Manipulating light transmission and absorption via an achromatic reflectionless metasurface

Abstract

Freely switching light transmission and absorption via an achromatic reflectionless screen is highly desired for many photonic applications (e.g., energy-harvesting, cloaking, etc.), but available meta-devices often exhibit reflections out of their narrow working bands. Here, we rigorously demonstrate that an optical metasurface formed by two resonator arrays coupled vertically can be perfectly reflectionless at all frequencies below the first diffraction mode, when the near-field (NF) and far-field (FF) couplings between two constitutional resonators satisfy certain conditions. Tuning intrinsic loss of the system can further modulate the ratio between light transmission and absorption, yet keeping reflection diminished strictly. Designing/fabricating a series of metasurfaces with different inter-resonator configurations, we experimentally illustrate how varying inter-resonator NF and FF couplings can drive the system to transit between different phase regions in a generic phase diagram. In particular, we experimentally demonstrate that a realistic metasurface satisfying the discovered criteria exhibits the desired achromatic reflectionless property within 160–220 THz (0–225 THz in simulation), yet behaving as a perfect absorber at ~ 203 THz. Our findings pave the road to realize meta-devices exhibiting designable transmission/absorption spectra immune from reflections, which may find many applications in practice.

Introduction

Freely controlling transmission, reflection, and absorption of light wave through a thin screen is highly desired in photonics [1,2,3,4,5,6]. For instance, while perfect absorption is favored in energy harvesting, perfect transmission is desired in applications related to sensing and cloaking. In many scenarios, reflection is undesired as it can cause efficiency reduction and failure of the expected functionality (say, cloaking). Unfortunately, switching between perfect light transmission and perfect absorption is extremely challenging in thin-screen systems where energy transports in different channels are usually coupled in a complex way.

Metasurfaces, ultrathin metamaterials composed by subwavelength microstructures exhibiting tailored electromagnetic (EM) responses, offer a new platform to control light waves [7,8,9,10,11]. Many fascinating effects were realized based on metasurfaces, including perfect absorption [12,13,14,15,16], perfect transparency [17,18,19,20,21], and many others [22,23,24,25,26,27,28]. However, strong reflections always appear in such meta-devices at frequencies out of their narrow working bands in which perfect absorption/transparency is achieved. Recently, A few attempts appear to construct reflectionless metasurfaces based on different approaches such as antireflection coating [29, 30], anomalous Brewster effect [31, 32], specular reflection reduction [33], etc. One typical design approach is to put into a unit cell two resonators, exhibiting electric and magnetic responses, respectively, so that their back scatterings can exactly cancel each other under certain condition (e.g., the Kerker condition [34,35,36,37]). However, since two created resonance modes usually exhibit distinct frequency dispersions, in most cases the Kerker condition is satisfied only at a single frequency, making the constructed metasurfaces exhibiting narrow working bands [6, 38,39,40,41]. Despite of many efforts devoted to expanding the reflectionless bandwidths through structural optimizations [32, 33, 42,43,44,45], such an issue remains intrinsic, and new mechanism is highly desired for designing achromatic reflectionless metasurfaces.

In this work, we propose a new strategy to realize achromatic reflectionless metasurfaces (see Fig. 1). Distinct from metasurfaces consisting of resonators placed on the same plane [46,47,48,49], here we study a bi-layer system containing two arrays of resonators placed on different planes. The inter-layer distance is a new parameter to tune both NF and FF couplings between two resonators (see inset in Fig. 1), which is lacked in previously studied systems. Base on coupled-mode-theory (CMT) calculations, we first establish a generic phase diagram showing how the reflection property of such system varies against the inter-resonator NF and FF couplings. We find that the Kerker condition can be rigorously satisfied at all frequencies below the first diffraction mode, as long as the NF and FF coupling strengths between two resonators meet a set of conditions. Moreover, adding Ohmic losses to the system does not violate the Kerker condition, but rather efficiently re-allocate light power between transmission and absorption channels. We design/fabricate a series of bi-layer metasurfaces and experimentally illustrate how varying their geometric configurations can drive them move inside the phase diagram via modulating two coupling strengths. We finally realize a metasurface satisfying the Kerker criterion, and experimentally demonstrate that it exhibits perfect absorption around 203 THz within an ultra-wideband reflectionless frequency band (160–220 THz in experiment, 0–225 THz in simulation). Many applications can be expected based on our reflectionless platform, with a tunable absorber numerically demonstrated as a particular example.

