Abstract
We consider light scattering by an anisotropic defect layer embedded into anisotropic photonic crystal in the spectral vicinity of an optical bound state in the continuum (BIC). Using a resonant state expansion method we derive an analytic solution for reflection and transmission amplitudes. The analytic solution is constructed via a perturbative approach with the BIC as the zeroth order approximation. The solution is found to describe the collapsing Fano feature in the spectral vicinity of the BIC. The findings are confirmed via comparison against direct numerical simulations with the Berreman transfer matrix method.
Introduction
Theoretical insight into resonant response from optical systems, including photoniccrystalline resonators^{1} and resonant metasurfaces^{2}, is of big importance in photonics^{3,4}. Very unfortunately only a few systems generally allow for a tractable analytic solution providing intuitively clear and mathematically exact picture, such as, e.g., the celebrated Mie–Lorenz theory^{5}. Thus, in the field of optics the resonant scattering quite often can only be understood in terms of the temporal coupled mode theory (TCMT)^{6,7,8}. The TCMT is a phenomenological approach that maps the scattering problem onto a system of field driven lossy oscillators. Mathematically, the problem is cast in the form of a system of linear differential equations. The coefficients of the system account for both “internal” modes of the resonant structure as well as for the coupling of the “internal” modes to incoming and outgoing waves. The interaction with the impinging light is understood in terms of “coupled modes” which are populated when the system is illuminated from the farzone. The elegance of the TCMT is in its simplicity and the relative ease in establishing important relationships between the phenomenological coefficients solely from the system’s symmetries and conservation laws^{7,8,9,10}. However, despite its numerous and successful applications, the TCMT generally relies on a set of fitting parameters. Moreover, the mathematical foundations of the TCMT remain vague since the theory neither gives an exact definition of the “coupled mode”, nor a clear recipe for such a “coupled mode” to be computed numerically.
Historically, the problem of coupling between the system’s eigenmodes to the scattering channels with the continuous spectrum has attracted a big deal of attention in the field of quantum mechanics^{11,12,13}. One of the central ideas was the use of the Feshbach projection method^{12,14} for mapping the problem onto the Hilbert space spanned by the eigenstates of the scattering domain isolated from the environment. Such approaches have met with a limited success in application to various wave related setups, including quantum billiards^{15,16,17}, tightbinding models^{18}, potential scattering^{19}, acoustic resonators^{20}, nanowire hetrostructures^{21} and, quite recently, dielectric resonators^{22}. Besides its mathematical complexity there are two major problems with the Feshbach projection method: First, the eigenmodes of the isolated systems are in general not known analytically; therefore, some numerical solver has most often to be applied. Furthermore, the computations of such eigenmodes requires some sort of artificial boundary condition on the interface between the scattering domain and the outer space. Quite remarkably the convergence of the method is shown to be strongly affected by the choice of the boundary condition on the interface^{16,23,24}.
In the recent decades we have witnessed the rise of efficient numerical solvers utilizing perfectly matched layer (PML) absorbing boundary conditions^{25,26}. The application of perfectly matched layer has rendered numerical modelling of wave propagation in open optical, quantum, and acoustic systems noticeably less difficult allowing for direct fullwave simulations even in three spatial dimensions. On the other hand, the application of PML also made it possible to compute the eigenmodes and eigenfrequencies of wave equations with refletionless boundary conditions. Such eigenmodes come under many names including quasinormal modes^{4}, Gamow states^{27}, decaying states^{28}, leaky modes^{29}, and resonant states^{30}. The availability of the resonant states has naturally invited applications to solving Maxwell’s equations via series expansions giving rise to a variety of resonant state expansion (RSE) methods^{4}. One problem with the resonant states is that they are not orthogonal in the usual sense of convolution between two mode shapes with integration over the whole space^{13,31}. This can be seen as a consequence of exponential divergence with the distance from the scattering center^{4}. Fortunately, both of the normalization and orthogonality issues have recently been by large resolved with different approaches, most notably through the PML^{32}, and the fluxvolume (FV)^{28,30} normalization conditions.
