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Abstract
Negative refraction (NR), self-collimation (SC), and zero refraction (ZR) effects of photonic crystals play an important role in beam steering. In this work, we report a multifunctional beam steering concept in photonic crystals, i.e., integrating two or three of the NR, SC, and ZR effects together at the same frequency. We find the square-lattice dielectric ring photonic crystal is an ideal candidate to realize the switchable function of ZR-SC while the square-lattice dielectric ring photonic crystal is more suitable for realizing the ZR-SC, ZR-NR, and ZR-SC-NR functions. The photonic band theory and an equivalent waveguide model are employed to explain these switchable functions in conventional and annular photonic crystals.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonic crystals (PCs) have been extensively studied since the initial conception proposed by John [1] and Yablonovitch [2] in 1987. Photonic crystal is an ideal platform for beam steering through building periodical Bloch states in alternating dielectric materials. Over the past decades, the studies of PCs’ effects for beam steering are mainly focused on several aspects. The first one is the band gap in perfect PCs and the Anderson localization in PCs with a kind of designed defects, accompanying with many useful applications, such as ominidirectional reflectors, filters, photonic crystal waveguides, and photonic crystal fibers. The second one is the unique band dispersion effects in PCs, including negative refraction (NR), self-collimation (SC), and slow light. For examples, the NR effect makes the refracted and incident light on the same side of the normal, which can be used to realize the “superlens” [3] and polarization beam splitters [4]. SC can generate non-diffractive propagation [5] with important applications in waveguide bends, beam splitters [6], and interferometers [7]. The last but not least one is the quantum-classical analogies in Dirac-like and topological PCs [8–16]. For instance, the Dirac-like point in PCs can result in zero refraction (ZR) [8]. Based on the ZR effect, light can propagate through the PCs without any phase delay which may have useful applications in subwavelength imaging [8], optical cloaking [8], and directional emission [9]. The topological PCs can support edge states for topologically protected refraction [11], unidirectional emission [13], and beam splitting [16].
These particular effects of PCs provide diverse beam steering methods which will play an important role in future compact, high-speed, and wireless optical communication. Considering the compact requirement in photonic integrated circuits, PCs with tunable beam steering functions are becoming an urgent demand. Some previous works have reported the trial of using phase-change materials to modulate the band-gap or filtering properties of conventional PCs [17,18]. Quite recently, several groups have investigated the tunability of topological photonic states propagating in kinds of PCs through optical pumping or using electro-optic/ phase-change materials [19–22]. Most of these works only consider the modulation or switch of single beam steering function, while switchable beam steering functions among several effects in PCs have been less concerned. In the present study, we would like to report a concept of actively switching the NR, SC, and ZR effects in PCs. Both conventional and annular PCs [23] will be comprehensively studied to show their possibility in integrating two or three of the NR, SC, and ZR effects together at the same working frequency for multifunctional beam steering purpose.
2. Model and methods
Figure 1 shows the schematic diagram of our proposed multifunctional beam steering model based on two-dimensional conventional/annular PCs. As an example, square-lattice arranged dielectric rods with a radius of R and refractive index nx are embedded in the air background (n0) in the x-y plane. a is the period constant. For two-dimensional PCs, the height of PCs is assumed infinite along the z direction. The dielectric rods are made of a special material whose refractive index or conductivity can be controlled artificially, such as phase-change materials. Since the working frequency of photonic crystal is dimensionless, without loss of generality, in our theoretical design we only considered the change of nx while the extinction coefficient kx of material was ignored. In practical application, it is important to note that a particular material with low loss should be carefully chosen for PCs working in different wavelengths. We will briefly discuss this issue in the last subsection. Based on the model in Fig. 1, switchable NR (I), SC (II), and ZR (III) effects can be realized at the same working frequency by varying nx within a certain range.
Fig. 1. Schematic diagram of the multifunctional beam steering model based on the NR (I), SC (II), and ZR (III) effects in two-dimensional PCs. The beam steering function can be switched by changing the refractive index (nx) of the constituted dielectric rods arranged in square lattice in the air background (n0).
