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Abstract
Zero-index medium has profound application for light manipulation. Certain types of dielectric photonic crystals (PCs) may have zero effective index since they form Dirac cone at the Γ point of their band structure. Although zero index photonic crystals provide a solution to impedance mismatch in photonic integrated circuits, its propagation modes strongly radiate to the surrounding environment, which hampers their application for high-density integration. In this paper, by an appropriate design of PC’s unit cell, toroidal dipole mode is excited at Dirac-point frequency through coupled Mie resonance to suppress radiative losses of other multipoles. The PCs with the Dirac-like dispersion at the Γ point can be mapped to an effective zero-index medium. The physical mechanism was utterly investigated by means of multipole decomposition and band structure analysis. Due to the non-radiation property of the toroidal dipole mode, the proposed photonic crystal slab process is low-loss based on numerical simulation. Moreover, its relatively simple design facilitates integration with future quantum photonic devices.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
When the refractive index of the material is zero, the phase velocity of the electromagnetic wave in the material is infinity, and the wavefront will not change, which achieves perfect spatial coherence. Based on this special property, zero-index materials have a series of interesting applications, including electromagnetic wave manipulation [1–3], electromagnetic cloaks [4–6], and electro-optical modulation [7–9]. To achieve zero-index materials, Liu et al. [10]have proposed a near-zero material in the microwave band through the complementary split-ring resonators. However, such zero-index metamaterials are usually associated with a large loss because of the intrinsic ohmic losses of metal, especially in the optical regime. To alleviate the ohmic losses, a promising platform for realizing zero-index is an integrated all-dielectric photonic crystal [11]. By adjusting the permittivity and geometric parameters of the dielectric cylinder in the photonic crystal structure, Huang et al. [12] found that the accidental degeneracy of electric monopole mode and magnetic dipole mode can be achieved at the center of the Brillouin zone. Dirac-like point (DLP) [13,14] can be formed therein. Electric monopole mode and magnetic dipole modes correspond to zero effective permittivity and permeability, which induce an impedance-matched zero index. Therefore, the photonic crystal can be equivalent to an isotropic two-dimensional zero-index material. When the height of the dielectric pillar is limited, the two-dimensional photonic crystal will become a three-dimensional photonic crystal slab (PhC slab). Since the propagation constants in the center of the Brillouin zone approach to zero, the in-plane wave can couple out of the slab upward and downward in the form of plane waves. Zero-index modes above the light-line can thus couple to plane waves traveling perpendicular to the slab [15]. In particular, the magnetic dipole modes at the DLP couple strongly to the out-of-plane radiation channel [16], which causes substantial out-of-plane radiation loss [17]. On the other hand, the electric monopole mode has no such out-of-plane radiation loss due to their inherent mode symmetry [18].
To suppress the radiation loss of the system, Tang et al. [19] have proposed a low-loss zero-index dielectric photonic crystal by introducing a resonance-trapped mode through changing the thickness of the PhC slab. The resonance condition for this design corresponds to the total destructive interference between the far-field radiation of the doubly degenerate magnetic dipole modes. This results in significant improvement of the Q-factor of the magnetic dipole modes. On the other hand, the out-of-plane radiation channels of the magnetic dipole modes will not be completely destroyed. Once the magnetic dipole modes are excited during the transmission of the PhC slab, out-of-plane radiation will still exist. To further suppress the associated radiation loss caused by the excitation of the magnetic dipole modes, the interactions between the zero-index modes should be considered. From a multipole view-point [20], the radiation of the electric dipole is concentrated within the in-plane, while the magnetic dipole corresponds to the high out-of-plane radiation. However, the electric dipole can’t effectively suppress the excitation of the magnetic dipole at the DLP frequency.
