1. Introduction

Optical absorption (OA), one of the most fundamental and important effects of light-matter interactions, plays a central role in optoelectronic devices such as photodetectors and photovoltaics. To achieve near-complete absorption of light, usually known as perfect OA (POA), a number of promising routes have been investigated based on, e.g. gradient structure [1–4], random medium [5–7], coherent optical absorption (CPA) [8–10], and critical coupling [11,12]. As an emerging field of optics, plasmonics exploits the unique optical features of metallic nanostructures to control light at subwavelength scales, which has been shown to possess extreme capabilities in stimulating the novel developments of optoelectronics. Plasmonics can also serve as an alternative platform for obtaining POA, but since the localized and leaky resonance nature of discrete plasmonic states, POA is usually limited by the operating bandwidth, incident polarization and angle, and efficiency. Accordingly, there is a great ongoing effort for pursuing plasmonic POA with robust performance [13–15], broad bandwidth [16–18], ultrathin structure design [13,19], and peculiar spectrum at terahertz (THz) [20–22] or infrared frequencies [23,24].

Graphene, a two-dimensional (2D) material first isolated in 2004 [25], has attracted tremendous attention over the past two decades owing to its unique optoelectronic properties. A monolayer graphene has been shown to exhibit a universal light absorption of about 2.3% [26], which is defined solely by the fine-structure constant. Since the atomic thickness, it is quite challenge, yet of significant importance to improve OA in graphene. It has been demonstrated that the OA of graphene layer can be strongly enhanced to about 50% by placing it on dielectric structures which host bound states in the continuum (BICs) [27–29], or the other forms of photonic resonances [30–33]. It is worth noting that in these works, graphene acts only as an absorbing constituent, while its own resonances play no role in OA. In such a situation, the OA of a thin material layer (not just graphene) can be simply evaluated by using classical electrodynamics [34,35], which shows that in a symmetric environment, the OA is universally limited to maximum 50% absorptance.

As is well known, graphene supports collective excitations of massless Dirac Fermions called graphene plasmons [36–38], which exhibit much tighter confinement and longer lifetime than usual noble-metal plasmons. They can be tailored by the shape and size of structures as usual plasmons, and moreover, possess a very attractive advantage of being highly tunable through electrostatic gating [39,40] and external magnetic field [41–43]. It is quite easy to think of that the scenarios for plasmonic POA in noble metals can be adopted to graphene, e.g. plasmonic POA was proposed in patterned graphene [34,44–46]. Here, a common trick is to place an additional reflective substrate such as metal layer, which can block the path of transmission, and thus makes the POA to be well illustrated by a one-port model. Nevertheless, it is still an open question whether it is possible to achieve plasmonic POA in graphene (or general 2D materials) without the aid of metal substrates.

In this work, we demonstrate plasmonic POA in a single array of folded graphene ribbons (FGRs). The whole structure can be placed simply in free space, while the asymmetric environments and metal substrates are not required. Under normal incidence, the OA of FGRs can reach as high as 50% absorptance, which is the fundamental OA limit of a thin film within symmetric environments. The obtained high OA stems from the local resonances of FGRs, e.g. a dipole state and a symmetry-protected BIC, which are tunable via folding angle. Moreover, under oblique incidence, these two states will lead to nearly complete OA at particular incident angles. The POA of the two states can be further tuned by changing Fermi level. The POA of the dipole state exhibits a continuous change in frequency, while that of the BIC is divided into three branches, due to Fano coupling to two guided modes. Finally, we show the achieved POA is well maintained even as the supporting substrate is introduced.

