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Abstract
Over the past few years, zero refraction (ZR) represents an important breakthrough in field of distinctive dynamic optical beam tuning for double-zero-index (media with zero relative permittivity and permeability) photonic crystals (PCs). Because accidental ZR effects for conventional PCs is sensitive to the structural or refractive index parameters, how to enhance the robustness of ZR design is vital for applications. Here, we report a pioneering wave property that reveals a non-accidental Dirac-like cone dispersion composed of low- or high-energy bands, corresponding to dual-insensitive ZR effects for both transverse magnetic and transverse electric polarizations based on the same annular PCs. The performance of non-accidental ZR is simultaneously based on refractive index-insensitivity and structural- insensitivity, attributed to a satisfactory filling ratio.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Artificial zero refraction (ZR) metamaterials [1–3] have attracted extensive attention, owing to their peculiar properties of controlling wave transmission [4,5]. By determining whether the relative permittivity (ɛ) and permeability (µ) can simultaneously or individually approach as high as zero, ZR materials can be divided into single zero refraction materials [6–10] (mu-near-zero (MNZ) material, µ = 0, ɛ ≠ 0; epsilon-near-zero (ENZ) material, ɛ = 0, µ ≠ 0) and double zero material (epsilon-and-mu-near-zero (EMNZ) material [11,12], ɛ = µ = 0). Between them, two single zero materials have been widely applied in the construction of perfect non-reflective waveguides [13,14], design of electromagnetic tunneling waveguides [15,16], and enhancement and modulation of light directional emission [17–19]. However, single zero materials have the following two limitations: one is a mismatch between the impedance and free space, leading to strong reflection at the interface [5], and the other is vanish of the group velocity, leading to the inability to transmit energy [20].
Furthermore, two-dimensional (2D) photonic crystals (PCs) composed of pure dielectric materials related to low-loss have unique energy band manipulation characteristics, which make PCs interesting and widely used [21–24], such as negative refraction, self-collimation, zero refraction, and slow light. As is well known, by easily changing the structural parameters and refractive index, PCs can achieve an accidental Dirac-like point, formed by the typical triple degeneracy of three energy bands at the center of the Brillouin zone with a specific spatial parameter under a specific permittivity parameter, which is defined as one of the most significant symbols of the ZR [11]. A Dirac-like cone dispersion (k = 0) in PCs differs from the electronic state double-degeneracy Dirac cone appearing in graphene that performs massless fermions with unique transport properties [25], accompanying with applications such as tunable optical absorbers [26] and optical switches [27,28]. The PCs proven to be EMNZ materials also have a near-zero refractive index [11,29], which produces a finite impendence and a non-zero group velocity to eliminate the limitations of single zero materials.
Zero-index PCs are a useful platform for realizing intriguing application prospects in large-area single-mode lasing [30], lensing [31], beam steering (or leaky-wave antenna) [32,33], and optical cloaking [34]. Most previous researches were mainly based on conventional zero-index PCs consisting of dielectric rods or air holes with triangular or square lattices, corresponding to an accidental Dirac-like triple degeneracy at Г point. When there is a small change of the geometric parameters or refractive indexes, a bandgap will easily appear in particularly. The disappearance of degeneracies leads to the disappearance of the ZR effects, greatly enhancing the difficulty in the integrated process technology of on-chip metamaterials. It is not difficult to find that zero-refracted conventional PC devices require a much higher machining precision. In our previous studies, two types of insensitive continuous ZR behaviors were identified: one is a refractive index-insensitive ZR effect in a 2D dielectric ring PC for transverse magnetic (TM) polarization [35], and the other is a structural-insensitive ZR effect in a 2D air ring PC for transverse electric (TE) polarization [36]. We have defined “non-accidental degeneracies” as: in the process of changing either refractive indexes or structural parameters at a relatively large range, not only Dirac cones always exist, but also relative permittivity (ɛeff) and permeability (µeff) are always near zero related to the ZR effects for PCs.
In this study, one direction is that square-lattice air hole PCs exhibit refractive index-insensitive ZR effects for the TM mode. In the other direction, square-lattice air ring PCs exhibit specific dual-insensitive ZR effects for both TM and TE polarizations at a fixed working frequency, regardless of whether a Dirac-like cone is fabricated by low or higher energy bands, only by changing the two parameters of structure and refractive index at the same time.
