Open Access
Add to BibSonomy
Abstract
In a hyperbolic metamaterial or metasurface (HMM or HMS), strong light-matter interaction occurs at the phase transition wavelength where the material behavior changes from metal to dielectric. As a result, the enhanced electric field can generate non-linear phenomena, such as optical bistability. In this work, we numerically investigate polarization-dependent dual-wavelength optical bistability from perovskite HMS consisting of alternating layers of MAPbBr3 perovskite and Au. Benefiting from the anisotropic property, the bistable operation wavelength for TE and TM polarizations are different. In particular, for TE polarization, strong light matter interaction resulting from the increment of incident light power not only shifts the phase transition wavelength but also enhances the transmission. Consequently, the bistable on-off contrast ratio is unprecedentedly enlarged. For TM polarization, strong light-matter interaction gives rise to an ultra-low bistable threshold. The proposed optical bistable states can serve as an activation function for optical neurons in a photonic neural network, with similar prediction accuracy across a variety of learning tasks as the classic activation functions ReLU and Sigmoid. Our work suggests a novel avenue towards the insertion of perovskite HMS into optical computing networks.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Compared to the traditional electronic integrated circuits (ICs), photonic ICs and photonic neural network manifest merits in signal transmission, data storage and computing with fast operating speed as well as low power consumption [1–3]. In photonic ICs and neural networks, the logic [4], switching [5,6], sensing [7], and storage [8–10] component units always require a stable photonic nonlinear process. This is where optical bistability [11], as known as the photonic analogue of the magnetic hysteresis loop, can accommodate and provide remarkable nonlinear manipulation. To achieve an optical bistable state, one requirement is the third-order nonlinear permittivity (Kerr effect) in the material [12–15]. As the Kerr effect is field intensity-dependent [16], to create strong field feedback and therefore generate bistability, one can always design a photonic cavity with high quality factor (Q factor) [10,17–23]. Epsilon-near-zero (ENZ) material has also been proposed to yield field enhancement in bistable devices [24–27]. Nevertheless, it is still an ongoing pursuit to explore novel cavities and materials that satisfy the requirement in photonic ICs and neural networks. One example of these materials is hyperbolic metamaterial or metasurface (HMS).
HMS is a unique class of anisotropic material that exhibits metal and dielectric properties simultaneously [28], enabling applications in super-resolution imaging [29–32] and spontaneous emission enhancement [33]. The typical design of HMS requires subwavelength constituent of metal and dielectric, enhancing strong light-matter interaction at the optical phase transition wavelength where material properties in selected directions change from metal to dielectric [34,29,35–40]. In this regard, when either the dielectric or metal constituent has nonlinear permittivity, one can observe optical bistability [41–43]. Therefore, it has been an ongoing pursuit to find suitable metal and dielectric materials with a large Kerr coefficient. Recently, lead halide perovskite has emerged and bloomed as a remarkable gain material (dielectric) for solar cells [44,45], LEDs [46] and lasers [47], due to its band gap tunability and solution processability [48]. Its usage in HMS is known as perovskite gain-assisted HMS, which manifests large anisotropic behavior [40]. Perovskites also show strong nonlinear permittivity on the order of 10−13 m2/V2 [49,50]. These findings motivate the insertion of perovskite HMS into non-linear photonic devices [51].
In this study, we design a perovskite HMS composed of alternating subwavelength layers of MAPbBr3 perovskite and Au. Benefiting from the anisotropy property of perovskite HMS, the phase transition wavelengths for TE and TM polarized light are different. Note that optical phase transition here means the material behavior changes from dielectric to metal. It occurs when the sign of real part of permittivity inverts. It can also happen at the metallic material plasma frequency (Below this frequency, it behaves as metal. Above this frequency, it behaves as dielectric). The phase transition wavelength is 520 nm for TE polarization, while it is 920 nm for TM polarization. Therefore, by strategically designing the dielectric component with a small ratio, electric field can be highly localized in the dielectric layer for both polarizations at different wavelengths, giving rise to the polarization-dependent dual-wavelength optical bistable phenomenon. Thanks to the strong nonlinear effects, the bistable power threshold for the TM polarization is $1.7{\; }mW/c{m^2}$, which is lower than that in previous studies [27]. Although the intrinsic loss in the large Au fraction attenuates transmission, nonlinear effects induce a 3-fold enhancement in transmission for TE polarization. Thus, the bistable contrast ratio is considerably large in comparison to previous works [42]. Finally, we apply the bistable nonlinear function as an activation function into a photonic neural network. It shows similar prediction accuracy as the classic activation functions ReLu and Sigmoid. Our work suggests an approach to implement HMS as a key component in photonic neural networks and other optical computing applications.
