Metamaterial Perfect Absorber Analyzed by a Meta-cavity Model Consisting of Multilayer Metasurfaces

We demonstrate that the metamaterial perfect absorber behaves as a meta-cavity bounded between a resonant metasurface and a metallic thin-film reflector. The perfect absorption is achieved by the Fabry-Perot cavity resonance via multiple reflections between the “quasi-open” boundary of resonator and the “close” boundary of reflector. The characteristic features including angle independence, ultra-thin thickness and strong field localization can be well explained by this meta-cavity model. With this model, metamaterial perfect absorber can be redefined as a meta-cavity exhibiting high Q-factor, strong field enhancement and extremely high photonic density of states, thereby promising novel applications for high performance sensor, infrared photodetector and cavity quantum electrodynamics devices.

The metamaterial perfect absorber (MPA), exhibiting nearly unity absorption within a narrow frequency range [1][2][3][4][5][6] , has attracted tremendous interest recently. The MPA possesses characteristic features of angular-independence, high Q-factor and strong field localization that have inspired a wide range of applications including electromagnetic (EM) wave absorption 3,7,8 , spectral 5 and spatial 6 modulation of light, selective thermal emission 9 , thermal detection 10 , infrared photo detection [11][12][13][14] , circularly polarized light detection [15][16][17] , and refractive index sensing for gas 18 and liquid [19][20][21] targets. The MPA typically consists of three layers: an array of metallic resonators (e.g., cross-type resonators 9,22 , split-ring resonators 3 or metallic nanoparticles 23 and chiral resonators 15,24 ) and a highly reflective layer (e.g. metallic film 3,9,22,23 or metallic mesh grid 3 ), separated by a subwavelength-thick dielectric film (spacer). The mechanism of MPA has been explained through a number of models including impedance matching 3,7,25 and destructive interference 26 . In the impedance matching mechanism, the entire three-layered structure is considered as a thin slab made from a homogeneous medium with frequency-dependent effective permittivity ∈ eff (ω) and effective permeability ( )  Meanwhile, the effective refractive index ω n ( ) eff at ω ω = 0 exhibits a large imaginary part,  ω ω ″ ′ n n ( ) ( ) eff 0 eff 0 , thus incident EM wave enters MPA without any reflection and then rapidly decays to zero inside the MPA. In the destructive interference mechanism, the resonator and the metallic film are considered as two decoupled surfaces, and destructive interferences of waves reflected multiple times by two surfaces lead to zero reflection 26,27 .
In this work, we use a transfer matrix method to assemble the MPA from three functional layers and obtain the phase and the amplitude conditions for the perfect absorption. Analyzing these conditions, we reinterpret the MPA structure as a meta-cavity bounded between a "quasi-open" boundary of a resonant metasurface and a "close" boundary of a metallic film. The Fabry-Perot (FP) modes are achieved when the phase condition is satisfied, which leads to a strong resonance with non-perfect absorption (NPA). However, the perfect absorption (PA) only occurs when both the phase and the amplitude conditions are satisfied simultaneously. Although PA has drawn significant attention, NPA are more commonly used in practical applications such as sensors [18][19][20][21]28 and detectors [10][11][12][13][14] . Both PA and NPA can be well explained by the meta-cavity model. With an improved retrieval method, we find that the resonant metasurface operates at off-resonance wavelengths, thereby acting as a homogenous thin film with high dielectric constant. The FP model redefines MPA as a meta-cavity with high Q-factor and extremely high photonic density of states (PDOS) where the FP modes are easily tailored by the geometric parameters of metasurface. Our work can pave the way for novel photonic, optoelectronic and cavity quantum electrodynamic (QED) applications 29 .

Results
Single-layer effective medium model. Our perfect absorber consists of three layers: an array of crosswire resonators, a dielectric spacer and a metal ground plane (MGP) as shown in Fig. 1a. We carry out a 3D fullwave simulation to solve Maxwell's equations and obtain numerical solutions by Computer Simulation Technology Microwave Studio that uses a finite integration technology 30 . The parameters used for simulation are as follows: period of cross-wire array, µ = p m 2 , length and width of the cross-wire, l m 1 7 µ = .