We employed a commercial software package ‘Bandsolve’ (RSoft Design Group) which is based on the plane wave expanded method to calculate the band structures and equi-frequency contours of PCs. A commercial software package ‘FDTD solutions’ (Lumerical Solutions Inc.) based on the finite-difference time-domain method was adopted to simulate the NR, SC, and ZR effects in PCs. For two-dimensional PCs, both the plane wave expanded and finite-difference time-domain methods were performed in a two-dimensional region without considering the height of PCs. These methods have been successfully used in our previous works to study the NR, SC, and slow light effects in conventional and annular PCs [24–26]. Moreover, in this study, only transverse-magnetic (TM) polarization was considered since the ZR effect mainly occurs for the TM polarization.
3. Results and discussion
3.1 Single NR, SC, and ZR functions in two-dimensional PCs
Before designing the multifunctional beam steering device which can integrate the NR, SC, and ZR effects together in one specific two-dimensional photonic crystal, it is necessary to first find which type of two-dimensional photonic crystal can support at least two of these effects. Here, we consider four typical two-dimensional PCs with different unit cells for a comparison, i.e., dielectric rod, air hole, air ring, and dielectric ring. Dielectric rod and air hole are conventional PCs, while the latter two are belong to annular PCs. For each type of two-dimensional photonic crystal, both triangular and square lattices were also studied.
Our simulation results show that the SC effect can be supported by all eight types of two-dimensional PCs, while the NR and ZR effects show quite different behaviors in conventional and annular PCs. For examples, except for the square-lattice air hole, all PCs can support the NR effect. However, it should be noted that the NR effect in triangular/square-lattice dielectric rod is realized in the TM-1 band with a structural rotation [27,28], which is different from the NR effect usually realized in the TM-2 band in other PCs without a structural rotation. Considering the SC and ZR effects usually occur in PCs without a structural rotation, in the following discussion, the integration of the NR effect with SC/ZR effect in triangular/square-lattice dielectric rod will be ignored. On the other hand, except for the triangular-lattice air hole, all conventional PCs can well support the ZR effect, while only the square-lattice dielectric ring can support the ZR effect among four types of annular PCs. All these results are summarized in Table 1. From Table 1, we can make a simple conclusion that it is not easy to integrate the NR effect with SC/ZR effect in conventional PCs and it is not easy to integrate the ZR effect with NR/SC effect in annular PCs.
Table 1. Comparison of the NR, SC, and ZR effects in conventional and annular PCs
3.2 Tunable beam steering functions in two-dimensional PCs
In this subsection, we have systematically studied the SC-NR, ZR-SC, ZR-NR, and ZR-SC-NR beam steering functions in two-dimensional PCs. It is found that the SC-NR function can be supported by the triangular-lattice air hole conventional PC and all types of annular PCs, while the performance of the particular conventional PC is better than annular PCs. Except for the triangular-lattice air hole, all conventional PCs can support the ZR-SC function. Among four types of annular PCs, only the square-lattice dielectric ring can realize the ZR-SC function. However, its performance is better than all conventional PCs. Moreover, among all types of conventional and annular PCs, only the square-lattice dielectric ring can realize the ZR-NR and ZR-SC-NR functions.
Here we give some examples to clearly explain the above results. Firstly, let’s understand why the triangular-lattice air hole can realize a better SC-NR function than other annular PCs. It is well-known that the group velocity’s direction of a refraction beam can be represented by the normal direction of the refraction point at the equi-frequency contour of PCs. Within a certain incident angles, the SC effect usually requires a flat equi-frequency contour to support non-diffraction beam propagation while the NR effect usually requires a convex equi-frequency contour to support negative refraction beam propagation [5]. Figure 2(a1) shows the equi-frequency contours of a triangular-lattice air hole photonic crystal (r = 0.4a) at normalized frequency 0.346×a/λ. This photonic crystal shows a standard hexagonal TM-2 equi-frequency contour (red dashed line) and a nearly circular TM-2 equi-frequency contour (red solid line) when nx=2.65 and nx=3 respectively at the same frequency. When the incident angle is 30 degrees, we can find the incident point “A” at the equi-frequency contour of light line as well as the refraction points “B” and ‘‘C” at the equi-frequency contours of photonic crystal. Consequently, this triangular-lattice air hole photonic crystal can well support the SC effect within an incident angle from -30 to 30 degrees, which can be verified from the simulation result by using a Gaussian beam source placed near the input surface [Fig. 2(a2)]. Meanwhile, since the equi-frequency contours of photonic crystal is circular and completely overlapped with the equi-frequency contour of light line, perfect all-angle single-beam NR effect (neff=-1) can be observed in this PC [Fig. 2(a3)]. Such high-quality SC-NR tunable phenomenon has also been verified in different triangular-lattice air hole PCs, where the normalized frequency and the refractive index’s tuning range are found both proportional to the air-hole’s radius.
