1. Introduction

Perfect absorption of electromagnetic waves is of great interest and importance in a variety of applications ranging from photovoltaics to stealth technologies [1–7]. Recently, coherent perfect absorbers (CPAs), as time-reversed lasers, have been proposed and increase absorption to 100% interferometrically [7–36]. Such CPAs provide a new way of absorption manipulation based on the phase difference of incident waves without modulating intrinsic nonlinearity and absorption coefficient. This makes CPAs particularly attractive to applications of transducers, modulators, and switches. Among many proposed schemes of CPAs, ultrathin conductive films (CFs) [29–36] are especially interesting because of the extraordinarily thin thickness and light weight. For example, graphene and indium tin oxide (ITO) films as conductive films (CFs) at low frequencies have shown broadband coherent perfect absorption characteristics due to the non-resonant nature [29–31]. In the standard scenarios of ultrathin CPAs, perfect absorption happens under the illumination of two beams of counterpropagating waves with the same amplitude and phase. The two beams interfere constructively and dissipate their energy completely inside the CFs. Such a two-channel scheme severely limits the applications of such CF CPAs.

In this work, we demonstrate a class of geometry-invariant multiple-channel CPAs that are realized by embedding arbitrarily shaped ultrathin CFs inside a zero-index medium (ZIM) host. We find that when the CFs are embedded inside the ZIM host, coherent perfect absorption can be obtained for waves incident from an arbitrary number of input channels oriented along different directions, as long as the total width of the channels equals to a critical value. Interestingly, such an effect is only determined by the length and material parameters of the CFs, but independent of their shapes and positions. The absorption can be manipulated by tuning the phase differences between different channels. Practical designs of such geometry-invariant multiple-channel CPAs have been demonstrated by using dielectric photonic crystals (PhCs) and photonic-doped ZIM. Furthermore, ultrathin CPAs based on ZIM are also demonstrated in waveguides. These findings indicate a unique switching mechanism from ZIM to CPAs.

2. Theory of CPAs by arbitrarily shaped CFs in ZIM

The concept of the proposed CPA by using M ($M \ge 1$) pieces of arbitrarily shaped CFs embedded in a ZIM host is illustrated by Fig. 1, in which transverse-electric (TE, out-of-plane electric fields)-polarized electromagnetic waves are normally incident onto the ZIM host from N ($N \ge 1$) different input channels. The amplitudes of all incident waves are assumed to be the same, i.e. ${E_0}$, while their phases can be different, as characterized by ${\varphi _n}$ in the $n$-th channel. The filling material in all channels is chosen to be free space. We assume that the relative permeability ${\mu _{\textrm{ZIM}}}$ of the ZIM is near zero (i.e. ${\mu _{\textrm{ZIM}}} \approx 0$), thus the electric field inside the ZIM host is almost a constant [37–40]. Upon matching the tangential electric field at the ZIM-channel interfaces, we arrive at the following relationship:

 

Fig. 1. Illustration of a geometry-invariant multiple-channel CPA by using arbitrarily shaped CFs in a ZIM host.

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Perfect absorption requires zero-reflection in all input channels, i.e., ${r_1} = {r_2} = \ldots = {r_N} = 0$. In this case, the phases of all incident waves should be the same, i.e., ${\varphi _1} = {\varphi _2} = \ldots = {\varphi _N}$, as implied in Eq. (1). Applying the Maxwell’s equation $\oint \textbf{H} \cdot d\textbf{l} = - i\omega \int\!\!\!\int {\textbf{D} \cdot d\textbf{S}}$ to the ZIM host, we have,

