Bandwidth bounds of coherent perfect absorber in resonant metasurfaces

Ming Kang; Ming Li; Jing Chen

doi:10.1364/OE.27.009004

1. Introduction

Absorption is an unavoidable phenomenon, besides the scattering, in various systems of wave. In general, absorption loss is believed to be the main factor impeding the utilizations of metamaterials and metasurfaces supporting localized resonances [1]. However, recently a class of synthetic systems with tailored absorption has recently generated huge interest in the optical and physical societies because they can reveal the virtue of absorption [2–8]. Among them the most representative one is the coherent perfect absorber (CPAs) [3–5]. A CPA refers to a resonator with critical losses that can perfectly absorb the entire incident light, and is believed to be the time-reversed counterpart of a laser operating at the threshold. Mathematically, the existence of CPA coincides with a zero eigenvalue of the scattering matrix S sitting at the real frequency axis. Physically, perfect absorption is due to a destructive interference in a critical-tuned two-port resonator [9–13]. Since the theoretical prediction by Chong et. al [3], most efforts were focused on the possible realization of CPAs in various systems such as photonic crystals and acoustic/optical metasurfaces [9–23].

The realization of CPA requires the presence of critical losses. As a critical underdamped resonator, the quality factor (Q factor) defined by the frequency-to-bandwidth ratio is a very important dimensionless parameter [24] of CPA. The bandwidth δf usually refers to the full-width half-maximum (FWHM) of the absorption peak. It is desirable to have a large Q factor to enhance the interaction between light and the resonator for potential applications such as in sensors and nonlinear optics. There are many approaches to increase the Q factor in photonic systems. For example, we can make use of avoided resonance crossing in a pair of coupled modes, where one of the eigen-modes has a reduced bandwidth [25–27]. On the contrary, a small Q factor, corresponding to a larger bandwidth, is also necessary to achieve an effective absorption within a broader frequency range. The detailed values and mechanisms of bounds in Q factor would help us to solve this tradeoff and provide us with a guideline in improving the performance of CPA toward a pre-defined application. Although there have seen a lot of theoretical and experimental verifications of CPAs at various frequency regimes, the bandwidth bond of CPA is rarely mentioned.

Here, we theoretically explore the bandwidth of CPAs within the scope of resonant metasurfaces by using the coupled mode theory (CMT), which is a very concise model to characterize resonant phenomena in various physical systems [28,29]. We show that in a typical resonant metasurface with weakly coupled bright and dark modes, there exist upper and lower bandwidth bounds that are determined by the dissipation loss rate. If the dissipation loss is very weak, it is better to increase the radiation loss rate of the bright resonant mode to narrow the bandwidth of CPA. In addition, the position of CPA in such a hybridized system is a robust phase singularity with a winding number of $\pm 1$ . The phase singularity implies an anomalous phase delay. To verify our theoretical explorations, we propose a metallic metasurface design that bears a sharp transition from positive to negative phase delay at the resonance of CPA. The numerical result is in good agreement with CMT. Our theoretical exploration can provide a guideline for designing CPAs in various resonant systems.

2. Theoretical conclusions from CMT

To derive the bounds on the bandwidth of CPA in a resonant metasurface, we adopt the simple CMT that can grasp main physical behaviors of various coupled resonators not limited to metasurfaces [30–35]. Without loss of generality, let us consider a metasurface exhibiting a mirror symmetry along the propagation direction $z$ . We further assume that the resonance of the metasurface is composed of a radiative resonant mode with a scattering loss rate of $γ_{s} > 0$ , and a non-radiative mode with a zero scattering. The complex amplitude of the coupled resonance is represented by $q = {(q_{r}, q_{n r})}^{T}$ , and ${| q_{r, n r} |}^{2}$ usually corresponds to the energy in each resonant mode, where the subscripts $r$ and $n r$ indicate the radiative and non-radiative modes, respectively. Such a kind of system with coupled bright and dark modes is a representative test-bed for electromagnetically induced transparency (EIT), Fano resonance and Autler-Townes effect [35]. To simplify our discussion on CPA, the resonant frequencies of the two modes are assumed to be equal, i.e. $f_{r} = f_{n r} = f_{0}$ . This assumption has been used to investigate slow light phenomena in the classical analogue of EIT effect [30,31,33]. Different from former investigations, here we focus on CPA associated with a zero eigenvalue of the so-called scattering matrix $S$ . Considering a near-field coupling $κ$ between these two resonant modes, and indicating the radiative coupling between the radiative mode and the input (output) light by $a (b)$ , we get two coupled equations [28–31]:

Ω q = K^{T} a, b = C a + K q .

The response matrix

Ω

, directive coupling matrix

C

between the input and the output fields, and the radiative coupling matrix

K

(

K^{T}

) between the output (input) field and the resonance in the metasurface are

Ω = (\begin{matrix} f - f_{r} + i γ_{s} + i γ_{d} & - i κ \\ - i κ & f - f_{n r} + i γ_{d} \end{matrix}), C = (\begin{matrix} 0 & - 1 \\ - 1 & 0 \end{matrix}), K = (\begin{matrix} \sqrt{γ_{s}} & 0 \\ \sqrt{γ_{s}} & 0 \end{matrix}),

respectively. Here γ_d is the intrinsic dissipation rate by the absorption in the resonator, which can be made extremely small, e.g. by working in the microwave regime or by using proper lossless media. The relationship between the input and output fields is then given by

b = S a

, where the scattering matrix

S

is formulated by

S = C + K Ω^{- 1} K^{T} .

Mathematically, the position of CPA corresponds to a zero eigenvalue of $S$ . Due to the mirror symmetry along the propagation direction, CPA only occurs for a symmetric eigenvector of ${(1, 1)}^{T}$ when the damping and coupling rates satisfy

f - f_{0} - i γ_{d} + \frac{i}{2} γ_{s} \mp \frac{1}{2} {(4 κ^{2} - γ_{s}^{2})}^{1 / 2} = 0

To be physically observable, the solution of

f

should be exactly at the real frequency axis by setting constraint conditions on

γ_{s}

,

γ_{d}

, and

κ

. In the strong coupling scenario of

4 κ^{2} - γ_{s}^{2} > 0

, by solving Eq. (4) it is readily to show that

γ_{s} = 2 γ_{d}

should be satisfied, and the frequencies of CPA are given by

f = f_{0} \pm 0.5 {(4 κ^{2} - γ_{s}^{2})}^{0.5}

. CPA in such a strong coupling scenario has been explored to imitate the phase transition in parity-time-symmetric Hamiltonian systems [36]. The strict requirement on

γ_{d}

and

γ_{s}

constrains the realization of CPA in a limited frequency range.

Here we are interested in the weak-coupling scenario of $4 κ^{2} - γ_{s}^{2} < 0$ , and would like to discuss the exact bandwidth and explore the bounds on the existing CPA because it is desirable to maximize\minimize the operational bandwidth δf by optimizing the resonator design in many potential applications. In this weak-coupling scenario, the null of Eq. (4) requires that both the real and imaginary parts are zero. It is then clear that the frequency of CPA is always $f_{0}$ because $4 κ^{2} - γ_{s}^{2} < 0$ . Starting from the extreme case of $κ = 0$ where the resonance of the metasurface is only dominated by the radiative dipolar mode, from Eq. (4) we can see that CPA at $f = f_{0}$ requires either a balanced scattering condition of $γ_{s} = γ_{d}$ [37] or a zero intrinsic dissipation of γ_d = 0. The latter case is physically impossible and should be neglected. Now, FWHM of the coherent absorption can be shown to be $δ f = 4 γ_{d}$ , which is only determined by the dissipation loss rate γ_d. The Q factor is then $Q = f / (4 γ_{d})$ .