2. Phase transitions and transmission of the hyperbolic metasurface
Figure 1(a) schematically shows the perovskite HMS composed of alternating subwavelength layers of MAPbBr3 perovskite and Au on a glass substrate. The inset shows one lattice unit, with dp and dg representing the thickness of MAPbBr3 perovskite and Au, respectively. The period of the lattice is P = dp + dg. The optical axis is along the Bloch vector [52] in the x direction. The incident light propagates along z direction with TE (when the electric field vector is along the optical axis) and TM (when the electric field vector is perpendicular to the optical axis) polarizations depicted. When each layer thickness is much smaller than the operation wavelength, which we aim to be close to MAPbBr3’s gain spectrum peak of 535 nm such that MAPbBr3 can compensate metal loss [53], the anisotropic permittivity in x, y and z directions can be approximated under the effective medium equations as follows:
Fig. 1. (a) Schematic of MAPbBr3 perovskite HMS. (b) Permittivity of Au and MAPbBr3. (c) Optical phase diagram of MAPbBr3 perovskite HMS with respect to the wavelength and Au fraction. Effective metal, effective dielectric, type I and type II HMS regions are depicted in yellow, red, green and blue, respectively. Only the linear permittivity is considered. (d) Nonlinear induced optical phase diagram with respect to electric field amplitude and wavelength when AF = 0.9. The third order permittivity is included. (e) Permittivity for ${\varepsilon _ \bot }$ and ${\varepsilon _{/{/}}}$ as a function of wavelength at AF = 0.9. The solid and dashed blue lines are real and imaginary permittivity of ${\varepsilon _{/{/}}}$, respectively. The solid and dashed red lines are real and imaginary permittivity of ${\varepsilon _ \bot }$, respectively. (f) Zoomed-in plot for (e) from 0.3 to 0.6 $\mu m$.
To achieve strong light-matter interaction in the metal/dielectric multilayers, a smaller fraction of the dielectric constituent is desired [56–58]. Thus, in our perovskite HMS, we allowed only a tiny ratio of MAPbBr3 perovskite in each lattice unit. The corresponding Au fraction thus must be close to 1. However, since metal intrinsic loss attenuates the field intensity, there is a trade-off between achieving strong field enhancement and suppressing metal’s intrinsic loss. Here, we design our non-linear HMS with a significantly large metal fraction of 0.9. The effective permittivity ${\varepsilon _ \bot }$ and ${\varepsilon _{/{/}}}$ are illustrated in Fig. 1(e). For ${\varepsilon _{/{/}}}$, phase transition from metal to dielectric occurs at 950 nm (solid blue curve). When the wavelength is smaller than 950 nm, it behaves as metal. In this scenario, only a tiny amount of light can be transmitted. When the wavelength is larger than 950 nm, it behaves as dielectric, which can transmit a large amount of the incident light. For ${\varepsilon _ \bot }$, it behaves as metal for almost the entire visible and near infrared frequency (Fig. 1(f)). In the zoomed-in plot (Fig. 1(f)), a kink can be identified at around 505 nm as an indicative of the phase transition at plasma frequency (solid red curve in Fig. 1(e)) [51]. Here, it is the effective plasma frequency approximated under effective medium theory in type II HMS supporting plasmonic mode. Such mode’s frequency is tunable with respect to metal fraction, and it can manifest tunable Purcell enhancement [33].
