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Abstract
Light redirection plays an important role in photonic integrated circuit system, which attracts much attention on account of thriving application prospects from microwave to visible frequency. By treating a two-dimensional photonic crystal array as a dielectric resonator with low effective index neff << 1, a new strategy of one-direction semi-enclosed meta-resonator is proposed for light redirection and splitting with a high efficiency beyond 90%. Instead of zero-index material, the phenomenon of significant collimating radiations with zero phase delay can also be achieved through a meta-resonator of low effective index to stretch the internal resonant field with a wavelength much longer than that in air. The geometrical dimensions and structural parameters of the meta-resonator offer a great design flexibility for modulating the operating frequency. The numerical simulation and experimental results perfectly coincide with the theoretical predictions. This strategy can also be extended to other artificial metamaterials and three-dimensional cases, which may offer fantastic possibilities to the development of integrated photonics.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light redirection is extremely important to route information in photonic integrated circuits (PIC). The development of metamaterial opened new ways to guide light and alter radiation behavior in large-scale PIC. Previous studies on light redirection have been focused on optical devices, such as zero-index metamaterials [1,2], phased array [3], plasmonic lenses [4], metasurfaces [5–8], etc. However, comprising metallic components to support light transfer in PIC with the intrinsic losses greatly weaken their optical functionalities. Due to the photonic band-gap and low losses, dielectric photonic crystals (PhCs) have been applied to guide light via the total internal reflection waveguides [9,10] by introducing line defects connected by T-junctions [11,12], Y-junctions [13,14], or directional couplers [15,16]. Nevertheless, the complex fabrication and scattering losses at the sharp turns limit their performance and potential applications.
Dielectric resonator [17,18] made of nonconductive dielectric material is usually designed to radio wave in microwave and millimeter wave regions with low dissipation factor. The confined waves bounce back and forth between the resonator sides to form standing waves oscillating with large amplitudes. Different from the metallic cavity resonator, the open circuit boundary condition is approximately satisfied for dielectric walls partially transparent to waves, leaking energy into the air [19]. The dielectric resonators with sufficiently high permittivity are good at storing electromagnetic (EM) energy, in contrast, those of low permittivity are better for radiation [17] with relaxed confinements. However, the condensed wavelengths in common materials make it difficult for optical applications in subwavelength scale. It seems as if reducing the size of resonator becomes the only way to face with this challenge.
According to the principle of scale invariance, PhC [20] can be designed at microwave [21], terahertz [22] to optical regimes [23] with feasible dimensions and low losses. The properties of zero-index [24], negative index [25] or any index even much less than 1 [26] can be achieved in PhC by properly designing photonic band structure. Here, based on the two-dimensional (2D) PhC array, we propose a novel method of one-directional semi-enclosed dielectric meta-resonators to achieve light redirection and splitting with high efficiency beyond 90% at a flexible operating frequency.
2. Theorical principle and analysis
Proper designed PhCs exhibit extraordinarily linear chromatic dispersion near the Brillouin zone center, which can be predicted from the characteristics of PhC band structure and its corresponding equal frequency contours (EFC) [26]. For a 2D square lattice PhC composed of GaAs cylinders with the parameters of ratio of radius to lattice constant r/a = 0.2, permittivity ɛr = 12.5, permeability μ = 1, the band structure map of transverse magnetic (TM) mode with the electric field E along the z-axis is shown in Fig. 1(a) with two inverse linear dispersion cones intersecting at the center point Γof the Brillouin zone. Making use of the effective medium theory beyond the long wavelength limit [27], the effective permittivity ɛeff, permeability μeff and refractive index neff were calculated, as in Fig. 1(b), which means the PhC can be regarded as a homogeneous medium of linear chromatic dispersion with a small effective index neff = λ0/λ <<1. The inset in Fig. 1(c) depicts the top-down round EFC of the upper linear dispersion cone with a 0.01 step for the normalized frequency extending from 0.541 to 0.6 (a /λ). In Fig. 1(c), with closing to the point Γ, the propagation wavelength λeff of the PhC increases gradually and becomes much longer than that λ0 in air, the smaller effective index, the longer wavelength.
Fig. 1. (a) Band structure of square lattice PhC with the parameters of ɛr = 12.5, μ = 1, a = 1 and radius r = 0.2a. (b) Calculated effective permittivity ɛeff, permeability μeff and refractive index neff versus normalized frequency. (c) Effective propagation wavelengths λeff in the PhC versus normalized frequency.
