## Abstract

Energy transfer is an essential light-matter interaction. The transfer efficiency is critical for various applications such as light-emitting, optical modulation, and the photoelectric effect. Two primary forms of light-matter energy transfer, including absorption and emission, can be enhanced in optical cavities. Both forms can reach an extremum inside the cavity according to the coupled-mode theory. Graphene conductivity at the terahertz frequency can be tuned from positive to negative, providing a suitable material to study switchable extremums of these two forms. We integrate graphene with a nested cavity where an infrared cavity is inserted in a terahertz cavity, thereby achieving terahertz perfect absorption at the static state and optimal gain under photoexcitation. Leveraging an inserted infrared cavity, we can elevate the working efficiency by strongly absorbing the infrared pump. We also numerically show the feasibility of electrically tunable extreme energy transfer. Our concept of the nested cavity can be extended to different materials and even to guided modes. A switchable synergy of loss and gain potentially enables high-contrast dynamic modulation and photonic devices with multiplexing functions.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Absorption and emission are two primary forms of light-matter energy transfer. Transfers with extreme efficiencies, i.e., perfect absorption and optimal gain, are often desired to maximize performance. For example, perfect absorption can vastly boost the working efficiency of absorptive photonic devices like solar cells [1], passive coolers [2], and detectors [3]. At the same time, the optimal gain can enhance the output of lasers [4] or amplifiers [5,6] and reduce the non-radiative energy loss [7].

Light interacts with loss medium or gain medium for multiple roundtrips in optical cavities, providing an excellent approach for extreme energy transfer [3,4]. Fig. 1(a) and Fig. 1(b) depict the schematics of perfect absorption and optimal gain in the presence of optical cavities. The reflection coefficients of the front and back mirrors are *r*_{1} and *r*_{2}, respectively. Due to the imperfect reflection of mirrors, there exists a radiative loss rate of *γ*_{r}. *α* and *g* in Fig. 1 are the intrinsic loss and gain rates, respectively. The absorption inside the cavity is then given by the coupled-mode theory [8],

Here *δ* is *α* or -*g*, *ω* is the angular frequency of the signal, and *ω*_{0} is the mode frequency of cavities. Assuming the signal is on resonance (*ω* = *ω*_{0}), the absorption *A* reaches two extremums (1 and -∞) when |*δ*| = *γ*_{r}. The first extremum, 1, appears when *γ*_{r} = *α*, corresponding to the perfect absorption. The second extremum, -∞, appears when *γ*_{r} = -*g*, denoting an optimal gain.

Direct bridging of perfect absorption and optimal gain in a single device may enable high-contrast dynamic modulation, photonic devices with multiplexing functions, and interesting physical phenomena [9–12]. However, optical cavities conventionally access only one extreme energy transfer. To access two extremums, we need to integrate optical cavities with a material with dynamic dielectric responses [13,14]. Graphene is a monolayer of graphite [15,16] with a dynamic conductivity under optical excitations and electrical injection [17–19]. As shown in Fig. 1(c), the electrons in the valence band can be excited to the conduction band. Because of the fast intraband scattering and relatively slow interband recombination, photoexcited carriers gather at the band edge, leading to a population inversion [17,20–23] even at room temperature [19,24]. This kind of carrier dynamic has been theoretically discussed [17,20,25], and the population inversion decaying in ∼100 fs scale has been observed [21,22,24,26]. Assuming that graphene is un-doped for a simple illustration, the real part of the terahertz (THz) conductivity can dynamically vary from positive to negative by altering the nonequilibrium Fermi levels *E*_{n} (also called “chemical potentials”) as shown in Fig. 1(d). Thanks to the dynamic conductivity of graphene, efficient absorbers, all-optical THz modulators [27–30], amplifiers [31–36], and even lasers [37–40] have been studied. The efficiency of light-graphene interaction is limited by the ultrathin thickness and limited density of states, especially at the low energy level. The contrast ratio is the key index for amplitude modulation, while perfect absorption [41] enables an optimal zero state. Besides, Auger recombination exists due to many-particle effects [42], even under the suppression of the Dirac dispersion of graphene bands. Therefore, radiative recombination can benefit from extreme gain when competing with Auger recombination. Meanwhile, most proposed all-optical THz modulators and emitters waste most optical excitations due to the limited pump absorption.