56
GaAs and ε = . , and thickness of cross-wire and MGP, . The gold used for cross-wire and MGP is described by the Drude model 31 with plasma frequency, ω = . × rad s 1 37 10 / p 16 and collision frequency rad s 4 08 10 / c 13 ω = . × . We use three different models to describe the perfect absorber: (i) A single-layer effective medium model where the entire structure ( Fig. 1a) is considered as a layer of homogeneous medium ( Fig. 1d) characterized by the effective permittivity  ( ) eff and effective permeability µ ( ) eff ; (ii) A three-layer effective medium model where the cross-wire resonator (Fig. 1b) is considered as a homogeneous effective film ( Fig. 1e) with  eff and eff µ on top of two layers of real films, the spacer and MGP; (iii) A transmission line model where the multiple-layer optical system (Fig. 1c) is modeled as a two-port network ( Fig. 1f) with input impedance z in and output impedance z out .
Since the MGP prevents the EM wave from propagating through, the perfect absorption is obtained when the reflection reaches zero at certain wavelength as shown in Fig. 2d. Among three models shown in Fig. 1, the single-layer effective medium model is widely used in various MPA works 1,3,7,25 , where the electric and the magnetic resonances lead to the impedance matching condition, , at the perfect absorption wavelength. Therefore, the incident EM wave propagates through the top surface of effective film in Fig. 1d without any reflection. Meanwhile, the effective film exhibits large imaginary effective refractive index,  n n eff e ff ″ ′ , so that the EM wave decays exponentially to zero before it exits from the bottom surface. To verify this interpretation, we use effective medium theory 32 to retrieve the effective permittivity ( eff  ) and effective permeability ( eff µ ) from the complex transmission and reflection coefficients, t and r. Due to the asymmetric structure of MPA along the propagation direction of incident light, the perfect absorption only occurs for incident illumination from the front-side (cross-wire) and the EM wave incident from the back-side (MGP) will be totally reflected. Strictly speaking, such extremely asymmetric propagation in MPA cannot be described by waves traveling in a simple medium with homogeneous permittivity and permeability. However, since we only consider the interaction of MPA with incident light from the front-side (cross-wire), we can use t and r for impinging light from the air-side to obtain eff  and µ eff as shown in Fig. 2a,b, respectively. We find a Lorentzian type resonance of  eff at wavelength of 5.13 μm, which can be ascribed to the first-order dipole resonance mode of the cross-wire driven by the electric Figure 1. Effective models for MPA. Schematic diagrams of (a) metamaterial perfect absorber (MPA) and its (d) equivalent single-layer film characterized by effective eff  and eff µ ; (b) perfect absorber with cross-wire, spacer and metal ground plate considered as a three-layer structure and (e) equivalent three-layer film model where the cross-wire is considered by a homogeneous film with  eff and µ eff . (c) View of perfect absorber along wave propagation direction. The MPA is surrounded by air and a dielectric substrate with optical impedances z s1 and z s2 at two interfaces, respectively. (f) A transmission line model uses a two-port network S to describe the MPA, where z 1 and z 2 are impedances of air and substrate, and z in and z out are input and output impedances of S, respectively.
field of the incident wave. For wavelength m 6 , λ µ > Re( ) eff  is negative. This Drude type response is attributed to the MGP, which exhibits the typical characteristic of permittivity of bulk metal, but with much lower plasma frequency due to diluted average electron density resulting from low volume-ratio of MGP inside the MPA unit cell 33 . The effective permittivity eff µ also shows a Lorentzian type resonance at wavelength of 5.94 μm driven by the magnetic field. This magnetic resonance is induced by the anti-parallel currents on the cross-wire and the MGP, and can be qualitatively explained by the method of image theory 7 . The solid lines in Fig. 2c are the real (red color) and the imaginary (blue color) parts of the effective impedance z eff , which show strong dispersive response as a function of wavelength.