Fig. 1

Schematics of the achromatic reflectionless bi-layer metasurface. Tailoring near-field and far-field couplings between two optically identical resonators placed on different phase planes (inset), one can realize a metasurface that is immune from reflections at all frequencies below the first diffraction mode, yet with light power switched between transmission and absorption channels dictated by the intrinsic loss

Results and discussion

Generic phase diagram of the bi-layer systems

We establish a phase diagram for the proposed bi-layer systems based on CMT analyses [49,50,51]. In our system, two layers of our system are both periodic arrays with subwavelength spacing, such that only the zero-order mode of transmission/reflection can survive. As depicted in the inset to Fig. 2a, the bi-layer system can be generically described by a 2-mode 2-port model, with time evolutions of the amplitudes a_{i = 1, 2} of two resonance modes satisfying the following equations:

$$\left\{\begin{array}{c}\frac{1}{2\pi}\frac{\partial }{\partial t}\left(\begin{array}{c}{a}_1\\ {}{a}_2\end{array}\right)=-i\left(\begin{array}{cc}{f}_0& \kappa \\ {}\kappa & {f}_0\end{array}\right)\left(\begin{array}{c}{a}_1\\ {}{a}_2\end{array}\right)+\left(\begin{array}{cc}-{\Gamma}_r-{\Gamma}_i& X\\ {}X& -{\Gamma}_r-{\Gamma}_i\end{array}\right)\left(\begin{array}{c}{a}_1\\ {}{a}_2\end{array}\right)+\left(\begin{array}{c}{d}_{11}\\ {}{d}_{12}\end{array}\right){s}_1^{+}\\ {}\left(\begin{array}{c}{s}_1^{-}\\ {}{s}_2^{-}\end{array}\right)=\textbf{C}\left(\begin{array}{c}{s}_1^{+}\\ {}{s}_2^{+}\end{array}\right)+\left(\begin{array}{cc}{d}_{11}& {d}_{12}\\ {}{d}_{21}& {d}_{22}\end{array}\right)\left(\begin{array}{c}{a}_1\\ {}{a}_2\end{array}\right)\end{array}\right..$$

(1)

Fig. 2

Generic phase diagram for the bi-layer metasurfaces. a Reflectance of the 2-mode 2-port model (inset) calculated by Eq. (3) in the main text, as a function of frequency, θ_X and κ, with κ − Im(X) = 0 satisfied on the horizontal curved surface. Green dash lines labeled with #1 to #4 represent the cases of κ/Γ_r = 1, 0, − 0.6, − 1 with fixed θ_X = π/2, while those labeled with #4 to #7 represent the cases of θ_X = π/2, 3π/5, 3π/4, π with κ − Im(X) = 0 satisfied, respectively. Here, Γ_r/f₀ = 0.2. b-c Calculated reflection spectra |r|² and those contributed by two decoupled modes |r_±|² calculated by Eq. (3) for 6 systems corresponding to 6 dashed lines shown in a. d Upper panel: spectra of |r|² and |r_±|² calculated by Eq. (3) for the #4 system; Lower panel: computed spectra of transmittance and transmission phase for the #4 system

Here, we assume that the two modes are identical, with f₀, Γ_r and Γ_i describing their resonant frequencies, radiation and absorption damping, respectively. Meanwhile, κ and X represent NF and FF couplings between two modes, d_ji describes the coupling between the j-th external port and the i-th excited mode, and ${s}_j^{+\left(-\right)}$ denotes the amplitudes of incoming (out-going) waves from (to) the j-th port. $\textbf{C}=\left(\begin{array}{cc}{r}_0& {t}_0\\ {}{t}_0& -{r}_0\end{array}\right)$ denotes the scattering properties of the background which is set as vacuum at the moment (i.e. r₀ = 0 and t₀ = 1). According to energy conservation and time reversal symmetry, we prove that ${d}_{11}={d}_{21}=i\sqrt{\Gamma_r}$, ${d}_{12}=i\sqrt{\Gamma_r}\exp \left(i{\theta}_X\right)$ and ${d}_{22}=i\sqrt{\Gamma_r}\exp \left(-i{\theta}_X\right)$, where θ_X measures the phase difference between radiated far fields from two excited resonators (see Fig. 2a). Finally, we find that the FF inter-resonator coupling X is not an independent parameter, but can be expressed as X =  − Γ_r exp(iθ_X) [49]. All derivation details can be found in Section 1 of Supplementary Information (SI). In what follows, we consider the lossless case (i.e. Γ_i = 0) first, where only four independent parameters (f₀, Γ_r, κ and θ_X) are relevant. We note that all these parameters are assumed as frequency-independent constants in CMT [49,50,51] derived under the high-Q approximation.