Equation (4) describes the condition of CPA, which requires a complex ${\varepsilon _{\textrm{CF}}}$ as ${\varepsilon _{\textrm{CF}}} ={-} \frac{S}{{Ld}}{\varepsilon _{\textrm{ZIM}}} + i\frac{W}{L}\frac{1}{{{k_0}d}}$. If the ZIM is a kind of double-zero ZIM whose ${\varepsilon _{\textrm{ZIM}}}$ is also near zero, the required permittivity of the CFs would be simplified to ${\varepsilon _{\textrm{CF}}} = i\frac{W}{L}\frac{1}{{{k_0}d}}$, indicating a pure imaginary permittivity. Interestingly, conductive materials, such as metal, graphene and ITO, naturally exhibit large imaginary permittivity at low frequencies [30,31]. For a conductive material with conductivity ${\sigma _0}$, its relative permittivity is described as ${\varepsilon _r} = 1 + i\frac{{\sigma (\omega )}}{{{k_0}}}{Z_0} \approx i\frac{{{\sigma _0}}}{{{k_0}}}{Z_0}$, where ${Z_0}$ is the impedance of free space. Usually, we characterize such ultrathin CFs by using the sheet resistance ${R_s}$ that is defined as ${1 / {({{\sigma_0}d} )}}$. In this way, the condition of CPA turns into

Equation (5) indicates that the proposed CPA is determined by the total width of all input channels W, and the total length L and sheet resistance ${R_s}$ of all CFs. This condition is clearly independent of the size and shape of the ZIM host, as well as the shape and position of CFs, revealing a unique kind of geometry-invariant multiple-channel CPA.

The condition of CPA can also be comprehended from the photonic doping theory [43–46]. In the doping scenario, the CFs would change the permittivity of the ZIM host, while maintaining the near-zero permeability. According to the photonic doping theory [43], the permittivity of the host is changed to ${\varepsilon _{\textrm{host}}} = \frac{1}{{{\varepsilon _0}{E_0}S}}\int\!\!\!\int_{\textrm{CF}} {\textbf{D} \cdot d\textbf{S}}$, which reduces to ${\varepsilon _{\textrm{host}}} = {{{\varepsilon _{\textrm{CF}}}Ld} / S}$ when considering the constant electric fields. This indicates that the CF-embedded ZIM can be regarded as a “diluted” conductive material. In such a conductive material with ${\varepsilon _{\textrm{host}}}$ and ${\mu _{\textrm{ZIM}}} \approx 0$, perfect absorption of incident waves requires ${\varepsilon _{\textrm{host}}} = {{iW} / {({{k_0}S} )}}$, as implied in Eq. (3). Consequently, we obtain the condition for CPA as ${R_s} = {{{Z_0}L} / W}$, which coincides with Eq. (5).

Next, we perform numerical simulations by using software COMSOL Multiphysics to verify this geometry-invariant multiple-channel CPA. We first consider a two-channel CPA composed of a double-zero ZIM host (${\varepsilon _{\textrm{ZIM}}} = {\mu _{\textrm{ZIM}}} = 0.001$) and a S-shaped CF ($d = {{{\lambda _0}} / {100}}$, $L = 2{\lambda _0}$, ${R_s} = 2{Z_0}$). Two TE-polarized waves with ${\varphi _1} = {\varphi _2}$ and the same amplitude of 1 V/m are incident onto the ZIM from the left and right input channels, whose widths are ${w_1} = 0.7{\lambda _0}$ (left) and ${w_2} = 0.3{\lambda _0}$ (right), respectively. These parameters satisfy the CPA condition in Eq. (5). Figure 2(a) presents the simulated electric-field distribution, clearly showing that almost all incident waves are absorbed by the CFs inside the ZIM host. The outside boundaries are set as perfect magnetic conductors (PMCs). Moreover, we calculate the absorptance by varying the length L and sheet resistance ${R_s}$ of the CF, as plotted in Fig. 2(b). It is clearly seen that near-perfect absorption is obtained along the line of ${{{R_s}} / {{Z_0}}} = {L / W}$ (black dashed lines), as predicted by Eq. (5). For comparison, we have removed the CF in Fig. 2(c). We see that standing waves with doubled amplitude at antinodes are formed in the two input channels, indicating the total reflection. Interestingly, the perfect absorption maintains if the CF is divided into several pieces as long as the total length is unchanged. As an example, when we divide the original CF in Fig. 2(a) into two pieces (i.e. S-shaped and A-shaped CFs), near-perfect wave absorption is still clearly seen, as verified by the electric-field distribution in Fig. 2(d). Clearly, the shape and position of CFs do not affect the wave absorption.