In this paper we propose a RSE approach to the problem of light scattering by an anisotropic defect layer (ADL) embedded into anisotropic photonic crystal (PhC) in the spectral vicinity of an optical bound state in the continuum (BICs). The optical BICs are peculiar localized eigenmodes of Maxwell’s equations embedded into the continuous spectrum of the scattering states^{33,34}. One remarkable property of the BICs is the emergence of a collapsing Fano features induced by the highquality resonant states in the spectral vicinity of the BIC proper^{35,36,37,38,39,40}. The sensitivity of the optical response to the parameters of the incident light allows for a fine control of Fano lineshapes making the optical BICs an efficient instrument in design of narrowband optical filters^{41,42,43,44}. Although BICs are ubiquitous^{33,34} in various optical systems, the system under scrutiny is the only one allowing for an exact fullwave analytic solution for an optical BIC^{45}. By matching the general solution of Maxwell’s equation within the ADL to both evanescent and propagating solutions in the PhC^{46,47,48,49} we find the eigenfield and eigenfrequency of the resonant mode family limiting to the BIC under variation of a control parameter. Next, for finding the scattering spectra we apply the spectral representation of Green’s function in terms the FVnormalized resonant states^{30}. This is a well developed approach which has already been applied to both two^{50} and three^{51}dimensional optical systems. The approach has also been recently extended to magnetic, chiral, and bianisotropic optical materials^{52} as well as potential scattering in quantum mechanics^{53}. Remarkably, so far RSE methods have been mostly seen as a numerical tool. Here we show how a perturbative analytic solution can be constructed in a closed form within the RSE framework. Such a perturbative solution uses the BIC as the zeroth order approximation and, very importantly, is capable of describing the collapsing Fano resonance^{35,36,37,38,39,40} in the spectral vicinity of the BIC. We shall see that the analytic solution matches the exact numerical result to a good accuracy.
The system
The system under scrutiny is composed of an ADL with two anisotropic PhC arms attached to its sides as shown in Fig. 1a. Each PhC arm is a onedimensional PhC with alternating layers of isotropic and anisotropic dielectric materials. The layers are stacked along the zaxis with period \(\Lambda \). The isotropic layers are made of a dielectric material with permittivity \(\epsilon _o\) and thickness \(\Lambda d\). The thickness of each anisotropic layer is d. The anisotropic layers have their principal dielectric axes aligned with the x, yaxes with the corresponding permittivity component principal dielectric constants \(\epsilon _{e}\), \(\epsilon _{o}\), but the principal axes of the ADL are tilted with respect of the principal axes of the PhC arms as shown in Fig. 1a. Propagation of the monochromatic electromagnetic waves is controlled by Maxwell’s equations of the following form^{45}
where \(\varvec{E}\) is the electric vector, \(\varvec{H}\) is the magnetic vector, \(k_0 = \omega /c\) is the wave number in vacuum with c as the speed of light, and, finally, \({\hat{\epsilon }}\) is the dielectric tensor. The orientation of the ADL optical axis is determined by the unit vector
as shown in Fig. 1a. Since the reference frame is aligned with the optical axes in the PhC, the dielectric tensor is diagonal everywhere out of the ADL. Given that \(\varvec{a}\) is specified by the tilt angle \(\phi \), in the ADL it takes the following form
In this paper we restrict ourselves with the normal incidence, i.e. the Bloch wave vector is aligned with the zaxis in Fig. 1a. The dispersion of waves in the PhC arms depends on polarization. For the xpolarized waves (xwaves) the dispersion relationship is that of a onedimensional PhC^{46,47,48,49}
where K is the Bloch wave number,
and the Fresnel coefficient \(r_{o e}\) is given by
Equation (4) defines the band structure for the xwaves with the condition \(\cos {(K \Lambda )} = 1\) corresponding to the edges of the photonic band gap in which the wave propagation is forbidden. In Fig. 1a we demonstrate the transmittance spectrum of the system with 20 bilayers in each PhC arms; the overall system being submersed in air. One can see that for the xwaves the transmittance is zero within the band gap. On the other hand the PhC arms are always transparent to the ypolarized waves (ywaves) with dispersion \(k_o=\epsilon _o k_0\). Notice, though, that the ywaves transmittance exhibits a sharp dip at the center of the band gap. This dip is due to a high quality resonant mode predicted in^{45}. Although the line shape is symmetric, the dip, nonetheless, can be attributed to a Fano resonance as the transmittance reaches zero at the center of the band gap indicating a full destructive interference between two transmission paths. In this paper we set a goal of finding the analytic solution describing the Fano anomaly in the band gap.
