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Abstract
Terahertz (THz) microcavities have garnered considerable attention for their ability to localize and confine THz waves, allowing for strong coupling to remarkably enhance the light-matter interaction. These properties hold great promise for advancing THz science and technology, particularly for high-speed integrated THz chips where transient interaction between THz waves and matter is critical. However, experimental study of these transient time-domain processes requires high temporal and spatial resolution since these processes, such as THz strong coupling, occur in several picoseconds and microns. Thus, most literature studies rarely cover temporal and spatial processes at the same time. In this work, we thoroughly investigate the transient cavity-cavity strong-coupling phenomena at THz frequency and find a Rabi-like oscillation in the microcavities, manifested by direct observation of a periodic energy exchange process via a phase-contrast time-resolved imaging system. Our explanation, based on the Jaynes-Cummings model, provides theoretical insight into this transient strong-coupling process. This work provides an opportunity to deeply understand the transient strong-coupling process between THz microcavities, which sheds light on the potential of THz microcavities for high-speed THz sensor and THz chip design.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microcavities are capable of significantly enhancing light-matter interactions through their ability to strongly confine light, leading to progresses in various fields such as lasers [1–3], sensors [4–7], frequency combs [8–10], Bose-Einstein condensation [11,12], and emerging terahertz (THz) technologies [13,14]. Strong coupling is typically employed in microcavities to further localize and regulate terahertz waves, as well as enhance light-matter interactions. Recent theoretical and experimental studies have explored the phenomena of strong-coupling terahertz cavities, including Rabi splitting [15–18] and Fano resonance [19–21] in the frequency domain. Such cavities are widely used in various applications including nonlinear frequency conversion [22,23], THz sources [24–26], real-time detection [27,28] and optical communications [29–31].
The development of microstructures, such as microcavities and metamaterials, has revealed wave physics properties that underlie well-established classical theories. The study of light-matter interactions in these fields has generated significant interest in the utilization of time as a special degree of freedom [32]. This is because, in quantum theory, time is the only quantity that is not associated with an observable quantity and is being merely treated as a variable. When a physical system involves time evolution, we may use different physical models to describe the state of the physical system at different times. By studying the temporal evolution process of the system, we may establish connections between different physical models and quantities, thereby deepening our understanding of them. As a result, studying time-domain processes provides the possibility of understanding the connections between physical quantities and the transformation between physical models. Therefore, investigating the transient time-domain processes of light-matter interaction, such as strong coupling, can provide further understanding of the physical basis and time evolution of these processes.
However, investigating the transient cavity-cavity strong coupling processes at THz frequency in the field of microcavities has proven to be challenging due to the high spatiotemporal resolution requirements on the picosecond time scale and tens of micrometres in space [33–35]. Unfortunately, most THz wave detection systems presently available do not meet these strict requirements at the same time. From a theoretical perspective, a transient process of the THz wave cannot be described by the usual approximation of the time-independent Helmholtz equations, which have been commonly used to describe steady-state processes, limiting theoretical description on strong coupling phenomena in THz microcavities. Nevertheless, studying the THz transient strong-coupling process can significantly advance the development of THz ultrafast photonics and THz strong-coupling applications, including high-speed THz communications and integrated platform development. To reveal the ultrafast transient phenomena at THz frequency, a phase-contrast time-resolved imaging system has been constructed [36], which can achieve spatial resolution of approximately 5 $\mathrm{\mu}$m and temporal resolution of 0.1 ps. This high precision platform offers the possibility to study various THz transient processes [34,37,38].
In this work, we investigated the transient cavity-cavity strong coupling at THz frequency based on a phase-contrast time-resolved imaging system. Accordingly, a Rabi-like oscillation between two microcavities was found through direct observation of periodic energy exchange. The whole process of the transient cavity-cavity strong coupling was recorded, and the results suggest that the period of the Rabi-like oscillation is directly related to the coupling rate of the strong-coupling system, which is determined by the interval of the spectrum splitting in the frequency domain. We numerically solved the time-dependent Maxwell equations describing this transient process using finite-difference time-domain method. Additionally, we provided a theoretical interpretation of the phenomenon based on the Jaynes-Cummings model. The results show good agreement between the experiment, simulation, and theoretical model.