 

Fig. 2. (a) Simulated electric field-distribution in a two-channel CPA composed of a double-zero ZIM host and a S-shaped CF under the illumination by TE-polarized waves. (b) Absorptance as functions of the length L and sheet resistance ${R_s}$ of the CFs. The black dashed line denotes the condition of CPA. (c) Simulated electric field-distribution in the absence of the CFs. (d) Simulated electric field-distribution when the S-shaped CF in (a) is divided into two pieces (i.e. S-shaped and A-shaped CFs) while keeping the total length unchanged. (e) Absorptance as the function of phase difference ${\varphi _2} - {\varphi _1}$ for the two-channel model in (d). (f) Distribution of the amplitude of electric fields when the two incident beams are out-of-phase. (g) Simulated electric field-distribution when the left input channel in (d) is divided into two channels while keeping the total width unchanged. (h) Absorptance as functions of phase differences ${\varphi _2} - {\varphi _1}$ and ${\varphi _3} - {\varphi _1}$ for the model in (g). (i) Distribution of the amplitude of electric fields when ${\varphi _2} - {\varphi _1} = \pi$ and ${\varphi _3} - {\varphi _1} = \pi$. The relevant parameters in (a), (c)–(i) are $W = {\lambda _0}$, $L = 2{\lambda _0}$ and ${R_s} = 2{Z_0}$.

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Such a two-channel CPA provides an efficient way to control the absorption via tuning the phase difference between incident waves [7]. To show the phase-difference control effect, in Fig. 2(e), we plot the absorptance for the model in Fig. 2(d) as a function of the phase difference ${\varphi _2} - {\varphi _1}$. Here, the absorptance is evaluated based on $A = 1 - {{\sum\limits_{n = 1}^N {\frac{{{P_{b,n}}}}{{{P_{0,n}}}}{w_n}} } {\bigg / } W}$, where ${P_{0,n}}$ and ${P_{b,n}}$ denote the time-averaged power flow of incidence (forward direction) and reflection (backward direction) in the $n$-th channel, respectively. The maximal (minimal) absorptance occurs when the phase difference is 0 (${\pm} \pi $). It is found that the maximal absorptance can reach 100%, while the minimal absorptance is non-zero, because the widths of the two input channels are different. The minimal absorptance can be derived as ${A_{\textrm{min}}} = 1 - {{4{w_1}{w_2}} / {{{({{w_1} + {w_2}} )}^2}}}$ according to Eqs. (1) and (2), which indicates zero absorptance when ${w_1} = {w_2}$ [26]. The distribution of the amplitude of electric fields in Fig. 2(f) shows clear interference patterns due to the incident and reflected waves in input channels, confirming the low absorption.

Superior to the traditional CPA of the simple two-channel configuration [7], the proposed geometry-invariant CPA allows more than two input channels, thus providing extra degrees of freedom to tune the absorption. As an example, in Fig. 2(g), we divide the left channel in Fig. 2(d) into two channels while keeping the total width unchanged. The parameters of CFs remain unchanged. Figure 2(h) presents the simulated electric-field distribution in such a three-channel configuration, also showing near-perfect absorption of all incident waves. Moreover, we change the phase differences ${\varphi _2} - {\varphi _1}$ and ${\varphi _3} - {\varphi _1}$, and plot the corresponding absorptance contour in Fig. 2(f). The results show the perfect (minimal) absorption at the 0 (${\pm} \pi $) phase difference, indicating the efficient control of wave absorption. The zero absorption in the out-of-phase situation is confirmed by the standing waves in input channels, as observed from the distribution of the amplitude of electric fields in Fig. 2(i).