Resonant eigenmode and bound state in the continuum
The resonant states are the eigenmodes of Maxwell’s equations (1) with reflectionless boundary conditions in the PhC arms. The equation for resonant eigenfrequencies can be obtained by matching the general solution in the ADL to the outgoing, both propagating and evanescent, waves in the PhC arms. Let us start from the general solution in the ADL.
General solution in the ADL
First, the unit vector along the propagation direction is defined as
where the symbol ± is used to discriminate between forward and backward propagating waves with respect to the orientation of the zaxis. The ADL supports two types of electromagnetic waves of different polarization. The ewaves with wavenumber \(k_e=\epsilon _e k_0\) are polarized along the director \(\varvec{a}\), Eq. (2), while the owaves with wavenumber \(k_o=\epsilon _o k_0\) have their electric vector perpendicular to both \(\varvec{a}\) and \(\varvec{\kappa }\), as seen from Fig. 1. The electric and magnetic vectors of the ewave can be written as
where \(E_{e}^{\scriptscriptstyle (\pm )}\) are unknown amplitudes. At the same time for owaves we have
where \(E_{o}^{\scriptscriptstyle (\pm )}\) are again unknown amplitudes. The general solution of equations (1) in the ADL, \(\ z \in [d,\ d]\), is written as a sum of the forward and backward propagating waves
General solution in the PhC arms
The general solution of Maxwell’s equations (1) in the PhC arms is also written as a sum of forward and backward propagating waves. For the xwaves the field components \(E_x\) and \(H_y\) in the isotropic layer with the cell number m, \(\ z \in [d + m \Lambda ,\ (m + 1) \Lambda ]\), are written as
In the anisotropic layer with the cell number m, \(\ z \in [(m+1) \Lambda ,\ d + (m + 1) \Lambda ] \), we have
By applying the continuity condition for the tangential field components the solutions (11) and (12) are matched on the boundary between the anisotropic layer in the \((m1)_{\mathrm{th}}\) cell and the isotropic layer in \(m_{\mathrm{th}}\) cell, \(\ z = d + m \Lambda \), as well as on the boundary between the layers in the \(m_{\mathrm{th}}\) cell, \(\ z = (m + 1)\Lambda \). This gives us a system of four equations for four unknowns, \(A^{\scriptscriptstyle (+)}, A^{\scriptscriptstyle ()}, B^{\scriptscriptstyle (+)}, B^{\scriptscriptstyle ()}\). After solving for \(B^{\scriptscriptstyle (+)}\) and \(B^{\scriptscriptstyle ()}\), this system can be reduced to the following two equations
where \(r_{oe}\) is given by Eq. (6). One can easily check that Eq. (13) is only solvable when K satisfies the dispersion relationship (4).
In contrast to the xwaves, for the outgoing ywaves in the right PhC arms the solution is simple
Notice that so far we have not written down the solution in the left PhC arm. The direct application of that solution can be avoided by using the mirror symmetry of the system. Here, in accordance with ref^{45} we restrict ourselves with the antisymmetric case
The generalization onto the symmetric case is straightforward.
Wave matching
Now we have all ingredients for finding the field profile of the antisymmetric resonant eigenmodes. By matching equation (10) to both Eqs. (11) and (14) on the interface between the ADL and the right PhC arm and using Eqs. (13, 15) one obtains eight equations for eight unknown variables \(E_{e}^{\scriptscriptstyle (+)}, E_{e}^{\scriptscriptstyle ()}, E_{o}^{\scriptscriptstyle (+)}, E_{o}^{\scriptscriptstyle ()}, A^{\scriptscriptstyle (+)}, A^{\scriptscriptstyle ()}, C^{\scriptscriptstyle (+)}, K\). After some lengthy and tedious calculations one finds that the system is solvable under the following condition
where
Taking into account Eq. (5) we see that the above formulae represent the transcendental equation for complex eigenvalues, \(k_0\) of the Maxwell’s equations (1). The analytic solution for the electromagnetic field within the ADL can be evaluated through the following formulae
Substituting (18) into Eq. (10) one finds the profile of the resonant eigenmode within the ADL
The amplitude A has to be defined from a proper normalization condition. We mention that in limiting case \(\phi \rightarrow 0\) the \(\zeta \rightarrow \infty \) and fields \(E_x = (A/n_e) \sin {(k_e z)}\), and \(E_y = 0\) coincide with exact solution for BIC (8), (21) from our previous work^{45}. The obtained eigenfield are plotted in Fig. 1a. One can see that though the ycomponent is localized due to the band gap, the xcomponent grows with the distance from the ADL as it is typical for resonant eigenstates^{4}.