2. Design and simulation
The geometric design of the strong-coupled microcavities is illustrated in Fig. 1(a). The entire structure was fabricated on $x$-cut 50 $\mathrm{\mu}$m-thick Lithium Niobate (LN) wafers, with several rectangular shapes carved and drilled out from them. The long side of the carved slot was aligned parallel to the optical axis ($z$-axis) of the LN crystal. In each wafer, 7 air slots with a width of 100 $\mathrm{\mu}$m and period of 200 $\mathrm{\mu}$m were etched to form a Bragg grating mirror. Two identical Bragg grating mirrors with this specific interval formed a 1D Fabry-Perot microcavity. Another air slot, carved in the center of the microcavity, effectively divided it into two identical microcavities, as shown in Fig. 1(a). The length of both two microcavities are fixed at $L$ = 300 $\mathrm{\mu}$m for all circumstances. The gap between the two microcavities was determined by the width of the center air slot, which varied (30 $\mathrm{\mu}$m, 50 $\mathrm{\mu}$m, 70 $\mathrm{\mu}$m, 90 $\mathrm{\mu}$m) in different LN wafers to modify the coupling rate of the two microcavities. The dimensions of the coupled microcavities remained constant, and the height of all air slots was $h$ = 1000 $\mathrm{\mu}$m. All rectangular air slots were fabricated using the femtosecond laser direct writing technology [39](details in Appendix). Fig. 1(b) displays an image of a fabricated sample under a microscope, while the employed experimental setup used for exciting and detection can be found in Fig. 1(c).
Fig. 1. Depiction of the samples used in the experiment and the experimental setup. (a) Schematic illustration of the samples. (b) Microscope image of the sample with the gap of two microcavities $d$ = 50 $\mathrm{\mu}$m. The samples used in the experiment have four different microcavity gap values of 30 $\mathrm{\mu}$m, 50 $\mathrm{\mu}$m, 70 $\mathrm{\mu}$m and 90 $\mathrm{\mu}$m. The length of both two cavities was $L$ = 300 $\mathrm{\mu}$m, while the length of air slots on the side was $m$ = 100 $\mathrm{\mu}$m. The distance between the two adjacent air slots was $n$ = 100 $\mathrm{\mu}$m. The height of the microcavity structure was $h$ = 1000 $\mathrm{\mu}$m. (c) Schematic of the experimental setup for generating and detecting THz waves.
In the experiment, the THz wave was generated and detected by femtosecond laser pulses in one of the microcavities through impulsive stimulated Raman scattering (ISRS) [40]. The laser pulse had a center wavelength of 800 nm, a repetition rate of 1 kHz, and a duration of 120 fs. It was split into a pump pulse (450 $\mathrm{\mu}$J) and a probe pulse (50 $\mathrm{\mu}$J), which were delayed from each other. The pump pulse was focused by a cylindrical lens onto the left microcavity to generate THz waves that propagate along $y$-axis (shown in Fig. 1(a)). Some of the THz wave generated in the left microcavity transmitted the central air slot and reached the right microcavity, where they were transmitted in both microcavities and reflected by the Fabry-Pérot mirrors. The amplitude of the THz wave at each time and position $E$($y, z, t$) was measured using a phase-contrast time-resolved imaging system. When a THz wave propagates through a crystal, the refractive index of the crystal is modulated due to an electrooptic effect, which creates a spatial refractive index distribution $n$($y, z, t$) proportional to the amplitude of the THz wave [37]. The spatial refractive index distribution varies with the transmission of the THz wave in the crystal. A probe pulse, which was frequency-doubled to 400 nm, was used to irradiate the LN crystal along the $x$-direction. As a result, the probe pulse passing through the LN crystal exhibited a corresponding phase distribution $\mathrm{\phi}$($y, z, t$) due to the refractive index distribution inside the crystal. Based on theoretical calculations, the THz field induces such a relative phase shift inside the crystal:
In numerical simulation, we conducted simulation settings based on the above experimental parameters and collected data through simulated detectors (details in Appendix). In experiment, a phase-contrast imaging technique was used to transform the phase distribution of the probe pulse into an intensity distribution $I_\mathrm{\phi}(y, z, t$), which was collected by a charge-coupled device (CCD) camera. (as shown in Figs. 3(b) and (c)) Through this method, the intensity distribution of the THz wave in the LN crystal at any time could be obtained, depending on the optical path difference between the probe pulse and pump pulse. A delay line was used to change the optical path difference, enabling the measurement and recording of the intensity distribution of the THz wave in the LN crystal at different times, revealing the temporal evolution process of the THz wave in the crystal.
