1. Introduction

Circuit elements, [e.g., resistors (R), inductors (L), and capacitors (C)] can be effectively and flexibly used to design complex micro-electronic devices and circuitries for information processing, stealth and communication operating in radio and microwave frequency domains [1–3]. Their effective and flexible usage to optical domain is eagerly demanding. Introducing new paradigms and feasible methods to bring more circuit functionalities into the optical domain would represent an important advance in nano-electronics technology.

Engheta et al. first proposed that plasmonic or nonplasmonic nanostructures, when properly designed and judiciously arranged, could behave as nanoscale lumped circuit elements for the terahertz, infrared (IR) and visible wavelength regions [4, 5]. This new concept of metamaterial-inspired optical nanocircuitry, dubbed “metatronics”, has been applied to various wavelength regimes which are supported by a series of theoretical analysis and numerical simulations [6–9]. With the birth of “metatronics”, interests in combining optical guiding devices, as optical interconnects, with classical circuits is always high [10], since it is still impossible to perform all classic circuit operations in the optical domain.

In this letter, the first try to combine the transmission line theory in microwave domain with lumped nanocircuit concepts to design a super broadband absorber device is successfully demonstrated. Metamaterial absorbers are used broadly in thermal detectors [11], imaging [12], security detection [13] and stealth devices [14]. After extracting the equivalent circuit of the absorber, the absorption spectrum can be effectively and quickly evaluated through our theoretical method. The feasibility is confirmed by the consistency with the FDTD calculation. Unlike conventional absorbers, the highly efficient absorption of incident light was explained by the destructive interference between reflected waves from the electrical ring resonator (ERR) and in addition, the metallic substrate, our proposed structure has no metallic substrate but a dielectric substrate instead. The incident light beyond the absorption band can be transmitted and almost there is no light reflected. Therefore, an optical window in NIR region, beyond a pure absorber, could be realized as well. Furthermore, such a broadband absorber possesses the angle-insensitive and polarization-independent properties. All of the above characteristics make such an absorber have promising prospects in applications.

2. Design and methods

The unit cell structure of the proposed truncated pyramid absorber is presented in Fig. 1(a). We adopt a square lattice for our sample in order to acquire dual polarization working performance. The lateral lattice constant p is chosen to be 350nm, which is much smaller than the wavelength discussed here. Each unit cell consists of four layers of 150-nm-thick (h) nano-square (NS) patch from bottom to top from bottom to top with side width (w) being 300, 250, 200, and 150nm, respectively. In our prototype, the material used for NS is indium-tin-oxide (ITO), and the permittivity is modeled using the Drude dispersion theory [15]: ${\epsilon }_{\text{ITO}}={\epsilon }_{\infty }-{\omega }_p^2/({\omega }^2+i{\omega }_{\tau }\omega )$with${\epsilon }_{\infty }=3.91$,${\omega }_p=2.65\times {10}^{15}rad/s$and ${\omega }_{\tau }=2.05\times {10}^{14}rad/s$, where${\omega }_p$ is the bulk plasma frequency and ${\omega }_{\tau }$is the relaxation rate. All the layers are separated by a SiO₂ dielectric layer with a fixed distance D = 600nm, as well as the bottom substrate. In the analysis, a TM plane wave source is used to illuminate this nano structure from the top side vertically (along k direction), and the electric field is parallel to the x direction.

 

Fig. 1 The schematic (a) of the proposed absorber, equivalent lumped circuit elements (b), and equivalent circuit (c) of the single NS layer.