When $κ$ is not zero, the bright and dark modes would couple with each other. Now CPA occurs at a real frequency of $f = f_{0}$ under the satisfaction of a critical coupling of

κ_{c}^{2} = γ_{d} (γ_{s} - γ_{d})

According to Eqs. (3) and (5), we can find the frequencies where the output intensity of output field b is half of the incident field a = (1, 1)^T. The difference between them is defined as FWHM of CPA. Under the simple assumption of

γ_{s} > γ_{d}

, FWHM of the coherent absorption is found to be

δ f = \sqrt{2} \sqrt{- 4 γ_{d}^{2} + 8 γ_{d} γ_{s} - γ_{s}^{2} + \sqrt{16 γ_{d}^{4} - 64 γ_{d}^{3} γ_{s} + 88 γ_{d}^{2} γ_{s}^{2} - 16 γ_{d} γ_{s}^{3} + γ_{s}^{4}}}

We can see that δf is determined not only by the dissipation loss rate

γ_{d}

, but also by the scattering one

γ_{s}

. For a selected system working at a desired frequency, the dissipation loss is generally fixed by the geometry of the structure and the constituent media forming it. Unlike the isolated radiative resonance of

κ = 0

, we could engineer FWHM in weak-coupling scenario by optimizing the radiation loss rate with the help of resonator design, i.e. to modify the brightness of the radiative mode. When we select the scattering loss rate as the varied parameter, according to Eq. (6), there exists an upper bound in FWHM

δ f_{\max} = 4 γ_{d} \sqrt{2 + 5^{1 / 2}} \approx 8 .233 γ_{d}

when

γ_{s} = γ_{d} (3 + 5^{1 / 2}) \approx 5 .236 γ_{d} .

When $γ_{s} > > γ_{d}$ , FWHM is approximated given by $δ f = 4 γ_{d} + 16 γ_{d}^{2} / γ_{s}$ . Together with the scenario of κ = 0 as discussed above, we can see that CPA only occurs at $γ_{s} = γ_{d}$ , and the lower bound of FWHM according to Eq. (6) is

δ f_{\min} = 4 γ_{d} .

Since the resonant frequency of CPA is fixed at

f = f_{0}

, we can also utilize the Q factor to characterize the bandwidth. Figure 1 shows the effective Q factor of f₀/δf versus

γ_{s} / γ_{d}

and is confined by the upper and lower bounds predicted by Eqs. (7) and (8). For potential applications we can engineer the value of

γ_{s} / γ_{d}

to obtain the desired functionality within the bounded region.

Fig. 1 (a) Evolution of the effective Q factor versus γ_s (in the unit of f₀) when the dissipation loss rate $γ_{d} = 0.02 f_{0}$ . Dashed lines represent the upper and lower bounds. The green arrow tip illustrates the position of a CPA when $γ_{s} = 0.2 f_{0}$ . (b) The phase profile of Eq. (4) with a phase singularity at $(f_{0}, κ_{c} = \sqrt{γ_{d} (γ_{s} - γ_{d})})$ in 2D space formed by $f$ and $κ$ .

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Note that in the two-dimensional (2D) parameters space formed by $f$ and $κ$ the solution of CPA satisfying Eq. (4) is a phase singularity with a winding number of $\pm 1$ [see the example shown in Fig. 1(b)]. The sign of the winding number is determined by the sign of the coupling strength $κ$ . The existence of this robust phase singularity, which is immune to γ_d, γ_s and κ, implies an anomalous phase delay around CPA [38–40]. The phase delay $τ_{d} = d ϕ / d f$ for the symmetric eigenvalue $ξ = Α e^{i ϕ}$ of the scattering matrix $S$ at the resonant frequency $f_{0}$ is given by

τ_{d} |_{f = f_{0}} = \frac{2 γ_{s} (κ^{2} - γ_{d}^{2})}{[κ^{2} + γ_{d} (γ_{d} + γ_{s})] [κ^{2} - γ_{d} (γ_{s} - γ_{d})]} .

The second term of the denominator is the one that gives the critical condition of Eq. (5), and consequently, τ_d indeed changes its sign when f crosses the frequency f₀ of CPA.