Perturbative solution
Equations (16, 17) are generally not solvable analytically. There is, however, a single tractable perturbative solution in the case of quarterwave optical thicknesses of the layers
where \(\omega _{\scriptscriptstyle PBG}\) is the center frequency of photonic band gap, and \(\lambda _{\scriptscriptstyle PBG}\) is the corresponding wavelength. In our previous work^{45} we found an exact solution for \(\phi =0\). Here by applying a Taylor expansion of equations (16, 17) in powers of the tilt angle \(\phi \) we found approximate solutions for both resonant eigenfrequency and resonant eigenomode. By writing the resonant eigenfrequency as
where both \({\tilde{\omega }}\) and \(\gamma \) are real and positive, and substituting into Eqs. (16, 17) one finds
where \(q = n_{o}/n_{e}\). Notice that the imaginary part of \(\omega \) vanishes if \(\phi =0\). Thus, the system supports an antisymmetric BIC with the frequency \(\omega _{\scriptscriptstyle BIC}=\omega _{\scriptscriptstyle PBG}\). That BIC was first reported in our previous work^{45}. We address the reader to the above reference for detailed analysis of the BIC and the plots visualizing its eigenfields. For the further convenience we also introduce the resonant vacuum wave number as
where the complex valued \(\alpha \) is implicitly defined by Eq. (22) and \(k_{\scriptscriptstyle BIC}=\omega _{\scriptscriptstyle BIC}/c\). Finally, expanding (19) into the Taylor series in \(\phi \) we find the following expression for the resonant eigenmode profile within the ADL
Notice that \({\varvec{E}}_r\) can be handled as \(2\times 1\) vector since \(E_z=0\).
Normalization condition
There are several equivalent formulations of the FV normalization condition^{30,50,52}. Here we follow^{50}, writing down the FV normalization condition through analytic continuation \({\varvec{E}}(z,k)\) of the resonant eigenmode \({\varvec{E}}_r(z)\) around the point \(k=k_r\)
At the first glance the “flux” term in Eq. (25) differs from that in^{50} by the factor of 2; this is because the “flux” term is doubled to account for both interfaces \(z=\pm d\). For the resonant eigenmode \(\varvec{E}_r(z)\) found in the previous subsection the amplitude A corresponding equation (25) can be found analytically we the use of Eq. (24). This would, however, result in a very complicated expression. Fortunately, we shall see later on that in our case we do not need the general expression for A in the second order perturabtive solution consistent with Eq. (22). By a close examination of Eq. (24) one can see that the the Taylor expansion of A can only contain even powers of \(\phi \). Thus, for the further convenience we can write
assuming that F and B are such that the normalization condition (25) is satisfied up to \({\mathscr{O}}(\phi ^4)\).
Scattering problem
Let us assume that a monochromatic ywave is injected into the system through the left PhC arm. The scattering problem can now be solved through the following decomposition of the electric field within the ADL
where the direct contribution is simply the electric field of the incident wave
with intensity \(I_0\), while \({\varvec{E}}_{\scriptscriptstyle res}\) can be viewed as the contribution due to the resonant pathway via exitation of the resonant eigenmode \({\varvec{E}}_r\). Substituting Eq. (27) into Eq. (1) one obtains the inhomogeneous equation
with
Equation (29) can be solved with the use of Green’s function
that is defined as the solution of Maxwell’s equations with a delta source
where \(\widehat{{\mathbb {I}}}_{2}\) is the \(2\times 2\) identity matrix. According to^{50} Green’s function can expanded into the orthonormal resonant eigenmodes as
Of course we do not know the full spectrum \(k_n\), since Eqs. (16, 17) are not solved analytically. We, however, assume that the contribution of all eigenfields except \({\varvec{E}}_{\scriptscriptstyle res}\) is accumulated in the direct field. Thus, in the spectral vicinity of the quasiBIC we apply the resonant approximation taking into account only the eigenmode associated with the BIC
The resonant field can now be calculated from Eq. (31) with the resonant Green’s function Eq. (34) once the FV normalization condition (26) is applied to the quasiBIC eigenmode. The analytic expression for the resonant field reads
