Metasurface virtual absorbers: unveiling operative conditions through equivalent lumped circuit model

Virtual absorption concept has been recently introduced as a new phenomenon observed in electromagnetics and optics consisting of an undefined energy accumulation within a finite volume of material without dissipation. The anomalous behaviour is achieved by engaging the complex zero scattering eigenmodes of the virtual absorbing system by illuminating it with a proper complex frequency ω = ωr + jωi, whose value is strictly determined by the system characteristics. In this paper, we investigate on the position of the zero-pole scattering pairs in the complex frequency plane as a function of the input impedance of the metasurface-based lossless virtual absorber. We analytically derive the conditions under which a properly modulated monochromatic plane wave can be virtually absorbed by the system and stored within its volume. The analysis is developed by modelling the propagation of a normally impinging plane wave though its equivalent transmission line model terminated on an arbitrary reactive load, which in turn models the input impedance of the metasurface-based system under consideration. The study allows to determine a priori whether the metasurface-based system can support the virtual absorption or not by evaluating the time-constant from its equivalent circuit.


Introduction
The localization and confinement of electromagnetic energy in a finite volume without dissipation has been extensively studied by the scientific community in the last years, by exploiting the anomalous interaction between light and artificial metastructures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this framework, virtual absorption concept [20][21][22][23][24] represents one of the most appealing technique for storing and releasing electromagnetic energy, due to passive and simple structures that can support the desired energy accumulation. Indeed, originally the virtual absorption has been achieved by illuminating a lossless dielectric slab of finite thickness with two perfectly synchronized signals at its opposite sides, demonstrating the possibility to achieve zero total scattering and energy accumulation when the illuminating signals have a proper complex temporal frequency r i j    =+ [20]. This special temporal excitation is fundamental for engaging the complex zero scattering eigenmodes of the system and enabling its anomalous electromagnetic behaviour of zero scattering and energy storing.
More recently, to overcome the strictly requirements of pure lossless materials and perfect coherent illumination of the structure, we proposed a metasurface-based virtual absorber [25,26], able to store energy from an impinging plane wave under arbitrary illumination condition, and without the use of dielectric materials. As shown in Fig. 1a, the proposed structure consists of a 1D cavity filled by vacuum and bounded by a reflector and a metasurface. The metasurface is designed to be penetrable by the external illuminating field, allowing the cavity to interact with the external environment.
In [25], exploiting the transmission line equivalence in Fig. 1b, we analytically derived the required complex frequency of the illuminating plane wave as a function of the surface impedance mts X of the metasurface, the electrical size of the cavity, polarization and incidence angle of the illuminating wave. In [25] we also reported an effective design guideline of a metasurface-based virtual absorber for a given frequency, noticing that the values of metasurface reactance was strongly linked with the electrical size of the cavity (eq. 5 in [25]). This suggests that the virtual absorption phenomenon does not occur for any pair of metasurface and cavity impedances.
The aim of this paper is to identify the conditions in terms of equivalent lumped impedances modelling the metasurface and cavity that enable the virtual absorption. Therefore, the entire analysis is reduced to the circuital response of the transmission line model shown in Fig. 1b Fig. 1c. Such a representation significantly simplifies the analysis by reducing the problem of virtual absorption to a very well-established concept in microwave circuit, that is the perfect impedance matching [27]. Indeed, when the transmission line in Fig. 1c is matched under complex excitation, the corresponding metasurface-bounded open cavity in Fig. 1a is virtually absorbing the impinging electromagnetic wave, allowing us to easily identify the operative bounds for the proposed system. By exploring the scattering zero-pole pairs and their position in the complex frequency plane as a function of such reactances, we analytically derive the conditions under which the system can support virtual absorption and the corresponding operative bounds.
The paper is organized as follows. In Sec. II we introduce the analysis method by focusing our attention on the response from the load in terms of reflection coefficient when an excitation signal with complex frequency is propagating along the feeding transmission line. Here, we investigate the two possible configurations for the input reactance

Operative conditions of Metasurface virtual absorber through TL model
Let us consider an arbitrary metasurface virtual absorber as shown in Fig. 1a. Under plane wave excitation from freespace, the propagation and interaction with the virtual absorber can be modelled as shown in Fig. 1b The overall input impedance in Z is a frequency dependent quantity and is purely imaginary. Being 0 Z a real valued impedance, the system turns out to be always mismatched, since values. Each pair occurs periodically over the real frequency axis, as shown in [20,21], due to the fact that the cavity exhibits the same input reactance for different cavity sizes [29].
On the contrary, using lumped-elements representation (Fig. 1c) where