2. Models and methods
Figure 1 shows a 2D PC with square lattice dielectric rods (nx) in air holes that form air rings. The lattice constant is a, and air rings of refractive index (n0) with outer radius R and inner radius r in the x-y direction are embedded in the dielectric medium background (n1). It should be noted that the height of the PC can be theoretically infinite in the z-direction. Usually, an approximately ideal 2D PC is obtained provided that the height is much higher than lattice constant of PCs [5]. Further, the TM polarization mode (an electric field is parallel to air holes or air rings), and TE polarization mode (an electric field is perpendicular to air holes or air rings) were considered in this study.
The band structures of the PCs were efficiently calculated by using the commercial software package ‘Bandsolve’ (RSoft Design Group), which employs Maxwell’s wave equation with the plane wave expansion method. We also used the 2D finite difference time-domain ‘FDTD’ solution to simulate the results of the transmission behavior of plane waves for both TM and TE polarizations. To verify the ZR effects, further simulations using a numerical software COMSOL Multiphysics were adopted. At the same time, a Dirac-like cone dispersion was a requisite to realize of a zero-index system; consequently, the effective wavelength could approach infinity, making it possible to numerically calculate the effective parameters of ɛ and µ in 2D PCs using the S-parameter inversion technique reported by Smith et al. in 2005 [37].
3. Results and discussion
3.1 Continuous triple degeneracy for air hole PCs with square lattice in TE and TM mode
At first, we should clarify the relationship between the triple degeneracy of the energy band and the ZR phenomena in 2D PCs. Take a square-lattice air hole PC for the TE-polarized beam as an example, clearly, this type of PC shows unusual non-accidental degeneracy-induced Dirac points at the Brillion center by changing the refraction index n1 from 3 to 7, when R = 0.3135a, as plotted in Figs. 2(a1) and 2(a2). Meanwhile, Figs. 2(b1) and 2(b2) depict a three-dimensional (3D) dispersion surface formed by the TE-3, TE-4, and TE-5 bands. However, when n1 is only fixed at 3 and the normalized frequency is 0.431 × a/λ, the corresponding simulation shows that the plane wave propagates without any spatial phase change, demonstrating a well-defined perfect ZR performance, as presented in Fig. 2(c1). Nevertheless, Fig. 1(c2) reveals such a bad ZR effect in which the plane wave is cluttered when n1 = 7 and the normalized frequency is 0.1929 × a/λ. By comparing Fig. 2(c1) with 2(c2), we can find Dirac cones always exist from n1 = 3 to n1 = 7 for the TE mode, a square-lattice air hole PC shows a perfect ZR effect when n1 is only fixed at 3. As a result, the continuous Dirac cone linear dispersions are necessary, but insufficient for the continuous ZR effects, continuous Dirac cone linear dispersions at k = 0 do not exactly indicate continuous ZR effects.
Fig. 2. Photonic 2D band structures, 3D dispersion surfaces at the Dirac point, FDTD simulations of the ZR (Hz distribution) for TE mode of a square-lattice air hole PC when n1 = 3, n0 = 1, R = 0.3135a at the normalized frequency 0.431 × a/λ [(a1)–(c1)] and n1 = 7, n0 = 1, R = 0.3135a at the normalized frequency 0.1929 × a/λ [(a2)–(c2)].
It should be emphasized that most previous studies were carried out that conventional 2D PC models consisting of dielectric rods with square or triangular lattices faced extreme difficulty in accomplishing non-accidental ZR effects at different geometric parameters or refractive indexes [11,38]; basically, the degenerate phenomena may disappear by changing the refractive index or filling ratio, which results in a relatively higher machining accuracy for zero-refracted electromagnetic functional waveguides.
Notably, in comparison with the PC for TE mode in Fig. 2, a square-lattice air hole PC for TM-polarized beam also performs non-accidental triply degenerate Dirac-like cone dispersions at Г point, at a fixed outer radius by increasing refractive index n1. Subsequently, we can also observe a standard property of the Dirac cone, which in turn demonstrates TM-2, TM-3, and TM-4 degeneracy, as plotted in Figs. 3(a1) and 3(a2). Additionally, Figs. 3(b1) and 3(b2) show the 3D band diagram formed by the TM-2 and TM-4 bands, accompanying with a flat TM-3 band passing through the Dirac-like cone. A Dirac cone is a necessary and insufficient condition for the ZR effects. To illustrate this idea visually, we further simulated the propagation properties of the plane wave inside the PC waveguide. Surprisingly, such a robust ZR effect can always exist as n1 varies from 3 at the normalized frequency 0.441 × a/λ [Fig. 3(c1)] to 4 at the normalized frequency 0.3353 × a/λ [Fig. 3(c2)], respectively, when R = 0.44a. Under these conditions, 2D conventional PCs of square-lattice air holes for TM polarization light exhibit an intriguing continuous refractive index-insensitivity, which is in good agreement with continuous non-accident zero-index metamaterials.