Here, we numerically propose a nested cavity to numerically demonstrate switchable extreme energy transfer between THz waves and graphene. Figure 1(f) shows our concept of the nested Bragg cavity, where graphene is inserted in both THz cavity and IR cavity. When the IR pump is small, our nested Bragg cavity simultaneously performs as THz perfect absorber and IR strong absorber, thereby efficiently using the pump and achieving THz extreme energy transfer simultaneously. Our calculation shows that our nested cavity has combined functions of two cavities working individually. The carrier heating induced by photoexcitation is taken into consideration. The required IR intensity to modulate THz absorption from 1 to 0 is about 0.17 MW cm^{-2}, achieving a high contrast ratio and low excitation power. By further raising the pump strength to 0.61 MW cm^{-2}, our nested Bragg cavity works as an amplifier and reaches extreme amplification of 14.9 dB when the cavity gain rate compensates for the radiative loss rate.

## 2. Results

Figures 2(a)-(b) illustrates the schematic structure of our nested cavity. Both THz and IR cavities are Bragg-type cavities, consisting of the front distributed Bragg reflector (DBR), back DBR, and cavity layer. DBR periodically stacks two types of dielectric layers with an optical length of a quarter of the wavelength, thereby forming a constructive interference in reflection [43]. In this work, we choose SiO_{2}/Air DBR for the THz cavity and Si/SiO_{2} DBR for the IR cavity (See **Methods** for the dielectric response of all materials used in this work.). Since THz wavelength is ∼200 times of IR wavelength, the THz response of the nested cavity is almost identical to the individual THz cavity. In this work, we only consider a linear response of the THz field.

Our nested cavity aims to realize the functions of the IR cavity and THz cavity separately. To show the individual response of IR and THz cavities and the influence of being bound together. We start with a separate design of IR perfect absorber for 1550 nm and THz perfect absorber for 1 THz at normal incidence. Here, the condition of perfect absorption is set to be absorption *A* greater than 99%. The polarization of IR and THz waves are both set to be *p*-polarization, and graphene is set to be one layer in optimization flow. The cycles of DBRs are the only undetermined geometry parameters. We use the particle swarm optimization (PSO) algorithm to find suitable numbers (see **Methods** for technical details about PSO). The cycles of IR front and back DBRs are chosen as 2 and 27 after optimization. The cycle of THz front and back DBRs are selected as 3 and 35. Figure 2(c) plots the absorption spectra of the individual THz cavity without IR cavity at *E*_{n} = 0 as a function of the incident angle. We can obtain perfect absorption at 1 THz with normal incidence. The IR response of the individual IR cavity is illustrated in Fig. 2(d). Similar to the THz cavity, we can realize approximate perfect absorption at 1.55 µm with normal incidence. The slight imperfection is attributed to the discrete value choice for the cycles of DBRs.

To insert IR cavity into THz cavity without disturbing the optical length of THz cavity layer and still realize an absorption peak at 1 THz with normal incidence, we adjust the thickness of SiO2 cavity layer appropriately. Figure 2(e) illustrates the absorption spectra *A* of the nested cavity with a very similar shape to Fig. 2(c), proving that inserted IR cavity has little impact on the THz response of the THz cavity. The Q-factor at 1 THz is ∼244. When increasing the IR pump intensity, the THz conductivity of graphene decreases, inducing a tunable THz response of the nested cavity. Considering the ultra-thin thickness of graphene, the mode frequency of the THz cavity remains almost constant with the IR pump. Figure 2(f) plots the absorption spectra of the nest cavity in the IR range. Compared with Fig. 3(d), there is a rugged pattern modulating the original absorption spectra, and the wavelength of perfect absorption is slightly shifted. This pattern is mainly attributed to the interference caused by THz DBR.

Because the transmission of the nested cavity at 1THz and 1550 nm vanishes at normal incidence, the reflectance *R* nearly equals 1-*A*. Figure 2(g) depicts the reflectance in log scale, i.e., 10log_{10}(*R*), of our nested cavity as a function of *E*_{n}. When there is one graphene layer, the nested cavity performs as a perfect absorber without the IR pump. However, due to the limited gain rate provided by one graphene layer, there is no optimal gain at *E*_{n}<100 meV. We then increase the number of layers to 2. The nested cavity performs as a perfect absorber at *E*_{n} = 3 meV, as marked by the orange circle in Fig. 2(g). When the THz conductivity equals zero at *E*_{n} = 32.5 meV, the nested cavity performs as a total reflector marked by the purple circle in Fig. 2(g). For amplitude modulation applications, this means a zero-to-unity modulation without insertion loss. The red circle marks out a giant gain of 14.9 dB marked by at *E*_{n} = 87.3 meV. Figure 2(h) shows the IR reflectance spectra as a function of *E*_{n}, with the reflectance below 0.1. The variation of the IR reflectance is mainly caused by carrier heating.