2. Structure and model

Graphene ribbon (GR) is the simplest patterned graphene, yet of particular interest for studying plasmon confinement, propagation, and hybridization [47–49]. As the electric field is incident along the width direction of GR, localized plasmons of different orders will be excited across the width. Besides the tunability mentioned above, these plasmonic resonances can also be strongly modulated through mechanical vibration [50–52] and folding [53,54], which emerges as a new degree of freedom for tuning graphene plasmons. In other words, the plasmons bound to 2D horizontal plane can be effectively tuned by taking advantage of structural change in the vertical direction. Inspired by this idea, we investigate OA in an array of FGRs, as shown in Fig. 1. Each FGR is folded along its central axis to have a folding angle $\theta $ (planar GR as $\theta = 0^\circ $), which is supposed to be positive or negative as folding the left half to $+ z$ or ${-}z$ direction, respectively. It is embedded in a homogeneous dielectric medium with a relative dielectric constant ${\varepsilon _s}$. For simplicity, the surrounding dielectric medium is considered as air, namely ${\varepsilon _s} = 1$, unless otherwise specified. This simple environment setting does not affect any of the fundamental conclusions, and can make the study focus on the light-matter interactions in FGRs. Furthermore, the geometric parameters, e.g. the width W of each FGR and the periodicity P of the lattice, are set as constants: $W = 10\; \mathrm{\mu m}$, $P = 25\; \mathrm{\mu m}$, throughout the work.

 

Fig. 1. The schematic diagram of an array of folded graphene ribbons. Each ribbon has a folding angle $\theta $ (to x-axis) and a width W, and the lattice constant is P. The wave of transverse magnetic polarization is obliquely incidence with opposite incident angles, corresponding to forward incidence ($+ \varphi $) and reverse incidence ($- \varphi $).

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In the terahertz (THz) and infrared frequency regions, the highly doped graphene behaves like a 2D electron gas, for which a frequency-dependent 2D surface conductivity ${\sigma _g}(\omega )$ can be employed to characterize its optical response [55,56]. By convention, the three-dimensional permittivity ${\varepsilon _g}(\omega )$ of graphene is introduced via an equivalent form [57–59]

To explore light-matter interactions explicitly, full wave simulations are carried out based on the frequency domain modeling in a commercial finite element software COMSOL MULTIPHYSICS. In practice, FGRs are treated as transition boundary conditions with permittivity ${\varepsilon _g}(\omega )$ and thickness $\varDelta $, respectively; periodic boundary conditions are set in the x direction, and two ports are implemented in the z direction (see Fig. 1). A THz plane wave of transverse magnetic (TM) polarization (magnetic field $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} \parallel \hat{y}$) is incident from port 1, and is then received at port 2. As a result, the absorption spectrum (absorptance as a function of frequency) presented in the manuscript is directly extracted from the built-in S parameters via $\textrm{Absorptance} = 1 - {|{{S_{11}}} |^2} - {|{{S_{21}}} |^2}$ [64,65].

3. Result and discussion

3.1. Plasmonic OA of FGRs at normal incidence

To start, we first investigate plasmonic OA of FGRs when the THz wave is normally incident. The absorption spectra of FGRs with different folding angles are shown in Fig. 2(a). It is seen that, in a planar GR ($\theta = {0^\circ }$), the absorption spectrum is dominated by two absorption peaks, which correspond to 38.4% absorptance at 4.53 THz and 26.3% absorptance at 9.17 THz, respectively. As comparing to extended graphene layers (about 2.3% absorptance), the OA of GRs has been greatly improved. Referring to their mode profiles (Mode-I and Mode-III in Fig. 2(b)), the two absorption peaks are caused by dipole and hexapole resonances, respectively. This means localized graphene plasmons are the key factors of OA enhancement, while in extended graphene layers, due to momentum mismatch, the propagating surface plasmons can not be excited directly [66–68]. Even they were excited by using a coupler, their limited density of states (divergent for localized plasmons) will seriously hamper OA enhancement.

 

Fig. 2. The OA of FGRs under normal incidence. (a) The OA spectra of FGRs with different folding angle $\theta $. Two dashed lines and yellow streak indicate the evolution of the three modes. (b) The electric field norm distributions of the three modes at $\theta = 0^\circ $ (planar) and $\theta = 90^\circ $ (folded), respectively. The white curves with arrows present the following of electric field lines. (c) The OA of the three modes as a function of folding angle $\theta $. The maximum 50% absorptance is marked by two dashed lines at $\theta = 80^\circ $ and $100^\circ $, respectively.

