Topological properties of bound states in the continuum in geometries with broken anisotropy symmetry

Samyobrata Mukherjee, Jordi Gomis-Bresco, Pilar Pujol-Closa, David Artigas, and Lluis Torner

Phys. Rev. A 98, 063826 – Published 18 December 2018

Abstract

Waveguiding structures made of anisotropic media support bound states in the continuum (BICs) that arise when the radiation channel of otherwise semileaky modes is suppressed. Hitherto, only structures with optical axes aligned in symmetric orientations inside the waveguide plane, where BICs appear as lines in the momentum-frequency dispersion diagram, have been considered. Here we address settings where such symmetry is broken and unveil a number of fundamental different features. Weak and strong symmetry-breaking regimes are identified, corresponding to azimuthal and polar optical axes orientation asymmetries, respectively. The azimuthal symmetry breaking is found to still preserve the existence loci of BICs in the momentum-frequency dispersion diagram as lines. However, all possible BICs become interferometric, while the polarization separable states that occur in symmetric settings cease to exist. The polar symmetry breaking has stronger effects and transforms the BICs' existence loci from lines to points, which correspond to full-vector states that exist at discrete values of the optical axis orientation for a given wavelength. Such transformation results in fundamental changes in the topological properties of the radiated field around the BICs.

Received 25 June 2018

DOI:https://doi.org/10.1103/PhysRevA.98.063826

Physics Subject Headings (PhySH)

Integrated optics Metamaterials Photonics Topological effects in photonic systems

Atomic, Molecular & Optical

Authors & Affiliations

Samyobrata Mukherjee¹, Jordi Gomis-Bresco¹, Pilar Pujol-Closa¹, David Artigas^1,2,*, and Lluis Torner^1,2

¹ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²Department of Signal Theory and Communications, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain

^*david.artigas@icfo.eu

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Vol. 98, Iss. 6 — December 2018