The strong IR absorption fulfills our goal of the efficient usage of the IR pump. We can alter the operation wavelength and angle of perfect absorption and optimal gain by adjusting the IR pump strength according to Fig. 2(e). The required IR power for one graphene layer at a stable state is given by the conservation of carrier number [17],

*ω*

_{pump}is the IR photon energy,

*A*

_{inter}is the pump absorption contributed by interband transitions,

*τ*

_{R}is the recombination time of electron-hole pairs, and

*n*is the carrier density in the conduction band. Calculations of the relation between

*E*

_{n}and

*T*

_{e}can be seen in the

**Method**section. The required IR intensity to modulate THz absorption from 1 to 0 is about 0.17 MW cm

^{-2}, and a pump strength of 0.61 MW cm

^{-2}is required to reach an optimal gain of 14.9 dB.

## 3. Discussion

We have numerically calculated two extremums of energy transfer between THz waves with graphene in a nested Bragg cavity, where the DBRs are used to tailor the radiative loss. This work aims to conduct a proof of concept for our nested cavity to realize two extreme energy transfer cases in one single device. The intrinsic graphene without unintentionally doping [19] is assumed here for a simple illustration. In principle, the population inversion and optical gain are possible in doped graphene [21–23], but the net gain is hampered by the material loss caused by doping [44]. The relatively high IR power is mainly due to the high photon energy of the IR pump (∼0.8 eV) and the fast recombination of electron-hole pairs (∼100 fs). Carrier heating hampers population inversion and eventual gain [19]. We may use a CO_{2} laser as the pump source with a smaller photon energy of 0.12 eV (10.6 µm) than 0.8 eV, thus reducing the energy waste and weakening consequent carrier heating. The active material can also be other materials [12,45,46] with longer non-radiative lifetimes like HgCdTe [42] to reduce the required pump intensity. The back mirrors can also adopt metals, polar materials with Reststrahlen bands [47]. Some highly reflective meta-structures are also good candidate reflectors to replace DBRs. Furthermore, our concept of the nested cavity for extreme energy transfer can also be extendable to guided modes by designing waveguide cavities [48,49].

Here, we adopt an Au reflector as THz back mirror as a template, as illustrated in Fig. 3(a). Distinct from DBR, Au reflector induces a phase change of ∼π on the reflected wave. In this case, the optical length of the cavity layer is a quarter of the wavelength. Similar to the previous procedure, we reduce the thickness of the SiO_{2} cavity layer to realize high absorption at 1 THz and insert IR cavity. Figure 3(d) shows the dynamical THz reflectance of the nested cavity using an Au reflector. We can obtain two extremums of -43.6 dB and 14.1 dB by integrating three layers of graphene into the cavity. Figure 3(c) shows the THz of our nested cavity using Au reflector at *E*_{n} = 20 meV, with a Q-factor of ∼ 104 for 1-*R*. The difference between 1-*R* and the graphene absorption, as well as the drop of Q-factor compared to the nested Bragg cavity, is attributed to the absorption in the Au reflector. To increase the absorption contribution of graphene, we can stack multilayer graphene or apply other low-loss reflectors.

The conductivity of graphene can also be dynamically controlled through electrical gating. We replace the IR cavity with a field-effect structure to form a THz cavity that provides electrical tuning on graphene conductivity. As shown in Fig. 3(d), we adopt a configuration of the dual-gate graphene channel field-effect transistor (DG-GFET) that was experimentally examined to have a broadband THz emission [19]. Using the same optimization flow as the nested cavity, we determine the cycle of the front DBR to be 4 and the cycle of the back DBR to be 22. The gate voltage *V*_{g} is set to be 5 V. The energy transfer between THz field and graphene then varies at different Drain-Source voltage *V*_{d}. The THz response of the THz cavity at *V*_{d}=0 is illustrated in Fig. 3(e). Perfect absorption is realized at normal incidence with a Q-factor of 1136. Fig. 3(f) calculates the reflectance of the THz cavity at different *V*_{d}. The energy transfer changes from perfect absorption at *V*_{d}=0 to optimal gain at *V*_{d}=47.5 mV.