Fig. 2. Equi-frequency contours and FDTD simulation (|E| distribution) of the SC and NR effects for a triangular-lattice air hole photonic crystal (r = 0.4a) [(a1)–(a3)], a triangular-lattice air ring photonic crystal (R = 0.48a, r = 0.48R) [(b1)–(b3)], and a triangular-lattice dielectric ring photonic crystal (R = 0.47a, r = 0.75R) [(c1)–(c3)]. In (a2)–(c2), a Gaussian beam source with a width of 5a is placed about 2a away from the input surface. In (a3)–(c3), a plane-wave beam source with a width of 5a is placed about 5a-7a away from the input surface. The arrows represent the main direction of group velocity. The cutoff values at both input and output surfaces are 0.5r and 0.5R for conventional and annular PCs, respectively.
As a comparison, Fig. 2(b1) shows the equi-frequency contours of a triangular-lattice air ring photonic crystal (R = 0.48a, r = 0.48R) at normalized frequency 0.415×a/λ. This PC still shows good SC effect (nx=2.75) from the equi-frequency contour and simulation results [Fig. 2(b2)]. However, the NR effect (nx=3) becomes poor because TM-2 equi-frequency contour is smaller than the light line and TM-3 equi-frequency contour is very close to the light line. Due to the shrunken TM-2 equi-frequency contour as compared to the light line, the effective range of incident angle of single-beam NR effect will be narrowed. At the same time, the TM-3 equi-frequency contour will bring either dual NR effect [24] or positive-negative mixed refraction when the incident angle is larger than a certain value. As shown in Figs. 2(b1) and 2(b3), the maximum incident angle of single-beam NR effect is only about 17 degrees. Similar good SC effect [Figs. 2(c1) and 2(c2)] and poor NR effect [Figs. 2(c1) and 2(c3)] can be found in the triangular-lattice dielectric ring photonic crystal (R = 0.47a, r = 0.75R) at normalized frequency 0.425×a/λ. The maximum incident angle of single-beam NR effect is about 18 degrees. The annular PCs are not suitable to support a good SC-NR function due to its squeezed band structures and hybrid equi-frequency contours [23,25,26].
Secondly, let’s understand why the conventional PCs can support the ZR-SC function but its performance is poor. Previous studies have shown that the ZR effect usually occurs in PCs when there is an accidental degeneracy of TM-2, TM-3, and TM-4 bands at Γ point [8,9]. Figure 3(a1) presents the band diagram of a square-lattice dielectric rod photonic crystal (r = 0.245a, nx=2.46). This photonic crystal exhibits a typical degeneracy of TM bands to form a Dirac-like point “D”. The corresponding simulation shows standard ZR at normalized frequency of 0.635×a/λ, i.e., the plane wave can propagate without phase delay inside the photonic crystal [Fig. 3(a2)]. When the refractive index of dielectric rods becomes 1.92 and 2.2, the photonic crystal shows an ideal square and circular TM-2 equi-frequency contour respectively at the same frequency [Fig. 3(a3)]. For a similar reason, only a poor ZR-SC function can be verified in a typical square-lattice air hole photonic crystal (r = 0.48a), as shown in Figs. 3(b1)–3(b4).
Fig. 3. Photonic band structures, equi-frequency contours, and FDTD simulation of the ZR (Ez distribution) and SC (|E| distribution) effects for a square-lattice dielectric rod photonic crystal (r = 0.245a) [(a1)–(a4)] and a square-lattice air hole photonic crystal (r = 0.48a) [(b1)–(b4)]. In (a2) and (b2), a plane-wave beam source with a width of 10a is placed about 10a away from the input surface.
Lastly, let’s understand why the square-lattice dielectric ring can realize better ZR-SC, ZR-NR, and ZR-SC-NR functions than other PCs. From Fig. 3, it is not difficult to find that for conventional PCs the ZR effect usually works at a relatively high frequency above the light line, while the SC and NR effects prefer to work within a much lower frequency region. Therefore, how to reduce the working frequency of ZR is critical for realizing the ZR-SC, ZR-NR, and ZR-SC-NR functions. As compared to conventional PCs, annular PCs become an ideal candidate to solve this problem since their air bands can be effectively suppressed due to the hybrid bands of constituted unit cells. As indicated from Table 1, only the square-lattice dielectric ring can support the ZR effect among four types of annular PCs. This is because the other three types of annular PCs do not show degeneracy of TM bands at Γ point due to the existence of band gaps.