To further reduce the loss of the system, the toroidal dipole has been introduced in the metamaterial design [21], since the toroidal dipole can suppress the excitation of other multipoles [22,23]. This approach provides a new idea for us to design a low-loss photonic crystal slab. The toroidal dipole can be regarded as the higher-order term of the electric dipole, and their far-field radiation characteristics are quite similar. It indicates that the toroidal dipole will realize the zero effective permittivity [24]. Furthermore, He et al. [25] have reported toroidal dipole bound states in the continuum [26], and such toroidal bound states in the continuum can be turned into ultrahigh-Q resonances with a dominant toroidal dipole excitation. In a word, the toroidal dipole can be used to replace the electric dipole in the zero-index mode, which will have the potential for suppressing the associated radiation loss caused by the excitation of the magnetic dipole modes.
In this paper, we achieve a low-loss zero-index photonic crystal slab with $C_{4 v}$ square symmetry. Because of the near-field coupling between the magnetic Mie-type resonance [27], the toroidal dipole can be excited in the unit cell. The band of the PhC slab forms Dirac-like dispersion near the $\Gamma$ point, which can be considered as a material with $\varepsilon =0$ and $\mu =0$. By using the Cartesian multipole decomposition [28], one can quantitatively identify the multipole character of each eigenmode. The numerical simulation results indicate that the toroidal dipole is the main multipole radiation contribution in the eigenmode at the DLP. We simulate the plane wave transmission throughout the PhC slab. The numerical simulation results indicate that the PhC slab represents the perfect phase-matching property and low loss at the DLP frequency. By extracting the radiation powers of multipoles within the unit cell, we found that the toroidal dipole plays the dominant role near the DLP frequency and the magnetic dipole is significantly suppressed. It indicates that the toroidal dipole can effectively eliminate the radiation loss of the system. Our approach will further promote the large-scale integration of zero-index materials in photonic devices.
2. Design
We implement our design in the structure shown in Fig. 1(a), the PhC slab is composed of the cross arranged silicon cylinders and silicon square prisms with the thickness $h=180~\mu\textrm{m}$. The inset shows the structure of the unit cell (in the dotted line) with lattice period $a=116 ~\mu\textrm{m}$. Here, the cylinder radius $r=24 ~\mu\textrm{m}$ and the side length of square prism $l=a \sqrt {2} / 2$. This design can be easily fabricated using conventional nanofabrication processes.
Fig. 1. (a) Three-dimensional schematic diagram of a PhC slab, the thickness $h=180 ~\mu\textrm{m}$. The inset shows the structure of the unit cell (in the dotted line), the unit cell period $a=116 ~\mu\textrm{m}$, the cylinder radius $r=24 ~\mu\textrm{m}$, the side length of square prism $l=a \sqrt {2} / 2$. The material of the PhC slab is silicon with a relative permittivity $\varepsilon =11.7$ and relative permeability $\mu =1$. (b) Schematic diagram of electromagnetic field distribution in the unit cell. The blue arrow represents the directions of displacement current (electric displacement), and the red arrow represents the directions of the magnetic field. (c) Schematic diagram of the formation of toroidal dipole moment. The black arrow shows toroidal dipole moment T in the z-axis direction, the blue loop shows the displacement current j, and the red loop shows the magnetic dipole moment m.
For the photonic crystal slab, depending on the structure of the unit cell and the polarization of the incident electric field, the electromagnetic scattering produced by the displacement current can cause multiple resonance modes (known as the Mie resonance). As shown in Fig. 1(b), there will be the displacement current oscillating in the opposite direction within the square prism and the cylinder. Due to the special design, the magnetic dipole moments become aligned head to tail by near-field coupling, and the magnetic field appears as a dynamic vortex state. The multipole moments in the proposed design can be conceptually depicted in Fig. 1(b). Such a state is characterized by zero magnetic and electric multipole moments, and a toroidal dipole moment oscillating along the axis of the unit cell shown in Fig. 1(c).