## 4. Conclusion

In conclusion, we have proposed a nested Bragg cavity integrated with graphene that achieves extreme THz absorption and amplification through strongly absorbed IR excitation. The required IR intensity to modulate the nest cavity from perfect absorption to optimal gain is 0.61 MW cm^{-2}. The reflector that forms the nest cavity can also be replaced by an Au reflector, with a similar qualitative performance as the nested Bragg cavity. In addition, the graphene conductivity can also be dynamically controlled by electrical gating. Our results can inspire the exploration of extreme physics in light-matter energy transfer and serves as a good platform for high-contrast optical modulation and photonic devices with multiple functions.

## 5. Methods

#### 5.1 Dielectric response of materials

With optical excitation, the dynamic conductivity of graphene is [50],

*q*is the elementary charge,

*E*is the energy,

*ħ*is the reduced Planck’s constant,

*k*

_{B}is the Boltzmann constant,

*T*

_{e}is the temperature, and

*γ*

_{intra}(

*E*) is the energy-dependent intra-band momentum relaxation rate. The lattice temperature is set to be the ambient temperature. The first and second terms on the right side of Eq. (3) corresponds to the conductivity contribution from intraband transition and interband transition, respectively. The carrier distribution will change with photoexcitation. To consider the influence of photoexcitation on the intra-band momentum relaxation rate, we can resolve

*γ*

_{intra}(

*E*) and replace the original constant scattering rate

*τ*

^{-1}with

*γ*

_{intra}(

*E*). We adopt a usual expression for it,

*ζ*is a fitting number of order unity and

*E*

_{i}is a fitting number of energy unit. We use a typical value of 4 for

*ζ*and 100 meV for

*E*

_{i}.

Photoexcitation will alter the electron temperature and the momentum relaxation time of graphene [51,52]. The characteristic time of the ultrafast photoexcitation and consequent relaxation is at the level of 1 ps. In this work, the IR pulse duration is assumed much longer than this time scale. The carriers quickly redistribute via scattering processes like electron-electron (e-e) scattering [22,52]. The characteristic e-e scattering rate (on a time scale of ∼10 fs) is usually much faster than the interband transition rate. This condition breaks at pretty high signal intensity, which is not in the range of this work. Therefore, the carrier distribution in the conduction band under long-pulse photoexcitation can be described as a stable solution. As mentioned above, carriers in the conduction band and valence band follow separate Fermi-Dirac distribution functions. Two parameters remain undetermined for a given pump intensity, e.g., the electron temperature *T*_{e} and the nonequilibrium Fermi level *E*_{n}. Since we work at a stable state, we can use energy conservation and carrier cconservation [17,52] to obtain the relation between *T*_{e}, *E*_{n}, and the pump intensity *I*_{pump}.

The conservation of carrier energy provides,

*A*is the IR absorption,

*τ*

_{R}is the characteristic recombination time of electron-hole pairs,

*τ*

_{P}is the relaxation time through intra-band phonon emission,

*l*is the number of the graphene layer, and Ξ

_{e}is the carrier energy in the conduction band per unit area. The left side of Eq. (5) is the absorbed IR power. The first term at the right side of Eq. (5) is the inter-band recombination, and the second term is carrier cooling through phonon emission.

*E*

_{0}is the initial Fermi level. In this work,

*E*

_{0}equals zero. Ξ

_{e}(

*E*

_{0},

*T*

_{e}) and Ξ

_{e}(

*E*

_{n},

*T*

_{0}) are the quasi-final states for inter-band recombination and intra-band relaxation. These two processes drive graphene towards an equilibrium state of (

*E*

_{0},

*T*

_{0}). The coefficient “2” is because the carrier energy distributions in the conduction and valence bands are symmetric about the Dirac point. Ξ

_{e}(

*E*

_{n},

*T*

_{e}) is expressed as

*Li*

_{s}(

*z*) is the polylogarithm function. The conservation of carrier number provides,

*n*is the carrier density in the conduction band. The left side of Eq. is the generation rate of electron-hole pairs per area, while the right side is the recombination rate of electron-hole pairs per area.