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To improve OA further, creating asymmetric environments, or even total internal reflections, are commonly implemented [34]. Here, a completely different route, namely mechanical folding, is introduced. As each GR is folded to have folding angle $\theta = {30^\circ }$, the OA of dipole mode is enhanced, while that of hexapole mode is suppressed. Meanwhile, there is a new peak emerging (marked by Mode-II), which is absent in the absorption spectrum of a planar GR. As increasing folding angle to $\theta = {60^\circ }$, the OA of Mode-I and Mode-III increases slightly and decreases slightly, respectively; and the OA of Mode-II increases rapidly, which becomes larger than that of Mode-III. As $\theta = {90^\circ }$, the Mode-I and Mode-II will result in even larger OA, which reach nearly 50% absorptance, e.g. 48.7% and 48.4% absorptance, respectively. However, as $\theta = {120^\circ }$, the OA of Mode-I decreases dramatically to 27.0% absorptance, and that of Mode-II remains almost unchanged. Meanwhile, the absorption peak of Mode-III will be merged into that of Mode-II gradually.

For interpretation, the electric field patterns of three modes are shown in Fig. 2(b). In planar GRs, the dipole (Mode-I) and hexapole (Mode-III) possess net dipole moment, thus are bright under plane wave excitations. The Mode-II, consisting of two antiparallel dipoles, can be considered as a quadrupole mode, of which the induced charges at the center and at the two ends are of opposite sign (see the direction of electric field lines). In FGRs, taking $\theta = {90^\circ }$ as an example (very similar for other folding angles), the number of hotspots of three modes are maintained, but the flowing of electric field lines have changed, which indicates their local field distributions are different from those in planar GRs, and thus causes different coupling strengths to the incident plane wave. The change of Mode-II is significant. Since folding, the two decomposed dipoles will not be aligned, and not be equally excited. As increasing $\theta $, the one at the left side will become weaker gradually, and completely dark at $\theta = 90^\circ $. At this moment, the excitation of Mode-II originates from the other decomposed dipole bound to the right side of FGRs. Obviously, this is a typical symmetric-protected BIC [69], enabled by folding rather than structural change.

The detailed change of OA is summarized in Fig. 2(c). For the OA of Mode-I and Mode-II, they both can reach maximum 50% absorptance at $\theta = {80^\circ }$ and $\theta = {100^\circ }$, respectively. The former one increases gradually to maximum absorptance and then decreases rapidly to roughly zero absorptance, while the latter one increases rapidly to maximum absorptance and then stays at a high absorptance. It is also seen that the OA of Mode-III is much weaker, and maximum 29% absorptance can be achieved at $\theta = {140^\circ }$, slightly improved in contrast to 26.3% in planar GRs. Therefore, mechanical folding can be an alternative degree of freedom to tune OA of more than the dipole mode in GRs, and the way to achieve maximum 50% absorptance is much more flexible than those by using substrates.

Besides absorptance of the three modes, their frequencies also undergo changes as increasing the folding angle $\theta $. It is seen that Mode-I and Mode-III exhibit a redshift, and Mode-II shows a blueshift. As well known, the resonance frequency of plasmons in planar GRs can be well described by a simple formula [70–72]

 

Fig. 3. (a) Resonance frequencies of the three modes as a function of folding angle $\theta $. The circles and curves are obtained from simulations and empirical formula Eq. (3), respectively. (b) The Q-factor of the Mode-II as a function of ${\alpha ^{ - 2}}$, with the asymmetry parameter $\alpha = \sin ({|\theta |/2} )$. The circles, dots and blue curve present the results from simulation, linear formula, and Eq. (5), respectively.