3. Results and analysis
We will analyze the experimental and numerical simulation results from two perspectives. In the frequency domain, the strong coupling of two microcavities produces the Rabi splitting, which can be observed in Fig. 2. The simulation results for various samples are shown in Fig. 2(a), while the experimental results are displayed in Fig. 2(b). Adjusting the width of the air slot between the two microcavities changes the distance between the two peaks in the spectrum. The change in coupling rate under different air slots can be calculated from peaks in the spectrum, as shown in Fig. 2(c). Based on the above results, we can also calculate that the cavities are strong-coupled. (details in Appendix) The experimental results are in good agreement with the numerical simulation results.
Fig. 2. Depiction of the experimental and numerical simulation results in the frequency domain. (a) and (b) present the simulated results and experimental results of the samples with microcavity gap values of 30 $\mathrm{\mu}$m, 50 $\mathrm{\mu}$m, 70 $\mathrm{\mu}$m and 90 $\mathrm{\mu}$m. The simulation results are shown in (a) and the experimental result in (b). (c) Peak position of the Rabi splitting for both the theoretical and experimental results. The coupling rate is calculated from the frequency peaks of each coupling system (details in Appendix). The frequency error arises from the measurement of frequency peak, while the coupling rate error comes from its calculation.
Fig. 3. Depiction of the experimental and numerical simulation results in the time domain. (a) The simulated results of the relative intensity variation of the THz wave in strong-coupling microcavities with microcavity gap values of 30 $\mathrm{\mu}$m, 50 $\mathrm{\mu}$m, 70 $\mathrm{\mu}$m and 90 $\mathrm{\mu}$m. The white boxes identify the location of the air slots in the structure, and the zero point of the spatial position in the figure is located in the middle of the gap between the two cavities. (b) and (c) represent the experimental results of the relative intensity variation of THz wave in strong-coupling microcavities with microcavity gap values of 30 $\mathrm{\mu}$m and 70 $\mathrm{\mu}$m. The white dotted line boxes indicate the approximate time during which the THz wave energy stays in the two microcavities, respectively.
Regarding the time domain, we can observe the Rabi-like oscillation of the cavity-cavity strong-coupling system when the THz wave is excited in one of the microcavities, as shown in Fig. 3. Both the numerical simulation results (Fig. 3(a)) and the experimental results (Figs. 3(b) and 3(c)) indicate that the THz wave is first localized in the microcavity where it was generated. For instance, in the 30 $\mathrm{\mu}$m-gap sample, as the terahertz wave needs to propagate from the excitation microcavity into the boundary of the other microcavity for the mode of Rabi splitting occurring after strong coupling to be fully established, the first localization of the terahertz wave energy in the excitation microcavity lasts in time for about 20 picoseconds due to this effect. Then the THz energy is transmitted from one microcavity to the other with a period of about 30 ps. Different coupling rates resulting from different center air slot widths lead to the formation of periods of transferring THz wave energy between the two microcavities. Together with the frequency domain results, we demonstrate that a stronger coupling rate between the two microcavities leads to a shorter period of the Rabi-like oscillation, as indicated in Table 1. Specifically, Fig. 4 illustrates the electric field intensity of the THz wave and the period of the Rabi-like oscillation in the microcavity. These data were collected at the geometric center of the empty microcavity. Figures 4(a) and (b) show the simulated and experimental results of two microcavities with a microcavity gap of 30 $\mathrm{\mu}$m, while Figs. 4(c) and (d) show the results with a microcavity gap of 70 $\mathrm{\mu}$m. These results also support the conclusion that stronger coupling rates between the two microcavities result in a shorter period of the Rabi-like oscillation.