At the perfect absorption wavelength λ = 5.94 μm ( = . + . i 2 432 24 175 eff  and µ = . + . i 1 003 25 517 eff ), the effective impedance is found to be z i 1 02 0 03 eff = . + . , which matches with the impedance of air ( has a large imaginary part (i.e., ″ = . n 24 849). These calculated effective constitutive parameters very well support the single-layer effective medium model. Specifically, the matched impedance allows the incident wave passing through the top surface of the effective thin film (Fig. 1d) without any reflection. Meanwhile, the intensity of the wave is I I e , + . and = . + . n i 0 338 7 49 eff . To the best of our knowledge, this second absorption peak (λ = 1.79 μm) has not been thoroughly studied in the past. As will be shown later, it results from the second-order dipole resonance of the cross-wire. The effective impedance of the MPA can also be obtained by the effective circuit model 34 . Figure 1f shows an effective circuit for the MPA, where the MPA is described as a two-port element S in connection with loads z 1 (air) and z 2 (substrate). We calculated the input impedance z in based on transmission line theory and plotted it as triangle (real part) and circle (imaginary part) symbols in Fig. 2c. Although z in was calculated with complete different methodology (see details in Methods section), it shows excellent agreement with z eff obtained from single-layer thin film model. Using the retrieved effective permittivity and permeability ω µ ω  ( ) and ( ) eff eff , we performed full-wave simulation for the single-layer effective thin film (Fig. 1d). Figure 2d shows that the simulated reflection (green-circle) completely overlaps the reflection from the original absorber structure (blue-solid line) as expected. The single layer effective medium model can provide fundamental insight for MPA (e.g., reflection and transmission), however this model cannot provide the microscopic mechanism of the absorption, cannot very well explain NPA and can miss some important features such as ultra-thin thickness.
Three-layer effective medium model. To better understand the underlying mechanism that the wave travels and decays inside the MPA, we have studied the transmission and reflection of waves at each constituting layers and used a transfer matrix method to obtain the overall reflection/absorption properties of MPA. We considered the cross-wire as a metasurface and described it as a homogeneous thin film with effective permittivity  eff and effective permeability eff µ . The effective parameters were calculated using transmission and reflection coefficients of an air/cross-wire/spacer configuration through an improved retrieval method to take account of the asymmetric structure due to different incoming and outgoing medium (air and spacer 35 ). As shown in Fig. 3a, the effective permittivity of cross-wire shows a Lorentzian type electric resonance at wavelength of λ µ = . m 4 75 r , which is shorter than the electric resonance obtained by the single-layer model ( µ . m 5 13 ). The resonance at µ . m 4 75 is purely electric since the effective permeability (Fig. 3b) takes a constant value of Re( ) eff µ = 0.8 except a typical anti-resonance at λ r due to the periodicity effect 36 , we found the second-order electric dipole resonance mode of the cross-wire, which is much weaker and has negligible influence on the value of  eff . With those values of eff  and eff µ , we used a thin film of the same thickness to replace the cross-wire in the absorber structure. 3D Full-wave simulation of the three-layer thin film model (Fig. 1e) was carried out. The simulated reflection (red-solid curve) is plotted in Fig. 2d, which matches very well with simulation of actual MPA structure (blue-solid) at the first absorption wavelength ( m 5 94 λ µ = .