Light transmission through the HMS is numerically simulated using commercial full wave solver COMSOL, shown in Fig. 2. To accommodate the effective medium theory, the metasurface unit is designed to be 70 nm, and the height is set to be 50 nm. Note that the transmission from a 50 nm thick Au thin film is also included for comparison. For TE polarization (Fig. 2(a)), as the HMS with 0.9 Au fraction behaves as a metal for the entire visible and near-infrared frequency (as illustrated by the solid red curve in Fig. 1(e), ${\varepsilon _{/{/}}}$ is negative), when the incident power is low, the transmission coefficient for the perovskite HMS is similar to that for the Au film (blue and yellow curves). The small bump at 505 nm wavelength in Fig. 2(a) corresponds to the kink in the Fig. 1(f). If only considering the linear permittivity, the transmission is independent of the incident power. However, when nonlinear permittivity is introduced into perovskite, the transmission behavior is significantly changed by different power densities. When the incident power is increased, an enhancement of transmission accompanied by a wavelength peak red-shift to 760 nm can be observed (green, purple, and magenta curves). The electric field distributions at peak wavelength of 532 nm for $2\; W/c{m^2}$ incident power density and 760 nm for $9\; W/c{m^2}$ incident power density are illustrated in Fig. 2(b) and (c), respectively. When the incident power density is low ($2\; W/c{m^2}$), the electric field is uniformly distributed in both the Au and MAPbBr3 layers (Fig. 2(b)). Meanwhile, when the power density is high ($9\; W/c{m^2}$), the electric field is highly localized in MAPbBr3 (Fig. 2(c)), indicating the strong enhancement raised by non-linear effects. Such strong field enhancement gives rise to optical phase transition at this polarization, where its behavior changes from metal to dielectric (Fig. 1(d)), and the transmission coefficient is consequently enlarged.
Fig. 2. Transmission spectra and electric field profiles for perovskite HMS. The lattice period is 70 nm. The thickness of MAPbBr3 and Au layers are 7 nm and 63 nm, respectively. MAPbBr3 and Au layers are marked by arrows. (a) Transmission for TE polarization at different incident power densities. (b)-(c) Field distribution for TE. (b) At power density of $2\; W/c{m^2}$ (532 nm wavelength). (c) At power density of $9\; W/c{m^2}$ (760 nm wavelength). (d) Transmission for TM polarization at different incident power densities. (e)-(f) Field distribution for TM. (e) At power density of $1 \times {10^{ - 2}}\; \mu W/c{m^2}$ (820 nm wavelength). (f) At power density of $3mW/c{m^2}$ (946 nm wavelength).
The transmission behavior for TM polarized light is different, as illustrated in Fig. 2(d). When the power density is low, one can see a peak at 820 nm indicative of the phase transition from metal to dielectric, in agreement with the effective permittivity calculation (solid blue curve in Fig. 1(e)). By increasing the power density, the transmission peak shifts to longer wavelengths like in the case of TE polarization, but the magnitude remains at a similar level. Further, in stark contrast with the TE polarization, electric field is highly localized in the MAPbBr3 layer even at the low power density of $1 \times {10^{ - 2}}\; \mu W/c{m^2}$ (Fig. 2(e)), and it is enhanced at the high power density of $3\; mW/c{m^2}$ (Fig. 2(f)).
Benefiting from the strong light-matter interaction, optical bistable states are generated from the nonlinear effects for both TE and TM polarizations. For normally incident TE polarized light of 660 nm, the transmission coefficient (red line in Fig. 3(a)) gradually increases as the increment of power density from $2{\; }W/c{m^2}$ to $10{\; }W/c{m^2}$, experiencing a significant jump at $8.1{\; }W/c{m^2}$ as an indication of nonlinearity-induced transmission change, in agreement with predictions in Fig. 2(a). Above $8.1{\; }W/c{m^2}$, the transmission coefficient gradually increases when the power density is further increased to $10{\; }W/c{m^2}$. However, when the power density is decreased from $10{\; }W/c{m^2}$, the transmission comes down from a different route (blue curve in Fig. 3(a)), experiencing a sudden drop at $4.2{\; }W/c{m^2}$ when a phase transition occurs. Finally, it enters a gradual decreasing region again when the power continues to decrease. The area enclosed by the two curves forms the bistable states as a photonic analogue of the magnetic hysteresis loop. If the HMS is employed as an optical switch or a memory component, the contrast on-off ratio in the bistable region is higher than 0.6.
Fig. 3. Polarization dependent dual-wavelength optical bistability. The solid red and blue curves represent the transmission behavior with respect to the power increment and decrement, respectively. (a) TE polarization at normal incidence. (b) TM polarization at normal incidence. (c) TE polarization at oblique incident angle of 10$^\circ $. (d) TM polarization at oblique incident angle of 10$^\circ $.