3. Designs of geometry-invariant CPAs

The key in designing the proposed geometry-invariant CPAs lies in the realization of double-zero ZIM. In the following, we will show two approaches to obtain effective double-zero ZIM. The first approach is based on a dielectric PhC with Dirac-like dispersion. Figure 3(a) illustrates the unit cell (upper inset) and the first Brillouin zone (lower inset) of the dielectric PhC. It is composed of a square array of dielectric rods, which has the advantage of low material loss. The relative permittivity of dielectric rods is 7.5 and the radius of rods is $0.23a$, where a is the lattice constant of the square array. Figure 3(b) presents the band structure of this PhC for the TE polarization, showing a Dirac-like dispersion at Γ point around the normalized frequency ${{fa} / c} = 0.604$ ($f$ is the eigen-frequency, and c is the speed of light in vacuum). It has been demonstrated in Ref. [47–51] that this kind of PhC can be homogenized as an effective double-zero ZIM. For verification, in Fig. 3(c), we retrieve the effective parameters ${\varepsilon _{\textrm{eff}}}$ (solid lines) and ${\mu _{\textrm{eff}}}$ (dashed lines) of the PhC nearby the Dirac-like cone based on eigen-field averaging [52–54], clearly showing that ${\varepsilon _{\textrm{eff}}} \approx {\mu _{\textrm{eff}}} \approx 0$ at $fa/c = 0.604$. Note that there exists a flat band at the Dirac-like point, which corresponds a longitudinal mode. Such a mode is a deaf mode for normal incidence, and couples weakly even to oblique incident waves [47–51]. Here, a working frequency slightly lower than the longitudinal mode is chosen to avoid its influence.

 

Fig. 3. (a) Illustration of the unit cell (upper) and the first Brillouin zone (lower) of a dielectric PhC for the effective double-zero ZIM. (b) Band structure of the PhC for the TE polarization. (c) Effective parameters ${\varepsilon _{eff}}$ (solid lines) and ${\mu _{eff}}$ (dashed lines) of the PhC. [(d) and (g)] Schematic graphs of a (d) two-channel, (g) three-channel CPA composed of the PhC with CFs. [(e) and (h)] Simulated electric-field distributions in the (e) two-channel, (h) three-channel CPA when a CF is placed at the position 2. [(f) and (i)] Absorptance as a function of phase difference (f) ${\varphi _2} - {\varphi _1}$ for the two-channel model, (i) ${\varphi _3} - {\varphi _2}$ for the three-channel model, when the CF is alternately placed at position 1 (blue), 2 (red) and 3 (green). The normalized working frequency in (d)–(i) is $fa/c = 0.604$.

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Figure 3(d) shows the schematic graph of a two-channel CPA by using this PhC as the effective double-zero ZIM host. The PhC is composed of 16 ${\times} $ 16 unit cells. The outermost unit cell boundary has a distance of ${{{\lambda _0}} / 4}$ away from the surrounded perfect electric conductor (PEC) walls, so that the PEC walls can effectively mimic the PMC boundaries [26,51]. The input channels are surrounded by perfect matched layers (PMLs) to adjust the width of incident beams. In this two-channel model, the total width of input channels is $W = 26a$. A CF with ${R_s} = 0.31{Z_0}$ and $L = 8a$ is chosen, so that the CPA condition in Eq. (5) can be satisfied. The CF is alternately placed at positions 1–3, as illustrated in Fig. 3(d). These positions are along unit cell boundaries of the PhC, so as to ensure the validity the effective medium description. Figure 3(e) presents the simulated electric field-distribution at $fa/c = 0.604$ when the CF is placed at position 2, showing that almost all incident waves are absorbed by the CF. Moreover, in Fig. 3(f), we calculate the absorptance as the function of phase difference ${\varphi _2} - {\varphi _1}$ when the CF is alternatively placed at position 1 (blue lines), 2 (red lines) and 3 (green lines). The results clearly show the phase-controllable absorption and the robustness to the position and shape of the CF.

