The Design of Highly Reflective All-Dielectric Metasurfaces Based on Diamond Resonators

1. Introduction

Metasurfaces provide a way to manipulate light by precisely arranging sub-wavelength structures and fine-tuning the parameters of these structures on a flat substrate [1,2,3]. By sophisticatedly designing the 2D-arrays, metasurfaces show the capability of controlling the phase, amplitude, and polarization of the light [4,5], resulting in a wide range of applications, such as imaging [6], holography [7], quantum optics [8], spectrometry [9], and structured light [10]. One important application of interest is the highly reflective mirrors, which often use all-dielectric nanomaterials taking their advantages of the low-loss property and enhanced resonance [11]. Such devices are also referred to as perfect reflectors because they can achieve near-unity reflectivity at certain wavelengths by properly selecting the geometries of resonant nano-structures [12,13,14,15] and precisely arranging their positions [16]. For example, Slovick et al. has presented a structure of square-latticed Si cubes on ${\mathrm{SiO}}_2$ substrate with over 99.999% reflectivity and less than 0.001% absorptivity at $1.5$ $\mathsf{\mu }$$\mathrm{m}$ in simulation [12]. Based on the principles of this work, cylindrical silicon resonators were experimentally demonstrated to achieve a reflectance of 99.9% at 1450 $\mathrm{n}$$\mathrm{m}$ [13]. Further, large-scale dielectric reflectors with frustums in hexagonal lattices applying the same material were fabricated and demonstrated to show 99.7% reflectivity at 1530 $\mathrm{n}$$\mathrm{m}$ [14].

To better guide the design of the highly reflective metasuraces, one can mainly follow the principles of guided-mode resonance (GMR) [17,18,19]. The GMR gives a physical picture of how the incident light diffracts and propagates in a grating waveguide slab. By optimizing the parameters of the grating layer, it is possible to select the diffracted light at certain mode to achieve the constructive interference [17], which comes from the electromagnetic resonances within the nanostructures of the grating [20,21,22]. In order to have a deeper insight of the physics, the effective medium theory [12,13,14,23,24,25,26] and the multipole decomposition method [27,28,29] are often used to help the analysis. Following the effective medium theory, one can first calculate the effective permittivity and permeability via the scattering-parameters (S-parameter) [12,25], and correspondingly acquire the effective impedance and refractive index, which directly show the properties of the grating layer. On the other hand, electromagnetic resonances in the grating layer result in the scattered fields of electric and magnetic multipole. Through the multipole decomposition, the scattering cross-sections (SCSs) of different resonant modes can be quantitatively calculated, which reveals the contributions of various electromagnetic dipoles, quadrupoles, or higher-order multipoles [28].

In recent years, metasurface reflectors have also been demonstrated to show a high laser induced damage threshold (LIDT) under high-power lasers, using single-crystal diamonds (SCDs) as the material [15]. Because such materials have excellent optical and thermal properties: relatively high refractive index ($n=2.4$), wide bandgap ( $5.5$ $\mathrm{e}\mathrm{V}$), and most importantly, the highest thermal conductivity ( 2200 $\mathrm{W}$/$\mathrm{K}$/$\mathrm{m}$ at room temperature) [30]. Atikian et al. fabricated a diamond-based metasurface by angled-etching nanostructures into "golf tee"-shaped columns, achieving over 98% reflectivity at 1070 $\mathrm{n}$$\mathrm{m}$ and damage-free operation under 10 $\mathrm{k}$$\mathrm{W}$ continuous-wave laser with the spot diameter at 750 $\mathsf{\mu }$$\mathrm{m}$. [15]. Another advantage of SCDs is that they have extremely low absorption at a broad range of light spectrum, possessing the potential for shorter wavelengths, e.g. visible or even ultraviolet light [30]. Therefore, the diamond-based metasurfaces have the potential to be used for high-power alkali lasers [31], whose typical wavelength is at 795 $\mathrm{n}$$\mathrm{m}$. Moreover, the etching (vertical and angled) of diamond pillars has been achieved in the study by Atikian et al [15], demonstrating the potential of more diamond-based metasurface designs in fabrication.

In this paper, we present the design of a diamond-based metasurface, which can achieve near-unity reflectivity at 795 $\mathrm{n}$$\mathrm{m}$. First, we utilize GMR as design principles to select the configuration and the substrate material of desired, and analyse the scattering properties of a single SCD resonator by multipole decomposition. Then, we select two configuration of cylinders in square lattice and frustums in hexagonal lattice, and by optimizing the geometries of cylinders in square lattice in simulation we can maximize reflectivity while minimizing absorption. Next, we adjust the geometries of the square-latticed cylinder to obtain dual-band reflectivity for certain applications. For the designs above, analyses of electromagnetic fields, effective medium properties and geometrical parameter sweepings are conducted to enhance the understanding of the optimized structures.