Similar bistable behavior can be observed for normally incident TM polarized light but at a different operation wavelength of 905 nm, in line with the prediction of the solid blue curve in Fig. 2(c). The kinks for the increasing and decreasing curves are at power densities of $2.5{\; \textrm{m}}W/c{m^2}$ and $1.7{\; \textrm{m}}W/c{m^2}$, respectively. Although the bistable contrast ratio is an order smaller than that with TE polarization, the threshold is three orders of magnitude lower. The operation power needed is still at the level of mW/cm2, which is much lower than that reported in literature [23,41]. The low power feature of the TM polarized bistable region makes it desirable in photonic neural networks and other photonic ICs. When the incident angle is slightly tuned from normal to 10$^\circ $ degree, the bistable states persist at the same operation wavelength for both TE and TM polarizations, as shown in Fig. 3(c) and (d), respectively. Notably, the corresponding bistable threshold and on-off contrast ratio still remain at similar values. This robustness against angle deviation makes the HMS a good candidate to operate in realistic photonic ICs and neural networks. Not only with the tuning of incident angle, but the bistable feature also remains when the lattice period varies, as long as it satisfies the effective medium theory, i.e., the lattice period is one order of magnitude smaller than the operation wavelength. This robustness on the lattice period increases the tolerance of the HMS to system disorders, defects, or fabrication variations.
Lastly, we apply our bistable HMS as a nonlinear activation function for an optical neuron in the emerging field of photonic neural network, which manifests merits in terms of energy consumption and operation speed compared to the conventional neural networks on central processing unit (CPU) or graphic processing unit (GPU). The optical nonlinear activation function of a neuron is crucial for the network to process complex tasks [59–62]. Tremendous research efforts have been devoted to exploring these optical nonlinear activation functions, and bistable devices are being considered as one of the potential options [63,64]. They are utilized in creating high-precision, per-pixel modulation arrays, thereby allowing for accurate manipulation of light in free space [63,65]. Moreover, the innate nonlinear characteristics of bistable devices make them highly effective in processing time series data that contains complicated nonlinear patterns. To seamlessly integrate into the sophisticated optical deep neural network architectures, one requirement of an optical neuron is compatibility with existing waveguiding systems and photonic ICs. As perovskite HMS has been experimentally demonstrated on the silicon on insulator (SOI) platform [40], it can accommodate well the integration requirement. In particular, the polarization-dependent bistable behavior provides a new degree of freedom in optical computing [64] as it allows to run two channel computing simultaneously in a single network.
We implement the TM polarization bistable features as an all-optical nonlinear activation function into a fully connected deep neural network simulation depicted in Fig. 4(a). Note TE polarization bistable states can also be applied with the same scheme. Our simulations employed MNIST, Fashion-MNIST, and CIFAR-10 databases for a comprehensive study on a variety of tasks. The neural network is composed of a flat image as the input layer, two hidden layers with 400 and 200 nodes, respectively, and 10 output nodes for classification purposes. The simulation results indicate that our bistable nonlinear activation function comes closer in performance to classic digital nonlinear activation functions like ReLU and Sigmoid, which are widely adopted in different network architectures. For MNIST, the classic digital nonlinear activation functions achieved an accuracy of 98%, for Fashion MNIST 88%, and for CIFAR-10 52%. Our bistable nonlinear activation function reached an accuracy of 96% for MNIST, 86% for Fashion-MNIST, and 45% for CIFAR-10. In comparison, neural networks without a nonlinear activation function showed recognition rates of only 92% for MNIST, 83% for Fashion-MNIST, and 37% for CIFAR. Note that the activation function ReLU and Sigmoid are more popular for conventional neural network on CPU and GPU. In photonic neural network, the activation function should be provided by a photonic structure. The results indicate our proposed nonlinear HMS exhibits great potential to serve as activation component unit in a realistic photonic neural network.
Fig. 4. (a) A schematic of optical neural network with bistable states in perovskite HMS. (b)-(d) Comparison of prediction accuracy for linear function (black), classic nonlinear function ReLU (red) and Sigmoid (blue) as well as our proposed nonlinear function from perovskite HMS (green). The simulation result shows our bistable nonlinear has similar performance with classic digital nonlinear activation functions ReLu and sigmoid, 98% for MNIST dataset, 88% for FashionMNIST dataset, and 51% for CIFAR-10 dataset comparing to the linear activation function with 90%, 83%, and 40%, respectively. The corresponding datasets are (b) CIFAR, (C) MNIST and (d) Fashion-MNIST.
