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Abstract
Graphene is an attractive two-dimensional material for nonlinear applications in the THz regime, since it possesses high third order nonlinearity and the ability to support tightly confined surface plasmons. Here, we study 2D-patterned graphene-patch metasurfaces for efficient third harmonic generation. The efficiency of the nonlinear process is enhanced by spectrally aligning the fundamental and third harmonic frequencies with resonances of the metasurface, leading to spatiotemporal energy confinement in both steps of excitation at ω and radiation at 3ω. This precise resonance alignment is enabled by the 2D-patterning; it is achieved by modifying the dispersion of the underlying plasmons and, thus, the spectral positions of the supported standing wave resonances. Efficiencies as high as −20dB (1%) for input intensity 0.1 MW/cm2 are achieved. Moreover, we verify that the efficiency does not deteriorate when finite-size metasurfaces are used in place of ideal periodic systems. Our results highlight the potential of graphene-based metasurfaces for nonlinear applications.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The development of the three-dimensional metamaterials [1,2] and their two-dimensional counterparts, known as metasurfaces [3,4], has enabled unique electromagnetic properties, not possible with natural materials, and intriguing phenomena. Metasurfaces (MSs) are artificial thin-film materials composed of a single layer of periodically arranged subwavelength elements termed “meta-atoms”. Despite being electrically thin, they can interact strongly with incident radiation, due to the resonant nature of their meta-atoms, and allow control over the amplitude [5,6], phase [7–9], and polarization [10,11] of light, leading to a broad range of interesting linear phenomena.
At the same time, the resonant nature of MSs makes them interesting candidates for observing nonlinear phenomena. Resonant response is associated with high local fields and strong confinement of energy in both space (effective mode volume) and time (quality factor), thus boosting the efficiency of nonlinear processes [12]. Recently, strategies for obtaining sharp resonances, such as employing electromagnetic induced transparency (EIT) [13,14], Fano resonances [15], bound states in the continuum (BICs) [16–18], anapoles [19] and broken symmetry [20,21] have been proposed in the literature. All approaches are based on the engineering of resonances with the aim of achieving high radiation quality factors and sharply-resonant response. Thus, there is a constantly increasing interest for the study of nonlinear metasurfaces [15], with the goal of not only miniaturization, but moreover lowering the power requirements of conventional, bulky nonlinear crystals. To this end, materials with high intrinsic nonlinearity are a necessary prerequisite. The judicious combination of high material nonlinearity with strong resonant response can produce high effective nonlinear properties and significantly boost the efficiency of nonlinear phenomena.
A prominent example of novel photonic materials, graphene, a single layer of carbon atoms arranged in a hexagonal lattice, has attracted considerable interest for a plethora of practical applications [22–25], in part due to the ability of controlling its conductivity by changing the chemical potential with a gate voltage [26,27]. Graphene’s exceptional properties as a tunable material [28–30] extend to the technologically attractive THz regime [31], enhanced by the capability to support tightly-confined surface plasmons at these frequencies [32–34]. In addition, it possesses high third-order nonlinearity (nonlinear surface conductivity $\sigma ^{(3)}$) across a broad range of frequencies [35–37]. If momentarily viewed as a thin bulk material, the nonlinear index $n_2$ would acquire values of $\sim 10^{-13}$ m$^2$/W [36], which exceeds those of typical semiconductors, nonlinear polymers and chalcogenide glasses by orders of magnitude, providing a rough comparative assessment with conventional nonlinear materials. However, note that graphene should be naturally modeled as a current sheet (infinitesimal thickness) described by linear and nonlinear surface conductivities; this modeling approach is followed throughout this work. Characteristic of the strength of graphene’s nonlinearity is that a single unpatterned monoatomic graphene layer has enabled the generation of frequencies up to seventh order, as experimentally demonstrated by Dragoman et al. at the microwave [38] and by Hafez et al. at the THz regime [39].