Equation (1) can be analytically solved through standard CMT analyses. The identical symmetry possessed by two matrixes in the first line of Eq. (1) ensures that they can be simultaneously diagonalized with an orthogonal transformation ${\tilde{a}}_{\pm }=\left({a}_2\pm {a}_1\right)/\sqrt{2}$. After the transformation, we get two decoupled hybridized modes with

$${\tilde{f}}_{\pm }={f}_0\pm \left(\kappa +{\Gamma}_r\sin {\theta}_X\right),\kern0.5em {\tilde{\Gamma}}_{\pm }={\Gamma}_r\left(1\pm \cos {\theta}_X\right)$$

(2)

being their resonant frequencies and radiation damping, respectively. We note from Eq. (2) that the shift in frequency is determined by the NF coupling κ and the Hermitian part of FF coupling Im(X), while the shift in radiation damping is solely dictated by the non-Hermitian part of FF coupling Re(X). Reflection coefficient of our metasurface can then be analytically derived as (see Sec. 1 in SI)

$$r={r}_{+}+{r}_{-},{r}_{+}=\frac{-{\Gamma}_r}{2}\frac{{\left(1+{e}^{i{\theta}_X}\right)}^2}{-i\left(f-{\tilde{f}}_{+}\right)+{\tilde{\Gamma}}_{+}},{r}_{-}=\frac{-{\Gamma}_r}{2}\frac{{\left(1-{e}^{i{\theta}_X}\right)}^2}{-i\left(f-{\tilde{f}}_{-}\right)+{\tilde{\Gamma}}_{-}},$$

(3)

where r₊ and r₋ denote the scatterings due to two independent modes.

We employ Eq. (3) to study how the reflectance R = |r|² varies against κ and θ_X with all single-mode properties (f₀ and Γ_r) fixed. Fixing θ_X at 0, π/4 and π/2, respectively, we illustrate on three vertical planes in Fig. 2a how the obtained reflectance spectrum R(f) changes versus κ. While two branches of reflection peaks appear on the planes at π/4 and π/2, only one branch exists on the plane at θ_X = 0 since one mode becomes completely dark (${\tilde{\Gamma}}_{-}=0$) dictated by the symmetry, which is also called a bound state in continuum [36, 52]. Set κ as 3 different values on the θ_X = π/2 plane, we depict in Fig. 2b the calculated reflection spectra R(f) (solid lines) and those contributed by two decoupled modes |r_±(f)|² as defined in Eq. (3) (dotted lines). We find that varying κ modifies two peak positions keeping the bandwidths of two decoupled modes unchanged, in consistency with Eq. (2). The final reflection spectrum R(f) is thus significantly modulated by κ due to the interference between two modes (See Fig. S2 for more details in Sec. 2 of SI). Meanwhile, Eq. (2) predicts that the two hybridized modes exhibit identical resonance frequencies in the cases of κ =  − Γ_r sin θ_X, corresponding to a curved surface in Fig. 2. Choosing three typical cases on the curved surface, we depict in Fig. 2c how changing θ_X further modulates the final reflection spectrum. As shown in Fig. 2c, now the R spectrum is an interference of two degenerate modes exhibiting different radiation damping, in consistency with Eq. (2). In particular, in the case of θ_X = π/2 (the #4 line in Fig. 2a), the two degenerate modes are out of phase and of identical bandwidths, and therefore, their interference leads to complete cancellation of reflections within the entire considered frequency range as shown in Fig. 2d (i.e., R(f) ≡ 0). Meanwhile, it is noticeable that the transmission phase covers a whole 2π range. Such a rigorous zero-reflection solution can be re-phrased as the following condition

$$\left|X- i\kappa \right|=\left|-{\Gamma}_r\cos {\theta}_X-i\left(\kappa +{\Gamma}_r\sin {\theta}_X\right)\right|=0$$

(4)

which states that the total effective coupling (including both Hermitian and non-Hermitian parts) that we define as ξ = |X − iκ| between two original modes must be exactly zero.