The unique band characteristics of square-lattice dielectric ring provide necessary conditions for realizing an integrated function of ZR with the SC or NR effect. As an example, Fig. 4(a) shows the three-dimensional band diagram of a square-lattice dielectric ring photonic crystal (R = 0.48a, r = 0.38R) when the refractive index of dielectric rings is 3. We can observe a standard Dirac-like cone formed by TM-2 and TM-4 bands, accompanying with a flat TM-3 band passing through the Dirac-like cone to form a triple degeneracy state at normalized frequency 0.3705×a/λ. This photonic crystal can support perfect ZR effect as shown in Fig. 4(b1). As compared to the conventional PCs in Fig. 3, the equi-frequency contour (n = 2.54) of square-lattice dielectric ring is much closer to the light line, resulting in better SC performance [Figs. 4(b1) and 4(c1)]. However, the equi-frequency contour (n = 2.65) of square-lattice dielectric ring is still far away from the light line and the disturbance of TM-3 band still exists [Fig. 4(c1)], resulting in poor NR performance. Anyway, the square-lattice dielectric ring can support better ZR-SC, ZR-NR, and ZR-SC-NR functions than conventional PCs.
Fig. 4. (a) Three-dimensional photonic band structure near the Dirac-like point for a square-lattice dielectric ring photonic crystal (R = 0.48a, r = 0.38R, nx=3); (b1, b2) FDTD simulation of the ZR (Ez distribution, nx=3) and SC (|E| distribution, nx=2.54) effects for a square-lattice dielectric ring photonic crystal (R = 0.48a, r = 0.38R) at normalized frequency 0.3705×a/λ; (c1, c2) Equi-frequency contours for two square-lattice dielectric ring PCs (c1: R = 0.48a, r = 0.38R; c2: R = 0.46a, r = 0.52R); (d) Dependences of inner radius and working frequency on the outer radius of dielectric ring for realizing the ZR-SC-NR function.
The performance of the ZR-SC-NR function can be further improved by carefully designing the structural parameters and increasing the refractive index of dielectric rings. Figure 4(c2) shows the equi-frequency contours of a square-lattice dielectric ring photonic crystal (R = 0.46a, r = 0.52R). Perfect ZR effect can be realized at normalized frequency 0.24×a/λ when the refractive index of dielectric rings is 5.05, while much better SC and NR effects can be supported at the same frequency when the refractive index is 4.2 and 4.5, respectively. More importantly, we have found that the performance of the ZR-SC-NR function is robust. Taking the dielectric ring in Fig. 4(a) as an example, the ZR-SC-NR function can be realized within a large variation range of refractive index from 1.4 to 8. When the refractive index is fixed at 3, there always exists a certain range of r to support the ZR-SC-NR function as R increases from 0.4a to 0.48a, while the corresponding working frequency is inversely proportional to R and proportional to r, respectively [Fig. 4(d)]. Such robust performance is actually due to the non-accidental ZR effect in the square-lattice dielectric ring photonic crystal [29], which is quite different from the accidental ZR effect in conventional PCs, i.e., there is only one specific structure to support the ZR effect when the refractive index is fixed.
We have employed an equivalent waveguide model to further explain the non-accidental ZR effect in the square-lattice dielectric ring photonic crystal. Among four typical unit cells, namely, dielectric rod, air hole, air ring, and dielectric ring, only the dielectric ring can be equivalent to a planar rectangular waveguide as shown in Fig. 5. Such planar rectangular waveguide is infinite in z direction with thickness h = R-r along y direction. The length of the upper and lower planes along x direction is 2πR and 2πr respectively. Based on this model, the degeneracy problem of TM-2, TM-3, and TM-4 eigen modes at Γ point is similar to that in planar rectangular waveguide when TM-polarized plane wave (electric field along z direction) is normally incident to the waveguide. A planar rectangular waveguide usually behaves like an infinite isotropic homogeneous medium which can easily support mode degeneracy within a wide variation range of waveguide’s refractive index. Meanwhile, for a fixed outer radius there exists a certain range of inner radius to make the equivalent model effective. A relative small inner radius will gradually turn the planar rectangular waveguide into a triangular prism waveguide, resulting in an accidental ZR effect. A relative big inner radius will gradually turn the planar rectangular waveguide into free space which can not support the ZR effect at all.