) and is slightly off at the second absorption wavelength ( m 1 79 λ µ = . ). We ascribe the small discrepancy at λ µ = . . − . , respectively. The role of the cross-wire can be completely understood as a thin film with these  eff and eff µ . In spite of minor discrepancy at short wavelengths, the excellent agreement between the three-layer thin film model and actual MPA proves that the cross-wire layer and the MGP layer are decoupled, i.e. neither the existence of MGP will affect the resonance of cross-wire nor the resonant current on the cross-wire will affect MGP where the latter only plays a role of a reflecting mirror. Therefore, the transfer matrix method as a common approach to study multiple-layered structure can be used to obtain the overall property of MPA from each layer and to investigate the absorption mechanism of EM waves inside the MPA. The overall reflection of the three-layer structure can be calculated by multiplying the transfer matrix of each layer, = ⋅ ⋅ M M M M 1 2 3 , as given below (more details can be found in our previous works 27,35 ). In Eq. 1, r 12 and r 21 are the reflection coefficients of the cross-wire from front (air)-and back (spacer)-side, respectively. r 23 is the reflection coefficient of the MGP. n k t s s β = ⋅ ⋅ is the propagating phase in the dielectric layer, where n s , t s and k are the refractive index, the spacer thickness and the wave vector in free space, respectively. α = − t t r r 21 12 21 12 , where t 12 and t 21 are the transmission coefficients through the cross-wire along forward (air/ cross-wire/spacer) and backward (spacer/cross-wire/air) directions, respectively. Although α is strictly equal to 1 at the interface between two homogenous media, α ≠ 1 for the cross-wire around the resonance wavelength. This is caused by the structural asymmetry depending on the propagation direction of incident light. For perfect absorption (i.e., r 0 = ) according to Eq. 1, we obtain the following conditions for the amplitude and the phase, respectively: The coefficients r 12 , r 21 , t 21 and t 12 are obtained from simulation of the air/cross-wire/spacer configuration and r 23 is obtained from the simulation of the spacer/MGP configuration. Around the perfect absorption wavelength λ = 5.94 μm, both the amplitude condition in Eq. 2 ( r r ) 12 23 α = ⋅ and the phase condition θ π = ( ) are satisfied simultaneously as shown in Fig. 3c and d, respectively. For wavelengths below 5.94 μm, θ decreases as the wavelength decreases and finally reaches −π at λ = 1.79 μm. Meanwhile, the amplitude condition is also satisfied at λ = 1.79 μm, thereby leading to the second absorption peak. By carefully examining Fig. 3d, we find that the amplitude condition ( r r 12 23 α = ⋅ at λ = 5.61 μm) and the phase condition θ π = ( at λ = 5.88 μm) are not satisfied at exact same wavelength. This explains the fact that the ref lection shown in Fig. 2d does not reach zero completely ( = = . × − r R min( ) 3 68 10 min 2 4 at λ = 5.94 μm). The phase condition in Eq. 3 suggests a Fabry-Perot cavity-like mechanism. The cross-wire acts as a "mirror" that reflects light due to its high effective permittivity and forms a cavity together with the MGP. The incident wave enters the MPA and is reflected multiple times by MGP and cross-wire. The FP cavity resonances establish when the round-trip phase is fulfilled with the following condition: where r ( ) 21 φ and r ( ) 23 φ are the phases of reflection coefficients at the interfaces of the spacer/cross-wire and the spacer/MGP (Fig. 3d), respectively. When γ equals to m 2 π, the waves that travel multiple round-trips between cross-wire and MGP are in phase and hence interfere constructively, thereby leading to resonant cavity modes. As the wavelength decreases, γ (blue curve in Fig. 3d) increases monotonically and reaches the values of 0 and 2π at wavelength of 5.94 μm and 1.82 μm, respectively (matching very well with two perfect absorptions wavelengths). As shown in Fig. 3d, the main contribution to γ comes from φ r ( ) 21 , which increases from π . 0 89 to π . 2 63 as the wavelength decreases from 5.94 μm to 1.82 μm. The significant change of φ r ( ) 21 is caused by  eff , which changes from i 64 72 7 08 . + . to . + . i 4 19 1 28 in the same wavelength range as shown in Fig. 3a. Note that the phase condition of Eq. 4 only guarantees the FP cavity modes, resulting in the absorption peaks. To achieve perfect absorption, the amplitude condition in Eq. 2 needs to be satisfied at the same wavelength as well. Nevertheless, Eqs 3 and 4 well explain the NPA peaks of a variety of resonator/spacer/MGP structures that have been widely used in sensing and detecting applications 10-14, 18-21, 28 where the absorption peaks do not necessarily reach 100% (i.e., R = 0). The ultra-thin thickness of MPA can be also explained by the cavity model. As shown in Fig. 3d, at the first perfect absorption wavelength λ = 5.94 μm where γ = 0, the phase r ( ) 23 φ π ≈ − and φ r ( ) 21 is slightly less than π, i.e., φ π δ = − r ( ) 21 with δ π  . To achieve 0 γ = , it is required β δ ≈ /2, thus the spacer thickness t s is found to be . In our simulations, we obtained δ π = .
0 11 and thus t m 0 0275 0 108 , which is close to the actual thickness being the wavelength inside the spacer medium. This thickness is extremely thin in comparison to a regular FP cavity with size of λ /2 s . Note that the key fact that leads to this ultra-small size of cavity is r ( ) 21 φ being slighly smaller than π due to high effective permittivity of the cross-wire layer.
