Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2022; 6(3): 270-281
Published online June 25, 2022 https://doi.org/10.3807/COPP.2022.6.3.270
Copyright © Optical Society of Korea.
Qingyue Zhang^{1,2}, Weitao Mao^{1}, Qiuling Zhao^{1,2} , Maorong Wang^{1,2}, Xia Wang^{1,2}, Wing Yim Tam^{1,3}
Corresponding author: sdqlzhao@163.com, ORCID 0000-0003-2701-5344
^{¶} Current affiliation: Nantong Academy of Intelligent Sensing, Nantong 226010, China
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Based on the transfer-matrix method (TMM), we report the characteristics of the interface states in one-dimensional (1D) composite structures consisting of two photonic crystals (PCs) composed of binary dielectrics A and B, with unit-cell configurations ABA (PC I) and BAB (PC II). The dependence of the interface states on the number of unit cells N and the boundary factor x are displayed. It is verified that the interface states are independent of N when the PC has inversion symmetry (x = 0.5). Besides, the composite structures support the formation of interface states independent of the PC symmetry, except that the positions of the interface states will be varied within the photonic band gaps. Moreover, the robustness of the interface states against nonuniformities is investigated by adding Gaussian noise to the layer thickness. In the case of inversion symmetry (x = 0.5) the most robust interface states are achieved, while for the other cases (x ≠ 0.5) interface states decay linearly with position inside the band gap. This work could shed light on the development of robust photonic devices.
Keywords: Interface states, Inversion symmetry, Photonic crystals, Robustness, Topological protection
OCIS codes: (050.5298) Photonic crystals; (080.6755) Systems with special symmetry; (350.5500) Propagation
In the past few decades the novel concept of topological states has attracted intensive research interest, and has been exploited widely in many fields of modern physics [1, 2] and materials science [3, 4], due to unique boundary states and the related propagation properties. Inspired by these encouraging ideas, topological photonics has emerged as a hot research topic to manipulate the behavior of light, promising a new variety of high-performance photonic devices [5–7]. Tremendous amounts of research [3–8] have shown that photonic topological protection can be realized in photonic crystals (PCs), waveguides, nanocavities, coupled resonators, metamaterials, etc. In PCs, topological states manifest as interface states [9–11] in periodic as well as quasiperiodic composite structures, which are topologically protected and hence robust against defects, edges, or disorder owing to experimental nonuniformity or finite size [12, 13]. Thanks to the structural flexibility and experimental advances, topological PCs provide an excellent platform for potential applications in enhancing nonlinearity [14–16], lasing [17, 18], sensing [19], waveguiding [20], etc.
Xiao
In this paper, we systematically characterize the robustness of the interface states in 1D composite structures consisting of two binary layered PCs, with and without inversion symmetry, employing the transfer-matrix method. The inversion-symmetric PC’s interface states are well defined, deep inside of the photonic band gaps, and insensitive to system size. Thereafter, the layer thickness of the PCs is combined with random noise to analyze the robustness of the interface states, confirming that the inversion-symmetric PC’s interface states are the most robust. Overall, the robustness of the non-inversion-symmetric PC’s interface states has a linear dependence on the positions of the interface states inside the band gaps, which yields practical insight into the development of photonic devices by fine tuning the interface states.
The transfer-matrix method (TMM) is a common method to model 1D photonic structures [26]. It involves two matrices: one for the propagation of light in each medium (propagation matrix), and another for the transmission across the boundary (matching matrix) [25]. The propagation matrix relating to the change of electric fields in the forward and backward directions (E_{+}_{j}, E_{−}_{j}) when light passes through a homogeneous
where
where
The top inset of Fig. 1 depicts the composite photonic structure with PC I on the left and PC II on the right, in which a boundary is generated in the middle and indicated by the vertical green dashed line. The boundary layer is controlled by the boundary factor
The transfer matrix of the unit cell for the ABA (from left to right) configuration,
where the matrix elements are
Here
Similarly, the transfer matrix of the unit cell for the BAB configuration,
where the matrix elements are
Note that
Specifically, when the individual PC has inversion symmetry,
The dispersion relation of PC I (and also for PC II) can be calculated using the Bloch theorem with the following equation [27, 28]:
where
The transfer matrix T_{PCI} for PC I in air with
where the matrix elements are
Furthermore, the transmission (
Similarly, the transfer matrix for PC II with
where the matrix elements are
Note that
From Eqs. (9), (11), and (12) we can obtain the reflection coefficients, and hence the reflectance of PC I and PC II respectively
For
Finally, the transfer matrix for the 1D composite structure can be described as
The reflection coefficient is
On the basis of Eq. (15), the reflectance of the composite structure can be expressed as
where
The interface states, visible as reflection dips inside the band gaps, are clearly observed in the middle of the odd band gaps, as indicated by the black arrows in Fig. 1(d) for the first and third band gaps. Similar interface states can also be observed for 0 <
We first study the dependence of the interface states on the number of unit cells
Considering two PCs as components, interface states will exist in the composite structure when the sum of reflection phases of the two individual PCs is equal to zero,
When
Additionally, according to Eq. (19) the imaginary part of
Figure 3(a) shows clearly that |
at the interface states, as shown in Fig. 3(b). According to Eqs. (10) and (13), the conditions given by Eqs. (21) and (22) can be simplified as
relating the physical parameters of PC I and PC II. Particularly, these conditions for interface states are independent of
For non-inversion-symmetric cases,
According to Eqs. (18) and (19), the reflectance of interface states in the composite structure can be described as
When
For practical applications, randomness is inevitable in the preparation of photonic structures. The layer thickness is most susceptible to minor errors due to fabrication uncertainties, even as it has a crucial role in regulating the interface states in PCs. Accordingly, the robustness of the interface states against random thickness is significant, for both simulations and experiments. To study the robustness of the interface states of the 1D composite structures described above, we add Gaussian noise to the layer thickness.