Fig. 4. Depiction of the specific experimental and numerical simulation results in the time domain for determining the period of energy oscillation. (a) and (b) present the numerical simulation and experimental results of the relative intensity variation of the THz wave in the strong coupling state of two microcavities with a microcavity gap of 30 $\mathrm{\mu}$m, while (c) and (d) present the results with a microcavity gap of 70 $\mathrm{\mu}$m. The change of the electric field intensity with time shown in these figures was collected from the central position of the microcavity on the right side of each sample.
Table 1. Experimental results of the energy transfer period between two microcavities. The unit of gap, frequency and period are micrometers, THz and picoseconds.
We analyze the cavity-cavity strong-coupling process in both frequency and time domains. In the frequency domain, the Rabi splitting phenomenon occurs, and the Jaynes-Cummings is used for analysis. (details in Appendix) However, the structure dispersion of the LN subwavelength waveguide [42] and the frequency-dependent reflectivity of the DBRs cause the two split frequencies to have different effective refractive indices and reflected phase changes. As a result, the two split frequency peaks are located at asymmetric positions around the original resonant mode of the single microcavity, as shown in Fig. 2. Furthermore, our extracted results demonstrate that the resonance frequency values of the microcavities split into two different frequencies when the two microcavities are coupled. These two new energy levels exist in both microcavities. The results show that the frequency splitting of the two modes becomes large as the coupling rate increases, which occurs when the distance between the two coupled structures becomes smaller. The experimental and simulation results are in good agreement with the theoretical expectations. The second part of our experimental results focuses on the time-domain process of strong coupling. In the time domain, the process of energy transfer between two microcavities is interpreted as "Rabi-like" oscillations because it involves the oscillation of energy transfer between two pairs of degenerate energy levels: |$c_{1}$, $\omega _{1}\rangle \leftrightarrow$|$c_{2}$, $\omega _{1}\rangle$ and |$c_{1}$, $\omega _{2}\rangle \leftrightarrow$|$c_{2}$, $\omega _{2}\rangle$, where $c_{1}$ and $c_{2}$ represent two coupled microcavities, $\omega _{1}$ and $\omega _{2}$ represent two mode frequency in each cavity generated by strong coupling. The resonant energy levels with the same frequency in both microcavities are discrete. Therefore, the energy transfer between the two microcavities can be understood as THz phonon-polariton oscillations between energy levels with the same resonant frequency in different microcavities. This process shares the physical nature of Rabi oscillation, namely "Rabi-like" oscillation to describe our observations.
Here, a semi-classical theory was used to explain the Rabi-like oscillation of the coupled microcavities. According to the time-resolved detection, THz waves do not form standing waves in microcavities but rather propagate steadily forward or transmit and reflect between the two microcavities. Therefore, the behavior of the THz waves can be described as traveling waves. Additionally, simulation results extracted from nanosecond durations indicate that the wave packet length of THz waves increases slightly as they propagate in the microcavity. However, the multi-level quantum system used to describe Rabi-like oscillation cannot be modified by introducing dispersion and other conditions, so we resort to a semiclassical physical description. By considering the superposition of two THz waves with similar frequency values and at the same direction of transmission, the transmission form of the two THz waves in a single microcavity can be written as follows:
where $A_{1}$ and $A_{2}$ are the amplitudes of the two THz waves, $\omega$ represents the resonance frequency of the microcavity and $\Omega$ denotes the coupling rate of the two microcavities. The superposition result of these two THz waves in a single microcavity can be written as follows:When THz waves propagate, they will also attenuate due to material absorption and other dissipations, which can be described as follows:
4. Conclusion
In conclusion, we thoroughly investigated the transient cavity-cavity strong coupling at THz frequency using a phase-contrast time-resolved imaging system and found a Rabi-like oscillation of THz waves in strong-coupling microcavities. We found that the period of the Rabi-like oscillation of the coupling microcavities in the time domain is directly related to the coupling rate, while in the frequency domain, the extent of the frequency splitting is positively correlated with the coupling rate. Based on the Jaynes-Cummings model, we proposed an interpreted theoretical model that was verified by performing numerical calculations. From this time-domain phenomenon, it was demonstrated that the periodic energy exchange between the structures participating in coupling is one of the internal manifestations of the strong coupling for strong-coupling systems. Our findings provide a transient perspective to understand the origins of the strong coupling effect in a THz microcavity system. The approach allows for the recognition of any transient manipulation, which can be easily transferred from one microcavity to another. By employing a suitable design, this result may enable time-dependent signal transfer over long distances in the future. Moreover, our findings could be further applied to highly sensitive biosensing, ultrafast communications, THz signal processing technologies and the development of future THz transient sub-wavelength on-chip integration.