Here, we propose graphene-based metasurfaces with an engineered spectrum of resonances for efficient third harmonic generation (THG). Specifically, we exploit multiple MS resonances, properly aligned with the fundamental (FF) and third harmonic (TH) frequencies, a strategy which has been termed double-resonant enhancement [40,41]. Importantly, the short wavelength of propagating graphene plasmons (large effective index or slow phase velocity) [42,43], enables the design of metasurfaces supporting higher-order standing wave resonances, while the lattice constant remains subwavelength even for higher harmonic frequencies, avoiding diffraction effects. Graphene-ribbon metasurfaces, i.e., 1D-patterned graphene structures, have been studied in the literature for nonlinear phenomena and THG in particular [41,44–46]. In this work, we examine structures with two-dimensional (2D) graphene patterning, rectangular and cross patches, in order to further confine the surface plasmons and produce hot spots with stronger field intensities and to allow isotropic response. In addition, the presence of an additional geometric parameter (the finite width of the underlying graphene strip waveguide), allows control over the dispersion of graphene plasmons and, consequently, the ability to adjust the resonant frequencies in the neighborhood of the fundamental and third harmonic. Having as a reference the THG conversion efficiency (CE) of −26dB or 0.25% (at input intensity of 0.1 MW/cm$^2$) reported in [44], we propose structures with increased conversion efficiency by exploiting the two-dimensional patterning and consequent field enhancement and by carefully designing the structures so that the fundamental (FF) and third-harmonic (TH) frequencies fall in close proximity to strong resonances. We are able to report conversion efficiencies as high as −20dB or 1%.
The paper is organised as follows: In Section 2 we study a nonlinear graphene metasurface composed of simple square/rectangular patches. We examine the effect of patch dimensions on the THG conversion efficiency and we investigate the correlation with the spectral positions of the structure resonances. In Section 3, we extend the study to an alternative design of graphene crosses, that is characterized by isotropic response (identical response for the two linear polarizations). Finally, in Section 4, with an eye on practical considerations we investigate the impact of a finite metasurface extent using Gaussian beam excitation and compare with the infinite periodic system (plane wave excitation). Our conclusions are summarized in Section 5.
2. Rectangular graphene patch metasurface
In this section, we propose a rectangular-patch multiresonant graphene metasurface for efficient third harmonic generation. We find the geometric parameters for optimum generation efficiency and unveil the underlying physics by identifying the resonances mediating the generation process.
2.1 Metasurface structure
The metasurface structure is depicted in Fig. 1; it is composed of graphene patches on a dielectric substrate with thickness $h=6.3~\mu$m and relative electric permittivity $\varepsilon _{\mathrm {r}}=4.41$, backed by a gold backreflector. The lattice constant of the metasurface is $a=a_x=a_y=3.88~\mu$m (square periodicity). The thickness of the backreflector is considered to be considerably larger than gold’s skin depth at THz frequencies and thus transmission is zeroed out.
Fig. 1. Metasurface composed of graphene rectangular patches ($w_x \times w_y$). Graphene meta-atoms have been placed in a square periodicity with period $a=3.88\mathrm ~\mu$m, on top of a dielectric slab with relative permittivity $\varepsilon _{\mathrm {r}}=4.41$ and thickness $h=6.3~\mu$m, backed by a gold backreflector. The electric field is parallel to the $x$-axis. Due to the short wavelength (high $k_x$) of supported graphene plasmons, the metasurface can support higher order resonances (standing waves along the $x$ axis) while the lattice periodicity remains subwavelength at their resonant frequency, avoiding higher diffraction orders. Insets on the right: Examples of one-, two-, and three-half wavelength resonances (Re$(E_x)$ distribution on a $xz$ plane), as obtained from eigenvalue simulations. The extent of the graphene patch is marked with a black solid line. Note that the two-half-wavelength resonance in the middle panel (and all even orders in general) are dark to a normally-incident $E_x$-polarized plane wave.
The metasurface will be illuminated with a linearly polarized, normally-incident plane wave with the electric field along the $x$ axis. The length and width of each patch are $w_x$ and $w_y$, respectively. Upon illumination with $E_x$ radiation, graphene surface plasmon polaritons (SPPs) are excited on the graphene patches and propagate along the $\pm x$ axis, since SPPs are TM polarized. They form standing waves that fit inside the patch length $w_x$, as shown in the insets in Fig. 1 taken at an $xz$ plane. They depict resonances of one-, two- and three-half-wavelengths inside $w_x$ and are obtained by eigenvalue simulations. Note that the two-half-wavelength resonance in the middle panel (and all even orders in general) are dark to a normally-incident $E_x$-polarized plane wave, i.e., they would not be excited in this case, but we can solve for them via eigenvalue simulations. The backreflector further enhances the strength (quality factor) of the resonances [44,47,48], which is important for attaining high conversion efficiency. In addition, it can also act as an electrode for tuning graphene’s Fermi level.