*n*(

*E*

_{n},

*T*

_{e}) is expressed as

Combining Eq. (7) and Eq. (8), we can numerically obtain the relation between *E*_{n}, *T*_{e} and *I*_{pump}, as illustrated in Figs. 4(a) and 4(b). The distribution function is plotted in Fig. 4(c) when *E*_{n} = 87.3 meV. The carrier heating contributes to photoconductivity depending on the doping level [53–55], where photoconductivity is the photo-induced variation of conductivity. However, a negative conductivity can be obtained only when population inversion occurs.

The refractive index of SiO_{2} is set as 2 at the THz frequency and 1.44 at the IR frequency. The refractive index of Si is set as 3.4 at the THz frequency and 3.48 at the IR frequency. The Au permittivity is described by the Drude model [56],

*ɛ*

_{∞}= 1 is the high-frequency dielectric constant,

*ω*

_{p}= 1.372×10

^{16}rad s

^{-1}is the plasma frequency, and

*Γ*= 0.405×10

^{14}rad s

^{-1}is the Drude damping rate.

In the THz cavity integrated with the field-effect transistor, we adjust the thickness of SiC layer to support resonance at 5 THz, and the number of graphene layers is 2. *V*_{g} controls the injection level. We set *V*_{g} to be 5 V, thereby enabling a decent gain coefficient and reducing the risk of breakdown. *V*_{d} is set to be a variable parameter to control the energy transfer between graphene and THz wave. If there is initial doping in graphene, we can individually control *V*_{g1} and *V*_{g2} to obtain the expected carrier distribution. Other parameters remain unchanged compared to the Ref. [19]. Figure 4(d) plots the relation between graphene conductivity and *V*_{d}.

#### 5.2 Transfer matrix method

Since we use layered structure in this work, we can apply the widely used transfer matrix method (TMM) to calculate the response of our nested cavity. Based on Maxwell’s equations, TMM describes electromagnetic (EM) waves as field vectors containing two elements and the propagation of EM waves in layers as a transfer matrix. Two common expressions of field vectors are: two elements are tangent components of electric and magnetic fields; two elements are forward and backward propagating electric fields or magnetic fields. Here we use the first form [*E*_{t}; *H*_{t}], enabling a straightforward graphene model. The transfer matrix for layer *j* is given by

*δ*=

_{j}*n*d

_{j}*cos*

_{j}*θ*/c,

_{j}ω*η*=

_{j}*η*

_{0}

*n*cos

_{j}*θ*for s polarization and

_{j}*η*=

_{j}*η*

_{0}

*n*/cos

_{j}*θ*for p polarization. Here

_{j}*n*is the refractive index of layer

_{j}*j*,

*η*

_{0}is the optical admittance in free space which is given by

*η*

_{0}

^{2}=

*ɛ*

_{0}/

*µ*

_{0}, and

*θ*is the propagation angle which can be retrieved from the law of refraction. The reflection and absorption can be easily calculated. Here we give the transfer matrix of graphene in this work. Graphene is modeled as a conducting boundary [57]. Due to continuity of the tangent components of the electric field and magnetic field, we can obtain that

_{j}#### 5.3 Particle Swarm Optimization (PSO) algorithm

To reach perfect absorption for THz and IR cavities, we apply the PSO algorithm [58,59] to select suitable DBR cycles. The PSO algorithm iteratively improves the parameters to optimize the figure of merit (FOM). Each iteration, there is a population of candidate parameters in search space corresponding to the position of particles. These particles also have respective velocities according to the optimal local position. The position and velocity of particles are updated based on the speed of the last iteration and the optimal local position of this iteration, respectively. The PSO algorithm ends when FoMs of all particles converge. In this work, the optimization flow based on PSO is exhibited in Fig. 5. The population of each iteration is set to be 20. Since our cavity has no transmission, we set FoM to be 1-min(*R*).