Another important characteristic feature of MPA is the angular independence of the incident wave. This is very useful in thermal imaging/sensing and energy harvesting applications to receive off-normal incident lights. We performed simulations of actual MPA structure, single-layer and three-layer thin film model with the polar angle θ (in spherical coordinates as displayed in the inset of Fig. 4) of incident wave varying from °0 to 80°. The reflection spectra are shown in Fig. 4a-c for TE wave and Fig. 4a-f for TM wave, respectively. As shown in Fig. 4a, the wavelength of absorption peak at λ = 5.94 μm for TE wave does not shift as the incident angle increases from °0 to °80 , showing good angular independence. Reflections of both the single-layer (Fig. 4b) and the three-layer (Fig. 4c) models well match with that of actual MPA structure (Fig. 4a). For TM wave, the reflection of the MPA is blue-shifted as θ is increased by ~4% as shown in Fig. 4d. Both the single-layer and the three-layer models also show the blue-shift with ∆ λ =7.4% and 9.1%, respectively, which is slightly higher than actual MPA. The three-layer model can qualitatively explain the angular dependence for TE and TM modes. The cross-wire layer exhibits an electric resonance excited by the electric field of incident wave. For TE mode, the electric field (See the inset of Fig. 4a) is always parallel to the cross-wire in spite of changing the polar angle θ. Thus, the resonance of cross-wire does not change with incident angle. This leads to the angular independence of absorption for the entire MPA structure. For TM mode, however, the electric field direction changes with θ (see the schematic diagram as shown in the inset of Fig. 4d), resulting in shifting the wavelength of absorption peak. At short wavelengths (λ < p), incident waves become significantly diffracted through the cross-wires and lead to shifting and splitting of the second absorption peak at λ=1.79 μm as indicated in Fig. 4a and d. These diffractions do not exist for two effective film models as shown in Fig. 4b,c,e and f. In addition, we observed new absorption peaks at around 2.75 μm and 4.75 μm (Fig. 4d) for TM wave, which becomes more significant for θ >°50 . These absorption peaks (reflection dips) result from resonance modes of cross-wires and MGP induced by the vertical component of electric field (E z ), which is nonzero for TM wave and is increased with θ.
Higher order FP modes. Equations 3 and 4 indicate that the phase conditions are satisfied when β increases by an integer multiple of 2π, thereby leading to a series of FP cavity modes. The increment of π 2 in β corresponds to an increase in the thickness of spacer (t s ) by /2 s λ . For the first absorption peak at λ = 5.94 μm shown in Fig. 2d, the increment of thickness is calculated as m /2 1 97 and the corresponding n-th order thickness is given is the thickness of the original MPA. Figure 5a shows the reflection spectra for the three lowest order thickness, µ = . for 1 st order thickness). We attribute the minor discrepancy in wavelengths to the slight changes of cross-wire resonances due to different separation between cross-wire and MGP. Figure 5b shows the reflections of three-layer thin film model using the same thickness. The results very well match the reflections of actual MPA (Fig. 5a). We recognize these resonances as the three lowest order FP cavity modes, which is confirmed by observing the x-component of electric field E x inside the MPA structure. The right-side figures in Fig. 5c-e show the real part of E x along z-direction inside the spacer at the wavelength of absorption peaks, where E x is evaluated along a straight line in z-direction below the center of cross-wire (from z 0 = , the bottom of cross-wire to t z sn = , the spacer thickness). For the first-order thickness t m 0 09 s1 µ = .