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To establish our equivalent circuit model for this whole device, the first step is to extract a single layer of NS patch, a free standing 200nm width square patch suspended in air for example, and to present its equivalent nanocircuit element, as shown in Fig. 1(b). When such a nano-square array is illuminated by an optical signal from the top side vertically (along z direction) with electric field E polarized parallel to the x direction, the optical displacement current $\partial D/\partial t$ “flows” along both the unit cells and air gaps [5], and here time-harmonic behavior with time convention ${\text{e}}^{-i\omega t}$ is assumed. The nano-square array first acts as a “parallel” combination of lumped elements R, L and C₁, and then in series with a capacitance C₂, as shown in Fig. 1(b). In addition, because at optical frequencies, the real of ITO material’s permittivity $\mathrm{Re}({\epsilon }_{\text{ITO}})<0$ and the imaginary part $\mathrm{Im}({\epsilon }_{\text{ITO}})\ne 0$, the NS would act as a parallel combination of a nano-inductor L and a nano-resistor R (Fig. 1(b)). Following the general capacitor impedance formula$Z_c=i/(\omega C)$, with the capacitance$C=\epsilon a/b$, and a, b being the element’s two dimensions, the effective lumped impedance of the NS can be written as$Z_{\text{NS}}=iw/(\omega hw{\epsilon }_{\text{ITO}})$. Similarly, due to the permittivity of air gap is${\epsilon }_{air}=1$, larger than zero, therefore, the parallel capacitive impedance of the air gap with nano-square cell can be written as$Z_{c1}=iw/(\omega hg{\epsilon }_{\text{air}})$, and the serial capacitive impedance of the air gap with nano-square cell is given as $Z_{c2}=ig/(\omega h(w+g){\epsilon }_{\text{air}})$. Thus, the total impedance for this layer is $Z=Z_{\text{NS}}\Vert Z_{c1}+Z_{c2}$, where “||” is the parallel symbol (e.g., $A\Vert B=AB/(A+B)$).

Above all, the equivalent circuit of this single layer of NS patch is shown in Fig. 1(c), where ${\eta }_0$ is the intrinsic impedance of air. Within the transmission line theory [16], the light reflection coefficient from the nano-square array is derived as:$S_{11}=(Z{\Vert {\eta }_0-\eta }_0)/\left(Z\Vert {\eta }_0+{\eta }_0\right)$. The transmission coefficient can be obtained as${\text{S}}_{21}=S_{11}+1$, which is exactly valid only if the thickness of the array is negligible. Then, the reflectance and transmittance of the incident optical signal are obtained as $\text{R=}{\left|S_{11}\right|}^2$ and $\text{T}={\left|S_{21}\right|}^2={\left|2Z/(2Z+{\eta }_0)\right|}^2$. Naturally, the absorptance is obtained as$\text{A=1-T-R}$, which is obviously dependent on the size of the patch (w, h, g), the constituent materials properties (${\epsilon }_{\text{ITO}}$), and the wavelength of the illumination source ($\lambda =2\pi c/\omega$).

As for a confirmation to the above methods, a direct comparison to the corresponding numerical calculation results using three-dimensional (3D) finite difference time domain (FDTD) method [17–20] is implemented. A single cell containing a periodic structure is simulated with periodic boundary conditions along both the x and y directions, and the perfect match layer (PML) boundary condition is used along the propagation of electromagnetic waves (z axis). The mesh size is set as 5nm in all three directions, and the accuracy of the simulation results is guaranteed as convergent via the reduction of the mesh size by half.

3. Results and discussions

Following the above formulas from the equivalent circuit theory, the calculated absorptance, transmittance and reflectance are shown in Fig. 2(a). One sharp absorptance peak with efficiency up to 50% occurs at around 2000nm. The corresponding FDTD solutions are shown in Fig. 2(b). It is found that the theoretical and numerical results are well consistent with each other, especially the central wavelength and absorption efficiency, which confirms the effectiveness of our equivalent circuit theory. The electric field spatial distribution, corresponding to this absorptance peak at 2000nm wavelength in Fig. 2(b), is shown in Fig. 2(c). The NS, working as an antenna, localizes the incident light, which results in a significant electrical field resonant enhancement and guidance of most light through the air gap [19, 20].

 

Fig. 2 (a) Theoretical calculation from the equivalent nanocircuit theory and numerical calculation from FDTD algorithm to the absorptance, reflectance, and transmittance when light propagate to a single NS layer sample with side width w = 200nm. (c) Spatial distribution for the electric field at 2000nm. (d) Theoretical (circled) and numerical (squared) absorption peaks and the corresponding wavelengths versus w.