3. Metasurface design and numerical verification

In order to put the theoretical conclusion of CMT made in the former section into realistic systems, a resonant metallic metasurface as schematically shown in Fig. 2(a) is proposed. The dissipation loss rate of the metal, such as copper used here, is very small in the microwave region. Now it is difficult to satisfy the condition of $γ_{s} = γ_{d}$ in the metallic metasurface to approach the upper bound of Q factor. However, instead of searching for a very small radiative loss, we can switch to the scenario of a large $γ_{s} / γ_{d}$ value in our metasurface design according to the analysis present in Section 2. A large $γ_{s} / γ_{d}$ value could provide a FWHM of $δ f \approx 4 γ_{d} + 16 γ_{d}^{2} / γ_{s}$ that can still approach the lower bound of Eq. (8). For example, if the value of $γ_{s} / γ_{d}$ equals 1000, δf is about $1.004 δ f_{\min}$ .

Fig. 2 (a) Schematic view of a unit cell of the designed metasurface. (b) - (e) Fitted CMT parameters versus the transverse displacement $t_{x}$ . The discrete dots (red sphere) are obtained by fitting COMSOL results with CMT. These parameters are then fitted by using polynomial fitting (solid lines).

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In our structure design shown in Fig. 2(a), the metal of the resonant metasurface is copper with a deep subwavelength thickness of 0.5 mm and an electric conductivity of $5.7 \times 10^{7}$ S/m. The normally incident plane wave is linearly polarized with electric field E oscillating in the $y$ direction. Each unit cell is a rectangular lattice with a dimension of $d_{x} \times d_{y} = 10 \times 8$ mm². Width of each metallic strip is $w = 1$ mm. The non-radiative mode is supported by the middle two metallic strips oriented along the $x$ direction, where the length of each strip is $b = 6$ mm with a gap of $g = 1$ mm. The radiative mode is generally from the outer two metallic strips oriented along the $y$ direction, where the length of each strip is $a = 6.6$ mm with a gap size of $2 p = 8$ mm. The tunable parameter here is the transverse displacement $t_{x}$ of the center metallic strips. When $t_{x}$ is varied, not only the coupling strength $κ$ but also some other parameters such as the resonant frequency $f_{0}$ and the scattering loss rate $γ_{s}$ are modified.

To analyze the performance of CPA, we firstly numerically calculate the complex transmission and reflection coefficients by using COMSOL Multiphysics. The frequency regime is from 18.5 to 22.5 GHz, and the transverse displacements $t_{x}$ varies from 0 to 0.48 mm with a step of 0.02 mm. Then, we retrieve a set of CMT parameters (red spheres in Fig. 2) by fitting the numerical reflection results with the analytical formulae of Eqs. (1)-(3). Finally, the discrete CMT parameters are fitted by using a polynomial fitting method to reveal simple relation between them and $t_{x}$ . The fitted polynomial curves (in the unit of GHz) versus t_x (in the unit of mm) are displayed in Fig. 2 as solid lines, as: $κ = 1.554 t_{x} + 2.806 t_{x}^{3} - 10.08 t_{x}^{5}$ , $f_{r} = 20.58 + 1.195 t_{x}^{2} + 0.7527 t_{x}^{4} + 18.02 t_{x}^{6}$ , $f_{n r} = 20.58 - 2.902 t_{x}^{2} - 13.64 t_{x}^{4} + 7.066 t_{x}^{6}$ , and $γ_{s} = 5.236 + 0.6653 t_{x}^{2} - 4.925 t_{x}^{4} + 4.379 t_{x}^{6}$ .

The fitting shows that an increased transverse displacement $t_{x}$ always leads to a blue (red) shift in the resonance of the radiative (non-radiative) mode. The scattering loss rate γ_s shows neglectable deviation from 5.24 GHz, and the dissipation loss rate γ_d is a constant of 0.005GHz, three orders smaller than γ_s.