The above equation constitutes the perturbative solution of the scattering problem with the accuracy up to \(\mathcal{O} (\phi ^3)\). Notice that the terms dependant on B also vanish as \(\mathcal{O} (\phi ^3)\). Thus, in evaluating the FV normalization condition (25) we can restrict ourselves to \(\phi =0\) in which case the eigenmode is a BIC. Further on we safely set \(B=0\) in all calculations. The BIC is localized function decaying with \(z\rightarrow \pm \infty \). Since the division into the scattering domain and the waveguides is arbitrary and the flux term is vanishing with \(z\rightarrow \pm \infty \), one can rewrite the normalization condition for the BIC as follows
We mention in passing that the equivalence between Eqs. (25) and (36) can also be proven by subsequently applying the Newton–Leibniz axiom and Maxwell’s equations (1) to the flux term in Eq. (25). The integral in Eq. (36) is nothing but the energy stored in the eigenmode up to a constant prefactor. This integral for the system in Fig. 1a has been evaluated analytically in our previous work^{45}. As a result the normalization condition (36) yields
Finally, the reflection and transmission coefficients can be found through the following equations, correspondingly
and
where \({\varvec{e}}_y= \left\{ 0, 1 \right\} ^{\dagger }\) and \({\varvec{E}}\) is computed through Eqs. (27, 28, 35). In Fig. 2 we compare the perturbative analytic solution against direct numerical simulations with the Berreman transfermatrix method^{54}. One can see that deviation is no more than 2% even for relatively large angle \(\phi = 10\) deg. In Fig. 2 one can see a typical Fano transmission pattern with interference between two pathways^{3}. The direct pathway is due to the incident ordinary wave penetrating through the ADL from the left to the right PhC arm. The second pathway is the resonant excitation of the quasiBIC. Finally, the origin of the collapse of the Fano resonance^{35,36,37,38,39,40} at the exact normal incidence is easily explained by the denominator \(k_0k_r\) in Eq. (35) which contains a vanishing resonant width \(\gamma \).
Conclusion
In this paper we considered light scattering by an anisotropic defect layer embedded into anisotropic photonic crystal in the spectral vicinity of an optical BIC. Using a resonant state expansion method we derived an analytic solution for reflection and transmission amplitudes. The analytic solution is constructed via a perturbative approach with the BIC as the zeroth order approximation. The solution is found to accurately describe the collapsing Fano feature in the spectral vicinity of the BIC. So far the theoretical attempts to describe the Fano feature induced by the BIC relied on phenomenological approaches such as the Smatrix approach^{38}, or the coupled mode theory^{39}. To the best of our knowledge this is the first fullwave analytic solution involving an optical BIC reported in the literature. We believe that the results presented offer a new angle onto the resonant state expansion method paving a way to analytic treatment of resonant scattering due to optical BICs. In particular we expect that the resonant approximation can be invoked to build a rigorous theory of nonlinear response^{55}. Recently, the Fano resonance induced by an optical BIC in a liquid crystal ADL has been reported experimentally in^{56} at the Brewster angle. We believe that extending our theoretical approach to the Brewster BIC could be an interesting topic for the future studies. Nowadays, the BICs in photonic systems have already found important applications in enhanced optical absorbtion^{57}, surface enhanced Raman spectroscopy^{58}, lasing^{59}, and sensors^{60}. We speculate that analytic results are of importance for a further insight into localization of light as well as the concurrent phenomenon of collapsing Fano resonance.
Data availability
The data that support the findings of this study are available from the corresponding author, P.S.P., upon reasonable request.
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Acknowledgements
This work was supported by Russian Foundation for Basic Research project No. 195252006. This project is also supported by by the Higher Education Sprout Project of the National Chiao Tung University and Ministry of Education and the Ministry of Science and Technology (MOST No. 1072221E009 046MY3; No. 1082923E009003MY3).
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D.N.M. and T.I.V. conceived the idea presented. P.S.P. and D.N.M. have equally contributed to the analytic results. P.S.P. ran numerical simulations. D.N.M., P.S.P., K.P.C. and T.I.V. have equally contributed to writing the paper.
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Pankin, P.S., Maksimov, D.N., Chen, K. et al. Fano feature induced by a bound state in the continuum via resonant state expansion. Sci Rep 10, 13691 (2020). https://doi.org/10.1038/s41598020706542
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