## 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.

## References

**1. **J. Müller, B. Rech, J. Springer, and M. Vanecek, “TCO and light trapping in silicon thin film solar cells,” Sol. Energy **77**(6), 917–930 (2004). [CrossRef]

**2. **A. P. Raman, M. Abou Anoma, L. Zhu, E. Rephaeli, and S. Fan, “Passive radiative cooling below ambient air temperature under direct sunlight,” Nature **515**(7528), 540–544 (2014). [CrossRef]

**3. **M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-Integrated Graphene Photodetector,” Nano Lett. **12**(6), 2773–2777 (2012). [CrossRef]

**4. **N. V. Proscia, H. Jayakumar, X. Ge, G. Lopez-Morales, Z. Shotan, W. Zhou, C. A. Meriles, and V. M. Menon, “Microcavity-coupled emitters in hexagonal boron nitride,” Nanophotonics **9**(9), 2937–2944 (2020). [CrossRef]

**5. **N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics **3**(12), 715–719 (2009). [CrossRef]

**6. **A. Capua, O. Karni, G. Eisenstein, V. Sichkovskyi, V. Ivanov, and J. P. Reithmaier, “Coherent control in a semiconductor optical amplifier operating at room temperature,” Nat. Commun. **5**(1), 5025–5027 (2014). [CrossRef]

**7. **S. I. Bogdanov, O. A. Makarova, X. Xu, Z. O. Martin, A. S. Lagutchev, M. Olinde, D. Shah, S. N. Chowdhury, A. R. Gabidullin, and I. A. Ryzhikov, “Ultrafast quantum photonics enabled by coupling plasmonic nanocavities to strongly radiative antennas,” Optica **7**(5), 463–469 (2020). [CrossRef]

**8. **J. R. Piper and S. H. Fan, “Total Absorption in a Graphene Mono layer in the Optical Regime by Critical Coupling with a Photonic Crystal Guided Resonance,” ACS Photonics **1**(4), 347–353 (2014). [CrossRef]

**9. **K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. **108**(15), 153901 (2012). [CrossRef]

**10. **Y. Hadad, J. C. Soric, and A. Alu, “Breaking temporal symmetries for emission and absorption,” Proc. Natl. Acad. Sci. U. S. A. **113**(13), 3471–3475 (2016). [CrossRef]

**11. **V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. **88**(3), 035002 (2016). [CrossRef]

**12. **F. Hu, L. Li, Y. Liu, Y. Meng, M. Gong, and Y. Yang, “Two-plasmon spontaneous emission from a nonlocal epsilon-near-zero material,” Commun. Phys. **4**(1), 1–7 (2021). [CrossRef]

**13. **Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. **19**(19), 3077–3083 (2009). [CrossRef]

**14. **A. H. Castro 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]

**15. **M. Mittendorff, S. Winnerl, and T. E. Murphy, “2D THz Optoelectronics,” Adv. Opt. Mater. **9**(3), 2001500 (2021). [CrossRef]

**16. **F. H. Koppens, D. E. Chang, and F. Javier García de Abajo, “Graphene plasmonics: a platform for strong light–matter interactions,” Nano Lett. **11**(8), 3370–3377 (2011). [CrossRef]

**17. **V. Ryzhii, M. Ryzhii, and T. Otsuji, “Negative dynamic conductivity of graphene with optical pumping,” J. Appl. Phys. **101**(8), 083114 (2007). [CrossRef]

**18. **M. Ryzhii and V. Ryzhii, “Injection and population inversion in electrically induced p–n junction in graphene with split gates,” Jpn. J. Appl. Phys. **46**(No. 8), L151–L153 (2007). [CrossRef]

**19. **D. Yadav, G. Tamamushi, T. Watanabe, J. Mitsushio, Y. Tobah, K. Sugawara, A. A. Dubinov, A. Satou, M. Ryzhii, and V. Ryzhii, “Terahertz light-emitting graphene-channel transistor toward single-mode lasing,” Nanophotonics **7**(4), 741–752 (2018). [CrossRef]

**20. **A. Satou, F. T. Vasko, and V. Ryzhii, “Nonequilibrium carriers in intrinsic graphene under interband photoexcitation,” Phys. Rev. B **78**(11), 115431 (2008). [CrossRef]

**21. **T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. **108**(16), 167401 (2012). [CrossRef]

**22. **I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. Turcu, E. Springate, A. Stohr, A. Kohler, U. Starke, and A. Cavalleri, “Snapshots of non-equilibrium Dirac carrier distributions in graphene,” Nat. Mater. **12**(12), 1119–1124 (2013). [CrossRef]