Furthermore, we add a new input channel in the right side, so that it becomes a three-channel model, as illustrated in Fig. 3(g). Now, the total width of all input channels is $W = 36a$. In order to satisfy the CPA condition, the sheet resistance of the CF is changed to ${R_s} = 0.22{Z_0}$ while keeping the length unchanged. Still, we see near-perfect absorption when the three incident waves are in-phase, as shown by the electric field-distribution in Fig. 3(h). The absorptance as the function of phase difference ${\varphi _3} - {\varphi _2}$ for the CF alternately placed at position 1 (blue lines), 2 (red lines) and 3 (green lines) also confirms the controllable and geometry-insensitive absorption characteristics in such a three-channel CPA [see Fig. 3(i)].

The second approach to realize the double-zero ZIM is by using photonic-doped single-zero ZIM with ${\varepsilon _{\textrm{ZIM}}} = 1$ and ${\mu _{\textrm{ZIM}}} \approx 0$. Such single-zero ZIM can be realized based on metamaterials possessing magnetic resonance in practice [55–57]. Compared the double-zero ZIM based on PhCs, single-zero ZIM based on metamaterials can be more compact, and is of practical possibility in designing subwavelength CPAs. Generally, the relative permeability of the single-zero ZIM can be described by a Lorentz model as ${\mu _{\textrm{ZIM}}} = 1 - {{({\omega_p^2 - \omega_0^2} )} / {({{\omega^2} - \omega_0^2 + i\omega \gamma } )}}$, where $\omega$, ${\omega _p}$, ${\omega _0}$ and $\gamma$ are the angular frequency, plasma frequency, resonance frequency and damping constant, respectively. Here, we set ${\omega _0} = 0.8{\omega _p}$ and $\gamma = {10^{ - 4}}{\omega _p}$, so that ${\mu _{\textrm{ZIM}}} \approx 0$ when $\omega \approx {\omega _p}$. According to the photonic doping theory [26,27,43–46], a defect inside such a single-zero ZIM won’t affect the near-zero ${\mu _{\textrm{ZIM}}}$, but can efficiently tune the permittivity, which can be derived as [43]

 

Fig. 4. (a) Schematic graph of a CPA composed of CFs embedded in a photonic-doped single-zero ZIM host. (b) Effective relative permittivity ${\varepsilon _{eff}}$ of the photonic-doped ZIM with respect to the ${\varepsilon _d}$ of doping defect. (c) Simulated electric field-distribution in a two-channel CPA. The single-zero ZIM host is doped with a dielectric defect. (d) Simulated electric field-distribution when the positions of CF and doping defect are changed. (e) Absorption spectra of the CPA models in (c) and (d), as denoted by red solid and blue dotted lines, respectively. (f) Absorptance of the CPA model in (c) as a function of the $\mu ^{\prime\prime}$ of the ZIM host.

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Figure 4(b) plots the ${\varepsilon _{\textrm{eff}}}$ with respect to the ${\varepsilon _d}$ based on Eq. (6) by setting $S = 2.277\lambda _0^2$. We see that ${\varepsilon _{\textrm{eff}}} \approx 0$ occurs at ${\varepsilon _d} = 14.26$, thus effective double-zero ZIM is obtained. Note that due to the oscillation of Bessel functions in Eq. (7), there exist infinite solutions of ${\varepsilon _d}$ for the effective double-zero ZIM if its value can be infinitely large.

Next, we illuminate the ZIM by two TE-polarized waves incident from the left and right channels onto the ZIM, as shown in Fig. 4(c). The total width of the two channels is $W = 0.7{\lambda _0}$. A CF with $L = 0.7{\lambda _0}$ and ${R_s} = {Z_0}$ is placed inside the ZIM to absorb the incident waves. The working frequency is $\omega = {\omega _p}$, and the relative permittivity of the doping is ${\varepsilon _d} = 14.26$, so that the single-zero ZIM behaviors as an effective double-zero ZIM. Figure 4(c) presents the simulated electric field-distribution. Near-perfect absorption of all incident waves is seen, which can also be observed when the positions of the CF and doping defect are changed, as shown in Fig. 4(d). This confirms the position-independence characteristic. Moreover, in Fig. 4(e), we plot the absorption spectra of the models in Figs. 4(c) and 4(d), as shown by the red solid and blue dotted lines, respectively. The near-perfect absorption is observed at $\omega = {\omega _p}$ for both models, further confirming the position-independent CPA. One may also notice absorption peaks away from the frequency ${\omega _p}$, which are evidently position-dependent. They originate from resonances inside the ZIM and defect as the ${\mu _{\textrm{ZIM}}}$ is not near-zero.