We now discuss the role played by the absorption loss. Figure 3a and b depict, respectively, the calculated reflectance R on two θ_X − f planes (with κ =  − Γ_r sin θ_X fixed) and two κ − f planes (with θ_X = π/2 fixed) with different absorption damping parameters (Γ_i = 0 and Γ_i = 0.5 Γ_r). Obviously, absorption loss does not affect the desired zero-reflection property of the system, as long as Eq. (4) is met (dashed lines in Fig. 3a-b, more cases are presented in Figs. S3-S4 of SI). In fact, turning on Γ_i in Eq. (1) does not violate the symmetry possessed by our system, and thus reflections from two decoupled modes still exactly cancel each other. Rigorously solving Eq. (1) with Γ_i present, we get the following analytical expressions (see Sec. 1 in SI for detailed derivations):

$$r=0;\kern0.5em t=1-\frac{2{\Gamma}_r}{-i\left(f-{f}_0\right)+{\Gamma}_r+{\Gamma}_i}.$$

(5)

Fig. 3

Influences of Γ_i on the optical properties of the model satisfying Eq. (4). a Reflectance of the model system versus frequency and θ_X in the cases of Γ_i/Γ_r = 0 and Γ_i/Γ_r = 0.5, respectively, with κ/Γ_r fixed as − 1. b Reflectance of the model system versus frequency and κ/Γ_r in the cases of Γ_i/Γ_r = 0 and Γ_i/Γ_r = 0.5, respectively, with θ_X fixed as π/2. c Calculated spectra of transmittance (orange lines) and absorbance (blue lines) of the model systems with κ/Γ_r =  − 1 and θ_X = π/2, and with Γ_i/Γ_r taking 4 different values. d Transmission phase coverage of the model system versus Γ_i/f₀ and Γ_r/f₀. Four symbols labeled with #1–4 correspond four cases studied in c. Upper and lower insets depict the calculated spectra of transmission phase for the systems labeled with #1 and #4, respectively. In all cases studied, we have Γ_r/f₀ = 0.2

Remarkably, we find that the expression of t is exactly the same as that of the reflection co-efficient of a 1-port 1-mode model, widely used to describe the metal/insulator/metal (MIM) system [13]. Therefore, all interesting physical behaviors revealed in the MIM systems are also expected here [13]. Indeed, increasing Γ_i can gradually enhance the absorbance A until the critical-damping condition (Γ_i = Γ_r) is reached at which perfect absorption happens (A(f₀) = 1), while further increasing Γ_i in the region of Γ_i > Γ_r will decrease the absorbance (see blue lines in Fig. 3c). Along varying Γ_i, however, we find distinct behaviors exhibited by the transmission phase ϕ_t(f) = arg[t(f)] in two different cases with Γ_i < Γ_r and Γ_i > Γ_r, respectively (see insets to Fig. 3d). Color map in Fig. 3d depicts the variation range Δϕ_t of transmission phase ϕ_t(f) upon frequency changing, as the functions of Γ_i and Γ_r. The critical-damping line (Γ_i = Γ_r) separates the whole Γ_i − Γ_r phase diagram into two parts, which are the under-damped region (Γ_i < Γ_r) with Δϕ_t covering a whole 2π range and the over-damped region (Γ_i > Γ_r) with Δϕ_t less than π. In all these cases studied, we have A(f) + T(f) = 1 strictly satisfied (see Fig. 3c), since the reflection channel of our system is completely blocked as long as Eq. (4) is satisfied. Therefore, tuning Γ_i in our system can efficiently re-allocate the power of light between transmission and absorption channels, in an ideal achromatic reflectionless platform.

Experimental demonstration of the phase diagram

We now experimentally realize a series of bi-layer metasurfaces exhibiting different NF and FF couplings, starting from designing two optical modes exhibiting identical resonance frequency f₀ and radiation damping Γ_r. Since in reality it is still challenging to fabricate a free-standing sample with two resonators on two sides of a thin dielectric spacer, here we choose to design our bi-layer metasurface in such a configuration that one plasmonic resonator (an Au bar) is on top of a SiO₂ substrate while another Au bar is buried inside the substrate (see Fig. 4a). As the two resonators are now in different dielectric environments, they must possess different geometries in order to exhibit identical optical responses. Moreover, the presence of a dielectric substrate changes the background scattering matrix C and the coupling matrix d_ji in Eq. (1). Despite of these differences with the ideal case, we still analytically proved the following two conclusions for such metasurfaces (see Sec. 3 in SI): 1) as the Kerker condition (Eq. (4)) is met, such a metasurface in the lossless condition only exhibits the background reflectance within the entire frequency band below the first diffraction mode; 2) perfect absorption can still happen as the critical damping condition is met.