**23. **I. Gierz, M. Mitrano, J. C. Petersen, C. Cacho, I. C. Turcu, E. Springate, A. Stohr, A. Kohler, U. Starke, and A. Cavalleri, “Population inversion in monolayer and bilayer graphene,” J. Phys.: Condens. Matter **27**(16), 164204 (2015). [CrossRef]

**24. **S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B **85**(3), 035443 (2012). [CrossRef]

**25. **D. Svintsov, V. Ryzhii, A. Satou, T. Otsuji, and V. Vyurkov, “Carrier-carrier scattering and negative dynamic conductivity in pumped graphene,” Opt. Express **22**(17), 19873–19886 (2014). [CrossRef]

**26. **J. C. Johannsen, S. Ulstrup, F. Cilento, A. Crepaldi, M. Zacchigna, C. Cacho, I. C. Turcu, E. Springate, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, and P. Hofmann, “Direct view of hot carrier dynamics in graphene,” Phys. Rev. Lett. **111**(2), 027403 (2013). [CrossRef]

**27. **Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nat. Photonics **10**(4), 227–238 (2016). [CrossRef]

**28. **F. Javier García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics **1**(3), 135–152 (2014). [CrossRef]

**29. **M. Rahm, J. S. Li, and W. J. Padilla, “THz Wave Modulators: A Brief Review on Different Modulation Techniques,” J. Infrared, Millimeter, Terahertz Waves **34**(1), 1–27 (2013). [CrossRef]

**30. **Q. Xing, C. Song, C. Wang, Y. Xie, S. Huang, F. Wang, Y. Lei, X. Yuan, C. Zhang, and L. Mu, “Tunable terahertz plasmons in graphite thin films,” Phys. Rev. Lett. **126**(14), 147401 (2021). [CrossRef]

**31. **T. Low, P.-Y. Chen, and D. Basov, “Superluminal plasmons with resonant gain in population inverted bilayer graphene,” Phys. Rev. B **98**(4), 041403 (2018). [CrossRef]

**32. **A. F. Page, F. Ballout, O. Hess, and J. M. Hamm, “Nonequilibrium plasmons with gain in graphene,” Phys. Rev. B **91**(7), 075404 (2015). [CrossRef]

**33. **T. Guo, L. Zhu, P.-Y. Chen, and C. Argyropoulos, “Tunable terahertz amplification based on photoexcited active graphene hyperbolic metamaterials,” Opt. Mater. Express **8**(12), 3941–3952 (2018). [CrossRef]

**34. **Q. L. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano **6**(5), 3677–3694 (2012). [CrossRef]

**35. **I. Kaminer, Y. T. Katan, H. Buljan, Y. Shen, O. Ilic, J. J. López, L. J. Wong, J. D. Joannopoulos, and M. Soljačić, “Efficient plasmonic emission by the quantum Čerenkov effect from hot carriers in graphene,” Nat. Commun. **7**(1), ncomms11880 (2016). [CrossRef]

**36. **F. Rana, J. H. Strait, H. Wang, and C. Manolatou, “Ultrafast carrier recombination and generation rates for plasmon emission and absorption in graphene,” Phys. Rev. B **84**(4), 045437 (2011). [CrossRef]

**37. **A. R. Davoyan, M. Y. Morozov, V. V. Popov, A. Satou, and T. Otsuji, “Graphene surface emitting terahertz laser: Diffusion pumping concept,” Appl. Phys. Lett. **103**(25), 251102 (2013). [CrossRef]

**38. **V. Ryzhii, A. A. Dubinov, T. Otsuji, V. Y. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express **21**(25), 31567–31577 (2013). [CrossRef]

**39. **V. V. Popov, O. V. Polischuk, S. A. Nikitov, V. Ryzhii, T. Otsuji, and M. S. Shur, “Amplification and lasing of terahertz radiation by plasmons in graphene with a planar distributed Bragg resonator,” J. Opt. **15**(11), 114009 (2013). [CrossRef]

**40. **R. Jago, T. Winzer, A. Knorr, and E. Malic, “Graphene as gain medium for broadband lasers,” Phys. Rev. B **92**(8), 085407 (2015). [CrossRef]

**41. **S. Thongrattanasiri, F. H. Koppens, and F. J. G. De Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. **108**(4), 047401 (2012). [CrossRef]