2.2 Electromagnetic response and third harmonic conversion efficiency
To examine the effect of the patch size on the conversion efficiency, we let $r_{x}$ and $r_{y}$, defined according to $w_x = r_{x}a$ and $w_y = r_{y}a$, vary from 0.65 to 0.95 and and 0.1 to 0.95, respectively. For every combination of $w_x$ and $w_y$, we calculate the THG conversion efficiency. Details regarding the nonlinear simulations and graphene linear and nonlinear properties can be found in the Appendix. In each case, the operating (fundamental) frequency is selected to match with the the absorption peak of the fundamental (half-wavelength standing wave) resonance of the graphene meta-atom. The results are depicted in Fig. 2. Varying $w_x$ changes the length of the underlying finite SPP waveguide segment and modifies the frequencies of the standing wave resonances that are formed. In addition, varying $w_y$ (width of the underlying SPP waveguide) modifies the dispersion of the propagating surface plasmon and thus provides an additional degree of freedom in this case of 2D-patterned graphene metasurfaces over 1D-patterned graphene ribbons [44]. Note that the standing wave resonances are not anticipated at integer multiples of the fundamental mode due to strong material dispersion of graphene at THz frequencies (dependence of $\sigma ^{(1)}$ on frequency $\omega$). Thus, $w_y$ is an essential degree of freedom toward adjusting the resonant frequencies and aligning them with the fundamental and third harmonic. Additional advantages of 2D over 1D patterning, is that we can further confine the plasmons and produce hot spots with even stronger intensities boosting the conversion efficiency, as well as the ability for polarization insensitive response in the $x$ and $y$ axes. The latter will be further discussed in Section 3.
Fig. 2. Conversion efficiency (for input intensity of 0.1 MW/cm$^2$) of the third harmonic generation process under continuous wave plane wave illumination (normal incidence) as a function of patch size: $w_x = r_{x}a$ and $w_y = r_{y}a$. The operating (fundamental) frequency is for each point selected to match the absorption peak corresponding to the fundamental (half-wavelength standing wave) resonance of the structure. The optimum conversion efficiency is obtained for $r_{x}=0.84$ and $r_{y}=0.69$ (marked with a black dot) and equals −20dB (1%).
The maximum THG conversion efficiency (for input intensity of 0.1 MW/cm$^2$) is found to be approximately −20dB (1%) and is obtained for a rectangular patch with $r_{x}=0.84$ and $r_{y}=0.69$. The point is marked with a black dot in Fig. 2. Note that there is a relatively wide dimension range for which the CE remains close to maximum, indicating that the design is not overly sensitive to deviations from the optimal dimensions. To unveil the physical mechanism behind the highest CE point we plot the linear plane-wave scattering spectrum (power coefficients) in Fig. 3. The operating frequency ($\omega _{\mathrm {FF}}$) is selected exactly at the absorption peak of 2.32 THz (black arrow), which is associated with the fundamental resonance of the structure. (The same is exercised for all parameter combinations depicted in Fig. 2). The second black arrow in the absorption spectrum of Fig. 3, depicts the TH frequency $\omega _{\mathrm {TH}}=3\omega _{\mathrm {FF}}=$6.96 THz, where radiation of the generated third harmonic will take place. It can be seen that the third harmonic is also nicely aligned with a pronounced absorption peak of the linear spectrum.
Fig. 3. Linear plane-wave scattering spectrum (power coefficients) for the optimum point in Fig. 2 ($r_{x}=0.84$, $r_{y}=0.69$). The operating frequency $\omega _{\mathrm {FF}}$ (marked with a black arrow) is aligned with the first absorption peak, which is associated with the fundamental resonance of the patch. The resulting third harmonic $\omega _{\mathrm {TH}}=3\omega _{\mathrm {FF}}$ is also marked, and is also nicely aligned with an absorption peak. Via eigenmode analysis, we identify the two resonances that are associated with the said absorption peaks (red dashed arrows) and mediate the the interaction of the incident and generated fields with the metasurface. The respective mode profiles can be found in Fig. 4.