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In addition, since varying the NS geometry would change the impedance of the nanocircuit elements, and subsequently different optical response can be achieved, meaning that such an absorber could be flexibly controlled. In other words, the above demonstrated single NS layer can be thought as a wavelength-selective absorber. As shown in Fig. 2(d), with the increment of w, the central wavelength has a red-shift and simultaneously the absorption peak efficiency increases. Such a red shift can be interpreted as [21]: The larger width of NS, the bigger inductor L induced between two adjacent air gaps, which hence leading to a larger resonant wavelength. The reason for the absorption peak’s increment is that: The larger width of NS means larger energy-collecting area, which leading to an enhanced guidance of light.

The above single NS layer structure has been successfully demonstrated to take a role of absorber, however its lower absorption efficiency and narrower bandwidth makes it still on a long way to a practical application. As done for a third-order Butterworth bandstop filter [22], a cascading multilayer structure of truncated pyramids, as shown in Fig. 1(a), is proposed and designed to realize a practical absorber. A direct extension to the design and analysis procedure for a single NS layer model can be done, since the equivalent circuit model for truncated pyramid multilayer structure is easily extracted. It is confirmed that such a specific multilayer structure indeed behaves like a broad-band absorber with almost 100% absorption efficiency.

Here, we take a triple layer structure as an example, and each layer is separated by a finite distance D (Fig. 3(a)). Here D is set as one quarter of the central wavelength of the incident light. This is based on the consideration that because quarter wavelength sections of line between the layers act as admittance inverters to effectively convert alternate shunt resonators to series resonators, the transmission line sections are ${\lambda }_c/4$ long at the center wavelength ${\lambda }_c$ in order to convert the triple layers into series of branches [23]. Therefore, the corresponding equivalent circuit can be modeled as that shown in Fig. 3(b), where $\eta =\sqrt{\mu /\epsilon }$ is the intrinsic impedance of SiO₂ dielectric layers, and Zi (i = 1, 2, 3) is the impedance of each NS layer. Within transmission line theory [16], Z(3) is the effective impedance at the top interface of the third NS layer, which can be given by $Z(3)=Z_3\Vert \eta$.

 

Fig. 3 (a) Schematic diagram of S-parameters used for theoretical calculation, and (b) equivalent nanocircuit model for the proposed truncated pyramid triple layer absorber.

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Similarly, for the interface at the second or first layer, the effective impedances can be written as $Z(2)=Z_2\Vert ({\eta }^2/Z(3)),Z(1)=Z_1\Vert ({\eta }^2/Z(2))$. Then, the reflectance of this pyramid triple layer structure is given by:

After the reflectance R from the top surface is obtained, one need to calculate the transmittance through the bottom substrate. Separating the whole structure into 4 independent regions from the left to right (Fig. 3(a)) in correspondence to the three different NS layers, the transmittance T after the final layer can be obtained step by step. In this case, the theoretical S-parameters (S₁₁, S₂₁, …) for each layer are:

Since it is a long distance between the adjacent NS layers, we can neglect the dielectric losses through the SiO₂ material just for the convenience of simplification. Therefore, the reflected power P_iR and transmitted power P_iF from each NS layer are evaluated as follows:

Solving the Eqs. (2) and (3) simultaneously, the total transmittance of the structure is finally expressed as,

Obviously, the absorptance A for this truncated pyramid multilayer structure can be evaluated form Eqs. (1) and (4) as: A = 1-R-T. The corresponding absorption spectrum (black dashed line) and transmission spectrum (red dashed line) are shown in Fig. 4(a). As for triple layers replace single layer absorber, the absorptance bandwidth becomes obviously wider and also the efficiency is almost larger than 90%. The corresponding FDTD numerical results (solid lines) indicate a good agreement between our equivalent circuit modeling and the numerical simulation, except for a slight deviation at the top absorption efficiency, which could arise from our simple assumption that the SiO₂ material losses are negligible just for the convenience of calculation. However, anyway, the proposed synthesis circuit procedure is confirmed to be feasible enough to provide us a way to predict the responses of such absorbers, which can be naturally and easily extended to analyze other nano-optical devices.

