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Abstract
In this study, a one-dimensional (1D) two-material period ring optical waveguide network (TMPROWN) was designed, and its optical properties were investigated. The key characteristics observed in the 1D TMPROWN include the following: (1) Bound states in continuum (BICs) can be generated in the optical waveguide network. (2) In contrast to the BICs previously reported in optical structures, the range of the BICs generated by the 1D TMPROWN is not only larger, but also continuous. This feature makes it possible for us to further study the electromagnetic wave characteristics in the range of the BICs. In addition, we analyzed the physical mechanisms of the BICs generated in the 1D TMPROWN. The 1D TMPROWN is simple in structure, demonstrates flexibility with respect to adjusting the frequency band of the BICs, and offers easy measurement of the amplitude and phase of electromagnetic waves. Hence, further research on high-power super luminescent diodes, optical switches, efficient photonic energy storage, and other optical devices based on the 1D TMPROWN designed in this study is likely to have implications in a broad range of applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the introduction of bound states in continuum (BICs) [1] by Neumann and Wigner in 1929, it has been widely applied in the fields of quantum mechanics, acoustics, water waves, and optics [2–17]. The typical characteristics of BICs include: (1) Electromagnetic (EM) wave lies in continuum. (2) EM wave should be radiated but remains bounded in special optical structure [2]. BICs are found in a wide variety of material systems through a binding mechanism that is completely different from the traditional bound state [2]. In recent years, since photonic structures can be tailored in terms of their materials and structures, the unique properties of BICs in optical structures have led to the development of several devices, such as filters, low-loss fibers, lasers, and sensors.
Notably, BICs are primarily classified into two types: symmetrical protection and resonance-trapped BICs [2]. Ideal BICs exist only in lossless structures [6,18] or at extreme parameter values [7,11]. In general, resonance-trapped BICs can be realized via inverse construction in optical metasurfaces, waveguide structures, and photonic crystals [8–12]. However, since the inherent non-radiative property of ideal BICs is the disappearance of the resonance linewidth, BICs cannot be observed in the electromagnetic spectrum. Consequently, optical structures in practical applications are frequently designed using quasi-BICs [7,13–17], and optical properties in the range of BICs have not been studied.
Optical waveguide networks (OWNs) have been extensively studied over the last two decades and have been widely used to observe related optical phenomena, such as photonic localization [19,20], non-reciprocity of EM wave propagation [20,21,22,23,24], Bloch oscillation [25,26], Rabi oscillation [27], slow light effect [28] and ultrastrong extraordinary transmission and reflection characteristics [29–31]. Additionally, they can form an artificial photonic bandgap (PBG) structure that can limit and control the propagation of EM waves [25–28,32–44] and can produce ultrawide PBG [41]. Besides, the structural connection of the OWN is extremely flexible, and the flexible connection between the waveguide segments makes it easier to realize a rich symmetry and high-dimensional structures [32,33].
The prerequisite of investigating the properties of BICs in a special optical structure requires that the BICs generated must be nonzero line-width, and this case does not studied in the past literatures. Instantaneously, because most wave systems prohibit BICs in compact structures, so lots of theoretically proposed or experimentally observed BICs are implemented as extended optical structures. Therefore, in general, structures yielding BICs extend to infinity in at least one direction [2]. Under the conditions of the optical structure based on previous design, we design a 1D TMPROWN composed of lithium chloride (LiCl) which can saturate the condition of nonzero line-width.
In this study, we search for BICs in OWN systems in regards to ultrawide PBG and superior optical properties of the OWN system and explore the properties of the BICs generated in the OWN. Actually, the distribution range of the BICs is determined by adjusting the structural parameters of the 1D TMPROWN and calculating the photon mode and transmission. Subsequently, the mechanism of BIC generation is explained by calculating the distribution of the photonic localization in the waveguide segment. Since the amplitude and phase of the EM wave propagating through the OWN can be easily measured at each node, we anticipate the discovery of novel characteristics of BICs that are not found in other optical systems, which can serve as a solid foundation for the practical application of BICs.
This paper is organized as follows. Section 2 introduces the model and theory. In Section 3, we explain the causes of the formation of the BICs and investigate the unique optical properties of the 1D TMPROWN in the range of the BICs. Finally, the conclusions of this study are presented in Section 4.
