particular, one way is the use of anisotropic optical waveguides 28,30,35 . Another way is 1D-alternating PhC that supports continua specified by light polarization 29 having close correspondence with the spin model. The ADL conjugated with two 1D PhC arms is equivalent to the central layer with the tilted magnetic field (see Fig. 2a), while the 1D-alternating PhC arms respond for the continua split by polarization as shown in Fig. 2b. Similar to the spin model, we expect that the FW BICs of superposed polarizations are embedded into the TM continua of the 1D PhC arms specified by wave number k x or angle of incidence θ.
A fabrication of the 1D anisotropic PhC is technologically difficult because of the necessity to exploit highly anisotropic materials. The exploitation of low-anisotropic material would increase the total number of layers, and elongate the structure, leading to fabrication inaccuracies. Here we suggest a different fabrication-friendly photonic structure for observing FW BICs. In order to split the propagation bands into the polarization of light outside the ADL, we use semi-infinite 1D PhC arms (Fig. 2a) composed of alternating isotropic layers A and B with refractive indices n a and n b and thicknesses d a and d b . The polarizationdependent band structure in these 1D PhC arms is given in Supplementary Note 3 and shown in Fig. 2b.
The 1D PhC arms are conjugated with the uniaxial ADL with thickness L. The optical properties of the ADL are determined by the longitudinal n ∥ and transverse n ⊥ refractive indices with the unit vector of the direction of the optical axis a ¼ ðsinðϕÞ; cosðϕÞ; 0Þ. In Fig. 2b, the TM continuum (shown by red) is open with the appropriate choice of k x , while the TE continuum (shown by blue) is closed. Similar to the layer with a tilted magnetic field in the spin model, the ADL supports two orthogonal eigenmodes whose polarization vectors are tilted relative to the polarization vector of TM-wave light propagating in the 1D PhC arm. These modes can also be identified as the ADL channels. As a result, we obtain the one-by-one correspondence between the spin toy model in Fig. 1a and the present photonic system as illustrated in Table 1.
If a TM plane wave e i(kr−ωt) is injected, both eigenmodes of the ADL become observable as resonances in the TM reflectance spectra. Here wave vector k = (k x , 0, k z ) and k x is fitted into the propagation band of the 1D PhC shown by red in Fig. 2b. Figure 2c demonstrates how the reflectance depends on wavelength and anisotropy axis tilt ϕ. Similar to the spin model, one can plot the dependencies on the angle of incidence θ and the ADL thickness L. Both analytical and numerical routines are presented in Supplementary Notes 3, 4 and "Methods". Under variation of the parameters (Fig. 2c), the resonant width turns to zero similar to the spin behavior shown in Figs. 1b,d. Thereby, we realize the FW BICs with zero resonant widths as the result of complete destructive interference of TE and TM polarizations. The BICs are decoupled from the continua of the 1D PhC arms. The coupling can be easily tuned by rotating the optical axis given by the azimuthal angle ϕ. In this study, we observe two types of BICs: SP (at ϕ = 0, 90°) and nontrivial FW BICs (at ϕ ≠ 0, 90°). The value of the Q factor is plotted in Fig. 2d. Three BICs correspond to the infinite Q factor.
Experimental realization of Friedrich-Wintgen BICs. For experimental verification, the model was modified to facilitate fabrication and measurement (Fig. 3a) where the right 1D PhC arm was replaced by a gold mirror film with refractive index n M (thickness = 300 nm). The left 1D PhC arm consists of eight pairs of SiO 2 (d a = 145 nm) and TiO 2 (d b = 94 nm) layers. The ADL with length d = L/2 = 1.375 μm is filled by planar-aligned E7 nematic liquid crystal. Such a setup allows us to eliminate the right 1D PhC and optical prism collimating the light beam at the right of ADL, and control the angle ϕ of the anisotropy axis of the ADL through the voltage applied to the gold film. That setup brings unnecessary material losses in metals and sets the limit to the Q factor as shown in Fig. 2d even for low-loss metal. The number of BICs with the metal mirror case (Figs. 3b, 4a) is two times less than that in the symmetric setup (Fig. 2c). It stems from the fact that the metal mirror can support only antisymmetric solutions in the ADL. In the case of the ideal metal mirror (n M = i∞), the electric field has its node at the interface.  A vivid example of the difference in the 1D PhC band structures for the TE-and TM waves is the presence of a certain wave propagation direction in which the photonic bandgaps for the TM waves vanish at the Brewster's angle 36 . In Fig. 2b, the Brewster's direction is shown by the black solid line. In this situation, the angles of propagation in alternating layers A and B satisfy the relation θ a + θ b = π/2, where θ a ¼ arctanðn b =n a Þ. The radiation was introduced into the structure using a glass lens to implement the Brewster's effect for the 1D PhC arm. The sample was mechanically rotated to change the tilt angle ϕ of the optical axis relative to the plane of incidence. The measured reflectance spectra of the structure are presented in Fig. 4a, b right panel. The details of sample preparation and measurement techniques are presented in Methods. For comparison, the results of the numerical calculation with the Berreman matrix method 37 are shown in Fig. 4a, b left panel.