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As an important feature of symmetry-protected BICs, the excitation of Mode-II relies on the symmetry breaking of GRs, which is able to be qualified via the Q-factor change during the folding process. In dielectric systems (no material loss), the Q-factor of BICs is divergent, and is linearly proportional to inverse square of the asymmetry parameter $\alpha $ [69]; while in metallic systems, the Q-factor of BICs is limited due to the material loss, and will deviate from the linear formula existed in dielectric systems [73,74]. Actually, the linear formula can be improved by considering different roles of the material loss ${\gamma _m}$ and radiative loss ${\gamma _r}$. Specifically, the Q-factor is given by the total loss ${\gamma _t} = {\gamma _m} + {\gamma _r}$, equivalently ${Q^{ - 1}} = Q_m^{ - 1} + Q_r^{ - 1}$ [74], while since the intrinsic feature of ${\gamma _m}$, only ${\gamma _r}$ depends on the asymmetry parameter. In this sense, the correct formula can be obtained by simply replacing ${Q_r}$ by ${Q_r}{\alpha ^{ - 2}}$, and similar to ${Q_m}$, the parameter ${Q_r}$ is now a constant as well, which is determined by the structural design. Accordingly, we have

It is clear that a finite ${Q_m}$ will change the dependence of Q on the asymmetry parameter $\alpha $, and Eq. (5) will be reduced to the well-known linear formula $Q = {Q_r}{\alpha ^{ - 2}}$ as ${Q_m} \to \infty $, e.g. in absence of the material loss (${\gamma _m} = 0$). On the other hand, as ${\alpha ^{ - 2}} \to \infty $, $Q = {Q_m}$, namely, the Q-factor is limited by the material loss, rather than divergent.

In practice, the Q-factor of Mode-II can be calculated based on absorption spectra, e.g. the peak frequency divided by the full width at half maximum (FWHM), or alternatively, via eigenvalue analysis. The complex eigenvalue of Mode-II can be expressed as $\tilde{\omega } = {\omega _0} - i\gamma $, and then the Q-factor is defined to be $Q = {\omega _0}/2\gamma $ [75,76]. Considering that Mode-II is exactly embedded in the absorption spectrum as $\theta = 0^\circ $, the latter method is preferred, which will be employed throughout the work. On the other side, the asymmetry parameter $\alpha $ is determined by the folding angle $\theta $, e.g. being zero as $\theta = 0^\circ $ and reaching its maximum as $\theta = 180^\circ $. Thus, without loss of generality, $\alpha $ can be assumed to take a simple form like $\alpha = \textrm{sin}({|\theta |/2} )$. The Q-factor as a function of the folding angle $\theta $ is plotted in the inset of Fig. 3(b), from which the dependence of Q-factor on ${\alpha ^{ - 2}}$ can be extracted and is shown by the circles in Fig. 3(b). It is seen that for smaller ${\alpha ^{ - 2}}$ (larger $\theta $), the behavior of Q-factor can be well described by the linear formula (black dots). However, as ${\alpha ^{ - 2}} > 2$ (corresponding to $\theta < 90^\circ $, marked by the black arrow), the Q-factor curve will deviate from the linear formula gradually, and becomes flat finally. Such an observation is exactly illustrated by the improved formula Eq. (5) (see blue curve), where ${Q_m} = 29.3$ and ${Q_r} = 9.1$.