 

Fig. 4 (a) Theoretical calculations (dashed lines) from the equivalent nanocircuit theory and numerical calculations (solid lines) from FDTD algorithm to the absorptance A, reflectance R, and transmittance T when light propagate to truncated pyramid triple layer absorber, and (b) the corresponding electrical field distribution at a specific 2500nm wavelength. Each NS patch has a side width w of 300nm, 250nm, and 200nm from bottom to top.

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The enhanced electric field, as shown in Fig. 4(b), concentrates at both the lateral edges of the NS and the air gaps between two adjacent unit cells. This is the evidence of the existence of a strong electric dipole resonance, caused by the localized surface plasmon polaritons excited at the metal/dielectric interface; thus, the charges are accumulated, and the localized electric field is enhanced. This strong electric resonance couples efficiently to the incident light and dissipate the energy within the metals via ohmic losses [24]. After the third layer, nearly no light can be transmitted from our absorber.

It must be emphasized that, it is in principle possible to get a perfect absorber with a 100% absorption efficiency just by adding more NS layers. To demonstrate this point, one can stack one by one NS layers with suitable width to investigate the variance of the absorption efficiency. The numerical demonstration with the number of layers varying from 1 to 6 is shown in Fig. 5(a). The bottom NS patch layer 1 has a side width of 300nm, and adding one new NS patch upper, its side width decrease 50nm. With the increment of the number of NS layers, the absorption efficiency increases and simultaneously the bandwidth becomes wider,. When the number of NS layers exceeds 4, a super broad absorption band from 1.5μm to 3.5μm with efficiency larger than 80% can be obtained.

 

Fig. 5 Absorptance spectra for different numbers of NS layer (a), each layer has a side width of 300nm to 50nm from bottom to top in 50nm steps. (b) The absorptance (solid line) and transmittance (dashed line) spectra of 4 layers structure. The absorptance spectra as a function of the incident angle for TM (c) and TE (d) waves.

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Unlike conventional absorbers, our proposed structure has no metallic substrate but a dielectric one replaced, therefore the incident light with frequency beyond the absorption band can be transmitted. As presented in Fig. 5(b), the light has been completely absorbed in the near-infrared region (NIR) region and transmitted in other regions, while the light reflectance is smaller than 10% in the whole studied band (0.8μm ~4μm). An optical window in NIR region, beyond a pure absorber, could be realized as well. Moreover, it needs to be pointed out that, although in the above demonstration the input light is always normal incidence. The results at the cases of non-normal incidences are presented in Fig. 5(c) for TM waves and Fig. 5(d) for TE waves, where the truncated pyramid triple layer absorber is respectively illuminated by TM and TE plane waves at various incident angles from θ = 0° to 50° at a 10 degree step. The frequency response of the broadband absorber is stable, which indicates the non-sensitivity to the incident angles. What’s more, due to the clear four-axis symmetrical property, the broadband absorber is also insensitive to the polarization of the incident light.

4. Conclusions

In conclusion, we theoretically introduced a new paradigm and tool to design infrared broadband absorber device based on the equivalent circuit theory which is founded by extending the classic circuit functionalities in radio frequency domain to optical domain. A super broadband (1.5μm~3.5μm) absorber is successfully demonstrated. This new design paradigm makes the formerly complicated frequency responses be evaluated relatively easily. This point is confirmed by the consistency of the simple calculation from the EC theory to the rigorous FDTD simulation. This design paradigm may be considered as an informative step towards the realization of functional nanocircuit at visible or UV, and other relatively higher frequency domains. Moreover, as for the absorber designed here, on the one hand, the usual metal substrate is replaced by a dielectric one. The latter makes the incident light beyond the absorption band can be transmitted and almost there is nearly no light reflected, and an optical window in near-infrared region, beyond a pure absorber, could be realized as well. On the other hand, the gradually modified square patch design makes such an absorber device not sensitive to the incident-angle and light’s polarization. Such an absorber can find its applications in energy harvesting, IR detection, stealth and communication systems.