Fig. 3. Photonic 2D band structures, 3D dispersion surfaces at the Dirac point, FDTD simulations of the ZR (Ez distribution) for TM mode of a square-lattice air ring PC when n1 = 3, n0 = 1, R = 0.44a at the normalized frequency 0.441 × a/λ [(a1)–(c1)] and n1 = 4, n0 = 1, R = 0.44a at the normalized frequency 0.3353 × a/λ [(a2)–(c2)].
3.2 Continuous high-energy band triple degeneracy and zero refraction phenomena for air ring PCs with square lattice in TE and TM mode
Compared with conventional square-lattice air hole type PCs, annular PCs constructed by merging dielectric rods into air hole PCs have more flexible tuning degrees of freedom, which can simultaneously change the structural parameters and refractive indexes. Furthermore, annular PCs most likely achieve novel and strange optical characteristics. By numerical simulation, when R and r are 0.31a and 0.15R, respectively, the outer refractive index n1 is 3, and it is not difficult to observe that this annular PC has the same dispersion characteristics of the energy band, resulting in accidental degeneracy of the TE-3, TE-4, and TE-5 bands (nx = 6) at the normalized frequency 0.4281 × a/λ, as extracted from Fig. 4(a). When nx of the inner dielectric rods increases, a triply degenerate Dirac-like cone dispersion can always be engineered to exist from low-energy bands to higher energy bands. For example, when the inner refractive index increases to 25, the TE-3 band is suppressed to a lower frequency, which can lead to the intersection of bands 4, 5, and 6, as shown in Fig. 4(b). A similar triple degeneracy phenomenon can also be observed when nx is 35 (TE-6, TE-7, and TE-8) and 45 (TE-8, TE-9, and TE-10), as presented in Figs. 4(c) and 4(d). Simultaneously, the plane wave propagates without any phase delay inside the PC waveguide, and this PC exhibits such specific wave transport properties related to such perfect ZR performance when nx is set to 22 [Fig. 4(e)]. For almost all nx in between the values presented in Fig. 4, the degenerate points present for every possible permittivity, except a few values of nx which occasionally breaks triple degeneracy when the low energy band degeneracies switch to much higher energy band degeneracies (see Fig. 6(a1) for more details). In general, based on the same mechanism, further simulations have shown non-accidental degeneracies that represent ZR performances as nx increases from 2.5 to more than 100 at the fixed working frequency 0.4281×a/λ.
Fig. 4. Photonic 2D band structures for TE mode of a square-lattice PC with air rings when n1 = 3, R = 0.31a, r = 0.15R at the normalized frequency 0.4281 × a/λ for TE-3, TE-4 and TE-5 (a), for TE-4, TE-5 and TE-6 (b), for TE-6, TE-7 and TE-8 (c), for TE-8, TE-9 and TE-10 (d) as the value of nx increases from 6 to 45. (e) FDTD simulations of the ZR (Hz distribution) when nx = 22.
One direction is that the square-lattice air ring PC supports refractive index insensitivity for TE polarization; the other direction is whether it can also accomplish the same performance for TM polarization or not. As seen from Fig. 5(a), we can observe a standard Dirac cone dispersion formed by the TM-2, TM-3, and TM-4 bands (nx = 2) when R = 0.44a, r = 0.15R, n1 = 3, and n0 = 1 at the normalized frequency 0.441 × a/λ, corresponding to a similar property in a square-lattice air hole PC in Fig. 3(a1). In addition, as shown in Figs. 5(b)–5(d), as nx varies from 6 (TM-3, TM-4, and TM-5), 16 (TM-5, TM-6, and TM-7), to 25 (TM-8, TM-9, and TM-10), a similar Dirac-like cone can be found with the same structural parameters at normalized frequency 0.441 × a/λ (see Fig. 6(a2) for more details). Furthermore, it is necessary to confirm a zero-index system with accurate calculation of effective material parameters. Taking nx = 23 as an example, both relative permittivity (ɛeff) and permeability (µeff) are simultaneously equal to zero, which implies a double relative zero refractive index as anticipated, as shown in Fig. 5(e). However, compared to the non-accidental ZR effects (n1 = 3–4) for TM-polarized air hole PCs at different working frequency, the non-accidental ZR effects for TM-polarized air ring PCs are always at the fixed working frequencies. As a result, air ring PCs with square-lattices for both TM and TE polarizations are sufficiently characterized by robust impedance-matched zero-index metamaterials, because of non-accidental high-energy band triple degeneracy.