3.2. Plasmonic OA of FGRs at oblique incidence

To proceed, we now study plasmonic OA of FGRs as the THz plane waves is obliquely incident, along either forward directions ($+ \varphi $) or reverse directions ($- \varphi $) (see Fig. 1). For convenience, these oblique incidences are hereinafter denoted by forward incidences and reverse incidences, respectively. When the folding angle is fixed to be $\theta = {90^\circ }$, the absorption spectra of FGRs for different incident angle ($\varphi $) are plotted in Fig. 4(a). It is seen that as comparing to the normal incidence ($\varphi = 0^\circ $), the OA of FGRs exhibits substantial asymmetry for oblique incidences, manifested as different absorption spectra for forward incidences (blue curves) and reverse incidences (olive green curves). For a slightly oblique incidence, e.g. $\varphi = 10^\circ $, the two absorption spectra are dominated by two peaks, which are from Mode-I and Mode-II, respectively. Interestingly, for the forward incidence, Mode-I shows much stronger OA than Mode-II, and it is just the opposite for the reverse incidence. Moreover, another arresting observation is the OA of Mode-I and Mode-II are dramatically enhanced, 64.9% absorptance and 63.2% absorptance, respectively. Here, they both exceed 50% absorptance, the maximum OA for a normal incidence and also the maximum OA by an extended graphene layer. This achievement originates from the multiple scattering caused by the vertical and horizontal arms of the FGRs, and with respect to the horizontal arm, the environment is actually asymmetric.

 

Fig. 4. The OA spectra of FGRs under oblique incidence, as folding angle $\theta = 90^\circ $. (a) The OA spectra of FGRs with different incident angle $\varphi $. Two dashed lines mark the frequencies of Mode-I and Mode-II at $\varphi = 0^\circ $, respectively. (b) The electric field norm distributions at the frequencies of the four OA peaks at $\varphi = 40^\circ $. The white curves with arrows present the flowing of electric field lines. (c) Peak absorptance as a function of frequency and $\varphi $. The dashed line indicates the OA at normal incidence.

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As increasing incident angle to $\varphi = 20^\circ $, the absorptance of Mode-I and Mode-II under specific incidence are enhanced further, achieving 80.9% and 77.5%, respectively; in contrast, the OA of Mode-I (Mode-II) is further suppressed, under the forward (reverse) incidence. Meanwhile, it is also noted that Mode-II exhibits a slight redshift, Mode-III remains almost unchanged in both absorptance and frequency, and two very narrow absorption peaks (marked as Mode-IV and Mode-V) arise. Different from the other three modes, the frequencies of Mode-IV and Mode-V show strong dependence on incident angle $\varphi $, which will result in significant consequences below. As $\varphi = 30^\circ $, Mode-I and Mode-II achieve POA, with 93.9% and 90.4% absorptance, respectively, and a new POA peak with 90.0% absorptance emerges at frequency 8.00 THz. This POA peak is from the well-known Fano interaction between Mode-II and Mode-IV [77], where the former and the latter act as a continuum and a discrete state, respectively.

If increasing incident angle further, e.g. $\varphi = 40^\circ $, Mode-I and Mode-II (redshift to frequency 7.08 THz) achieve nearly complete OA, showing the absorptance of 97.6% and 97.8%, respectively; and Mode-IV and Mode-V will redshift further, to result in two POA peaks with high absorptance of 90.8% and 72.2%, respectively. As discussed above, these two OA peaks are from the Fano coupling of Mode-IV and Mode-V to Mode-II, respectively. Another clear evidence is the change of absorption curve of Mode-II, from a typical Lorentz shape to an asymmetric Fano shape. This is also the reason why the OA of Mode-II can be improved suddenly to be stronger than that of Mode-I. Because of this coupling, an absorption dip (marked by black circle) appears at frequency 7.32 THz, at which the transmittance attains 92.7%, making a transparency window between Mode-II and Mode-IV. As shown by the blue dashed line, the window is roughly located at the position of the absorption peak of Mode-II as $\varphi $ is very small, such as $\varphi < 20^\circ $, which in this sense, may be interpreted as an effect of electromagnetically induced transparency (EIT) [78–80]. Contrary to the changes of Mode-II as increasing $\varphi $, Mode-I possesses a constant resonance frequency at 4.28 THz, and its Lorentz resonance shape is well preserved, which indicates it is a pure dipole state without coupling to other states.