In addition, we investigate the effect of material loss of the single-zero ZIM by assuming a complex relative permeability as ${\mu _{\textrm{ZIM}}} = 0.001 + i\mu ^{\prime\prime}$. The absorptance of the model in Fig. 4(c) is re-calculated as a function of the $\mu ^{\prime\prime}$, as plotted in Fig. 4(f). It is seen that near-perfect absorption occurs at $\mu ^{\prime\prime} = 0$. But the absorption decreases quickly as the $\mu ^{\prime\prime}$ increases. This is because the electric field is no longer a constant inside the ZIM, thus leading to the failure of the theories of photonic doping and CPAs. Interestingly, when $\mu ^{\prime\prime} = 1.2$, the secondary maximal absorption (∼0.9) is seen. In order to find out the underlying physics, the energy dissipation in the CFs and the ZIM host is evaluated based on ${L_{\textrm{CF}}} = \frac{1}{2}\int\!\!\!\int_{\textrm{CF}} {\omega {\varepsilon _0}\varepsilon _{\textrm{CF}}^{\prime\prime} {{|{{{\textbf E}_{\textrm{CF}}}} |}^2}{\mkern 1mu} \textrm{d}S} $ and ${L_{\textrm{ZIM}}} = \frac{1}{2}\int\!\!\!\int_{\textrm{ZIM}} {\omega {\mu _0}\mu^{\prime\prime} |{{\textbf H}_{\textrm{ZIM}}}{|^2}\textrm{d}S} $ [58], respectively. We find that when the $\mu ^{\prime\prime}$ is small, the loss of CFs dominates, while the loss of the ZIM host dominates when the $\mu ^{\prime\prime}$ is large. The decrease of absorption for the $\mu ^{\prime\prime}$ larger than 1.2 originates from the decreased skin depth in the dissipative ZIM host.

4. Ultrathin CPAs by arbitrarily shaped CFs in single-zero ZIM

Due to the geometry-invariant characteristic of the proposed CPAs, we can reduce the size of the ZIM (e.g. its thickness ${d_{\textrm{ZIM}}} < < {\lambda _0}$), so that ultrathin CPAs can be realized by using curved CFs. Interestingly, in such ultrathin CPAs, only single-zero ZIM is required. According to Eq. (4), we find that the requirement of CPA can be simplified to ${\varepsilon _{\textrm{CF}}} \approx i\frac{W}{L}\frac{1}{{{k_0}d}}$ when $S < < W{\lambda _0}$. As a result, the CPA condition in Eq. (5) can be approximately applied to ultrathin CPAs with single-zero ZIM. In practice, such subwavelength ZIM with ${\mu _{\textrm{ZIM}}} \approx 0$ can be realized by using metamaterials possessing magnetic resonances [55–57]. In the following, we will take two waveguide models for verification.

Figure 5(a) presents the schematic graph of the first model. It is a PMC plane waveguide, in which plane waves are supported. In the center, there is a wavy CF ($L = 3.08{\lambda _0}$, ${R_s} = 0.514{Z_0}$) embedded in a single-zero ZIM slab (${\varepsilon _{\textrm{ZIM}}} = 1$, ${\mu _{\textrm{ZIM}}} = 0.001$). The thickness of the ZIM slab is set as ${d_{\textrm{ZIM}}} = 0.2{\lambda _0}$, much smaller than the wavelength, to fulfill the condition $S < < W{\lambda _0}$. Two counterpropagating TE-polarized waves are incident from the left and right ports simultaneously. The simulated electric-field distribution is displayed in Fig. 5(b), showing near-unity (>0.97) absorption. For comparison, the ZIM slab is removed in Fig. 5(c). Clearly, the absorption decreases and diffraction pattern occurs due to the wavy shape of the CF.