We next perform an eigenmode analysis in order to identify the resonances in the vicinity of the fundamental and third harmonic frequencies. This way, we can unequivocally determine the metasurface resonances that mediate the interaction of the incident and generated fields with the metasurface and the third harmonic generation process in total. The identified resonances (eigenvalues) are marked with dashed (red) arrows in Fig. 3(a). They are associated with well-defined absorption peaks and in close proximity to the fundamental frequency and third harmonic frequency. The corresponding electric field components of the mode profiles (eigenvectors) are depicted in Fig. 4(a),(b) (right panels). For comparison, the field distributions extracted from the time-harmonic simulations are also depicted on Fig. 4(a),(b) [left panels]. They are almost indistinguishable, suggesting that we have successfully identified the resonances mediating the third harmonic generation. The total quality factors of the two resonances are found equal to approximately 3.5 and 11, respectively. The temporal confinement of energy in the metasurface along with the strong spatial confinement by the tightly confined graphene plasmons boost the conversion efficiency. Note that for the correct calculation of the quality factor, the highly dispersive nature of graphene in the THz regime should be taken into account [49].
Fig. 4. (a) Comparison of field distributions (electric field components, absolute value) at the fundamental frequency, as obtained from scattering simulations (left) and from eigenmode analysis (right) for the optimum metasurface with $r_x=0.84$ and $r_y=0.69$. (b) The same as in (a) but for the third harmonic frequency. The excellent agreement between scattering and eigenvalue simulations testifies that the resonances mediating the interaction of the metasurface with the incident and generated field are unambiguously identified.
Finally, we need to verify that the selection of the operating frequency at the first absorption peak (2.32 THz) is indeed the optimal choice in terms of conversion efficiency. Specifically, in Fig. 5 we plot the CE for different values of operating frequency. Indeed, lower or higher operating frequencies lead to decreased CE. Aligning the operating frequency with the absorption peak associated with the first resonance is the optimum choice in our structure due to minimization of reflection and the full exploitation of the enhanced local fields that accompany the resonance.
Fig. 5. Calculated THG conversion efficiency (black curve) for the optimum structure ($r_x=0.84$ and $r_y=0.69$) for different choices of fundamental frequency (FF). Maximum THG CE corresponds to $\omega _{\mathrm {FF}}=2.32$ THz. The maximum of linear absorption is found at the same frequency (red curve).
3. Cross graphene patch metasurface
In the previous section (Section 2), the optimum patch geometry is found to be rectangular ($w_x\neq w_y$). As a result, the metasurface will not have the same response for the two linear polarizations (along the $x$ or $y$ axis). In this Section, we propose an alternative design that exhibits isotropic response, i.e., identical response for the two orthogonal linear polarizations (for normal incidence).
3.1 Metasurface structure
The rectangular patch discussed in the previous section, is copied and rotated by 90 degrees resulting in a cross geometry [Fig. 6(a)]. The length and width of each rectangle comprising the cross (they are both identical) is described by the parameters $w_x$ and $w_y$, as previously. The remaining parameters are kept constant: dielectric substrate with thickness $h=6.3~\mu m$, relative electric permittivity $\varepsilon _r=4.41$, lattice constant $a=a_x=a_y=3.88~\mu$m.
Fig. 6. (a) Graphene-cross isotropic metasurface composed of two identical rectangular patches ($w_x \times w_y$) positioned perpendicular to each other. The meta-atoms have been arranged in a square periodicity with period $a=3.88\mu m$, on dielectric slab with relative permittivity $\varepsilon _r=4.41$ and thickness $h=6.3~\mu$m backed by a gold backreflector. (b) THG conversion efficiency (for input intensity of 0.1 MW/cm$^2$) as a function of rectangle size: $w_x = r_{x}a$ and $w_y = r_{y}a$. The optimum conversion efficiency is obtained for $r_{x}=0.9$ and $r_{y}=0.6$ (marked with a black dot) and equals approximately −20dB (1%). The upper left part of the plot is left blank, since it corresponds to already covered geometrical parameters.