Fig. 4

Optical properties of two basic resonators and their NF and FF couplings. a A unit cell of the bi-layer metasurface containing one Au nano bar on top of a SiO₂ substrate and another Au nano bar buried inside the substrate at a depth h. Here the two resonators are of the same thickness (30 nm) and different lateral sizes, with their centers on different lateral positions on the xy-plane. b Spectra of reflectance/transmittance of the metasurface containing top resonators (with a₁ = 140 nm and b₁ = 340 nm) arranged in a hexagonal lattice with periodicity p = 600 nm, obtained by experiments (stars), simulations (circles) and CMT calculations (solid lines) with parameters computed by the LEM theory (f₁ = 203.32 THz, Γ_1r = 5.997 THz, Γ_1i = 6.140 THz). c Spectra of reflectance/ transmittance of the metasurface containing resonators (with a₂ = 320 nm and b₂ = 280 nm) buried inside the substrate at h = 236 nm arranged in a hexagonal lattice with periodicity p = 600 nm, obtained by experiments (stars), simulations (circles) and CMT calculations (solid lines) with parameters computed by the LEM theory (f₂ = 202.57 THz, Γ_2r = 4.985 THz, Γ_2i = 4.975 THz). White dashed lines and gray areas in b-c denote the frequencies and radiation damping of the resonant modes, calculated by LEM theory, and right panels in b-c are the SEM images of the fabricated samples with scale bars (white lines) of 500 nm. d LEM-calculated near-field coupling κ between two resonators versus their relative lateral configuration (d_x, d_y) with fixed vertical distance h = 236 nm. Points labeled with 1–4 represent the cases with α = 90^∘, 65^∘, 55^∘, 0^∘ on the circle $l=\sqrt{d_x^2+{d}_y^2}=345\ \textrm{nm}$. e LEM-calculated inter-resonator couplings (X and κ) as functions of h, with fixed α = 0^∘ and $l=\sqrt{d_x^2+{d}_y^2}=345\ \textrm{nm}$. Dashed lines labeled with #5–8 represent the cases with h = 140, 236, 450, 600 nm, respectively. The violet star represents the position of critical point with κ − Im(X) = 0 and θ_X = π/2 satisfied. Here, κ and X are scaled by $\sqrt{\Gamma_{1r}{\Gamma}_{2r}}$, and the SiO₂ substrate has a thickness of 500 μm in experiments and is treated as a semi-infinite medium in simulations

We fix the geometric structures of two basic resonators with the help of finite-element-method (FEM) simulations, and fabricate two single-layer metasurfaces according to our designs using the standard electron-beam lithography (EBL) method. As shown in the right panels of Fig. 4b-c, each metasurface contains resonators of one type arranged in a hexagonal lattice with periodicity 600 nm, while resonators in the second sample are buried inside the substrate at the depth h = 236 nm [53]. Illuminating two samples with y-polarized normally incident light, we experimentally measure their scattering spectra, with reference signals taken as those obtained with the sample replaced by a gold mirror (for reflection) and the quartz substrate (for transmission), respectively (Sec. 4 in SI). Figure 4b-c clearly show that the two systems exhibit nearly identical measured transmission/reflection spectra (stars), which are in good agreement with their corresponding FEM simulations (circles). We further employ a recently developed leaky-eigen-mode (LEM) theory [49] to directly compute the optical characteristics (i.e., f₀, Γ_r and Γ_i) of the resonance modes supported by two systems, based on their LEM wave-functions derived from Maxwell equations (see more details in Sec. 5 of SI). Put these LEM-computed parameters into the CMT, we obtain the reflection spectra of two metasurfaces (solid lines in Fig. 4b-c), which are in excellent agreement with simulation and experimental results.

With LEM wave-functions of two designed resonance modes known, we can employ them to directly compute the FF and NF coupling parameters (e.g., κ and X) between two resonators, which are arranged in different relative configurations in forming our bi-layer metasurfaces. Obviously, while the inter-layer distance h dictates the FF coupling, the lateral relative configuration between two resonators and h are collectively responsible for the NF coupling. Figure 4e depicts how the LEM-computed X and κ changes as a function of h with lateral positions of two bars fixed. We find both Re(X) and Im(X) vary periodically versus h, as expected, while κ decays as h increase since near field localize around the particles. In particular, we get Re(X) = 0 at h = 236 nm, which is very close to the prediction h = λ_n/4 (with λ_n = 985 nm being the resonance wavelength inside the dielectric substrate). The slight discrepancy is caused by the difference between the realistic structures and the ideal model. Fix the top bar at the unit-cell enter (0, 0) and put the second bar at (d_x, d_y) on the h = 236 nm plane, we employ the LEM-theory to calculate how κ varies against d_x and d_y, and depict the results as a color map in Fig. 4d. In particular, we find that κ can continuously change from a negative value to a positive one as the relative horizontal angle α between two bars varies from 0^∘ to 90^∘ (see the circle with radius $l=\sqrt{d_x^2+{d}_y^2}=345\ \textrm{nm}$ in Fig. 4d). These results suggest that we have enough tuning freedoms to design bi-layer metasurfaces exhibiting different κ and X, and in turn, different optical responses.