To characterize the optical properties of the designed PhC slab, the band structure of a square lattice PhC slab computed for the TM polarized waves with an electric field along the axis of the cylinder is shown in Fig. 2(a). It can be seen that the photonic bands form an accidental triple degenerate state at the $\Gamma$ point. Two photonic bands with linear dispersion compose an isotropic Dirac-like cone, and the other one is the photonic band with quadratic dispersion. This photonic band structure can be used to create an effectively zero-index material [29]. At the DLP frequency (0.95 THz), according to the equation $k_{x}^{2}+k_{y}^{2}=n_{\mathrm {eff}}^{2} k_{0}^{2}$, it can be deduced $n_{\mathrm {eff}}=0$, where $k_{x}$ and $k_{y}$ represent $x$, $y$ components of the wave vector; $k_{0}$ is the wave vector in free space; and $n_{\mathrm {eff}}$ is the effective refractive index of the photonic crystal slab. Furthermore, we show the electromagnetic field distribution related to each eigenmode at the $\Gamma$ point in Fig. 2(b). The top left subplot of the Fig corresponds to the toroidal dipole mode in Band1 in which the electric field of the toroidal dipole mode is concentrated in the central part of the cylinder and square prisms. The antiparallel currents are formed in the unit cell, where the magnetic mode is represented by a pair of antiparallel steady currents of equal strength separated by distance $l$. The top right subplot corresponds to the magnetic field of toroidal dipole mode. From the directions in the magnetic field distribution (black arrow), it can be found that the distinct vortex of the magnetic field is produced by the near-field coupling between the excited magnetic modes, which closed loops of the magnetic field confined well within the unit cell. Furthermore, the bottom left subplot corresponds to the x-direction magnetic dipole mode of Band2, while the bottom right subplot corresponds to the magnetic dipole mode along the y-direction for Band3.
Fig. 2. (a) Photonic bands for the quasi-TM modes computed with COMSOL. (b) Electric field (normalized z component $\textbf{E}_{z}$) and magnetic field (normalized absolute value $\vert \textbf{H} \vert$) over x,y cross section in the unit cell of the different bands at the $\Gamma$ point. The black arrow shows the directions in the magnetic field distribution. (c) The Q-factor for the three bands. (d) The Q-factor for the eigenmodes of the slab with various thicknesses h.
The effective Q-factor of the different eigenmodes can be calculated by the eigenfrequency and damping. Assuming that the time-harmonic electromagnetic field in the unit cell has a complex phase parameter
Furthermore, as shown in Fig. 3(a), the conical behavior of the bands near the DLP is presented in a 3D dispersion band diagram, which shows the connection between the Floquet propagation vector $k$ and frequency. Here, the Band1 and Band3 form two inverted cones meeting at the $\Gamma$ point and the band2 intersects these two bands at the DLP frequency. Figure 3(b) presents the isofrequency contours (bands of constant frequency) of Band1. The isofrequency contours originate from the DLP frequency, and the almost circular contour shows that the PhC slab is isotropic near $k=0$, mimicking a zero-index in all propagation directions.
Fig. 3. (a) Three-dimensional dispersion surfaces near the DLP frequency. (b) The equal frequency contour in the two-dimensional reciprocal space of the Dirac-like cone (corresponding to Band1 in Fig. 2(a)), the colorbar represents the frequency.