3. Conclusion
By strategically utilizing the anisotropic property in MAPbBr3 perovskite/Au HMS, we proposed polarization-dependent optical bistable states operating at visible and near-IR frequency. The strong nonlinear effects are induced by the field enhancement in MAPbBr3 perovskite. Although the transmission level is low as a result from a large Au ratio, the nonlinear effects enlarge the transmission by 3-fold, giving rise to a considerably high bistable on-off contrast ratio for TE polarization. For TM polarization, the strong nonlinear effects give rise to ultra-low bistable threshold. Those results suggest a promising route towards the insertion of perovskite HMS into photonic IC and neural networks. In this regard, we test the feasibility of the bistable function as the activation function in a photonic neural network. It shows high accuracy, comparable to classic activation functions. Our designed perovskite HMS can be readily implemented into the future optical computing network.
Funding
Fundamental Research Funds for the Central Universities (2022RC28); State Key Laboratory of Information Photonics and Optical Communications (IPOC2023ZT03).
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.
Reference
1. Zhitong Li, X.-W. Luo, and Q. Gu, “Topological on-chip lasers,” APL Photonics 8(7), 070901 (2023). [CrossRef]
2. G. Wetzstein, A. Ozcan, S. Gigan, et al., “Inference in artificial intelligence with deep optics and photonics,” Nature 588(7836), 39–47 (2020). [CrossRef]
3. T. Alexoudi, G. T. Kanellos, and N. Pleros, “Optical RAM and integrated optical memories: a survey,” Light: Sci. Appl. 9(1), 91 (2020). [CrossRef]
4. P. Keshtkar, M. Miri, and N. Yasrebi, “Low Power, High Speed, All-Optical Logic Gates Based on Optical Bistability in Graphene-Containing Compact Microdisk Resonators,” Appl. Opt. 60(24), 7234–7242 (2021). [CrossRef]
5. X. Hu, P. Jiang, C. Ding, et al., “Picosecond and low-power all-optical switching based on an organic photonic-bandgap microcavity,” Nat. Photonics 2(3), 185–189 (2008). [CrossRef]
6. C. Min, P. Wang, C. Chen, et al., “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. 33(8), 869–871 (2008). [CrossRef]
7. C.-W. Liu, Y. Liu, L. Du, et al., “Enhanced sensing of optomechanically induced nonlinearity by linewidth suppression and optical bistability in cavity-waveguide systems,” Opt. Express 31(6), 9236–9250 (2023). [CrossRef]
8. Y.-N. Li, Y.-Y. Chen, R.-G. Wan, et al., “Dynamical switching and memory via incoherent pump assisted optical bistability,” Phys. Lett. A 383(19), 2248–2254 (2019). [CrossRef]
9. E. Kuramochi, K. Nozaki, A. Shinya, et al., “Large-scale integration of wavelength-addressable all-optical memories on a photonic crystal chip,” Nat. Photonics 8(6), 474–481 (2014). [CrossRef]
10. K. Nozaki, A. Shinya, S. Matsuo, et al., “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012). [CrossRef]
11. H. M. Gibbs, S. L. McCall, A. C. Gossard, et al., “Controlling light with light: optical bistability and optical modulation,” Laser Spectroscopy IV, Springer Series in Optical Sciences, 21, 441–450 (Springer1979).