**42. **G. Alymov, V. Rumyantsev, S. Morozov, V. Gavrilenko, V. Aleshkin, and D. Svintsov, “Fundamental Limits to Far-Infrared Lasing in Auger-Suppressed HgCdTe Quantum Wells,” ACS Photonics **7**(1), 98–104 (2020). [CrossRef]

**43. **F. Hu, W. Jia, Y. Meng, M. Gong, and Y. Yang, “High-contrast optical switching using an epsilon-near-zero material coupled to a Bragg microcavity,” Opt. Express **27**(19), 26405–26414 (2019). [CrossRef]

**44. **T. Okumura, K. Oda, J. Kasai, M. Sagawa, and Y. Suwa, “Optical net gain measurement in n-type doped germanium waveguides under optical pumping for silicon monolithic laser,” Opt. Express **24**(9), 9132–9139 (2016). [CrossRef]

**45. **N. Lu, H. Guo, L. Li, J. Dai, L. Wang, W.-N. Mei, X. Wu, and X. C. Zeng, “MoS 2/MX 2 heterobilayers: bandgap engineering via tensile strain or external electrical field,” Nanoscale **6**(5), 2879–2886 (2014). [CrossRef]

**46. **S. Zhang, Z. Jin, X. Liu, W. Zhao, X. Lin, C. Jing, and G. Ma, “Photoinduced terahertz radiation and negative conductivity dynamics in Heusler alloy Co 2 MnSn film,” Opt. Lett. **42**(16), 3080–3083 (2017). [CrossRef]

**47. **A. Turner, L. Chang, and T. Martin, “Enhanced reflectance of reststrahlen reflection filters,” Appl. Opt. **4**(8), 927–933 (1965). [CrossRef]

**48. **I. B. Burgess, Y. Zhang, M. W. McCutcheon, A. W. Rodriguez, J. Bravo-Abad, S. G. Johnson, and M. Lončar, “Design of an efficient terahertz source using triply resonant nonlinear photonic crystal cavities,” Opt. Express **17**(22), 20099–20108 (2009). [CrossRef]

**49. **R. Sinha, M. Karabiyik, C. Al-Amin, P. K. Vabbina, DÖ Güney, and N. Pala, “Tunable room temperature THz sources based on nonlinear mixing in a hybrid optical and THz micro-ring resonator,” Sci. Rep. **5**(1), 9422 (2015). [CrossRef]

**50. **F. Jabbarzadeh, M. Heydari, and A. Habibzadeh-Sharif, “A comparative analysis of the accuracy of Kubo formulations for graphene plasmonics,” Mater. Res. Express **6**(8), 086209 (2019). [CrossRef]

**51. **A. Marini, J. Cox, and F. G. De Abajo, “Theory of graphene saturable absorption,” Phys. Rev. B **95**(12), 125408 (2017). [CrossRef]

**52. **S. A. Mikhailov, “Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model,” Phys. Rev. B **100**(11), 115416 (2019). [CrossRef]

**53. **S. A. Jensen, Z. Mics, I. Ivanov, H. S. Varol, D. Turchinovich, F. Koppens, M. Bonn, and K.-J. Tielrooij, “Competing ultrafast energy relaxation pathways in photoexcited graphene,” Nano Lett. **14**(10), 5839–5845 (2014). [CrossRef]

**54. **S.-F. Shi, T.-T. Tang, B. Zeng, L. Ju, Q. Zhou, A. Zettl, and F. Wang, “Controlling graphene ultrafast hot carrier response from metal-like to semiconductor-like by electrostatic gating,” Nano Lett. **14**(3), 1578–1582 (2014). [CrossRef]

**55. **A. J. Frenzel, C. H. Lui, Y. C. Shin, J. Kong, and N. Gedik, “Semiconducting-to-metallic photoconductivity crossover and temperature-dependent Drude weight in graphene,” Phys. Rev. Lett. **113**(5), 056602 (2014). [CrossRef]

**56. **M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. **24**(24), 4493–4499 (1985). [CrossRef]

**57. **T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter **25**(21), 215301 (2013). [CrossRef]

**58. **R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science (Cat. No.95TH8079), 39–43 (1995).

**59. **M. Kelly, “Particle Swarm Optimization algorithm,” retrieved https://github.com/MatthewPeterKelly/ParticleSwarmOptimization.