The layer thicknesses of materials A and B are written as
where δ
Here σ(δ
where
The robustness of interface states against different noise levels is investigated for both inversion (Figs. 4 and 5) and non-inversion symmetry (Figs. 6 and 7). Gaussian-distributed values of δ
where mean = <
For
In terms of both cutoff levels and cumulative-sum criteria, we map out the robust interface states for the composite structures of various
In summary, the characteristics of the interface states in composite photonic structures consisting of two PCs have been illuminated by means of the transfer-matrix method. Interface states could be realized at the centers of the photonic band gaps for the inversion-symmetric case, but off-center toward the band edges for the non-inversion-symmetric case. We further examined the robustness of the interface states against noise by adding randomness to the layer thickness, and found that the interface states for the inversion-symmetric case were robust against random noise, as they were deep inside the band gaps and topologically protected while they were more sensitive to random noise for the non-inversion-symmetric case. More importantly, the robustness of the interface states scaled linearly with the relative location inside the band gap with respect to that of the inversion-symmetric case, and became unstable near or at the band edges. This work may be helpful for corresponding experimental fabrication, and has potential value in interface-state-relevant applications such as narrow-band filters.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, which may be obtained from the authors upon reasonable request.
National Natural Science Foundation of China (grant numbers: 11874232, 12174211, 61905127); Hong Kong RGC grants (grant numbers: AoE P-02/12, C6013-18G).
Curr. Opt. Photon. 2022; 6(3): 270-281
Published online June 25, 2022 https://doi.org/10.3807/COPP.2022.6.3.270
Copyright © Optical Society of Korea.
Qingyue Zhang^{1,2}, Weitao Mao^{1}, Qiuling Zhao^{1,2} , Maorong Wang^{1,2}, Xia Wang^{1,2}, Wing Yim Tam^{1,3}
^{1}College of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China
^{2}Shandong Advanced Optoelectronic Materials and Technologies Engineering Laboratory, Qingdao University of Science and Technology, Qingdao 266061, China
^{3}Department of physics, William Mong Institute of Nano Science and Technology, Center for Metamaterial Research, Hong Kong University of Science and Technology, Kowloon 999077, Hong Kong, China
Correspondence to:sdqlzhao@163.com, ORCID 0000-0003-2701-5344
^{¶} Current affiliation: Nantong Academy of Intelligent Sensing, Nantong 226010, China
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Based on the transfer-matrix method (TMM), we report the characteristics of the interface states in one-dimensional (1D) composite structures consisting of two photonic crystals (PCs) composed of binary dielectrics A and B, with unit-cell configurations ABA (PC I) and BAB (PC II). The dependence of the interface states on the number of unit cells N and the boundary factor x are displayed. It is verified that the interface states are independent of N when the PC has inversion symmetry (x = 0.5). Besides, the composite structures support the formation of interface states independent of the PC symmetry, except that the positions of the interface states will be varied within the photonic band gaps. Moreover, the robustness of the interface states against nonuniformities is investigated by adding Gaussian noise to the layer thickness. In the case of inversion symmetry (x = 0.5) the most robust interface states are achieved, while for the other cases (x ≠ 0.5) interface states decay linearly with position inside the band gap. This work could shed light on the development of robust photonic devices.
Keywords: Interface states, Inversion symmetry, Photonic crystals, Robustness, Topological protection
In the past few decades the novel concept of topological states has attracted intensive research interest, and has been exploited widely in many fields of modern physics [1, 2] and materials science [3, 4], due to unique boundary states and the related propagation properties. Inspired by these encouraging ideas, topological photonics has emerged as a hot research topic to manipulate the behavior of light, promising a new variety of high-performance photonic devices [5–7]. Tremendous amounts of research [3–8] have shown that photonic topological protection can be realized in photonic crystals (PCs), waveguides, nanocavities, coupled resonators, metamaterials, etc. In PCs, topological states manifest as interface states [9–11] in periodic as well as quasiperiodic composite structures, which are topologically protected and hence robust against defects, edges, or disorder owing to experimental nonuniformity or finite size [12, 13]. Thanks to the structural flexibility and experimental advances, topological PCs provide an excellent platform for potential applications in enhancing nonlinearity [14–16], lasing [17, 18], sensing [19], waveguiding [20], etc.