Fig. 5. Photonic 2D band structures for TM mode of a square-lattice PC with air rings when n1 = 3, R = 0.44a, r = 0.15R at the normalized frequency 0.441 × a/λ for TM-2, TM-3, and TM-4 (a), for TM-3, TM-4, and TM-5 (b), for TM-5, TM-6, and TM-7 (c), for TM-8, TM-9, and TM-10 (d) as the value of nx increases from 2 to 25. (e) The relative permittivity (ɛeff) and permeability (µeff) calculated by effective material parameters.
Fig. 6. Variation of nx as a difference of inner radius r for realizing ZR functions for a square-lattice PC with air rings when n1 = 3, R = 0.31a in TE mode (a1), when R = 0.44a in TM mode (a2). Dependences of an air-ring filling ratio and working frequency on the inner radius r when nx = 3 in TE mode (b1) and TM mode (b2).
To optimize the structural errors of zero-index metamaterials during processing, is it possible to maintain ZR performances even when dual parameters related to refractive indexes and structures are changed at the same time?
3.3 Continuous dual-insensitive zero refraction effects for air ring PCs with square lattice in TE and TM mode
First, we systematically investigated the ZR effect for both the TM and TE polarization modes by adjusting the two parameters r and nx at the same time. It was remarkably found that continuous dual-insensitive ZR effects can be realized in 2D air ring PCs, as shown in Figs. 6(a1) and 6(a2). Basically, as nx varies from 2.5 to 100 in TE light, and from 1.4 to 60 in TM light, such an outstanding ZR performances can be achieved, corresponding to a continuous triply degenerate Dirac-like cone dispersion for air ring PCs consisting of different inner radii r. Notably, only three types of triple degeneracy are considered in this condition, there will be much higher energy bands constituting Dirac cones in fact. Occasionally a few specific combinations of nx and r may break the non-accidentally degenerated Dirac-like cone into a bandgap.
Next, we take n1 = nx = 3 as an example, only the inner radius r of dielectric rods is adjusted when R = 0.31a and n0 = 1 for the TE mode, and the working frequency gradually decreases when r ranges from 0.05R to 0.45R, as shown in Fig. 6(b1). When R = 0.44a and n0 = 1 for TM mode, the working frequency remains unchanged when r ranges from 0.05R to 0.25R, as shown in Fig. 6(b2) (black solid squares). Why is the variation range of r larger for the TE polarization? Because as is known to all, a dielectric rod PC tends to support TM polarization; nevertheless, an air hole PC is typically inclined to accomplish TE polarization, which implies an air ring PC also requires a more dielectric background related to a smaller filling ratio of air rings [defined as ${{\pi ({{R^2} - {r^2}} )} / {{a^2}}}$] in TE polarization (24%–30%) than in TM polarization (57%–60.7%), as given by Figs. 6(b1) and 6(b2) (hollow blue squares). The aforementioned rule also applies to different inner radii r, our calculation resulting in Figs. 6(a1) and 6(a2) undoubtedly confirms this theory.
4. Conclusion
In summary, we have found dual-insensitive ZR effects for square-lattice air ring PCs, corresponding to diverse refractive indexes and structural parameters at the fixed working frequency. There is always a relatively large range of r and nx to realize a perfect ZR function, attributed to the triple degeneracy of low or high energy bands, making it an alternative for intriguing potential applications, which show a practical value in high-transmission waveguides, and highly integrated photonic circuits.
Funding
Open Research Fund of Henan Key Laboratory of Special Protective Materials (SZK-FKT202105); National Natural Science Foundation of China (61875088, 62005128).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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