To explicitly depict the light propagation behavior through a finite PhC array, we made a model of 20 × 20 PhC array of above-mentioned lattice structure with the FDTD Solutions and manipulated a plane wave normally incident on it in the normalized frequency a/λ span from 0.5 to 0.6, as the diagram shown in Fig. 2(a). For clarity and better contrast, the horizontal reflectivity spectrum R (dot line) and transmission spectrum TH (dash line), as well as the unilateral transmittance spectrum Tv (solid line) along the y direction are illustrated together in Fig. 2(b). The pseudo-diffusive phenomenon [28,29] can be observed with a stop band near the Dirac point frequency of 0.541 in the horizontal TH spectrum, the remainder waves are reflected horizontally or radiated perpendicularly. Out of the stop band of TH, the incident wave can pass through the PhC block efficiently with some weak periodic oscillations arising from the weak field oscillations in the square PhC array, due to the boundaries of abrupt refractive index change from neff to 1. The effective index is small enough (neff <<1) with a much long propagation wavelength around the point Γ that a transmission band appears in the unilateral radiation spectrum Tv with two approximately similar resonance peaks less than 30%.
Fig. 2. (a) Schematic diagram of measurement system with incident beam through a square PhC array. (b) Transmission and reflection spectra through the pure 20 × 20 PhC array. (c) Schematic diagram of semi-enclosed square resonator with a high reflector at the end surface of PhC array.
When we place a high reflector (perfect electrical conductor PEC) at the end surface of the square PhC model, as shown in Fig. 2(c), the horizontally transmitted waves are reflected back to overlap with the incident waves resulting in pure standing wave resonances along the x-axis, without disturbing the incident beam and other resonant characteristics of the PhC array. Due to the half-wave loss of light at the PhC interfaces for light incident from low-index (neff << 1) to high-index media (reflector or air), nodes of near zero intensity will appear at the surfaces of PhC resonator for resonant modes with the side length L equaling integer multiples of half wavelength of L = mλeff/2 (m = 1, 2, 3…). Inside and outside the square meta-resonator, the propagation constants of EM wave can be resolved into rectangular components to satisfy the following conservation equations.
where k0=2π/λ0=ω/c denotes the intrinsic propagation constant. For different resonance modes, the horizontal internal component kix can be expressed as m denoting the resonance-order. Supposing the square PhC-based meta-resonator is a 2D rectangular dielectric resonator, we use the transcendental equation [18] to estimate the resonant frequencies in an iterative manner, expressed as the followingThe evolution of calculated resonant frequencies versus the dimension of square meta-resonator for different resonant modes are illustrated in Fig. 3(a), it is clear that the resonant frequency increases with the increasing resonant-order m for a fixed resonator dimension, and the increase of resonator dimension will lead to a monotonic decrease of resonant frequency, with the frequency differences between the adjacent resonant modes tending to be uniform.
Fig. 3. (a) Geometric influences of square meta-resonator on different resonant frequencies. (b) Detected vertical radiation T and horizontal reflection R spectra through the square PhC-resonators with different dimensions of 20 × 20 and 30 × 30.
3. Numerical simulation and discussion
An x-axis semi-enclosed square meta-resonator was modeled and numerical simulations have been made to verify the above theoretical results. The detected horizontal reflectivity R and bilateral vertical radiation T spectra through the PhC-based meta-resonators with different dimensions of 20 × 20 and 30 × 30 are shown in Fig. 3(b). The complementary spectra of T and R demonstrate that the intrinsic and propagation losses can almost be neglected in this semi-enclosed meta-resonator model. Seven resonance peaks appear in the vertical radiation spectrum of 20 × 20 meta-resonator with the maximum transmittance beyond 90% at the fundamental frequency of a/λ=0.5516, and the transmittance of resonant peak decreases gradually with the increase of resonance order m. In comparison, more resonant modes can be excited by the larger 30 × 30 meta-resonator within the same frequency range from 0.54 to 0.6 with the maximum efficiency appearing at the fundamental frequency of a/λ=0.5473. Obviously, regardless of the size of PhC array, for square meta-resonators the fundamental mode always is the optimal mode with the highest radiation efficiency in the vertical y-direction.
The comparison of theoretical ft and simulated fs frequency results at different resonant modes for the 20 × 20 and 30 × 30 meta-resonators are shown in Table 1, respectively, with the corresponding effective indexes at different resonant frequencies. That they perfectly coincide with each other demonstrates the PhC-based meta-resonator can still be regarded as a rectangular dielectric resonator even with low index much less than 1, so the transcendental Eq. (3) is valid to identify resonant modes and predict the resonant frequencies accurately.