3.2 Electromagnetic response and third harmonic conversion efficiency
In order to examine the effect of the geometry on the THG conversion efficiency, we vary the dimensions of the cross resonator by varying the dimensions of the underlying rectangles (concurrently, so that the structure remains symmetric). The results are depicted in Fig. 6(b), where we have covered the range 0.60 to 0.95 for $r_{x}$ and 0.1 to 0.90 for $r_{y}$. The upper left part of the plot is left blank since it corresponds to already covered material combinations. The maximum THG conversion efficiency is found again approximately −20dB (1%), for the point with $r_x=0.9$ and $r_y=0.6$, marked with a black dot. The optimum geometric parameters remain close to those of Section 2.
We next investigate the underlying resonances of the structure. The linear plane-wave scattering spectrum (power coefficients) for the cross with $r_x=0.9$ and $r_y=0.6$ is presented on Fig. 7. With black arrows, we mark the chosen operating (fundamental) frequency at 2.2 THz (coinciding with the first absorption peak) and the corresponding TH frequency at 6.6 THz, which is located very close to a well-defined strong absorption peak. The two absorption peaks are associated with eigenmodes at 2.2 and 6.9 THz, respectively, as identified by eigenvalue simulations (red dashed arrows). The quality factors of the two resonances are found equal to approximately 5.2 and 21.1, respectively. Again, the enhanced local fields that accompany the resonances are exploited to the fullest since we have positioned both the fundamental and the third harmonic frequency in the vicinity of metasurface resonances. This is further corroborated in Fig. S1 of the Supplement 1, where it is shown that the optimum choice for the operating (fundamental) frequency is at 2.2 THz, i.e., coinciding with the first absorption peak. A similar conclusion was drawn earlier, in Fig. 5, for the rectangular geometry.
Fig. 7. Linear plane-wave scattering spectrum (power coefficients) for the optimum point in Fig. 6(b) ($r_x=0.9$ and $r_y=0.6$). The operating frequency $\omega _{\mathrm {FF}}$ (marked with a black arrow) is aligned with the first absorption peak, which is associated with the fundamental resonance of the patch. The resulting third harmonic $\omega _{\mathrm {TH}}=3\omega _{\mathrm {FF}}$ is also marked, and is nicely aligned with a pronounced absorption peak. Via eigenmode analysis we identify the two resonances that are associated with the said absorption peaks (red dashed arrows).
4. Practical considerations: comparison of finite and infinite system
With an eye on practical considerations, we next investigate the impact of a finite metasurface extent, as would be the case in an actual sample. Thus far, we have been considering a perfectly periodic system that extends to infinity. Truncating the structure will possibly modify the position and linewidth of the resonances (truncation results in some light leakage in the perpendicular direction). Assessing the possible impact is important from a practical standpoint. The modeling of a finite system of many unit cells would be quite computationally costly in 3D; thus we choose to perform this comparison on a 2D system of graphene ribbons (inset in Fig. 8).
Fig. 8. Comparison of absorption spectra of a periodic system excited with a plane wave (red curve) and a finite system excited with a Gaussian beam (black curve), see insets. The ribbons have width $w=0.75a$, where $a=3.88~\mu$m, and are considered infinite along the lateral direction. The finite structure comprises 257 ribbons corresponding to 10$\lambda _0$ at 3 THz. The remaining geometric parameters are as previously: gold-backed dielectric substrate with $\varepsilon _{\mathrm {r}}=4.41$ and thickness $h=6.3~\mu$m.
The ribbon width is $w=0.75a$, where $a=3.88~\mu$m. The substrate is again a dielectric slab with $\varepsilon _{\mathrm {r}}=4.41$ and thickness $h=6.3~\mu$m, backed with a gold reflector. For the infinite periodic system, periodic boundary conditions and plane wave excitation are used, as exercised thus far. The finite system, on the other hand, comprises 257 graphene ribbons, such that the metasurface extent is 10 mm, i.e., ten times the free-space wavelength at the fundamental frequency (3THz). The finite system is excited with a Gaussian beam focused on the metasurface with a waist radius of $3\lambda _{\mathrm {FF}}=300~\mu$m.