2. Model and theory
2.1 1D TMPROWN
A schematic of the 1D TMPROWN is shown in Fig. 1, where the length of each waveguide segment is denoted by d. We ensure that the regions where not only obvious BICs can be generated, but also it is with extremely weak LiCl dispersion are located near the communication wavelength by repeatedly adjusting the length parameter d. Finally, we set d = 131µm. Let l1 and l2 represent the length ratios in this study, and l1 = l2. In Fig. 1, the blue solid lines at the entrance and exit represent the optical waveguide segments in vacuum with a refractive index of n0 = 1. The red solid lines in the unit cells represent the two-material optical waveguide segments composed of LiCl, whose refractive indices for the two sub-waveguides are:
Fig. 1. Schematic of the 1D TMPROWN. A component of 1D TMPROWN consists of three unit cells (each of which is represented by a single solid red line and red ring), an entrance, and exit, where EI, ER, and EO represent the input, reflective, and output EM waves, respectively. Each waveguide segment is of length d. The solid blue lines represent the optical vacuum waveguide segments at the entrance and exit. The solid red lines in the unit cells represent the two-material optical waveguide segments with refractive indices of n1 and n2 and length ratios of l1 and l2.
It is reported that in the frequency range 18.73–1763.48 THz (i.e., 0.17–16.00 µm), the dispersion relation for LiCl satisfies the following equation: [45]
The corresponding dispersion curve for LiCl is illustrated in Fig. 2.
Fig. 2. Dispersion curve of LiCl. (a) A total view in the range 18.73–1763.48 THz (0.1700 –16.0000 µm). (b) An enlarged view in the range 193.35–193.45 THz (1.5497-1.5505 µm).
It can be observed from Fig. 2 (a) that LiCl exhibits a low dispersion in the frequency range 193.35–1763.45 THz (0.1700–1.5505 µm). Thus, to inhibit the effect of workload and dispersion on the results and to investigate the photonic characteristics in the range of the BICs near the communication wavelength, we only use EM waves with frequencies (wavelengths) between 193.35–193.45 THz (1.5497–1.5505 µm), which correspond to the dispersion curve of the red solid line in Fig. 2 (b).
2.2 Generalized eigenfunction method and generalized Floquet–Bloch theorem
In this study, the generalized eigenfunction method [46] is used to calculate the transmissivity, reflectivity, and photonic localization, and the generalized Floquet–Bloch theorem [33] is used to obtain the dispersion relation and determine the distribution of photonic modes. The generalized Floquet-Bloch theorem is expressed as follows:
2.3 Two-material network equation
The wave equation of the EM wave in a two-material network can be expressed as: [47]
At x = l1dij, using the continuity of the wave function and its differential quotient, we obtain:
Using Eqs. (5) and (6), we obtain:
Let ψi and ψj be the wave functions at nodes i and j, respectively. Using the continuity of the wave functions, we obtain:
Using Eqs. (7)–(9), we obtain:
Using Eqs. (7), (8), (10), and (11) yields:
3. Results and discussions
3.1 Photonic modes
The photonic mode can be classified into three categories based on the topological translational periodicity of the EM wave propagating through the optical waveguide network: (I) when the structural Bloch vector K lies in the range of real number, the photonic mode corresponds to the ordinary propagation mode (OPM) [49]. (II) when the structural Bloch vector K is in the range of complex numbers and K = KI + ιKII (KII > 0), the photonic mode corresponds to the attenuation propagation mode (AMP) [20]. (III) when the structural Bloch vector K is in the range of complex numbers and K = KI + ιKII (KII < 0), the photonic mode corresponds to the gain propagation mode (GPM) [20].
We obtain the following dispersion relation for 1D TMPROWN with infinite unit cells based on the generalized Floquet–Bloch theorem and two-material network equation:
whereThe refractive index of the two materials used in the 1D TMPROWN is a real number. From Eqs. (18) and (19), we obtain the dispersion equation f(ν) for the entire frequency range. From Eq. (17), we can deduce that the structural Bloch wave vector K has both real and complex solutions for f(ν). In two-material networks, EM waves propagate mathematically as OPM and APM. Because of the lack of a gain mechanism in the 1D TMPROWN, EM waves propagate as OPM and APM.
3.2 Bound states in the continuum
To determine the BICs generated by the 1D TMPROWN designed in our study, we use the generalized eigenfunction method and generalized Floquet–Bloch theorem to calculate the transmissivity and photonic mode in the total frequency range illustrated in Fig. 2 (b). The results are shown in Fig. 3.
Fig. 3. Spectrum structures of the left incident EM wave propagating in the 1D TMPROWN near the communication wavelength. (a) Photonic modes distribution diagram. (b) Two-dimensional transmission spectra of the 1D TMPROWN with twelve unit cells between the frequency (horizontal ordinate) and nb (vertical ordinate). Three green solid lines labeled 1, 2, and 3 denote the boundaries of the two distinct photonic modes. Label conventions: There are five legends above the graph, from the left to right, they are OPM, APM, BIC, T < 0.01, and T≥0.01, labelled with red color, blue color, red-grid, white-grid, and white color, respectively.
Figure 3(a) demonstrates that for the 1D TMPROWN, the red and blue areas represent the three distinct types of Bloch photonic modes, the OPM and APM, respectively, which are classified by the corresponding structural Bloch wave vectors. In Fig. 3 (b), the three green solid lines labeled 1, 2, and 3 denote the boundaries of the two distinct photonic modes. The transmission coefficient of the white region is greater than or equal to 0.01; the transmission coefficient of the white-grid region is less than 0.01.