We choose 7 points on the phase diagrams (see Fig. 4d-e) to design the corresponding metasurfaces. Samples #1–4 exhibit identical inter-layer distance h and different relative orientation angle α (Fig. 4d), while samples #5–8 have the same value of α but with h changing from 140 to 600 nm (Fig. 4e). We note that samples #4 and #6 are the same, although they are on different variation paths. To check whether these samples meet the Kerker condition Eq. (4), we depict the positions of 7 samples on the ξ~{κ, θ_X} phase diagram (Fig. 5a). We see clearly that sample #4 (the green star) just locates at the ξ = 0 point meeting the Kerker condition Eq. (4). We note that surface roughness caused by the presence of nano-structures in the bottom layer does not modify the EM responses of our metasurface obviously (See Fig. S9 in Sec.6 of SI).

Fig. 5

Experimental characterizations on two series of metasurfaces with varying NF and FF couplings. a Calculated value of ξ = |X − iκ| (color map) versus κ and θ_X, with points labeled with #1–8 representing those shown in Fig. 4d-e. b-c LEM-calculated κ and θ_X for two series of samples. d Spectra of reflectance/transmittance of samples #1–4 with top-view SEM pictures (scale bars: 500 nm) shown in the right panel, obtained by experiments (stars), simulations (circles) and CMT calculations (solid lines) with parameters computed by the LEM theory, respectively. Here the bumps in d marked by blue dotted lines are SiO₂ bumps stemming from the deposition process corresponding to the location of the bottom resonators. e Spectra of reflectance/transmittance of samples #5–8 with side-view FIB-SEM pictures (scale bars: 500 nm) shown in the right panel, obtained by experiments (stars), simulations (circles) and CMT calculations (solid lines) with parameters computed by the LEM theory, respectively. Dotted lines and shaded areas in d-e represent the resonance frequencies and radiation damping of the hybrid modes calculated with the LEM theory, respectively, while the dotted-line rectangles in the sample images denote the positions of two bars. Here, κ and X are scaled by $\sqrt{\Gamma_{1r}{\Gamma}_{2r}}$

We fabricate these bi-layer samples with a two-step EBL process (see Sec. 7 in SI) and experimentally characterize their optical properties. The top-view scanning-electron-microscopy (SEM) pictures of samples #1–4 (see right panel in Fig. 5d) show that two resonators are in different relative lateral configurations, while the side-view Focus-Ion-Beam (FIB) - SEM pictures of samples #5–8 (see right panel in Fig. 5e) reveal that two resonators exhibit different vertical distances h, in consistency with Fig. 4d-e. Shine these samples with y-polarized normally incident light, we measure their reflection and transmission spectra (stars in Fig. 5d-e), which are in excellent agreement with FEM simulations (see circles in Fig. 5d-e).

We first discuss the sample series #1–4. We find from experimental results (Fig. 5d) that decreasing α mainly changes the frequency interval between two reflection peaks, but has negligible influences on their bandwidths. To understand the physics, we employ the LEM theory to compute (κ, θ_X) of samples #1–4. We find from Fig. 5b that these samples exhibit continuously varying κ and identical θ_X as α changes, which well explains the salient features revealed in the reflection spectra. Put the LEM-computed (κ, θ_X) into the CMT equations (see Sec. 3 in SI), we find that the CMT-calculated reflection spectra R(f) (solid lines in Fig. 5d) are in excellent agreement with both simulation and experimental results. Obviously, such a line-shape evolution is governed by essentially the same physics as that discussed in Fig. 2b for the model systems.

We next discuss the sample series #5–8. We find from the measured spectra (Fig. 5e) that changing h in this series modifies not only the frequency positions but also the bandwidths of two modes. These features can be well explained by the LEM-calculated (κ, θ_X) as varying h (Fig. 5c). Again, put the LEM-computed values (κ, θ_X) in to the CMT equations, we find that the CMT-computed reflection spectra R(f) (solid lines in Fig. 5e) are in good agreement with both numerical and measured results. Slight discrepancies between experimental and simulation results can be attributed to differences in material parameters in the double-layer metasurface and their single-layer counterparts, caused by fabrication imperfections. In particular, the calculated positions and bandwidths of two hybridized modes, labeled by the dashed lines and shaded areas in Fig. 5e, respectively, re-enforce our notion that changing h affects both the frequencies and bandwidths of the hybridized modes through modifying both NF and FF couplings between two resonators.