The calculated linear absorption spectra for the two systems are depicted in Fig. 8. Evidently, the two cases do not vary considerably, indicating that a metasurface extent of $10\lambda _0$ is adequate to guarantee good performance, provided, of course, that the width of the Gaussian beam illumination is nicely accommodated in the structure. Due to the similarity of the linear spectra, we also anticipate similar THG conversion efficiencies. Indeed, placing the fundamental frequency at the first absorption peak (3 THz), we calculate for the periodic system a CE −25.8 dB and a CE of −22.9 dB for the finite structure. In this particular case, the finite system possesses a higher conversion efficiency due to a resonance peak approaching the third harmonic frequency of $\sim 9$ THz.
5. Conclusions
We have investigated multiresonant graphene metasurfaces, operating at the THz regime, for efficient third harmonic generation. Two variants of 2D-patterned graphene patches have been considered: a rectangular geometry leading to anisotropic response, and a cross geometry resulting in isotropic response. The 2D patterning enables higher conversion efficiencies compared to 1D-patterned graphene stripes in the literature, since it allows to better tune the spectral positions of the supported modes and achieves stronger plasmon confinement. In both cases, the patches have been carefully designed to achieve maximum conversion efficiency. The highest efficiency for both MS designs has been found to be −20dB or 1%. These values refer to an input intensity of 0.1 MW/cm$^2$; the generated power in the third harmonic exhibits a cubic dependence on input power for third order phenomena and, consequently, the conversion efficiency increases for higher input power (see Supplement 1, Fig. S2). Such high efficiencies have been achieved by aligning the fundamental and third harmonic frequencies with two resonances of the structure, as verified with linear scattering spectra and eigenvalues simulations, providing physical insight and highlighting the underlying operation principles.
With an eye on practical applications, we have verified that the conversion efficiency does not deteriorate when finite-extent metasurfaces are used (Gaussian beam excitation), instead of infinite periodic structures. The designed metasurfaces clearly demonstrate that graphene, in combination with properly designed resonances, can provide a promising path for enhanced nonlinear phenomena and processes, further advancing current capabilities in nonlinearity-related applications.
Appendix
Third harmonic generation simulation
Third-harmonic generation (THG) is a third order nonlinear phenomenon in which three photons at fundamental frequency (FF) $\omega _{\mathrm {FF}}$ produce a single photon at third harmonic frequency (TH) $3\omega _{\mathrm {FF}}$. THG in the nonlinear metasurface under study is simulated using full wave electromagnetic simulations with COMSOL Multiphysics commercial software (Finite Element Method). In order to calculate the nonlinear response of the metasurface under continuous wave (CW) illumination, we decouple the nonlinear problem into two linear frequency domain simulations, at the fundamental and third-harmonic frequency [44,50]. This is generally not possible for pulsed excitation.
The first simulation is conducted at the fundamental frequency with plane wave excitation. Graphene is naturally modeled as an infinitesimally-thin material via a surface current boundary condition. Graphene is highly dispersive; its linear conductivity at the THz regime can be described by a Drude dispersion model:
where $D = {q^2\mu _c/\hbar^2}$, $\tau =0.5$ ps is the electron relaxation time, $q$ is the elementary charge, $\mu _{\mathrm {c}}=0.3$ eV is the chemical potential and $\hbar$ is the reduced Planck’s constant.The second step is to calculate the induced (linear and nonlinear) current density:
Note that Eq. (3) predicts a purely imaginary third order conductivity. More accurate expressions for $\sigma ^{(3)}$ have been proposed in the literature [54,55] involving the relaxation time $\tau$ and leading to complex values. Next, we investigate the impact of such a more accurate expression on the conversion efficiency. We consider the optimum design of the rectangular-patch geometry (see Section 2). We resort to Ref. [55] and focus on the intraband contribution to $\sigma ^{(3)}$ which is dominant at THz (and microwave) frequencies. The corresponding expression is of the form of Eq. (3) with the substitution $\omega _{\mathrm {FF}}\rightarrow \omega _{\mathrm {FF}}+1/\tau$. Evaluating at the operating (fundamental) frequency of 2.32 THz we find $\sigma ^{(3)}=(0.388+i1.515)\times 10^{-17}$ S(m/V)$^2$; the imaginary part is only slightly different compared to that predicted by Eq. (3) [$i1.583\times 10^{-17}$] and the newly-introduced real part is approximately four times smaller. Adopting this new value, we find that the impact on the conversion efficiency is negligible, only 0.2 dB difference with respect to the value of −20 dB reported in Section 2. In addition, we anticipate that the geometrical sweep appearing in Fig. 2 will not change qualitatively, i.e., the optimum geometrical parameter combination will remain practically unaffected.