There are no resonances in the TE-polarized spectra (Fig. 4c), indicating that the TE continuum is closed. The position and width of the resonances corresponding to microcavity modes in the TM spectra (Fig. 4a) depend on the angle ϕ. The blue shift of the wavelength of the microcavity mode localized in the ADL is qualitatively explained by a decrease in the effective refractive index n d of the ADL and consequently by its optical thickness. Under variation of the angle of the optical axis in the range of 0°≤ ϕ ≤ 90°, the effective refractive index of the ADL for the electric component E y of the localized TE mode takes the values between n ∥ ≥ n d ≥ n ⊥ .
The change in the spectral width of the microcavity mode is caused by the change in the coupling between the TE-polarized localized mode (the analog of the state with # j i) and the TMpolarized continuum (the analog of the state " j i) through the mixing of the polarizations in the ADL. The situation is qualitatively explained by the fact that, in the general case, in the ADL upon rotation of the optical axis a, there exist two types of eigensolutions: the extraordinary (e) (the analog of the state 2 j i) and ordinary (o) (the analog of the state 1 j i) waves. The electric field vectors of the e-and o waves are mutually orthogonal ((E e a) ≠ 0, (E o a) = 0) and, generally, make the nonzero contributions to the TE-and TM waves, thereby ensuring the coupling between them. One can see from the spectra that at certain values of the parameters, the resonance collapses that is an unambiguous signature of the BIC.
The collapse of the resonance mode width at ϕ = 0°and ϕ = 90°indicates the existence of the SP BICs. At ϕ = 0°, the ADL optical axis a = (0, 1, 0) is oriented along the y axis, and the localized TE mode with the electric field component E y excites the e wave with the electric field vector directed along the y axis and does not contribute to the continuum of the propagating TM waves. Inversely, the propagating TM wave with the electric field component E x excites only the o wave in the ADL, the electric field vector of which is directed along the x axis and does not contribute to the localized TE mode. Thus, the localized TE mode is decoupled to the continuum of the TM waves; it can be neither excited through the continuum nor decays into it, since the coupling between the localized mode and the continuum turns to zero. Similarly, at ϕ = 90°, the optical axis a = (1, 0, 0) of the anisotropic layer is oriented along the x axis, so the e wave makes the nonzero contribution to the TM-wave only, and the o wave contributes to the TE-mode only.
The collapse of the resonant mode width in the experimental spectra at ϕ = 55°and ϕ = 40°evidences for the existence of the FW BICs as shown in Fig. 4a and Supplementary Fig. 8. The occurrence of the FW BIC is explained by complete destructive interference of the e-and o waves at the output from the ADL. The rotation of the ADL optical axis a, which is analogous to the rotation of the direction of the magnetic field, changes the absolute value and direction of the e-wave vector k e = k e (ϕ), as well as the electric fields E e = E e (ϕ) and E o = E o (ϕ) of both the eand o waves. As a result, at certain angle ϕ at the ADL output (at z = d), the conditions E ey + E oy ≠ 0 and E ex + E ox = 0 are satisfied. The contribution to the TE mode is nonzero, and that to the propagating TM wave turns to zero, i.e., the resonant mode becomes a BIC again.