Achromatic reflectionless metasurface

We now focus on the sample #4 (or #6), corresponding to the case of ξ = 0 in Fig. 5a. Experimental results clearly show that the reflectance of this sample nearly maintains at the background value within the whole experimentally accessible frequency range (160 to 220 THz) except at the resonance frequency. In fact, simulation results indicate that the sample maintains at the background value in a frequency range (0–225THz) far beyond that experimentally accessible (Fig. 6b). We note that diffractions inevitably appear at frequencies above 225 THz, since the realistic metasurface is a periodic structure. We note that our metasurface is designed under normal incidence, and its achromatic reflectionless property is maintained only as the incident angle lies in a narrow range centered at 0°, which can be enlarged by structural optimizations. Achromatic reflectionless metasurfaces under oblique incidence are also designable, as long as the system still works below the first-order diffraction (see more details in Sec. 8 of SI).

Fig. 6

Experimental/numerical characterizations on the achromatic reflectionless metasurface with tunable loss. a Top-view SEM image of sample #4 (scale bar: 1 μm) with top-left inset depicting a zoom-in top-view SEM picture (scale bar: 500 nm) and top-right inset depicting a side-view FIB-SEM picture (scale bar: 500 nm), respectively. b Spectra of reflectance and absorbance of sample #4 within the wide frequency range of 0–225 THz, obtained by experiments (stars), simulations (circles) and CMT calculations (solid lines) with parameters computed by the LEM theory, respectively. c-e FEM-simulated spectra of c reflectance/transmittance, d absorbance and e transmission phase for the sample #4, assuming that its absorption damping parameters ${\gamma}_i^{(m)}$ (m = 1,2) take different values (The dotted/ solid/ dashed lines denote cases with ${\gamma}_i^{(m)}=0.1{\gamma}_0^{(m)}$, ${\gamma}_i^{(m)}={\gamma}_0^{(m)}$, and ${\gamma}_i^{(m)}=3{\gamma}_0^{(m)}$, respectively). Here, ${\gamma}_0^{(1)}=3.182\times {10}^{16}{s}^{-1}$ and ${\gamma}_0^{(2)}=3.672\times {10}^{16}{s}^{-1}$ denote the absorption damping parameters adopted in our FEM simulations corresponding to the experimentally realized sample, obtained by fitting with the measured spectra in Fig. 4

We next study the absorption properties of the fabricated sample. Computing the absorption spectrum A(f) using A = 1 − R − T, we find that the experimentally measured absorbance A reaches 96% at the resonance frequency 203.1 THz (Fig. 6b) at which the transmittance T approaches zero. At frequencies away from the resonance one, A diminishes and T increases but A + T = 1 maintains approximately. To understand the intrinsic physics, we employ the LEM theory to calculate the intrinsic damping parameters of two resonance modes (see Sec. 5 of SI), which confirms that the critical damping condition (Γ_i = Γ_r) is indeed approximately satisfied for present system which already satisfies the Kerker condition Eq. (4).

Moreover, we find that tuning the intrinsic loss in this sample can further modulate the ratio between light transmission and absorption, yet keeping reflection diminished. To demonstrate this point, we perform a series of FEM simulations assuming that the damping parameter ${\gamma}_i^{(m)}\left(m=1,2\right)$ in the Drude model ($\varepsilon \left(\omega \right)={\varepsilon}_{\infty }-{\omega}_p^2/\left({\omega}^2+i{\omega \gamma}_i^{(m)}\right)$) describing our plasmonic metals forming the m-th resonator (m = 1, 2) take different values. Since the two metallic resonators are in different dielectric environments, it is quite natural to expect that ${\gamma}_i^{(1)}\ne {\gamma}_i^{(2)}$. As shown in Fig. 6d, tuning ${\gamma}_i^{(m)}$ from $0.1{\gamma}_0^{(m)}$ to $3{\gamma}_0^{(m)}$ with ${\gamma}_0^{(m)}$ representing the experimental case, we find that the peak absorbance A first increase from 36% to 100% and then decreases, in consistency with the under-damping to over-damping transition discussed in Sec. 2 (see Fig. 3). Meanwhile, the reflectance keeps negligible in all these cases studied (black lines in Fig. 6c), which is again consistent with the notion that the reflection channel is blocked under the Kerker condition (Eq. (4)). Finally, transmission phase spectra of the system with different ${\gamma}_i^{(m)}$ (Fig. 6e) also exhibit the under-damping to over-damping transition, in consistency with Fig. 3d. These results suggest the possibilities to realize transmission-mode tunable meta-devices for phase modulations and wave-front controls, through controlling the intrinsic damping of constitutional materials via electric or optical means.