Funding
Hellenic Foundation for Research and Innovation (916).
Acknowledgments
This work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects to support Post-doctoral Researchers” (Project Number: 916, PHOTOSURF).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011). [CrossRef]
2. N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11(11), 917–924 (2012). [CrossRef]
3. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013). [CrossRef]
4. S. B. Glybovski, S. A. Tretyakov, P. A. Belov, Y. S. Kivshar, and C. R. Simovski, “Metasurfaces: From microwaves to visible,” Phys. Rep. 634, 1–72 (2016). [CrossRef]
5. N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013). [CrossRef]
6. G. Perrakis, O. Tsilipakos, G. Kenanakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Perfect optical absorption with nanostructured metal films: design and experimental demonstration,” Opt. Express 27(5), 6842–6850 (2019). [CrossRef]
7. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D. P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef]
8. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef]
9. O. Tsilipakos, L. Zhang, M. Kafesaki, C. M. Soukoulis, and T. Koschny, “Experimental implementation of achromatic multiresonant metasurface for broadband pulse delay,” ACS Photonics 8(6), 1649–1655 (2021). [CrossRef]
10. X. Liu, K. Fan, I. V. Shadrivov, and W. J. Padilla, “Experimental realization of a terahertz all-dielectric metasurface absorber,” Opt. Express 25(1), 191–201 (2017). [CrossRef]
11. O. Tsilipakos, A. Xomalis, G. Kenanakis, M. Farsari, C. M. Soukoulis, E. N. Economou, and M. Kafesaki, “Split-cube-resonator-based metamaterials for polarization-selective asymmetric perfect absorption,” Sci. Rep. 10(1), 17653 (2020). [CrossRef]
12. S. A. Mikhailov, “Theory of the giant plasmon-enhanced second-harmonic generation in graphene and semiconductor two-dimensional electron systems,” Phys. Rev. B 84(4), 045432 (2011). [CrossRef]
13. Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5(1), 5753 (2014). [CrossRef]
14. S.-X. Xia, X. Zhai, L.-L. Wang, and S.-C. Wen, “Plasmonically induced transparency in double-layered graphene nanoribbons,” Photonics Res. 6(7), 692–702 (2018). [CrossRef]
15. Z. Li, W. Liu, Z. Li, H. Cheng, S. Chen, and J. Tian, “Fano-resonance-based mode-matching hybrid metasurface for enhanced second-harmonic generation,” Opt. Lett. 42(16), 3117–3120 (2017). [CrossRef]
16. L. Carletti, K. Koshelev, C. D. Angelis, and Y. Kivshar, “Giant nonlinear response at the nanoscale driven by bound states in the continuum,” Phys. Rev. Lett. 121(3), 033903 (2018). [CrossRef]
17. Z. Liu, Y. Xu, Y. Lin, J. Xiang, T. Feng, Q. Cao, J. Li, S. Lan, and J. Liu, “High-q quasibound states in the continuum for nonlinear metasurfaces,” Phys. Rev. Lett. 123(25), 253901 (2019). [CrossRef]
18. K. Koshelev, Y. Tang, K. Li, D.-Y. Choi, G. Li, and Y. Kivshar, “Nonlinear metasurfaces governed by bound states in the continuum,” ACS Photonics 6(7), 1639–1644 (2019). [CrossRef]
19. G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Efficient third harmonic generation and nonlinear subwavelength imaging at a higher-order anapole mode in a single germanium nanodisk,” ACS Nano 11(1), 953–960 (2017). [CrossRef]
20. C. Cui, C. Zhou, S. Yuan, X. Qiu, L. Zhu, Y. Wang, Y. Li, J. Song, Q. Huang, Y. Wang, C. Zeng, and J. Xia, “Multiple fano resonances in symmetry-breaking silicon metasurface for manipulating light emission,” ACS Photonics 5(10), 4074–4080 (2018). [CrossRef]
21. J. F. Algorri, F. Dell’Olio, P. Roldán-Varona, L. Rodríguez-Cobo, J. M. López-Higuera, J. M. Sánchez-Pena, and D. C. Zografopoulos, “Strongly resonant silicon slot metasurfaces with symmetry-protected bound states in the continuum,” Opt. Express 29(7), 10374–10385 (2021). [CrossRef]
22. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]
23. F. H. L. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett. 11(8), 3370–3377 (2011). [CrossRef]
24. Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012). [CrossRef]