The field distributions for the TE-and TM-polarized waves near the FW BIC are shown in Figs. 2a and 3a. One can see that the localized field near the FW BIC has both TE-and TM components, in contrast to the TE-polarized SP BIC. It is worth noting that, in contrast to the SP BICs, which exist for every resonant branch at ϕ = 0°and 90°only, the number and position of the FW BICs may be arbitrarily tuned. This depends not only on the angle ϕ of rotation of the optical axis, but also on the thickness d and anisotropy n ∥ /n ⊥ of the ADL.
For the qualitative description, we analytically solved the eigenvalue problem with reflectionless boundary conditions, and found the dispersion equation for the microcavity modes (Supplementary Note 4). The solution has a complex eigenfrequency ω = ω r + iγ, meaning that the corresponding microcavity mode has the spectral position λ r = 2π/ω r and the quality factor Q = ω r /2γ. The analytical dispersion curves λ r (ϕ) fit well with the experimental and numerical spectra ( Fig. 4b and Supplementary  Figs. 2, 4, 9). The Q factor of the resonant mode is determined by two components: the material loss 1/Q M , which is the absorption of light by the metallic layer, and the TE-mode leakage into the continuum of TM waves 1/Q(ϕ): 1/Q = 1/Q M + 1/Q(ϕ). At 1/Q (ϕ) = 0, the total Q factor is limited by the material loss. The experimental Q factors were found to be lower by an order of magnitude in comparison with the theoretical Q factor limited by Q M (see Supplementary Fig. 10). The reason is the liquid-crystal layer thickness variation, which can potentially be eliminated by replacing liquid ADL by metasurface [38][39][40][41] at the price of tunability.
It should be noted that for the normal incidence 29 , the FW BICs are described by relation 2d(k e − k o ) = 2πm. Physically, it means that the FW BICs occur when the ADL acts as a full-wave phase plate 42 . The TE-polarized light incident onto such a fullwave phase plate preserves its original polarization at the layer output, without being converted into the continuum of the orthogonal TM waves. To the best of our knowledge, the FW BICs in a 1D PhC-layered structure were experimentally observed in this study for the first time.

Discussion
The proposed scheme has an important advantage over the previous schemes of BIC observation in layered media [28][29][30] , it requires the only one 1D PhC arm and a defect layer holding a liquid-crystal cell. In addition, the Brewster's angle is less than the angle of total internal reflection 28,30 , providing easier excitation and increased confinement of the radiation. The sensitivity of a liquid crystal to external influence 43-45 allows one to control the coupling between the continuum and localized modes by heating or application of electric or magnetic fields.
A decrease in the ADL thickness leads to the reduction of the number of leaky bands up to the single one. In the last case, we face with Tamm plasmon-polariton 46,47 widely used in photonics and optoelectronics 48,49 , as an optical analog of the electronic Tamm state in condensed matter physics 50 . The Tamm state is transformed into the BIC in the present setup 1DPhC arm+ADL +metal when the ADL is sufficiently thin.
Owing to the one-by-one equivalence between quantum mechanics and optics, the Brewster-tilted BICs have been observed experimentally in the 1D PhC with a defect anisotropic layer. The possibility of controlling of the Q factor of the quasi-BIC modes is demonstrated by rotating the optical axis of the liquid crystal. One can use an all-dielectric structure 1D PhC arm +ADL+1D PhC (Fig. 1a) in order to diagnose BIC with an extremely high Q factor. We propose to replace one of the PhC arms with a metallic mirror in order to facilitate fabrication, decrease the structure size, and govern the BIC by external fields applied to liquid crystal. The experimental data obtained from reflectance spectra of E7-liquid-crystal cell placed between a PhC and a gold mirror are in good agreement with the theoretical and numerical results. We underline that the 1D-layered structures pave the way to novel tunable high-quality devices both in spintronics and photonics.