12. R. Zhang, J. Fan, X. Zhang, et al., “Nonlinear optical response of organic-inorganic halide perovskites,” ACS Photonics 3(3), 371–377 (2016). [CrossRef]
13. P. Li, Y. Chen, T. Yang, et al., “Two-dimensional CH3NH3PbI3 perovskite nanosheets for ultrafast pulsed fiber lasers,” ACS Appl. Mater. Interfaces 9(14), 12759–12765 (2017). [CrossRef]
14. Y. Peng, J. Xu, H. Dong, et al., “Graphene-based low-threshold and tunable optical bistability in one-dimensional photonic crystal fano resonance heterostructure at optical communication band,” Opt. Express 28(23), 34948–34959 (2020). [CrossRef]
15. B. Atorf, H. Mühlenbernd, T. Zentgraf, et al., “All-optical switching of a dye-doped liquid crystal plasmonic metasurface,” Opt. Express 28(6), 8898–8908 (2020). [CrossRef]
16. A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348(6234), 516–521 (2015). [CrossRef]
17. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]
18. S. Wu, B. Jiang, Y. Liu, et al., “All-Dielectric photonic crystal with unconventional higher-order topology,” Photonics Res. 9(5), 668–677 (2021). [CrossRef]
19. M. Soljačić, M. Ibanescu, S. G. Johnson, et al., “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66(5), 055601 (2002). [CrossRef]
20. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]
21. Z. Li, X. W. Luo, D. Lin, et al., “Topological microlaser with a non-hermitian topological bulk,” Phys. Rev. Lett. 131(2), 023202 (2023). [CrossRef]
22. P. T. Bowen and D. R. Smith, “Coupled-mode theory for film-coupled plasmonic nanocubes,” Phys. Rev. B 90(19), 195402 (2014). [CrossRef]
23. Z. Huang, A. Baron, S. Larouche, et al., “Optical bistability with film-coupled metasurfaces,” Opt. Lett. 40(23), 5638–5641 (2015). [CrossRef]
24. M. Kim, S. Kim, and S. Kim, “Resonator-free optical bistability based on epsilon-near-zero mode,” Sci. Rep. 9(1), 6552 (2019). [CrossRef]
25. A. D. Neira, G. A. Wurtz, and A. V. Zayats, “All-optical switching in silicon photonic waveguides with an epsilon-near-zero resonant cavity,” Photonics Res. 6(5), B1–B5 (2018). [CrossRef]
26. J. Xu, Y. Peng, J. Jiang, et al., “Nonlinear optical bistability based on epsilon-near-zero mode in near-infrared band,” Opt. Lett. 48(12), 3235–3238 (2023). [CrossRef]
27. R. Wang, F. Hu, Y. Meng, et al., “High-contrast optical bistability using a subwavelength epsilon-near-zero material,” Opt. Lett. 48(6), 1371–1374 (2023). [CrossRef]
28. Z. Li and Q. Gu, “Topological hyperbolic metamaterials,” Nanophotonics 13(6), 825–839 (2024). [CrossRef]
29. J. S. T. Smalley, F. Vallini, X. Zhang, et al., “Dynamically tunable and active hyperbolic metamaterials,” Adv. Opt. Photonics 10(2), 354–408 (2018). [CrossRef]
30. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). [CrossRef]
31. S. Xiao, V. P. Drachev, A. V. Kildishev, et al., “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010). [CrossRef]
32. N. Fang, H. Lee, C. Sun, et al., “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef]
33. D. Lu, J. J. Kan, E. E. Fullerton, et al., “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014). [CrossRef]
34. L. Ferrari, C. Wu, D. Lepage, et al., “Hyperbolic metamaterials and their applications,” Prog. Quantum. Electron 40, 1–40 (2015). [CrossRef]
35. A. Poddubny, I. Iorsh, P. Belov, et al., “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013). [CrossRef]
36. Z. Guo, H. Jiang, and H. Chen, “Hyperbolic metamaterials: from dispersion manipulation to applications,” J. Appl. Phys. 127(7), 1 (2020). [CrossRef]
37. S. Bang, S. So, and J. Rho, “Realization of broadband negative refraction in visible range using vertically stacked hyperbolic metamaterials,” Sci. Rep. 9(1), 14093 (2019). [CrossRef]
38. P. Huo, S. Zhang, Y. Liang, et al., “Hyperbolic metamaterials and metasurfaces: fundamentals and applications,” Adv. Opt. Mater. 7(14), 1801616 (2019). [CrossRef]
39. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]
40. Z. Li, J. S. T. Smalley, R. Haroldson, et al., “Active perovskite hyperbolic metasurface,” ACS Photonics 7(7), 1754–1761 (2020). [CrossRef]
41. A. Husakou and J. Herrmann, “Steplike transmission of light through a metal-dielectric multilayer structure due to an intensity-dependent sign of the effective dielectric constant,” Phys. Rev. Lett. 99(12), 127402 (2007). [CrossRef]
