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
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 , passive coolers , and detectors . At the same time, the optimal gain can enhance the output of lasers  or amplifiers [5,6] and reduce the non-radiative energy loss .
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 r1 and r2, respectively. Due to the imperfect reflection of mirrors, there exists a radiative loss rate of γr. 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 ,
Here δ is α or -g, ω is the angular frequency of the signal, and ω0 is the mode frequency of cavities. Assuming the signal is on resonance (ω = ω0), the absorption A reaches two extremums (1 and -∞) when |δ| = γr. The first extremum, 1, appears when γr = α, corresponding to the perfect absorption. The second extremum, -∞, appears when γr = -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 En (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  enables an optimal zero state. Besides, Auger recombination exists due to many-particle effects , 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.
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 . In this work, we choose SiO2/Air DBR for the THz cavity and Si/SiO2 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 En = 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., 10log10(R), of our nested cavity as a function of En. 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 En<100 meV. We then increase the number of layers to 2. The nested cavity performs as a perfect absorber at En = 3 meV, as marked by the orange circle in Fig. 2(g). When the THz conductivity equals zero at En = 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 En = 87.3 meV. Figure 2(h) shows the IR reflectance spectra as a function of En, 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 ,
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  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 . 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 . We may use a CO2 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  to reduce the required pump intensity. The back mirrors can also adopt metals, polar materials with Reststrahlen bands . 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 SiO2 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 En = 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 . 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 Vg is set to be 5 V. The energy transfer between THz field and graphene then varies at different Drain-Source voltage Vd. The THz response of the THz cavity at Vd=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 Vd. The energy transfer changes from perfect absorption at Vd=0 to optimal gain at Vd=47.5 mV.
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.
5.1 Dielectric response of materials
With optical excitation, the dynamic conductivity of graphene is ,
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 Te and the nonequilibrium Fermi level En. Since we work at a stable state, we can use energy conservation and carrier cconservation [17,52] to obtain the relation between Te, En, and the pump intensity Ipump.
The conservation of carrier energy provides,
Combining Eq. (7) and Eq. (8), we can numerically obtain the relation between En, Te and Ipump, as illustrated in Figs. 4(a) and 4(b). The distribution function is plotted in Fig. 4(c) when En = 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 SiO2 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 ,
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. Vg controls the injection level. We set Vg to be 5 V, thereby enabling a decent gain coefficient and reducing the risk of breakdown. Vd 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 Vg1 and Vg2 to obtain the expected carrier distribution. Other parameters remain unchanged compared to the Ref. . Figure 4(d) plots the relation between graphene conductivity and Vd.
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 [Et; Ht], enabling a straightforward graphene model. The transfer matrix for layer j is given by
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).
The authors declare no conflicts of interest.
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.
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