Table 1. Theoretical ft and simulated fs resonant frequencies for 20 × 20 and 30 × 30 meta-resonators, and corresponding effective indexes at different resonant frequencies.
The numerical field simulation results for a plane wave normally impinging on a meta-resonator at the resonant frequencies are helpful to further understand the mechanism of light redirection and splitting via the square PhC-based meta-resonator. Figure 4(a) gives the corresponding E-field distribution of fundamental resonance excited by the 20 × 20 semi-closed meta-resonator, with a horizontal incident wave and two symmetrically vertical radiation beams at the frequency of a/λ=0.5516. The internal field distribution displays the significant feature of standing wave resonance with invariable phase throughout the field oscillation space, which means the unique transmission property of zero phase delay can be discretionarily realized by stretching the incident wave via an artificial meta-resonator of low effective index, instead of effective zero-index, to form an internal standing wave resonance with a wavelength much longer than that in air. In the external field, that the wavefronts of two radiation beams neatly parallel to the output surfaces indicates this one-direction semi-closed meta-resonator has a good characteristic of light collimation in radiation direction. E-field is along the z-axis like displacement current resulting in a large magnetic field along the azimuth, as shown in Fig. 4(b), like a magnetic dipole to radiate transversely. The corresponding Poynting vector pattern is shown in Fig. 4(c) to distinctly illustrate the process of light redirection and splitting. According to the boundary condition of invariable tangential component of E-field at the interfaces, the amplitude distributions of radiation beams are proportional to that of internal field with the maximum intensity at the central antinode position and the minimum at the border node positions. For the second resonant mode at a/λ=0.5584, the internal E-field resonance is divided into two oscillation zones with the opposite phases divided by a central node line along the y-axis (Fig. 4(d)). The opposite E-field results in two magnetic fields along the opposite azimuths in Fig. 4(e). Since the internal resonance field approximates a set of magnetic dipoles with a sinusoidal and continuous resonance mode along the x-axis, the meta-resonator of second resonance gives rise to two pairs of radiation beams with perfect directivity and symmetry, as the Poynting vector pattern shows in Fig. 4(f). The similar phenomena of EM field divided into equal zones with the opposite oscillations between the adjacent also were found at the third and fourth modes with m = 3 and 4, respectively, as shown in Figs. 4(g) and 4(h). In addition to perfect directivity and symmetry, a constant phase difference of π always exists between the adjacent radiated beams.
Fig. 4. (a) Real E-field pattern of fundamental mode at f = 0.5516. (b) Fundamental H-field and (c) Poynting vector distributions in x-y plane. (d) Real E-field pattern of second-order resonance at the frequency of f = 0.5584, (e) Second-order H-field and (f) Poynting vector patterns. (g) E-field patterns for the third- and (h) fourth-order resonances through the 20 × 20 PhC resonator.
The added high reflector at the end surface of the PhC array drastically reinforces the internal field oscillation, with the minimum amplitude of near zero at the node positions and maximum amplitude at the antinodes. In the ideal case, the horizontal propagation constant component -kx of reflected wave superimposes and counteracts the incident one kx to get a pure stationary wave along the horizontal direction with Σkx=0. Nevertheless, the remainder component ky arising from the small effective index of neff << 1 provides a vertical energy output route for internal field oscillation, and guarantees the self-collimation phenomena of radiation beams in the perpendicular direction.
The upper and lower boundary offers a weak confine to the internal field resonance of the meta-resonator due to their drastic change of refractive index from neff to 1, so the width change has some impact on the resonant frequencies. According to Eq. (3), the resonant frequency versus the width W of 20 × 20 meta-resonator is shown in Fig. 5(a) at different resonant modes. With decreasing the width W, all resonant frequencies of different resonant modes will experience a blue shift. By investigating the vertical radiation spectra of different widths, we demonstrate that the resonant peak of maximum transmission efficiency shifts from the fundamental mode to the high-order gradually with the width W decreasing from 20a to 13a, as shown in Fig. 5(b). Therefore, the geometric tailoring can offer the meta-resonator a high design freedom by varying the width or length elaborately, the optimal resonant frequency and resonant mode can be controlled at will.
Fig. 5. When the length of rectangular PhC resonator is constant with L = 20a, (a) the influences of width W on the resonant frequencies. (b) The evolvement of transmittance and resonance peak through the rectangular PhC resonator of L = 20a for different widths of W = 20a, 15a and 13a.