Based on the above fitted function for each parameter and Eqs. (1)-(3), the amplitude of reflection is in good agreement with that from numerical simulation, see Fig. 3. Furthermore, we can obtain the coherent absorption spectra, $A_{C} = 1 - {| t + r |}^{2}$ , which is shown in Fig. 4. Around $t_{x} = 0.1$ mm we get $f_{r} = 20.59$ GHz, $f_{n r} = 20.55$ GHz, $γ_{s} = 5.24$ GHz, and $κ = 0.1613$ GHz, respectively. Based on these discrete fitted parameters, the operational frequency of CPA is approximately given by $f_{c} = 20.55$ GHz. These fitted parameters verify the assumptions of $f_{r} \approx f_{n r}$ and the weak coupling criterion of $4 κ^{2} - γ_{s}^{2} < 0$ made in the former section. To clearly indicate the position of CPA, we plot the numerical and fitted coherent absorption spectra at $t_{x} = 0.1$ mm in Fig. 5(a). The value of the coherent absorption from fitting (numerical simulation) at 20.55GHz (20.554 GHz) is 0.9978 (0.9977), and can be assumed to be unity. Based on the coherent absorption spectra, we can obtain the fitted (numerical) FWHM of CPA. The effective Q factor of CPA, defined by $f_{0} / FWHM$ , can reach 1027 (1027), which is high enough for applications in sensors and nonlinear optics. The obtained Q factor is almost the same as that from our theoretical prediction, i.e. $f_{0} / (4 γ_{d} + 16 γ_{d}^{2} / γ_{s}) \approx 1030$ , and approaches the upper bound of $f_{0} / (4 γ_{d}) \approx 1034$ .

Fig. 3 (a) COMSOL and (b) fitted reflection coefficient versus $f$ and $t_{x}$ .

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Fig. 4 (a) Numerical and (b) fitted coherent absorption coefficient versus $f$ and $t_{x}$ .

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Fig. 5 (a) Numerical (scatter) and fitted (solid line) coherent absorption spectra at $t_{x} = 0.1$ mm. (b) Numerical (scatter) and fitted (solid lines) phase delay above ( $t_{x} = 0.12$ mm) and below ( $t_{x} = 0.08$ mm) CPA.

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Besides the large Q factor close to the upper bound, we also verify that the eigenvalue of the scattering matrix $S$ at the CPA in such hybridized system is a phase singularity, see Eq. (4). Around such a phase singularity, the phase delay $τ_{d}$ shows a transition from positive to negative when we change the coupling strength, see Eq. (9). Examples about the phase delays $τ_{d}$ below ( $t_{x} = 0.08$ mm) and above ( $t_{x} = 0.12$ mm) the critical coupling condition of CPA are shown in Fig. 5(b). The peaks of phase delay in these two conditions are opposite with each other. Note that when we vary the transverse displacement $t_{x}$ to tune the coupling strength κ, the resonant frequency of CPA also changes. The frequency of CPA is very close to the resonant frequency of the non-radiative mode, so the resonant peak\dip of the phase delay $τ_{d}$ below\above the critical coupling also exhibits a weak red shift.

4. Conclusion

In summary, we derive the bandwidth of CPA in a resonant metasurface by using CMT, and show that the upper and lower bounds are determined by the dissipation loss rate of the resonances inside. Weakly-coupled resonators with a giant radiative loss rate can be utilized to get a narrow bandwidth approaching the lower bound of FWHM. We numerically verify our analysis in a designed metallic metasurface structure in the microwave frequency range, where the dissipation loss rate is very low. The numerical results are in good agreement with CMT. Our investigation can be transferred to higher frequency regimes such as infrared and visible light.

Funding

National Natural Science Foundation of China (NSFC) (11574162, 11674244, 11874228); Program for Innovative Research in University of Tianjin (TD13-5077).

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Bandwidth bounds of coherent perfect absorber in resonant metasurfaces

Abstract

1. Introduction

2. Theoretical conclusions from CMT

3. Metasurface design and numerical verification

4. Conclusion

Funding

References

Cited By

Figures (5)

Equations (9)

Optics Express