Metasurface and cavity with same electrical behaviour
Let us now consider the case where , respectively, with L and C being the total inductance and capacitance measured at the load terminals.
The reflection coefficients for the two considered scenarios can be easily written as: where  assumes the values It is interesting to note that, regardless the reactive load type, the amplitude of the reflection coefficient exhibits two singularities: a pole is in the positive half-space of the imaginary frequency, whereas a zero is in the negative half-space.
Both exist for the same real frequency 0 r  = and are complex conjugate quantities. From the physical point of view, the pole represents the case when the load is reflecting more energy than the impinging one. In passive systems, this is as a special scattering condition for which the scattered field decays slower than the excitation field, giving rise to a virtual gain effect, as shown in [30]. On the contrary, the zero corresponds to a zero-reflection condition, which behaves as an indefinite accumulator for the illuminating signal, without dissipating its energy but rather storing it within the reactive load. The zero-scattering condition is kept if the complex frequency is such to approach the zero of the reflection coefficients. As soon as the frequency of the signal changes, the reflection coefficient assumes a non-zero value and energy leaks out from the reactive load.
By forcing (2) to zero, we obtain that the reflection coefficient vanishes at the complex frequencies: The complex frequencies To conclude, when metasurface and cavity exhibit the same equivalent electrical behaviour, the metasurface-based virtual absorbing condition cannot be achieved, since the propagation of the exciting field requires a non-zero frequency to propagate.

Metasurface and cavity with opposite electrical behaviour
Let us now consider the case where mts X and cav X have opposite equivalent electrical responses, i.e. 0 mts cav XX  . Therefore, the lossless transmission line is loaded by a reactive network modelled as parallel connection of an inductor L and capacitor C (Fig. 3a), leading to a load reactance: Substituting (4) into , we obtain: It is worth noticing that, at a first glance, eq. (6) seems to identify pure imaginary frequencies, as in the case reported in Sec. 2.1. However, this is true only if the argument of the square root in (6)  It is interesting now to plot the amplitude of the reflection coefficient (5) for the three distinct cases: , as a function of the complex frequency. The plots are reported in Fig. 3b-c-d. The reflection coefficient exhibits some singularities in the spectrum, that are located differently in each considered case: Fig. 3b) the reflection coefficient has two pairs of singularities, zeros and poles are symmetrically distributed and lying along the imaginary axis. All singularities have zero real frequency.
ii) ( 2 4LC  = - Fig. 3c) the two poles and the two zeros degenerate, allowing only one zero to be engaged. Again, the singularities are located along the axis 0 r  = , meaning that the excitation signal is not oscillating with a specific real frequency as desired for achieving matching in microwave networks.
iii) ( 2 4LC   - Fig. 3d), the reflection coefficient exhibits two zeros for complex frequencies with a non-zero real part.
This last case is of fundamental importance for our analysis because it demonstrates that it is theoretically possible to achieve zero reflection from a purely reactive load (i.e., parallel connection of an inductor and capacitor) under monochromatic excitation with a signal at frequency r  , provided that the amplitude of the wave follows the profile exp( . It is worth noticing that Fig. 3d reports two possible zeros symmetrically distributed with respect to the imaginary axis, but only the zero for positive real frequency can be excited. Hence the left pair of pole-zero can be neglected.

Results and verification
This section is dedicated to the validation of operative conditions for an arbitrary metasurface-based virtual absorber as varying signal whose amplitude grows in time, as shown in Fig. 4a (orange line). As expected, as long as the voltage signal is applied to the circuit, no reflections occur till the kick-off time 0 t . Being the entire circuit passive, energy cannot be dissipated, allowing only energy storing in the reactive load. As soon as the signal stops, the zero-reflection condition is not satisfied anymore, forcing the reactive load to release the stored energy, as shown in Fig. 4a (green line). From the electrical point of view, this corresponds to the charging and discharging behaviours of an RL circuit, where the resistive element controlling the damping factor is the free-space impedance 0 Z . However, this case cannot be implemented in a realistic metasurface virtual absorber, due to the impossibility to excite it with a propagating monochromatic electromagnetic field.
Let us now consider a capacitive cavity with electrical thickness , as shown in Fig. 4b (orange line). Again, the voltage signal is virtually absorbed by the reactive load as for the previous case, storing the energy in the LC reactive load, and controlling the releasing through the kick-off instant of time. It is worth mentioning that, in this case, the system has its realistic free-space counterpart, being the exciting signal oscillating in time and supporting propagation as electromagnetic wave [25].

Conclusion
In this paper, we have explored the analytical operation limits to get virtual absorption for a metasurface based absorber.
By investigating on its lumped circuit equivalent loads, we have also found the conditions enabling the perfect matching