Methods
Berreman's transfer-matrix method. To calculate the reflection spectra of the layered structure, the transfer-matrix method is used, which is generalized by Berreman to anisotropic media 37 , a detailed description of which is given in ref. 42 . The system of Maxwell's equations is written in the form of wave equation for the 4 × 1 vector field amplitudes J ¼ ðE whereΔ is a differential matrix of propagation whose elements are expressed in terms of elements of the permittivity tensor. If the permittivity tensor does not depend on z within the jth layer with a thickness of d j , then integration (1) gives a connection of fields on the right (z = z j + d j ) and the left (z = z j ) boundaries of the layer: Jðz j þ d j Þ ¼L j Jðz j Þ,L j ¼ e ik 0 d jΔj . That allows to relate fields at the entrance to the fields at the first layer of the structure and further to the exit from the last, Nth layer, in the form The field on the left boundary of the structure can be represented as a sum of the incident and reflected waves J(z 1 ) = J i (z 1 ) + J r (z 1 ); the field on the right boundary is the field of the transmitted wave Jðz 1 þ P N j¼1 d j Þ ¼ J t ðz 1 þ P N j¼1 d j Þ. Substituting these boundary conditions into (2), we link the amplitudes J i , J r , J t . After that, reflection coefficients R s,s , R s,p , R p,p , R p,s are naturally expressed, as well as transmission coefficients T s,s , T s,p , T p,p , T p,s through the elements of the matrix L. The s index corresponds to the TE wave, and the p index corresponds to the TM wave. The Berreman's method is implemented in the MATLAB software (license # 984723).
Finite-difference time-domain method. To simulate the Brewster-tilted BIC resonance with tunable Q factors, the Finite-difference time-domain (FDTD, Lumerical) method 51 is used. In the modeling, the BIC structure includes the substrate, PhC, alignment layer, anisotropic layer, and the gold layer. The refractive indices of the anisotropic layer are set to (x, y, z) = (n o , n o , n e ). Boundary conditions (BCs) in the form of the perfectly matching layers (PML) are set in the y direction, and the Bloch boundaries are set in the x direction. The Bloch BCs allow us to find the solution of the entire system by simulating the one-unit cell by a phase shift of the fields. The source of the plane wave is illuminated from the substrate in the Brewster angle (53.13°). The monitor of frequency-domain fields and power is used to calculate the reflectance spectrum of the structure. The schematic diagram of FDTD simulation is given in Supplementary Fig. 5.
Calculation parameters. To calculate the band structure, dispersion curves and reflectance spectra of the finite structure, frequency-dependent refractive indices for gold (Aurum) (n M = n Au ) 52 , silicon dioxide (n a ¼ n SiO 2 ) 53 , titanium dioxide (n b ¼ n TiO 2 ) 54 , E7 liquid-crystal mixture (n ⊥,∥ = n o, eE7 ) 55,56 , and poly(methyl methacrylate) (n c = n PMMA ) 57 were used. The tangent component of the wave vector is k x ¼ n in k 0 sinðθ in Þ, where the RI of the prism is n in = 1.52 and the angle of incidence in the prism satisfies the Brewster condition for the PhC θ in ¼ θ B ¼ arcsinððn b =n in Þ sinðθ b ÞÞ % 53:1 . The geometrical parameters of the layers used for the calculation correspond to the real structure described in the "Experimental setup" section. To obtain the realistic Q-factor and reflection spectra, the simulated spectra were averaged over the E7-layer thicknesses ±10 nm.
Experimental setup: fabrication and measurements. To fabricate the BIC sample, the gold film (thickness = 300 nm) and the PhC (8 pairs of SiO 2 (d a = 145 nm) and TiO 2 (d b = 94 nm)) are deposited on two substrates separately. Polymethylmethacrylate (PMMA) (d c = 200 nm) is spin-coated on the PhC as the alignment layer. In order to make a smaller gap, the plano-convex curvature substrate (f = 5000 mm) is used for the gold film. Then, the optical fixture is used to clamp the two substrates, and the Newton's ring will appear and make the cell gap (~1.375 μm) in the structure. Then, the anisotropic material (liquid crystal E7) is filled into the small gap. The schematic diagram of the experimental sample is given in Supplementary Fig. 6. In characterization, the Brewster-titled spectral measurement system is set up. The excitation light source is a halogen lamp. The light incidents from the prism in the Brewster angle. The pinhole (aperture stop) is used to control beam size, and the linear polarizer is used to set the polarization state (TE or TM). The sample can be rotated in azimuthal angles to measure reflectance spectra from ϕ = 0°to ϕ = 180°with the Brewster-angle incident cone. The schematic diagram of the experimental setup is given in Supplementary Fig. 7.

Data availability
All essential data are available in the paper and Supplementary Information. Further supporting data can be provided from the corresponding author upon request.