3. Multipole decomposition method
Although our design can be used to enhance the excitation of the toroidal dipole, other multipoles may also be excited simultaneously, especially for the electric dipole. Since the toroidal dipole and electric dipole emits radiation with the same angular momentum and parity properties, which is indistinguishable from them for any distant observer [30]. However, we can distinguish them by the difference between the two types of radiation power: The radiated power for electric and toroidal dipoles scales as, respectively, $\omega ^{4}$ and $\omega ^{6}$ [31]. For the convenience of calculation, the radiation power of the multipole moment is mainly carried out in the Cartesian coordinates, and a position vector can be defined as $\boldsymbol {r}=(x, y, z)$. The displacement current distributions $\boldsymbol {J}(\boldsymbol {r})$ can be obtained from the electric field distributions $\boldsymbol {E}(\boldsymbol {r})$ by
Figure 4(a) shows the normalized radiation powers of multipoles for the toroidal dipole mode. The radiation powers of the TD moments are the dominant ones. The radiation power of electric dipole ED is about ten times lower than that of toroidal dipole TD. Figure 4(b) and Fig. 4(c) show the normalized radiation powers of multipoles for the magnetic dipole modes. The radiation powers of the MD moments are the dominant ones. The excitation of the other higher-order multipoles (magnetic quadrupole MQ, electric quadrupole EQ) is due to some slight asymmetry of the electromagnetic field distribution in the eigenmodes. We also calculate the components of the TD moments and MD moments in the eigenmodes. As shown in Fig. 5(a), the ${\textrm{TD}_{z}}$ is the main component of the toroidal dipole moments. Because the direction of toroidal dipole moment is along the z-axis, the radiation of toroidal dipole TD is confined within the in-plane, prohibiting the radiation in the vertical direction. For the magnetic dipole moments, the main components are the ${\textrm{MD}_{x}}$ and ${\textrm{MD}_{y}}$ shown in Fig. 5(b), which represents the high out-of-plane radiation along the z-axis. Therefore, the excitation of the magnetic dipole mode will bring abundant radiation loss to the system.
Fig. 4. (a) The normalized radiation powers of multipoles for the toroidal dipole mode at the $\Gamma$ point. (b), (c) The normalized radiation powers of multipoles for the magnetic dipole modes at the $\Gamma$ point. Namely, toroidal dipole TD , electric dipole ED, magnetic dipole MD, magnetic quadrupole MQ, electric quadrupole EQ.
Fig. 5. (a) Normalized components of the TD moments in the eigenmodes. (b) Normalized components of the MD moments in the eigenmodes.
4. Verification
To confirm that the toroidal dipole mode can suppress the excitation of the magnetic dipole mode during the in-plane transmission, we use a plane wave to excite the PhC slab. The electric field direction of the plane wave is along the z-axis.
As shown in Fig. 6(a), the orange shaded region shows the transmission band gap of the PhC slab. Transmission is higher than −3.3 dB in the range of 0.94-0.96 THz, and the PhC slab shows the perfect phase-matching property at the DLP frequency. Figure 6(b) shows the multipole radiation powers of the unit cell within the slab. In the range of 0.94-0.96 THz (orange shaded region), the radiation power of the toroidal dipole TD shows a considerable enhancement. The radiation power of toroidal dipole TD at 0.95 THz is about 10 times higher than that of the electric dipole ED. In particular, the radiation power of the magnetic dipole MD is significantly suppressed in the range of 0.94-0.96 THz. The weak excitation of magnetic dipole modes corresponds to the transmission band in Fig. 6(a). It means that the radiation loss of the system can be reduced by the toroidal dipole mode. Beyond the orange shaded region, the radiation power of the magnetic dipole MD has a great increase and the TD moments are not the dominant ones. The excitation of the magnetic dipole mode will produce abundant radiation loss, according to Fig. 6(a), it is clear that the transmission gradually decreases far away from the DLP frequency. In Fig. 7(a), we show the electric field distribution over the cross-section in the plane of the PhC slab. At the DLP frequency, the electric field distribution in the entire slab shows the toroidal dipole mode, which corresponds to the average permittivity of zero and perfect spatial coherence. Figure 7(b) shows the electric field on the longitudinal section of the PhC slab, we added an air layer above and below the slab to illustrate the out-of-plane radiation.
Fig. 6. (a) Transmission and reflection of the PhC slab, the excitation source is a plane wave with the electric field direction along the z-axis. (b) Normalized multipole radiation powers of the unit cell within the PhC slab.