For illustration, field patterns for the four absorption peaks as $\varphi = 40^\circ $ are plotted in Fig. 4(b). As comparing to Fig. 2(b) (normal incidence), it is seen that Mode-I has almost the same field pattern, while Mode-II is more concentrated to the three hotspots (nodes), namely, the two sides and the folding axis; they both exhibit much larger local field enhancement (denser electric field lines), consistent with their greatly improved OA. By examining the flowing of electric field lines, it is very easy to find that the other two peaks also have three hotspots, with the same arrangement as those of Mode-II. However, different from Mode-II, the local field enhancement is much stronger at the folding axis than that at two sides. The field patterns at these two peaks reveal from the other side, their strong OA closely related to Mode-II. To gain a more thorough picture of OA at oblique incidences, the color map of absorptance is given by Fig. 4(c), in which the asymmetric OA is clearly seen. It is also worth noting that for reverse incidences, over 90% absorptance is achieved at 4.28 THz as $- 52^\circ \le - \varphi \le - 26^\circ $. For forward incidences, the strong OA is distributed in three discrete regions, and is redshifted as increasing $\varphi $. Meanwhile, due to hybridization, Mode-II and Mode-IV follow a typical anti-crossing frequency splitting, and in between, the EIT-like absorption dip is also redshifted and getting close to Mode-II as increasing $\varphi $.

To get a thorough understanding of Fano interaction, the eigen frequencies of the five modes are plotted in Fig. 5. It is seen that, as increasing incidence angle $\varphi $, the frequencies of Mode-I, Mode-II, and Mode-III remain almost constant, while those of Mode-IV and Mode-V decrease significantly. The field patterns of Mode-IV and Mode-V as $\varphi = 10^\circ $ (marked by the two vertical arrows) are given by the insets in Fig. 5(a). Obviously, Mode-IV and Mode-V possess the characteristics of propagating modes, which indicates they are guided modes along the FGR lattice [33,81–83], rather than localized modes bound to each FGR. The excitation of guided mode depends also on its own dispersion, which will be different for different lattice constant. Thus, as another strong evidence, Mode-IV and Mode-V experience a substantial frequency decrease as increasing P (see Fig. 5(b)). Moreover, as shown in Fig. 5(c), their frequencies remain almost constant as increasing Fermi level ${E_\textrm{F}}$, while those of the other three modes increase dramatically. This observation confirms further that Mode-IV and Mode-V are guided modes.

 

Fig. 5. Resonance frequencies of the five modes as a function of incidence angle $\varphi $ (a), lattice constant P (b), and Fermi level ${E_\textrm{F}}$ (c). The insets in (a) show the electric field norm distribution and the flowing of electric field lines of Mode-IV and Mode-V at $\varphi = 10^\circ $ marked by two vertical arrows.

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Besides the dependence on folding angle $\theta $ and incident angle $\varphi $, the OA of FGRs can also be actively tuned through electrostatic gating, which is a distinct advantage of graphene plasmonics. As shown in Fig. 6, for the reverse incidence, the POA of Mode-I is maintained, with its operating frequency blueshifted from 3.54 THz to 5.36 THz as increasing Fermi level ${E_\textrm{F}}$ from $0.4 \;\textrm{eV}$ up to $1.0\;\textrm{eV}$. For the forward incidence, a similar frequency blueshift of POA is observed, e.g. the frequency varies from 6.08 THz to 9.60 THz as $0.4\;\textrm{eV} \le {E_\textrm{F}} \le 1.0\; \textrm{eV}$. However, due to Fano coupling to Mode-IV and Mode-V, the POA of Mode-II is divided into three branches, which is consistent with Fig. 4(c).

 

Fig. 6. The tunability of POA of FGRs enabled by Fermi level ${E_\textrm{F}}$ as folding angle $\theta = 90^\circ $ and incident angle $\varphi = 40^\circ $, for the forward incident (left panel) and reverse incidence (right panel).

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Fig. 7. The OA spectra of FGRs as placed on $\textrm{Si}{\textrm{O}_2}$ substrate, for the forward incidence (a) and the reverse incidence (b), with incident angle $\varphi = 40^\circ $. The $\textrm{Si}{\textrm{O}_2}$ has a dielectric constant ${\varepsilon _r} = 3.9$ and a thickness $t = 0.1\mathrm{\;\ \mu m}$ as given by the insets. The parameters of FGRs remain unchanged, with folding angle $\theta = 90^\circ $. The two dark dotted curves present the OA spectra of free-standing FGRs.