4. Experiment results
Experiments have been implemented to demonstrate the above theorical results. To simplify the fabrication process, Al2O3 ceramic cylinders with the parameters of permittivity ɛr=10, loss tangent tanδ∼10−4, radius r = 3.3 mm, and height h = 8 mm were employed to compose a 9 × 9 array of lattice constant a = 15.5 mm to interact with the incident EM wave. In microwave band, metallic aluminum can be regarded as a perfect mirror with high reflectivity, so here a polished aluminum bar was placed at the end interface of the PhC-resonator as a high reflector to form a one-direction semi-closed square meta-resonator. As the setup shown in Fig. 6(a), a 2D electric field mapping system [30] with the PhC-based meta-resonator placed in the planar waveguide, which consists of two aluminum plates to ensure the TEM mode invariable along the z axis. In the Ku-band region (12∼18GHz), only the dominant mode of TE10 can propagate in the planar waveguide with the electric field perpendicular to the metallic plates. The receiving probe was fixed on the upper metallic plate and can be controlled to scan the internal and external field distributions in a step of 2mm. The emitting probe was fixed on the lower plate as an incident source. The emitting and receiving probes were connected to the output and input ports of the vector network analyzer (VNA, AgilentN5071C).
Fig. 6. (a) Photograph of experimental layout in the microwave measurement system. Experimentally measured real part of electric field (upper) and phase (lower) distributions at three lowest resonant frequencies of (b) 12.99 GHz, (c)13.13 GHz and (d)13.28 GHz.
When an approximate plane wave normally incident on the square meta-resonator, the E-field and phase distributions of three lowest resonant modes with m = 1, 2 and 3 were experimentally measured at the resonant frequencies of 12.99 GHz, 13.13 GHz and 13.28 GHz, respectively, to visualize the characteristics of internal field oscillations and external beam radiations. The upper one of Fig. 6(b) depicts the measured real part of E-field distribution of fundamental mode at the frequency of 12.99 GHz with a distinct central maximum amplitude and boundary minimums, as well as the lower phase pattern gives a uniform phase distribution within the meta-resonator sample, and two external radiation beams with neat wavefronts parallel to the output surfaces. These experimental results perfectly support the proposed theorical predictions for the fundamental resonance in Fig. 4. At the secondary resonant frequency of 13.13 GHz, in Fig. 6(c) the internal E-field is divided into two distinct oscillation zones with opposite amplitudes by a node line along the y-axis, and each zone has the similar resonant features as described at the fundamental mode, except a phase difference of π between two zones, which lead to the staggered wavefronts between two adjacent pairs of radiation beams. For the third resonant mode occurring at 13.28 GHz, even with a bit of disturbances arising from the small dimension in Fig. 6(d), it still is easy to make out three internal resonance zones corresponding to three pairs of external collimating radiation beams with opposite phases between the adjacent ones. All these experimental results agree well with the above mentioned theoretical and simulation results, which demonstrate that the proposed one-direction semi-enclosed rectangular meta-resonator with low effective index is significantly suitable to get efficient light redirection and splitting with constant 0 or π phase shifts and excellent directivity based on the internal field resonances with the stretched wavelength much longer than that in air.
5. Conclusion
In summary, we have proposed a unique scheme of light redirection and splitting with equiphase based on a meta-resonator composed of dielectric PhC array with the low effective index of neff<<1. The one-direction semi-enclosed design of the dielectric meta-resonator greatly enhances the intensity of internal E-field oscillations, which drives the magnetic field along the azimuth like a magnetic dipole to further generate beam directional radiation with high efficiency. The stretched field resonances with a wavelength much longer than that in air make it easier to achieve symmetric beam redirection and splitting with constant 0 or π phase delays. A series of numerical simulations and experiments perfectly coincide with the theoretical predictions. The geometric variation of the meta-resonator leads to a flexible tailoring and control for the working frequency. This novel meta-resonator mechanism offers an extraordinary design freedom to light redirection and splitting. We believe it would offer numerous possibilities of potential applications and open a novel avenue for the design of compact optical devices for photonic integrated circuits.
Funding
National Natural Science Foundation of China (11574311, 51532004, 61775121); China Scholarship Council (201804910210); Natural Science Foundation of Shandong Province (ZR2016FM03); Double First Class University Plan (111800XX62).
Disclosures
The authors declare no conflicts of interest.
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