Xiao
In this paper, we systematically characterize the robustness of the interface states in 1D composite structures consisting of two binary layered PCs, with and without inversion symmetry, employing the transfer-matrix method. The inversion-symmetric PC’s interface states are well defined, deep inside of the photonic band gaps, and insensitive to system size. Thereafter, the layer thickness of the PCs is combined with random noise to analyze the robustness of the interface states, confirming that the inversion-symmetric PC’s interface states are the most robust. Overall, the robustness of the non-inversion-symmetric PC’s interface states has a linear dependence on the positions of the interface states inside the band gaps, which yields practical insight into the development of photonic devices by fine tuning the interface states.
The transfer-matrix method (TMM) is a common method to model 1D photonic structures [26]. It involves two matrices: one for the propagation of light in each medium (propagation matrix), and another for the transmission across the boundary (matching matrix) [25]. The propagation matrix relating to the change of electric fields in the forward and backward directions (E_{+}_{j}, E_{−}_{j}) when light passes through a homogeneous
where
where
The top inset of Fig. 1 depicts the composite photonic structure with PC I on the left and PC II on the right, in which a boundary is generated in the middle and indicated by the vertical green dashed line. The boundary layer is controlled by the boundary factor
The transfer matrix of the unit cell for the ABA (from left to right) configuration,
where the matrix elements are
Here
Similarly, the transfer matrix of the unit cell for the BAB configuration,
where the matrix elements are
Note that
Specifically, when the individual PC has inversion symmetry,
The dispersion relation of PC I (and also for PC II) can be calculated using the Bloch theorem with the following equation [27, 28]:
where
The transfer matrix T_{PCI} for PC I in air with
where the matrix elements are
Furthermore, the transmission (
Similarly, the transfer matrix for PC II with
where the matrix elements are
Note that
From Eqs. (9), (11), and (12) we can obtain the reflection coefficients, and hence the reflectance of PC I and PC II respectively
For
Finally, the transfer matrix for the 1D composite structure can be described as
The reflection coefficient is
On the basis of Eq. (15), the reflectance of the composite structure can be expressed as
where
The interface states, visible as reflection dips inside the band gaps, are clearly observed in the middle of the odd band gaps, as indicated by the black arrows in Fig. 1(d) for the first and third band gaps. Similar interface states can also be observed for 0 <
We first study the dependence of the interface states on the number of unit cells
Considering two PCs as components, interface states will exist in the composite structure when the sum of reflection phases of the two individual PCs is equal to zero,
When
Additionally, according to Eq. (19) the imaginary part of
Figure 3(a) shows clearly that |
at the interface states, as shown in Fig. 3(b). According to Eqs. (10) and (13), the conditions given by Eqs. (21) and (22) can be simplified as
relating the physical parameters of PC I and PC II. Particularly, these conditions for interface states are independent of
For non-inversion-symmetric cases,
According to Eqs. (18) and (19), the reflectance of interface states in the composite structure can be described as
When
For practical applications, randomness is inevitable in the preparation of photonic structures. The layer thickness is most susceptible to minor errors due to fabrication uncertainties, even as it has a crucial role in regulating the interface states in PCs. Accordingly, the robustness of the interface states against random thickness is significant, for both simulations and experiments. To study the robustness of the interface states of the 1D composite structures described above, we add Gaussian noise to the layer thickness.
The layer thicknesses of materials A and B are written as
where δ
Here σ(δ
where
The robustness of interface states against different noise levels is investigated for both inversion (Figs. 4 and 5) and non-inversion symmetry (Figs. 6 and 7). Gaussian-distributed values of δ
where mean = <
For
In terms of both cutoff levels and cumulative-sum criteria, we map out the robust interface states for the composite structures of various
In summary, the characteristics of the interface states in composite photonic structures consisting of two PCs have been illuminated by means of the transfer-matrix method. Interface states could be realized at the centers of the photonic band gaps for the inversion-symmetric case, but off-center toward the band edges for the non-inversion-symmetric case. We further examined the robustness of the interface states against noise by adding randomness to the layer thickness, and found that the interface states for the inversion-symmetric case were robust against random noise, as they were deep inside the band gaps and topologically protected while they were more sensitive to random noise for the non-inversion-symmetric case. More importantly, the robustness of the interface states scaled linearly with the relative location inside the band gap with respect to that of the inversion-symmetric case, and became unstable near or at the band edges. This work may be helpful for corresponding experimental fabrication, and has potential value in interface-state-relevant applications such as narrow-band filters.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, which may be obtained from the authors upon reasonable request.
National Natural Science Foundation of China (grant numbers: 11874232, 12174211, 61905127); Hong Kong RGC grants (grant numbers: AoE P-02/12, C6013-18G).