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Figure 1
(a) Layout of the system: a generic layered waveguide system comprising cladding, film, and substrate. Light propagates along the $Y$ direction. The green and the blue arrows indicate the substrate and film optical axis (OA), respectively. The polar $θ_{f}$ and azimuthal $ϕ_{f}$ angles indicate the film OA orientation in spherical coordinates. The substrate OA orientation is determined by $θ_{s}$ and the azimuthal detuning by $Δ ϕ = ϕ_{s} - ϕ_{f}$ . (b) Momentum-frequency dispersion diagram for the leaky modes of an anisotropy-symmetric structure with a film with positive birefringence ( $n_{o f} = 1.5$ , $n_{e f} = 1.75$ ), a substrate with a negative birefringence ( $n_{o s} = 2.00$ , $n_{e s} = 1.25$ ), and air as a cladding $n_{c} = 1$ . The polar orientation of the OA are $θ_{s} = θ_{f} = 90^{\circ}$ . The ratio $d / λ$ is the normalized frequency and $k_{z}$ and $k_{y}$ are the mode momenta along the crystallographic axis (with the OA laying in the $Z$ direction), which can be related to the angle $ϕ_{f}$ between the OA orientation and the propagation direction as $ϕ_{f} = arctan (k_{z} / k_{y})$ . Two yellowish surfaces (B0 and B1) for the leaky modes are plotted, where dashed lines show the limits of the surface and the color indicates the decay length $L$ (in $μ m$ ). Then, blue lines stand for an infinite decay length, indicating the existence of BICs. The labels INT-BICs and PS-BICs refer to interferometric and polarization-separable BICs, respectively. The gray surface behind the leaky mode sheets is the light cone, given by $k_{0} \cdot n_{o s}$ ; the light gray–transparent surface limited by solid lines sets the leaky mode cutoff, given by $k_{0} \cdot n_{e s} (ϕ_{f})$ . Here $k_{0}$ is the free-space wave number.
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Figure 2
Existence loci of BICs in structures with broken anisotropy symmetry. The colored area in the figures shows the decay length of the leaky modes in the $ϕ_{f} - d / λ$ plane, where BICs appear as blue lines. The transition from the colored to the white area is the cutoff for the leaky modes. (a) Anisotropy-symmetric structure shown in Fig. 1, where the substrate and film OAs are aligned and in plane. (b) Structure with weakly broken-anisotropy symmetry in the azimuthal direction ( $Δ ϕ = - 0 . 5^{\circ}$ ). (c) Structure with strongly broken-anisotropy symmetry in the polar direction, by taking the OA of the film out of the interface plane ( $θ_{s} = 90^{\circ}$ , $θ_{f} = 85^{\circ}$ ) and keeping $Δ ϕ = 0^{\circ}$ (i.e., the projection of the film OA is parallel to the substrate OA). Note the transition from existence lines in (a) and (b) to dots in (c).
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Figure 3
Transformation from simple phase dislocation to screw phase dislocations. The parameters are the same as in Fig. 2. The color stands for the phase difference of the ordinary radiative channel with respect to the phase of the extraordinary confined wave in the substrate. The existence lines of BICs in (a) and (b) exhibit phase dislocation of $π$ . BICs in (c) are singularities with winding numbers $+ 1$ (clockwise increase of phase) and $- 1$ (counterclockwise increase of phase).
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Figure 4
Topological transformation of the phase of the radiative channel in a structure with a positive birefringent substrate and negative film. The parameters of the structure are $n_{e s} = 2.00$ , $n_{o s} = 1.25$ , $n_{e f} = 1.50$ , $n_{o f} = 1.75$ , $n_{c} = 1$ , $θ_{s} = θ_{f} = 90^{\circ}$ , and $Δ ϕ = 0^{\circ}$ . The color stands for the phase difference between the extraordinary radiative channel and the ordinary wave in the substrate. The lines of existence of BICs in an anisotropy-symmetric structure (a) and in structures with the azimuthal symmetry broken (b) exhibit phase dislocations of $π$ . Breaking the polar symmetry by taking the film optical axis out of plane precludes the existence of BICs (c). BICs occur as points when the polar symmetry is broken by taking both the film and substrate optical axes out of plane, resulting in one BIC with winding number $+ 1$ (d) or two BICs with winding number $- 1$ (e) depending on the relative orientation of the film and substrate OAs. When both polar and azimuthal symmetries are broken, the BIC loci can be tuned for any orientation of $ϕ_{f}$ (f).
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Figure 5
Spherical representation of the BIC existence loci as a function of the optical axis orientation in the guiding film. The film OA orientation is given by the azimuthal angle $ϕ_{f}$ and the polar angle $θ_{f}$ in spherical coordinates, while the parameters for the substrate ( $θ_{s}$ and $Δ ϕ$ ) are kept constant in each plot. Only the hemisphere in the range $ϕ_{f} = [0^{\circ}, 180^{\circ}]$ is depicted as the vertical axis of the sphere; $θ_{f} = 0^{\circ}$ is a symmetry axis. Film thickness is $d = 0.35 μ m$ ; hence the colors of the BIC existence lines indicate the value of the wavelength. Winding numbers are labeled as “+1” or “ $- 1$ ” and phase jumps are labeled with “=”. (a) Spherical representation for a structure with a positive film and a negative substrate. The indices of refraction are as in Fig. 1. The substrate OA is parallel to the interface plane ( $θ_{s} = 90^{\circ}$ , $Δ ϕ = 0^{\circ}$ ). (b) Same as in (a) but the substrate OA is out of the interface plane ( $θ_{s} = 89^{\circ}$ ), resulting in strong anisotropy-symmetry breaking even at the equatorial plane. In both cases $Δ ϕ = 0^{\circ}$ . (c) Structure with azimuthal anisotropy-symmetry breaking ( $Δ ϕ = - 0 . 5^{\circ}$ ) with the substrate OAs in the interface plane $θ_{s} = 90^{\circ}$ . (d) Combined azimuthal ( $Δ ϕ = 0 . 5^{\circ}$ ) and polar ( $θ_{s} = 89^{\circ}$ ) anisotropy asymmetries. The insets in (b)–(d) are a magnification of the area near $θ_{f} = 90^{\circ}$ and $θ_{f} = 90^{\circ}$ that show the splitting of the polarization separable BIC line in (a), shown as a white point, into two segments of interferometric BICs. (e) Spherical representation of a structure with negative film and positive substrate. The indices of refraction are as in Fig. 4. The substrate OA is parallel to the interface plane ( $θ_{s} = 90^{\circ}$ , $Δ ϕ = 0^{\circ}$ ). For this particular case, BICs cease to exist when the polar anisotropy symmetry is broken ( $θ_{f} \neq 90^{\circ}$ ). (f) Same as in (e) but now the substrate OA is out of the interface plane ( $θ_{s} = 100^{\circ}$ ).
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