In general, when an EM wave propagates as an OPM in a 1D TMPROWN, the transmission typically equals or exceeds 0.01. When an EM wave propagates as an APM, the transmissivity is typically less than 0.01. However, in the red-grid region in Fig. 3 (a), the transmissivity of the EM waves propagating as an OPM is less than 0.01. At this point, the red-grid area may be the PBG zone. To confirm the preceding conclusion, we calculated the transmission of the EM waves for systems of different sizes in the black-grid region. Considering the red-grid area near the communication wavelength in Fig. 3 (a) as an example, Fig. 4 shows the transmission when nb = 1.222.
Fig. 4. The transmission spectra of the left incident EM wave propagating in the 1D TMPROWN in the range of the black-grid area near the communication wavelength, with red, blue, and black solid lines representing the transmission for unit cell counts of 3, 6, and 12, respectively.
As shown in Fig. 4, the transmission of the system decreased exponentially as the number of unit cells increased. Consequently, we can assert that the PBG is formed in the red-grid area. Usually, in the continuous domain, the BIC domain is a special state, nevertheless it is local [2]. Therefore, we can conclude that the region where the EM wave propagates as the OPM and is in the PBG is the distribution range of the BICs, such as the red-grid area in Fig. 3 (a). This is the first key characteristic observed in the 1D TMPROWN.
To further investigate the physical mechanism of the BICs generated by the 1D TMPROWN, we calculated the distribution of the photonic localization of the 1D TMPROWN with three unit cells using the BICs range near the communication wavelength as an example. The results are shown in Fig. 5.
Fig. 5. Three-dimensional photonic localization intensity map of the BICs in the 1D TMPROWN with three unit cells. (a), (c), and (e): The distribution of the photonic localization for the three waveguide segments between the nodes 1 and 2, 3 and 4, and 5 and 6, respectively. (b), (d), and (f): The distribution of the photonic localization of the upper or lower arm between the nodes 2 and 3, 4 and 5, and 6 and 7, respectively. The green circle represents the node of the waveguide segment, while the red circle represents the midpoint.
As can be seen in Fig. 5, the photons in each waveguide segment of the 1D TMPROWN with three unit cells are localized in the waveguide in the form of a standing-wave-like, and the photonic localization at the node and middle of each waveguide segment is significantly small, which can be considered as a wave node. The results in Fig. 5 demonstrate that the BICs generated in the OPM region are created by the standing-wave-like effect caused by the propagation of the EM wave in the 1D TMPROWN.
It is accepted that BICs can be generated in a 1D TMPROWN because each waveguide segment is composed of two segments that can lead to the formation of a sudden refractive index gradient, causing EM waves in certain ranges of the OPM to generate a standing-wave-like pattern when propagating in the 1D TMPROWN. The following section explains the reason why a 1D TMPROWN can produce a range of BICs that are not only larger, but also continuous. Each unit cell in the 1D TMPROWN consists of a waveguide segment and ring. However, the waveguide segment and ring are two distinct coupling resonant cavities. In addition, the alternating arrangement of the waveguide segments and rings enhances the resonant effect of the photonic coupling. Therefore, when EM waves propagate in the 1D TMPROWN, they complete complex coherent superposition several times, which further strengthens the standing-wave-like pattern and finally results in multiple continuous BICs.
4. Conclusions
In this study, we designed a 1D TMPROWN composed of LiCl to investigate the BICs generated by EM waves propagating in OWN. The following key characteristics were observed in the course of the study: (I) A large range of continuous BICs generated in the network is observed. We can flexibly adjust the distribution range and frequency of the BICs by regulating the refractive index of the materials, refractive index difference, and waveguide line length and thus analyze the mechanism of the BICs caused by a standing-wave-like effect. (II) We have also observed that the photons in the range of BICs produce strong localization owing to the standing-wave-like effect. There are standing wave nodes at the nodes and in the middle of the waveguide segments. This is because the refractive index between the adjacent waveguide segments and that between the sub-waveguides are different. Therefore, phase mutation and formation of a wave node occur when an EM wave propagates at the junction of adjacent waveguide segments and two sub-waveguides. BICs in an appropriate frequency-band range can be selected because the proposed TMPROWN has a simple structure and can flexibly be adjusted the value of nb to meet practical needs. Additionally, the amplitude and phase of the EM wave propagating through the OWN can be easily measured at each node. Therefore, the 1D TMPROWN has potential implications in the design of all-optical devices. Considering many advantages of the 1D TMPROWN, we believe that 1D TMPROWN may have potential and wide application value following further in-depth research.
Funding
Key Project of DEGP (2022KTSCX166); National Natural Science Foundation of China (11674109, 61774062, 62175070).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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