25. S.-X. Xia, X. Zhai, Y. Huang, J.-Q. Liu, L.-L. Wang, and S.-C. Wen, “Multi-band perfect plasmonic absorptions using rectangular graphene gratings,” Opt. Lett. 42(15), 3052–3055 (2017). [CrossRef]
26. F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008). [CrossRef]
27. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4(7), 532–535 (2008). [CrossRef]
28. K. S. Novoselov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). [CrossRef]
29. A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]
30. E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105(9), 097401 (2010). [CrossRef]
31. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). [CrossRef]
32. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]
33. P. Tassin, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nat. Photonics 6(4), 259–264 (2012). [CrossRef]
34. T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014). [CrossRef]
35. S. A. Mikhailov, “Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model,” Phys. Rev. B 100(11), 115416 (2019). [CrossRef]
36. E. Dremetsika, B. Dlubak, S.-P. Gorza, C. Ciret, M.-B. Martin, S. Hofmann, P. Seneor, D. Dolfi, S. Massar, P. Emplit, and P. Kockaert, “Measuring the nonlinear refractive index of graphene using the optical kerr effect method,” Opt. Lett. 41(14), 3281–3284 (2016). [CrossRef]
37. T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94(6), 062219 (2016). [CrossRef]
38. M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97(9), 093101 (2010). [CrossRef]
39. H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018). [CrossRef]
40. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ2 and χ3 harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007). [CrossRef]
41. J. W. You, J. You, M. Weismann, and N. C. Panoiu, “Double-resonant enhancement of third-harmonic generation in graphene nanostructures,” Philos. Trans. R. Soc., A 375(2090), 20160313 (2017). [CrossRef]
42. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009). [CrossRef]
43. A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011). [CrossRef]
44. B. Jin, T. Guo, and C. Argyropoulos, “Enhanced third harmonic generation with graphene metasurfaces,” J. Opt. 19(9), 094005 (2017). [CrossRef]
45. Q. Ren, J. W. You, and N. C. Panoiu, “Large enhancement of the effective second-order nonlinearity in graphene metasurfaces,” Phys. Rev. B 99(20), 205404 (2019). [CrossRef]
46. J. W. You and N. C. Panoiu, “Tunable and dual-broadband giant enhancement of second-harmonic and third-harmonic generation in an optimized graphene-insulator-graphene metasurface,” Phys. Rev. B 102(12), 121403 (2020). [CrossRef]
47. N. A. Savostianova and S. A. Mikhailov, “Giant enhancement of the third harmonic in graphene integrated in a layered structure,” Appl. Phys. Lett. 107(18), 181104 (2015). [CrossRef]
48. N. A. Savostianova and S. A. Mikhailov, “Third harmonic generation from graphene lying on different substrates: optical-phonon resonances and interference effects,” Opt. Express 25(4), 3268–3285 (2017). [CrossRef]
49. T. Christopoulos, O. Tsilipakos, G. Sinatkas, and E. E. Kriezis, “On the calculation of the quality factor in contemporary photonic resonant structures,” Opt. Express 27(10), 14505–14522 (2019). [CrossRef]
50. T. Christopoulos, O. Tsilipakos, G. Sinatkas, and E. E. Kriezis, “Degenerate four-wave mixing in nonlinear resonators comprising two-dimensional materials: A coupled-mode theory approach,” Phys. Rev. B 98(23), 235421 (2018). [CrossRef]
51. J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014). [CrossRef]
52. S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” EPL 79(2), 27002 (2007). [CrossRef]
53. S. A. Mikhailov and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys.: Condens. Matter 20(38), 384204 (2008). [CrossRef]
54. J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature,” Phys. Rev. B 91(23), 235320 (2015). [CrossRef]
55. S. A. Mikhailov, “Quantum theory of the third-order nonlinear electrodynamic effects of graphene,” Phys. Rev. B 93(8), 085403 (2016). [CrossRef]