Appendix
Additional depiction of the specific experimental and numerical simulation results in the time domain
Here, we give the additional experimental and simulation results to further determine the period of oscillation, as shown in Fig. 5.
Fig. 5. Depiction of the specific experimental and numerical simulation results in the time domain for determining the period of energy oscillation. (a) and (b) present the numerical simulation and experimental results of the relative intensity variation of the THz wave in the strong coupling state of two microcavities with a microcavity gap of 50 $\mathrm{\mu}$m, while (c) and (d) present the results with a microcavity gap of 90 $\mathrm{\mu}$m. The change of the electric field intensity with time shown in these figures was collected from the central position of the microcavity on the right side of each sample.
Manufacturing process of the coupled microcavities
Prior to laser fabrication, 2 $\mathrm{\mu}$m Si$\mathrm {O}{}_{2}$ layers were precoated on each side of the LN wafer to prevent sputtered debris. An amplified Ti:sapphire laser operating at a central wavelength of 800 nm, 1 kHz repetition rate, and 120 fs duration was used to fabricate the LN wafer. 8 $\mathrm{\mu}$J laser pulses were focused by a 10$\times$ microscopic objective with 0.25 N.A. into the LN wafer, which was fixed to a programmable three-dimensional translation stage for carving and drilling the structures. After fabrication, the Si$\mathrm {O}{}_{2}$ layer was removed using hydrofluoric acid to eliminate sputtered LN debris produced during the machining process.
Detailed settings of numerical simulation
In the numerical simulation, the commercial software Lumerical FDTD Solution is employed to solve 3D time-dependent Maxwell equations. The 50 $\mathrm{\mu}$m-thick LN sample is located in the center of the simulation region, whose boundary is set as a standard perfect match layer. The shape and dimensions of the air slots and coupled cavities on the crystal are the same as the experiment design shown in Fig. 1, with the long side of the carved slot parallel to the optical axis ($z$-axis) of the LN crystal. To simulate the propagation of terahertz wave generated after the crystal is excited, two planar mode light sources are set in the geometric center of the right cavity. The light sources have the same area as the cross section of the crystal, are oriented in opposite directions, and have a central frequency of 0.36 THz.
Strong coupling system analysis and Jaynes-Cummings model
According to cavity quantum electrodynamics, our cavity-cavity system needs to meet the following relationship to be strong coupled:
$\omega$ denotes the resonant frequency of a single microcavity, while $\Omega$ represents the coupling rate of two microcavities. The attenuation coefficient of the microcavity is denoted by $\kappa$, and can be determined by measuring the full width at half maximum of the spectrum peak in the microcavity. $\Omega$ and $\omega$ can be calculated from experimental data using the Jaynes-Cummings model. After conducting relevant measurements and calculations, we have obtained reference values for the three physical quantities mentioned above:In quantum electrodynamics and the related disciplines, the Jaynes-Cummings model is frequently used to describe strong-coupling systems and phenomena. It is one of the few quantum models with analytical solutions for describing coupling systems.
By considering the strong coupling of the two identical single-mode microcavities under the steady state, the Hamiltonian of this coupling system can be written in a matrix form as follows:
where $\omega$ represents the resonate frequency of the microcavity and $\Omega$ denotes the coupling rate of the two microcavities. By using the following Schrodinger equation to solve the Hamiltonian,The results could be written as follows:
To calculate the coupling rate of every coupling-cavity system, it can be obtained from the above equation:
That is to say, we can calculate the coupling strength of the system by measuring the two frequency peaks of the coupling system.