, E Re( ) x decreases monotonically from positive value to zero as z increases from 0 (at the cross-wire/spacer interface) to µ . m 0 09 (at the spacer/MGP interface). Such E x distribution results from the boundary conditions. At the spacer/MGP interface, the high conductivity of MGP forms a "close" boundary where the tangential electric field is forced to be zero (E 0 t x , = ). At the cross-wire/spacer interface, the high effective permittivity ( = . ⋅ + . , a half-period of sine curve is added to E x field for the first order thickness (E x as shown on the right side of Fig. 5c) as shown in Fig. 5d, which results in forming the second-order FP cavity mode. Similarly, E x for µ = . t m 4 03 s3 (Fig. 5e) shows the fundamental mode plus a full-period of sine curve, leading to the third-order FP cavity mode. The three-layer thin film model gives the same E x profiles as shown in Fig. 5f-h. All the E x fields for actual MPA (Fig. 5c-e) and for effective thin film model (Fig. 5f-  horizontal direction with maximum at the center and zero at ends. This reveals a first-order dipole resonance mode of the cross-wires and matches with the Lorentzian resonance of eff ε shown in Fig. 3a. Figure 5c shows the current on the MGP flowing toward the opposite direction with the maximum current in the center region. The anti-parallel currents on the cross-wire and MGP in Fig. 5c induce a strong magnetic moment that is responsible for the magnetic resonance for single-layer thin film model shown in Fig. 2b. We also observe the same anti-parallel currents in the simulation for the three-layer thin film model as shown in Fig. 5f. Such current distributions can be well explained by the E x profile shown in Fig. 5f. The tangential electric field E t x , induces the electric current, . For the first-order cavity shown in Fig. 5c,f, E x in metafilm and MGP are in phase, and thereby result in anti-parallel currents. For the higher order cavities (e.g. Figure 5g,h for the second-and third-order cavities, respectively), when t s is increased by /2 s λ , the phase of E x in MGP increases by π and the current in MPG changes to the opposite direction.
We expect the FP cavity model for MPA to lead to novel applications. As one example, the MPA provides much stronger local electromagnetic fields enhancement than typical plasmonic resonators due to its large Q-factor. Such local field enhancement is always demanded to improve the performance of sensor/detector at infrared and THz regimes 11, 21, 37 . However, the region between resonator and MGP in the MPA, where the fields are mostly enhanced, are too thin to fit most sensing and detector devices. With the cavity model, we can design ( . Top views of current density on the cross-wire and MPG for (c) 1 st order, (d) 2 nd order and (e) 3 rd order MPA. Insets placed on the right side of (c)-(e) show the x-component of electric field E ( ) x along z-direction (propagation direction). Current density on the meta-film representing cross-wire and the MGP for (f) 1 st order, (g) 2 nd order and (h) 3 rd order. Insets on the right side of (f)-(h) display E x along z-direction for the three-layer thin film structures.
the higher order cavities that can fit in the required thickness of practical devices. For instance, infrared focal plane array device working at 6 μm wavelength 37 , is composed of an active layer (20 stacks of InGaAs/AlGaAs quantum wells) and top/bottom contact layers (n-doped GaAs). The total thickness is about 2 μm which is much larger than the first-order MPA thickness of 0.09 μm, but is very close to the second-order thickness of 2.06 μm. Furthermore, this thickness is tunable by varying the size of the resonator (cross-wire) to fit the requirement of existing devices. Another potential application is to use the high photonic density of states (PDOS) for cavity quantum electrodynamic (QED) applications 29,38,39 . PDOS is inversely proportional to the volume of a cavity. The first-order cavity of MPA has thickness of ~λ/44, much smaller than typical FP cavity (λ/2). This size is also much smaller than typical plasmonic resonators 29 which EM fields normally extend to distance of a few wavelengths. Our calculation shows the MPA cavity can improve the PDOS by ~80 times as compared to plasmonic resonators. Such high PDOS promises great potential in cavity QED applications 29,38,39 .

Discussion
In summary, we developed a multi-layered metasurface model for MPA. With a transfer matrix analysis, we found a FP cavity mechanism that models the MPA as a meta-cavity bounded between the "qusi-open" boundary of resonator and the "close" boundary of MGP. The FP cavity model well explains the characteristic features including ultra-thin thickness and angular independence. We also found the higher order FP cavity modes when the thickness of MPA (spacer) increases by multiples of a half-wavelength. The strong field enhancement of cavity resonances and high PDOS promise novel applications in sensor/detector and cavity QED device.

Methods
The effective impedance z in of the circuit model is calculated from the reflection coefficients Γ in and Γ out at the the input and output termial of S, using the following equations 34  For MPA, we only consider the input impedance z in as shown in Fig. 1f for the incident wave from the front side.