42. M. Kim, S. Kim, and S. Kim, “optical bistability based on hyperbolic metamaterials,” Opt. Express 26(9), 11620–11632 (2018). [CrossRef]
43. E. Azmoudeh and S. Farazi, “Ultrafast and low power all-optical switching in the mid-infrared region based on nonlinear highly doped semiconductor hyperbolic metamaterials,” Opt. Express 29(9), 13504–13517 (2021). [CrossRef]
44. A. Kojima, K. Teshima, Y. Shirai, et al., “Organometal halide perovskites as visible-light sensitizers for photovoltaic cells,” J. Am. Chem. Soc. 131(17), 6050–6051 (2009). [CrossRef]
45. W. Liu, Q. Lin, H. Li, et al., “Mn 2+ -doped lead halide perovskite nanocrystals with dual-color emission controlled by halide content,” J. Am. Chem. Soc. 138(45), 14954–14961 (2016). [CrossRef]
46. K. Lin, J. Xing, L. N. Quan, et al., “Perovskite light-emitting diodes with external quantum efficiency exceeding 20 per cent,” Nature 562(7726), 245–248 (2018). [CrossRef]
47. Z. Li, J. Moon, A. Gharajeh, et al., “Room-temperature continuous-wave operation of organometal halide perovskite lasers,” ACS Nano 12(11), 10968–10976 (2018). [CrossRef]
48. H. Wang, R. Haroldson, B. Balachandran, et al., “Nanoimprinted perovskite nanograting photodetector with improved efficiency,” ACS Nano 10(12), 10921–10928 (2016). [CrossRef]
49. J. Yi, L. Miao, J. Li, et al., “Third-order nonlinear optical response of Ch3Nh3Pbi3 perovskite in the mid-infrared regime,” Opt. Mater. Express 7(11), 3894–3901 (2017). [CrossRef]
50. S. Mirershadi, S. Ahmadi-Kandjani, A. Zawadzka, et al., “Third order nonlinear optical properties of organometal halide perovskite by means of the Z-scan technique,” Chem. Phys. Lett. 647, 7–13 (2016). [CrossRef]
51. G. W. Drake, Springer Handbook of Atomic, Molecular, and Optical Physics (Springer Science & Business Media2007).
52. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, et al., Photonic Crystal: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).
53. J. Moon, M. AlahBakhshi, A. Gharajeh, et al., “Quasi-CW lasing from directly patterned and encapsulated perovskite cavity at 260 K,” ACS Photonics 9(6), 1984–1991 (2022). [CrossRef]
54. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
55. A. Ishteev, K. Konstantinova, G. Ermolaev, et al., “Investigation of structural and optical properties of MAPbBr3 monocrystals under fast electron irradiation,” J. Mater. Chem. C 10(15), 5821–5828 (2022). [CrossRef]
56. F. Ding, Y. Yang, R. A. Deshpande, et al., “A review of gap-surface plasmon metasurfaces: fundamentals and applications,” Nanophotonics 7(6), 1129–1156 (2018). [CrossRef]
57. S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14(20), 9467–9476 (2006). [CrossRef]
58. T. Søndergaard and S. Bozhevolnyi, “Slow-plasmon resonant nanostructures: scattering and field enhancements,” Phys. Rev. B 75(7), 073402 (2007). [CrossRef]
59. C. Ma, J. Lambrecht, F. Laporte, et al., “Comparing different nonlinearities in readout systems for optical neuromorphic computing networks,” Sci. Rep. 11(1), 24152 (2021). [CrossRef]
60. Y. Zuo, B. Li, Y. Zhao, et al., “All-optical neural network with nonlinear activation functions,” Optica 6(9), 1132–1137 (2019). [CrossRef]
61. Jonathan George, “Electrooptic nonlinear activation functions for vector matrix multiplications in optical neural networks,” in Advanced Photonics 2018 (Optica Publishing Group, 2018).
62. J. Liu, Q. Wu, X. Sui, et al., “Research progress in optical neural networks: theory, applications and developments,” PhotoniX 2(1), 5–39 (2021). [CrossRef]
63. Q. Fu, W. Yu, G. Bao, et al., “Electrically responsive photonic crystals with bistable states for low-power electrophoretic color displays,” Nat. Commun. 13(1), 7007 (2022). [CrossRef]
64. J. Gosciniak, Z. Hu, M. Thomaschewski, et al., “Bistable all-optical devices based on nonlinear epsilon-near-zero (ENZ) materials,” Laser. Photon. Rev. 17(4), 2200723 (2023). [CrossRef]
65. Y. Li, R. Chen, B. Sensale-Rodriguez, et al., “Real-time multi-task diffractive deep neural networks via hardware-software co-design,” Sci. Rep. 11(1), 11013 (2021). [CrossRef]