A comprehensive comparison of two/three integrated functions of the NR, SC, and ZR effects of conventional and annular PCs is finally summarized in Table 2.
Table 2. Comparison of tunable beam steering functions in conventional and annular PCs
3.3 Practical application
In this section, we will briefly discuss two important factors which may influence the practical application of switchable beam steering functions based on the NR, SC, and ZR effects in two-dimensional PCs. One important factor is the tunable materials. In practical application, it is better to choose a material with large tunable range of refractive index/conductivity and low loss to construct the PCs. Commonly used tunable materials include electro-optic and phase-change materials. Typical electro-optic material, such as graphene, is usually used as surface coating [19] in PCs. Phase-change materials are suitable for either surface coating or solid PCs. Typical phase-change materials include vanadium dioxide (VO2) [18] and Ge2Sb2Te5 (GST) [20]. VO2 has large tunable range of refractive index in the near-infrared region but its loss is also large. GST has great difference in optical and electrical properties between amorphous and crystalline states, while its extinction coefficient is relatively small in the middle- and far-infrared bands. Meanwhile, the two phases of GST are stable at room temperature. People can switch quickly and inversely between the two phases, and can maintain any intermediate crystal state stably by accurately controlling the energy and duration of stimulation [30].
Another important factor is the transmission efficiency of the NR, SC, and ZR effects in PCs. For the proposed model in this work, it is critical to reduce the reflection at the interface of PCs. Efficient ways include optimizing the thickness of PCs and carefully designing the cutoff value at the interface of PCs. Take the triangular-lattice air hole photonic crystal (r = 0.4a, f = 0.346×a/λ) in Fig. 2(a1) as an example, the number of layers shows periodical influence on SC but brings less influence on NR. The transmittance of SC can approach as high as about 97% when the thickness along the y direction is 10/15/20 layers, while the transmittance of NR always maintains a low value around 50% [ Fig. 6(a)]. When the layer number is fixed to be 10, it is found that the cutoff value δy at the termination of photonic crystal shows quite different influence on the SC and NR effects [Fig. 6(b)]. In order to realize the SC-NR function with high performance, the optimized cutoff value δy is about 0.25a, under which condition both SC and NR effects show high transmittance up to about 69% simultaneously. Similarly, as shown in Fig. 6(c), the transmittance of SC is periodically affected by the number of layers while the transmittance of ZR seems inversely proportional to the thickness of square-lattice dielectric ring photonic crystal (R = 0.48a, r = 0.38R, f = 0.3705×a/λ). The optimal thickness is 11 layers and the corresponding cutoff value periodically influences the transmittance of SC and ZR [Fig. 6(d)]. To maximize the transmittance efficiency of the ZR-SC function, the optimized cutoff value can be chosen as δy=0 or δy=0.28a.
Fig. 6. Dependences of transmittance of the SC-NR/ZR-SC function on the thickness and cutoff value at the interface of PCs for (a, b) a triangular-lattice air hole photonic crystal (r = 0.4a) and (c, d) a square-lattice dielectric ring photonic crystal (R = 0.48a, r = 0.38R).
4. Conclusions
In conclusion, we have studied the NR, SC and ZR effects in conventional and annular PCs. It is found that the triangular-lattice air hole photonic crystal can support a better SC-NR function than other PCs, while the square-lattice dielectric ring photonic crystal can support a better ZR-SC function than other PCs. Only the square-lattice dielectric ring photonic crystal can realize the ZR-NR or ZR-SC-NR functions in all PCs. The outstanding performance of square-lattice dielectric ring photonic crystal in integrating ZR with SC and ZR effects is attributed to its suppressed photonic bands and non-accidental mode degeneracy characteristics. The proposed concept in this work can be used for designing multifunctional beam steering devices with potential applications in imaging, sensing, and photonic integrated circuits.
Funding
National Natural Science Foundation of China (61205042, 61675096); Six Talent Climax Foundation of Jiangsu (XYDXX-027); Fundamental Research Funds for the Central Universities (30919011106); Natural Science Foundation of Jiangsu Province (BK20141393).
Disclosures
The authors declare no conflicts of interest.
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