2. The Principles for Designing the Perfect Reflector

2.1. Guided-Mode Resonance

As mentioned before, GMR is a fundamental principle used to design metasurfaces with high reflectivity [18]. Figure 1 (a) presents the schematic of a dual-layer GMR, a grating layer ($n_{\mathrm{G}}$ layer) and a waveguide layer ($n_{\mathrm{W}}$ layer), with a normal incident light at the intensity of I and its reflected and transmitted light (in solid blue). The incident light is first diffracted by the grating layer, and is going to be coupled into the waveguide layer (in dashed blue) when meeting the resonance condition. Here, GMR occurs because the incident light and the waveguide mode constructively interference, resulting in the reflectivity R approaching near-unity and the transmission T near-zero. Otherwise, the light would transport directly through the two functional layers as well as the substrate ($n_{\mathrm{S}}$ layer) (in solid gray). There is another way of realizing GMR by using a a single-layer configuration, as illustrated in Figure 1 (b). The grating layer can also act as the waveguide as long as the higher refractive index of the grating $n_{\mathrm{H}}$ , the lower index $n_{\mathrm{L}}$ and optical thickness d of the grating structure are carefully optimized [19,32,33]. Here, $n_{\mathrm{G}}$ represents the effective refractive index of the same layer of the grating and the waveguide. The single-layer GMR configuration can simplify the metasurface structure and facilitate dissipating heat [15,19].

The metasurfaces, commonly seen as a semi-finite-thickness slab, can be treated as an effective homogeneous medium, with the effective complex permittivity ${\epsilon }^{*}$ and the effective permeability ${\mu }^{*}$ expressed as ${\epsilon }^{\prime }+i{\epsilon }^{″}$ and ${\mu }^{\prime }+i{\mu }^{″}$ respectively [34]. Following the effective medium theory, the perfect reflection can be achieved while the Eq. (1a) and Eq. (1b) are both satisfied.

$$\begin{array}{cc}\hfill & {\epsilon }^{\prime }/{\mu }^{\prime }<0\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill & {\epsilon }^{″}{\mu }^{\prime }={\epsilon }^{\prime }{\mu }^{″}\hfill \end{array}$$

(1b)

The first condition in Eq. (1a) requires the real parts of the permittivity and permeability to have opposite signs, which is easily satisfied at the resonate wavelength of electric or magnetic resonances. Although the second condition in Eq. (1b) appears more strict, it can be satisfied at certain conditons. Particularly, it is more easily to be satisfied in all-dielectric materials with low loss, since the imaginary part of ${\epsilon }^{*}$ and ${\mu }^{*}$ are approaching zero [12,13]. Therefore, we can verify whether our design is indeed a perfect reflector judging by the two conditions above, of which the permittivity ${\epsilon }^{*}$ and permeability ${\mu }^{*}$ can be extracted from the S-parameter via simulation [23,24,25,26].

As mentioned before, the SCDs are the perfect candidate as the grating layer for metasurface reflector that can endure high-power lasers, but the materials for the substrate should also be thoroughly considered in order to meet our purpose. We propose the single-layer GMR configuration, using SCDs as the resonant layer with the effective index denoted as $n_{\mathrm{G}}$. Here, the high refractive index equals to that of the SCDs, i.e. $n_{\mathrm{H}}=n_{\mathrm{SCD}}$, and the low refractive index is that of the air ($n_{\mathrm{L}}=n_{\mathrm{air}}$). And in this paper we have chosen the fused silica (silicon dioxide, ${\mathrm{SiO}}_2$) as the substrate mainly for the two reasons below. First, its refractive index is lower than that of SCDs, meeting the conditions of the GMR theory. And second, silica has relatively high LIDT due to its extremely low thermal expansion coefficient, making it possible for high-power-laser applications [35]. Last but not least, the diamond coating on ${\mathrm{SiO}}_2$ substrates can be achieved using Microwave Plasma Chemical Vapor Deposition technique (MPCVD) [36].

2.2. The Multipole Decomposition

To fully understand the behavior of light scattering and reflection in these resonant structures, it is essential to decompose the scattering field into multipole components for the complexity of electromagnetic field [29]. Multipole decomposition is a crucial method for analyzing the complex electromagnetic fields of scattering, which can separate the contributions of electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), magnetic quadrupole (MQ), and higher-order multipoles. It effectively simplifies the description of the field, facilitating the analysis of the field distribution, which helps guide the design of highly reflective metasurfaces [29,37,38].

