Finite-size effects on topological interface states in one-dimensional scattering systems

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux

Phys. Rev. A 98, 023838 – Published 20 August 2018

Abstract

Topological edge modes in one-dimensional photonic systems are usually studied considering the interface between two different semi-infinite periodic crystals (PCs) with inverted band structure around the Dirac point. Here we consider the case where the two PCs are finite, constituting an open scattering system, and we study the influence of the size of this finite structure on the interface mode by inspecting the complex resonances. First we show the complex resonance distribution corresponding to the band inversion around the Dirac point. Perturbations from the Dirac point display the emergence of the localized interface mode. We also report on a remarkable robustness of the interface mode as the system size varies, which persists even for the smallest possible size.

Received 23 December 2017

DOI:https://doi.org/10.1103/PhysRevA.98.023838

Physics Subject Headings (PhySH)

Photonic crystals Photonics Topological effects in photonic systems

Atomic, Molecular & Optical

Authors & Affiliations

P. A. Kalozoumis^1,2,*, G. Theocharis¹, V. Achilleos¹, S. Félix¹, O. Richoux¹, and V. Pagneux¹

¹Laboratoire d'Acoustique de l'Université du Maine, UMR CNRS 6613 Av. O. Messiaen, F-72085 LE MANS Cedex 9, France
²Department of Physics, University of Athens, 15771 Athens, Greece

^*pkalozoum@phys.uoa.gr

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Vol. 98, Iss. 2 — August 2018

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Images

Figure 1
(a) Schematic of an one-dimensional infinite photonic crystal. (b)–(d) Band structure of the photonic crystal for $n_{A} = 1.95$ , 2, and 2.05 respectively ( $n_{A^{'}} = 1$ ). The width $d_{A}$ has to change according to the relation $d_{A} = (τ - n_{A^{'}}) / (n_{A} - n_{A^{'}})$ in order to retain the midgap positions [27]. (e) Schematic of two attached semi-infinite periodic systems, the left one with $n_{A} = 1.95, n_{A^{'}} = 1$ and the right one with $n_{B} = 2.05, n_{B^{'}} = 1$ . (f) Spectrum of the systems shown in panel (e) with an interface mode inside the gap.
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Figure 2
(a) Schematic of scattering through a 1D finite periodic photonic crystal. (b) Panel of plots illustrating the transmittance $T$ from a periodic structure of 20 cells with refractive index values $n_{A} = 1.95$ (point-down triangle), $n_{A} = 2$ (square), $n_{A} = 2.05$ (point-up triangle). (c) Panel of plots showing the distribution of the poles of the scattering matrix when the refractive index $n_{A}$ varies within the range $[1.95, 2.05]$ (top to bottom). The three colored poles (circles and star) correspond to PTRs of the corresponding unit cell. The star pole is the one which is related to the origin of the interface state. The purple solid lines at $f_{i} = 0$ correspond to the continuum spectrum of the infinite system.
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Figure 3
(a) Schematic showing how a simultaneous decrease (increase) from the left (right) of the refractive index $n_{A}$ , from an initially periodic structure forms an interface. (b) Panel of transmittance plots for three different values of $δ n$ ( $δ n = 0$ , $δ n = 0.025$ , $δ n = 0.05$ ). (c) Trajectories of the poles of the scattering matrix in the complex-frequency plane as the $δ n$ increases up to the value 0.05. The initial value of the refractive index is $n_{A} = 2$ and corresponds to the periodic case ( $δ n = 0$ ), indicated by the dotted line and the yellow circles. The pole corresponding to the interface mode (central) is shown with red color.
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Figure 4
Size variation of a structure with an interface state. The size varies from a total number of 20 cells to the minimum possible number of two cells. (a) Scattering matrix pole distribution for four different system sizes. As the size decreases the pole bands depopulate (yellow dots). However, the interface mode, shown with the red star, remains almost unaffected with respect to $f_{r}$ . The imaginary part of the frequency $f_{i}$ decreases, rendering the state more leaky. (b) Transmittance counterpart of panel (a). For large $N$ , the interface state in the center of the gap is more pronounced and the gap more well defined. For smaller sizes the peak becomes wider, implying a smaller quality factor. Nevertheless, it is unaffected with respect to the $f_{r}$ , even for the smallest possible case of two cells.
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