Funding
National Natural Science Foundation of China (11874229, 11974192, 62205158); China Postdoctoral Science Foundation (2022M711709); Foundation of State Key Laboratory of Laser Interaction with Matter (SKLLIM2101); 111 Project (B23045).
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. Q.-T. Cao, R. Liu, H. Wang, et al., “Reconfigurable symmetry-broken laser in a symmetric microcavity,” Nat. Commun. 11(1), 1136–1142 (2020). [CrossRef]
2. L. He, Ş. K. Özdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser & Photonics Reviews 7, 60–82 (2013). [CrossRef]
3. C. Schneider, A. Rahimi-Iman, N. Y. Kim, et al., “An electrically pumped polariton laser,” Nature 497(7449), 348–352 (2013). [CrossRef]
4. W. Chen, Ş. K. Özdemir, G. Zhao, et al., “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017). [CrossRef]
5. L. He, Ş. K. Özdemir, J. Zhu, et al., “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011). [CrossRef]
6. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112(20), 203901 (2014). [CrossRef]
7. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, et al., “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010). [CrossRef]
8. B. Yao, S.-W. Huang, Y. Liu, et al., “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558(7710), 410–414 (2018). [CrossRef]
9. Q. Li, T. C. Briles, D. A. Westly, et al., “Stably accessing octave-spanning microresonator frequency combs in the soliton regime,” Optica 4(2), 193–203 (2017). [CrossRef]
10. B. Corcoran, M. Tan, X. Xu, et al., “Ultra-dense optical data transmission over standard fibre with a single chip source,” Nat. Commun. 11(1), 2568–2574 (2020). [CrossRef]
11. J. D. Plumhof, T. Stöferle, L. Mai, et al., “Room-temperature Bose-Einstein condensation of cavity exciton-polaritons in a polymer,” Nat. Mater. 13(3), 247–252 (2014). [CrossRef]
12. J. Klaers, J. Schmitt, F. Vewinger, et al., “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef]
13. E. Yüce, G. Ctistis, J. Claudon, et al., “All-optical switching of a microcavity repeated at terahertz rates,” Opt. Lett. 38(3), 374–376 (2013). [CrossRef]
14. C. Pan, Q. Wu, Q. Zhang, et al., “Direct visualization of light confinement and standing wave in THz Fabry-Perot resonator with bragg mirrors,” Opt. Express 25(9), 9768–9777 (2017). [CrossRef]
15. A. Bayer, M. Pozimski, S. Schambeck, et al., “Terahertz light–matter interaction beyond unity coupling strength,” Nano Lett. 17(10), 6340–6344 (2017). [CrossRef]
16. Q. Zhang, M. Lou, X. Li, et al., “Collective non-perturbative coupling of 2d electrons with high-quality-factor terahertz cavity photons,” Nat. Phys. 12(11), 1005–1011 (2016). [CrossRef]
17. M. Teich, M. Wagner, D. Stehr, et al., “Systematic investigation of terahertz-induced excitonic Rabi splitting,” Phys. Rev. B 89(11), 115311 (2014). [CrossRef]
18. S. Gröblacher, K. Hammerer, M. R. Vanner, et al., “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009). [CrossRef]
19. W. Cao, R. Singh, I. A. Al-Naib, et al., “Low-loss ultra-high-Q dark mode plasmonic Fano metamaterials,” Opt. Lett. 37(16), 3366–3368 (2012). [CrossRef]
20. R. Singh, I. A. Al-Naib, M. Koch, et al., “Sharp Fano resonances in THz metamaterials,” Opt. Express 19(7), 6312–6319 (2011). [CrossRef]
21. M. Manjappa, Y. K. Srivastava, L. Cong, et al., “Active photoswitching of sharp Fano resonances in THz metadevices,” Adv. Mater. 29, 1603355 (2017). [CrossRef]
22. X. Jiang, L. Shao, S.-X. Zhang, et al., “Chaos-assisted broadband momentum transformation in optical microresonators,” Science 358(6361), 344–347 (2017). [CrossRef]