The electric far-field scattered by an electric and a magnetic dipole (scalar projection) can be written as Eq. (2a) and Eq. (2b) [37].

$$\begin{array}{cc}\hfill & E^{\mathrm{ED}}\left(z\right)=E_p{\mathrm{e}}^{i\pi }{\mathrm{e}}^{i\left({\omega }_{\mathrm{ED}}t{\displaystyle \frac{z}{\left|z\right|}}-k_{\mathrm{ED}}z\right)}\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill & E^{\mathrm{MD}}\left(z\right)=E_m{\mathrm{e}}^{i\pi \theta \left[z\right]}{\mathrm{e}}^{i\left({\omega }_{\mathrm{MD}}t{\displaystyle \frac{z}{\left|z\right|}}-k_{\mathrm{MD}}z\right)}\hfill \end{array}$$

(2b)

Here, $E_{\mathrm{p}}$ ($E_{\mathrm{m}}$) represents the amplitudes of the fields scattered by electric (magnetic) dipoles. The phase factor $e^{\mathrm{i}\pi }$ accounts for a phase shift observed during dipole excitation at resonance. The function $\theta \left[z\right]$ is a Heaviside step function, i.e. $\theta \left[z\right]=0$ for $z<0$ and $\theta \left[z\right]=1$ for $z>0$, which describes the antisymmetric behavior of the magnetic dipole resonance [37]. The functions $E^{\mathrm{ED}}\left(z\right)$ and $E^{\mathrm{MD}}\left(z\right)$ present the properties shown as below.

$$\begin{array}{cc}\hfill & E^{\mathrm{ED}}(-z)=E^{\mathrm{ED}}\left(z\right)\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill & E^{\mathrm{MD}}(-z)=-E^{\mathrm{MD}}\left(z\right)\hfill \end{array}$$

(3b)

The incident field $E_0$ interact with the scattered field $E_p$ or $E_m$, ending up with interference. When $E_0=E_{\mathrm{p}}$ or $E_{\mathrm{m}}$, the destructive interference occurs in the forward (transmitted) direction, and a standing wave is generated in the backward (reflected) direction. Particularly, when only ED and MQ contribute to the scattering field, the node of the standing wave is exactly at the surface of the reflector, exhibiting the same behavior of a perfect electric conductor (PEC), and therefore it can also be defined as the generalized electric mirror. In contrast, when the contribution purely comes from MD and EQ, an anti-node (instead of a node) should form at the surface of the mirror, which is referred to as the generalized magnetic mirrors [22,37].

Here, we start from the simplest cylindrical geometry structure of diamond metasurfaces, seen in Figure 2 (a). According to [39], the resonance wavelengths of ED and MD in a cylindrical resonator depend on the aspect ratio ($AR$), which is defined as $AR=H/D$, where H and D represents the height and diameter of the cylinder respectively. Here, the simulations are conducted by 3D finite-difference time-domain solvers (FDTD, Lumerical Solutions, Inc.) and a total field scattered field (TFSF) source is used to analyze the scattering properties. The SCSs along with the wavelength $\lambda$ can be numerically calculated, and the peaks are shown at certain wavelengths, representing electric or magnetic resonance. By analysing the wavelength difference of the peaks, an optimized geometry is determined with $H=370\mathrm{n}\mathrm{m}$ and $P=900\mathrm{n}\mathrm{m}$, where P (pitch) represents the center-to-center distance between adjacent diamond columns. Figure 2 (b) illustrates the SCS for a diamond cylinder resonator without substrate, with $AR$ ranging from 0.8 to 1.6. The calculation of the SCS relies on Hinamoto’s work [27], and it requires the results of electric fields and refractive indices, which varies with the spatial position (x, y, z) and the light frequency f. Shown in Figure 2 (b), the resonant modes happen at the certain wavelength peaks (see Table 1 in the supplementary materials), forming either generalized electric mirror (generated by ED+MQ) or generalized magnetic mirror (generated by MD+EQ).

To further investigate the collective response of the diamond resonators, the reflective spectra at square-latticed configuration have been calculated, as shown in Figure 2 (c). Corresponding to the SCSs, these illustrate that the yellow points of high-reflectivity are mainly the results of MD+EQ, while the peaks in green are the results of both ED+MQ and MD+EQ. However, for $AR=1.2$ and $1.4$, there are sharp drops of the spectra at $\lambda /D$ around 2.1. This is because the scattering field of different poles interference destructively [40], resulting in the energy localized in the resonators without emitting electromagnetic radiation, which have a high Q-factor [41].