Conclusion

To summarize, we employ CMT analyses to rigorously demonstrate that an optical metasurface formed by two arrays of resonators can be perfectly reflectionless at all frequencies below the first diffraction mode, when the NF and FF couplings between two constitutional resonators satisfy certain conditions. Tuning the intrinsic loss of the system can further modulate the ratio between light transmission and absorption, yet keeping reflection diminished strictly. We design/fabricate a series of metasurfaces and experimentally illustrate how the reflection line-shape of such metasurface is tailored by inter-resonator NF and FF couplings. In particular, we identify a specific metasurface from the sample series and experimentally demonstrate that it is immune from reflections within an ultra-wide frequency range (experiment: 160–220 THz; simulations: 0–225 THz), yet behaving as a perfect absorber at ~ 203 THz.

Many future works can be stimulated from the present study. For example, realizing such reflectionless meta-devices can find immediate applications in sensing, cloaking, and energy harvesting. Moreover, modulating transmission phase of light via tuning the intrinsic loss in our meta-device (Fig. 6) can inspire tunable transmissive meta-devices for high-efficiency wave-front controls. Finally, realizing free-standing samples exhibiting background-free achromatic reflections in different frequency regimes are challenging and interesting future projects.

Methods

Simulations

All FEM simulations are performed with the commercial software COMSOL Multiphysics. Permittivity of Au is described by the Drude model $\varepsilon \left(\omega \right)={\varepsilon}_{\infty }-{\omega}_p^2/\left({\omega}^2+i{\omega \gamma}_i^{(m)}\right)$ with ε_∞ = 9 and ω_p = 1.367 × 10¹⁶s⁻¹. The damping parameter is set as ${\gamma}_0^{(1)}=3.182\times {10}^{16}{s}^{-1}$ for top resonators and as ${\gamma}_0^{(2)}=3.672\times {10}^{16}{s}^{-1}$ for the resonators buried inside the substrate, obtained by fitting with our experimentally measured optical spectra for corresponding metasurfaces (Fig. 4). The SiO₂ substrate was considered as a lossless dielectric with permittivity ${\varepsilon}_{{\textrm{SiO}}_2}=2.25$. We note that additional losses caused by surface roughness, grain boundary effects, existence of adhesion layer as well as dielectric losses, have been effectively considered in choosing the values of two parameters ${\gamma}_0^{(m)}\left(m=1,2\right)$ describing the damping rates of Au forming two resonators.

Fabrications

All bi-layer samples were fabricated using two-step EBL and lift-off processes. First, the positive resist MMA EL6 (200 nm) and PMMA A2 (80 nm) were successively spin coated on a SiO₂ substrate. Next, bottom bars (270 nm × 320 nm), 4 global alignment marks (100 μm × 10 μm) and 4 chief alignment marks (25 μm × 2 μm) were lithographed with EBL (JEOL 8100) at an acceleration voltage of 100 kV. After exposure, the samples were developed in the solution with a 3:1 mixture of isopropanol (IPA) and methyl isobutyl ketone (MIBK). 3 nm - thick Cr and 30 nm – thick Au layers were subsequently deposited using electron-beam evaporation. After standard lift-off process, SiO₂ with a desired thickness was deposited on the first metal array as a dielectric interlayer by magnetron sputtering. The top layer was fabricated using the same method but including a precise alignment process, where the gold alignment marks are applied to ensure the accurate stacking of the top layer. Finally, the reference area near the samples was prepared with a more alignment by direct writer, followed by depositing 5 nm – thick Cr and 150 nm – thick Au layers with magnetron sputtering and performing lift-off process. The top-view pictures of fabricated samples were obtained using SEM (Zeiss Sigma). Note that the bumps in Figs. 4c and 5d are SiO₂ bumps stemming from the deposition process corresponding to the location of the bottom Au bars. The side-view images of the samples were obtained using a dual-beam FIB-SEM which can simultaneously obtain the local sectioning (with the FIB) and imaging (with SEM) of the samples. All samples have lateral dimensions of 100 μm × 100 μm.

Optical characterizations

We used a homemade NIR microimaging system equipped with a broadband supercontinuum white light source (Fianium SC400), polarizers, a beam splitter, a CCD, a fiber-coupled grating spectrometer (Ideaoptics NIR2500) and a Princeton Instruments HRS-300 spectrograph with an InGaAs camera to characterize the optical properties of the fabricated samples.