23. X. Zhang, Q.-T. Cao, Z. Wang, et al., “Symmetry-breaking-induced nonlinear optics at a microcavity surface,” Nat. Photonics 13(1), 21–24 (2019). [CrossRef]
24. K. Rojan, Y. Léger, G. Morigi, et al., “Enhanced second-order nonlinearity for THz generation by resonant interaction of exciton-polariton Rabi oscillations with optical phonons,” Phys. Rev. Lett. 119(12), 127401 (2017). [CrossRef]
25. A. Kavokin, I. Shelykh, T. Taylor, et al., “Vertical cavity surface emitting terahertz laser,” Phys. Rev. Lett. 108(19), 197401 (2012). [CrossRef]
26. L. Masini, A. Pitanti, L. Baldacci, et al., “Continuous-wave laser operation of a dipole antenna terahertz microresonator,” Light: Sci. Appl. 6(10), e17054 (2017). [CrossRef]
27. L. Chen, D.-g. Liao, X.-g. Guo, et al., “Terahertz time-domain spectroscopy and micro-cavity components for probing samples: a review,” Frontiers of Information Technology & Electronic Engineering 20(5), 591–607 (2019). [CrossRef]
28. I.-C. Benea-Chelmus, Y. Salamin, F. F. Settembrini, et al., “Electro-optic interface for ultrasensitive intracavity electric field measurements at microwave and terahertz frequencies,” Optica 7(5), 498–505 (2020). [CrossRef]
29. Y. Todorov, A. Andrews, I. Sagnes, et al., “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009). [CrossRef]
30. Q. Zhang, J. Qi, Y. Lu, et al., “Cavity-cavity coupling based on a terahertz rectangular subwavelength waveguide,” Journal of Applied Physics 126(6), 063103 (2019). [CrossRef]
31. W. Withayachumnankul, C. M. Shah, C. Fumeaux, et al., “Terahertz localized surface plasmon resonances in coaxial microcavities,” Adv. Opt. Mater. 1, 443–448 (2013). [CrossRef]
32. E. Galiffi, R. Tirole, S. Yin, et al., “Photonics of time-varying media,” Adv. Photon. 4(01), 014002 (2022). [CrossRef]
33. Y. Lu, Q. Zhang, Q. Wu, et al., “Giant enhancement of THz-frequency optical nonlinearity by phonon polariton in ionic crystals,” Nat. Commun. 12(1), 3183–3190 (2021). [CrossRef]
34. Y. Lu, Q. Wu, H. Xiong, et al., “Observation of “frozen-phase” propagation of THz pulses in a dispersive optical system,” Laser & Photonics Reviews 15, 2000591 (2021). [CrossRef]
35. H. Xiong, Y. Lu, Q. Wu, et al., “Topological valley transport of terahertz phonon–polaritons in a LiNbO3 chip,” ACS Photonics 8(9), 2737–2745 (2021). [CrossRef]
36. C. A. Werley, Q. Wu, K.-H. Lin, et al., “Comparison of phase-sensitive imaging techniques for studying terahertz waves in structured LiNbO3,” J. Opt. Soc. Am. B 27(11), 2350–2359 (2010). [CrossRef]
37. Q. Wu, C. A. Werley, K.-H. Lin, et al., “Quantitative phase contrast imaging of THz electric fields in a dielectric waveguide,” Opt. Express 17(11), 9219–9225 (2009). [CrossRef]
38. Y. Lu, Q. Wu, Q. Zhang, et al., “Propagation of THz pulses in rectangular subwavelength dielectric waveguides,” Journal of Applied Physics 123(22), 223103 (2018). [CrossRef]
39. P. Sivarajah, C. A. Werley, B. K. Ofori-Okai, et al., “Chemically assisted femtosecond laser machining for applications in LiNbO3 and LiTaO3,” Appl. Phys. A 112(3), 615–622 (2013). [CrossRef]
40. Z. Chen, Modeling phonon-polariton generation and control in ferroelectric crystals, Master’s thesis, Massachusetts Institute of Technology (2009).
41. T. Feurer, N. S. Stoyanov, D. W. Ward, et al., “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007). [CrossRef]
42. C. Yang, Q. Wu, J. Xu, et al., “Experimental and theoretical analysis of THz-frequency, direction-dependent, phonon polariton modes in a subwavelength, anisotropic slab waveguide,” Opt. Express 18(25), 26351–26364 (2010). [CrossRef]