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3.3. Experimental feasibility of folded graphene ribbons

The experimental realization of graphene nanostructures has been pursued by a great number of approaches based on chemical, electrochemical, mechanical, and radiation-assisted methods [84]. Besides changing the surface morphology and building the layered structure, the mechanical folding along the out-of-plane dimension can emerge as a new degree of freedom for tuning optoelectronic properties of graphene, including plasmonic excitations [53,54]. Considerable effort devoted to graphene folding (or called graphene origami) can be traced back to 1995 by Ebbesen and Hiura, who were first envisioned “Graphite Origami”, and they observed the folding of graphite surface layers with accidental tearing through an atomic force microscope (AFM) tip [85]. Since then, people have successively achieved the folding of graphite sheets and graphene in experiments, but the operation is always common along with tears or damage [86–89]. To solve this issue, Mu et al. develop a self-folding technology based on function-designed nanoscale building blocks [90]; Joung et al. propose a self-assembly technology by transforming 2D graphene into a 3D structure [91], which has been used to design 3D optical devices [92,93]; Xu et al. implement a thermally responsive method to fold/unfold monolayer graphene into predesigned, ordered 3D structures [94]; Wang et al. make use of a tailored substrate to fold monolayer graphene [95]; most impressively, based on scanning tunneling microscopy (STM) tip, Chen et al. demonstrate a very innovative technique in 2019, to enable atomically precise folding and unfolding of graphene [96]. As comparing to these complex 3D structures, FGRs are simpler, which should be much easier to be fabricated.

On the other hand, it needs to be noted that the POA results discussed above originate from free-standing FGRs, while in reality, the substrates are generally required to support FGRs. Thus, without loss of generality, the silicon dioxide ($\textrm{Si}{\textrm{O}_2}$) is considered to be the substrate, as shown by the insets in Figs. 7(a) and 7(b). The substrate has a dielectric constant ${\mathrm{\varepsilon }_r} = 3.9$ and a thickness of $t = 0.1\mathrm{\;\ \mu m}$. It is clear that, the POA results are maintained for both forward and reverse incidences, yet with a slight redshift for each POA peak.

4. Conclusion

In summary, we demonstrate an unprecedented approach to achieve THz POA in free-standing FGRs for TM incident waves. Different from other works, the general total reflection condition, e.g. through placing a metal reflector to block the transmission channel, is not required. Specifically, the OA spectra of FGRs are dominated by several local resonances induced peaks, e.g. the two highest peaks caused by a dipole state (Mode-I) and a symmetry-protected BIC (Mode-II) that can be actively tuned by folding angle, with their frequencies following a simple analytic formula. As normal incidence, the OA limit of a thin film in symmetric environments, namely 50% absorptance, is easily achieved by Mode-I and Mode-II at appropriate folding angles. As oblique incidence, the OA spectra of forward and reverse incidence are asymmetric, with Mode-I (Mode-II) being much stronger in the latter (former) case. As increasing incident angle, both Mode-I and Mode-II can achieve nearly complete OA. During the procedure, the frequency of Mode-I remains constant, while that of Mode-II undergoes a gradual redshift, which originates from the Fano interaction between Mode-II and two guided modes. The two POA can be further tuned by increasing Fermi level, where the POA of Mode-I shows a continue blueshift, and that of Mode-II is divided into three branches. Finally, the achieved POA in FGRs can be well maintained even as the supporting substrate is introduced. This work provides a new and simple scenario to achieve POA in one-atom-thick graphene layers, which can be extended to study more rich physical effects in 2D materials. The strongly enhanced light-matter interactions in FGRs evince directly that the mechanical folding can be a powerful degree of freedom for photonic device applications.