 

Fig. 5. [(a) and (d)] Schematic graphs of exotic ultrathin CPAs realized by using (a) a wavy CF, (d) a circular CF embedded in single-zero ZIM in waveguides. Two counterpropagating TE-polarized waves are incident from the left and right ports simultaneously. [(b) and (c)] Electric field-distractions in the (b) existence, (c) absence of the single-zero ZIM for the model in (a). [(e) and (f)] Electric field-distractions in the (e) existence, (f) absence of the single-zero ZIM for the model in (d).

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In the second example, a circular CF ($L = 2.51{\lambda _0}$, ${R_s} = 0.42{Z_0}$) is immersed in a ring of single-zero ZIM (${\varepsilon _{\textrm{ZIM}}} = 1$, ${\mu _{\textrm{ZIM}}} = 0.001$), which is integrated with a PMC plane waveguide, as illustrated in Fig. 5(d). The thickness of the ZIM slab inside the waveguide is ${d_{\textrm{ZIM}}} = 0.2{\lambda _0}$. In this model, near-unity (>0.93) absorption is also observed, as shown by the simulated electric-field distribution in Fig. 5(e). However, when the ZIM is removed (Fig. 5(f)), the absorption is decreased to near-zero because waves generally cannot “see” the CF outside the waveguide.

5. Discussion and conclusion

Since the CFs considered here are electric conductive, the proposed CPAs are polarization-selective. Energy is dissipated in the CFs due to currents induced by the parallel electric fields of TE polarizations. While for transverse-magnetic polarizations with in-plane electric fields, there would be no absorption unless magnetic CFs are exploited instead.

When the conductivity of the CFs is increased (i.e. the sheet resistance is decreased), the absorption would decrease. In particular, when the conductivity is infinitely large, the CFs turn to be PECs with infinitely small skin depth. In such an extreme case, all incident TE-polarized waves will be totally reflected, because the embedded PECs will enforce the electric fields inside the ZIM to be zero. Consequently, the ZIM host behaves as an “enlarged” PEC [46].

Finally, we note that compared with the CPAs by using bulky absorptive defects in Ref. [26], the configuration proposed here has many advantages. The CPAs by using bulky absorptive defects require the synthetic control of shape and size of defects, as well as real and imaginary parts of their permittivities, leading to great challenges in practical designs. Interestingly, the CPAs via ultrathin CFs here only rely on the length and sheet resistance of the CFs, which can be easily controlled in experiments. In practice, there are plenty of choices of CFs, such as graphene [32–34,36], ITO films [31], thin structural metal films [35,59], etc. They can work as ideal CFs from microwaves, terahertz to optics. Moreover, subwavelength CPAs can be realized by using such ultrathin CFs, which are absent in the previous configuration [26].

In summary, we have proposed a unique kind of geometry-invariant CPAs by using arbitrarily shaped CFs embedded in ZIM hosts. Compared with traditional two-channel CPAs, the proposed CPAs allow multiple input channels, thus providing more flexible manipulation of absorption through the control of the phase differences between different channels. Moreover, we demonstrate that the proposed CPAs are independent of the shapes and positions of CFs, which is contrary to the traditional CPAs. This unique feature originates from the uniformly distributed electric fields inside the ZIM hosts. We find that the CPA condition is solely determined by the total width of all input channels, the phase differences between different channels, and the total length and sheet resistance of all CFs. In addition, we show two approaches to realize such CPAs by using dielectric PhCs and photonic-doped single-zero ZIM. Due to geometry-invariant characteristic, we further demonstrate ultrathin CPAs based on ZIM. These results reveal unique and flexible functionalities beyond traditional CPAs.