Fig. 7. (a) Electric field (normalized z component $\textbf{E}_{z}$) distributions over the large-area zero-index PhC slab, $z=0 ~\mu\textrm{m}$, the excitation source frequency is 0.95 THz. (b) Electric field intensity (normalized absolute value $\vert \textbf{E} \vert$) distributions on the longitudinal section of the slab at 0.95 THz. (c) Electric field intensity (normalized absolute value $\vert \textbf{E} \vert$) distributions on the longitudinal section of the slab at 0.92 THz. The thickness of the PhC slab is $h=180 ~\mu\textrm{m}$, the white arrow represents the incident plane wave and digital labels correspond to the number of units. The upper and lower parts of the PhC slab are air layers, and the leakage electric field is absorbed by the perfectly matched layer (PML).
It is clear that the electric field within the slab is uniform, and there is some near-field enhancement at the contact boundary of the slab and the air. Importantly, the electric field is strongly confined in the unit cell, and the radiation leakage in the air is very weak due to the excitation of the toroidal dipole. However, far away from the DLP frequency, the radiation powers of the MD moments become the dominant ones and the in-plane electromagnetic wave can be easily coupled into the air by the magnetic dipole modes. For example, as shown in Fig. 7(c), the obvious radiation leakage in the air can be observed at 0.92 THz. Here, the PhC slab can not be regarded as an impedance-matched zero-index material when the excitation source frequency is far away from the DLP frequency. The plane wave incident from the left is strongly reflected (highlighted yellow at the incident port 1) and the electric field distribution behaves as a standing wave. Finally, we use the effective medium theory [34,35] to extract the effective parameters of the PhC slab. As shown in Fig. 8(a), real parts of both the effective permittivity and permeability cross zero simultaneously and linearly at the DLP frequency, indicating an impedance matched zero index. We compute the in-plane propagation loss of the PhC slab according to the retrieved effective index of refraction via (Section S1 in Supplement 1)
Fig. 8. (a) Effective relative permittivity and relative permeability, which linearly cross zero at 0.95 THz. (b) Effective index and propagation loss of the PhC slab. When the real part of the effective index crosses zero, the loss curve reaches its valley $(\sim 0.16~\mathrm {dB} / \mathrm {mm})$.
5. Discussion
In summary, we have shown a way to further eliminate the out-of-plane radiation loss by introducing the toroidal dipole mode. We have confirmed that both the eigenmode and the in-plane transmission mode exhibit a dominant TD moment character by using the multipole decomposition approach. The toroidal dipole mode has been verified to suppress the excitation of the magnetic dipoles during the in-plane transmission. In our design, the transmission band gap is in the range of 0.94-0.96 THz for the plane wave. At the DLP frequency, we estimate the propagation loss is about $0.16 ~\mathrm {dB} / \mathrm {mm}$. The presented photonic crystal slab provides a new approach for phase-matching in linear and nonlinear optics [36]. It can also serve as a platform to explore fundamental quantum science such as photon entanglement [37].
Author contribution
Z. L. and H. L. proposed the idea. Z. L. performed the calculation, produced all the figures, and wrote the manuscript draft. R. Z. and X. S. contributed to the calculation tools. Z. Y. analyzed the data. H. L. and Y. L. lead the project and revised the whole manuscript thoroughly. J. W. and R. T. revised the paper.
Funding
Educational Commission of Hubei Province of China (Q20201006, Q20211008); Department of Education in Hubei Province (T2020001-030200301301002); Science and Technology Department of Hubei Province (2018CFB148, 2020CFB266); National Natural Science Foundation of China (NSFC12047501, NSFC41974195); Young Scientists Fund (NSFC11804087); The fund of Hongque Innovation Center (HQ202104001); Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing, Guilin University of Electronic Technology (GXKL06190202); Fundamental Research Funds for the Central Universities (CCNU19TS073, CCNU20GF004).
Disclosures
The authors declare no conflicts of interest.
Data availability
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.
Supplemental document
See Supplement 1 for supporting content.
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