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Low-Power Threshold Optical Bistability Enabled by Hydrodynamic Kerr Nonlinearity of Free Carriers in Heavily Doped Semiconductors
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Low-Power Threshold Optical Bistability Enabled by Hydrodynamic Kerr Nonlinearity of Free Carriers in Heavily Doped Semiconductors
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Cite this: ACS Photonics 2024, 11, 11, 4812–4817
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https://doi.org/10.1021/acsphotonics.4c01308
Published September 19, 2024

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Abstract

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We develop an efficient numerical model based on the semiclassical hydrodynamic theory for studying Kerr nonlinearity in degenerate electron systems such as heavily doped semiconductors. This model provides direct access to the electromagnetic responses of the quantum nature of the plasmons in heavily doped semiconductors with complex geometries, which is nontrivial for conventional frameworks. Using this model, we demonstrate nanoscale optical bistability at an exceptionally low-power threshold of 1 mW by leveraging Kerr-type hydrodynamic nonlinearities supported by the heavily doped semiconductor’s free carriers. This high nonlinearity is enabled by a strong coupling between metallic gap plasmons and longitudinal bulk plasmons in the semiconductor due to quantum pressure. These findings offer a viable approach to studying Kerr-type nonlinearity and lay the groundwork for developing efficient and ultrafast all-optical nonlinear devices.

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Introduction

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The optical Kerr effect is a third-order nonlinear phenomenon that allows modulating the refractive index of the materials according to the applied electric field intensity. (1) Exploiting such an effect to attain efficient and ultrafast photon–photon interactions has long been sought after for the potential applications in all-optical information processing and storage. (2−10) More recently, the advent of artificial intelligence has led to an exponential increase in the demand for low-power computation solutions, bringing renewed interest in integrated all-optical nonlinear systems.
Semiconductors represent in this context an interesting class of materials, as they are compatible with large-scale fabrication methods for integrated devices while also supporting some of the largest optical nonlinearities. (11,12) When heavily doped, semiconductors can undergo a transition from the quantum regime, where their response is governed by single-particle excitations, to the classical regime of collective plasmon oscillations. (12) Due to the hydrodynamic nature of nonequilibrium carriers, collective plasmons can exhibit an intrinsic and strong free carrier optical nonlinearity. (2,13−17) Although these properties occur, in principle, in all degenerate electron systems, they are significantly larger for materials with low equilibrium density and small effective mass, such as doped InP and InGaAs. (18) In fact, very recently, it has been experimentally shown by measuring third-harmonic generation from heavily doped InGaAs nanoantennas that free electron nonlinearities can easily overcome intrinsic χ(3) nonlinearities. (19) The drawback, however, is that free electrons inside a material mostly respond to external optical fields that oscillate below the plasma frequency as surface plasmons. This confines hydrodynamic nonlinear interactions at the material surface, (13,14,17) severely limiting the Kerr effect.
In this context, longitudinal bulk plasmons (LBPs) – charge density waves occurring in the bulk of spatially dispersive materials (20−22) – become very relevant. The relative distribution of the electrons over a positively charged background generates their characteristic longitudinal electric fields. (23) LBPs have been observed in thin metal films where they naturally occur in the ultraviolet energy range. (24,25) More recently, LBP resonances have been observed in transparent conducting oxide materials in the infrared, such as doped cadmium oxide. (26) Existing above the plasma frequency, the LBP sacrifices the typical field confinement associated with surface plasmons for a much larger active volume for nonlinear interactions, unlocking a new mechanism for low-power Kerr nonlinearity at the nanoscale.
From a theoretical perspective, LBP modes can be excited in finite size systems only if the response of the material is nonlocal. (21,27) One way to account for nonlocality in degenerate electron systems is to model the classical free electron dynamics incorporated with density-dependent energy potentials that encapsulate their quantum properties. This approach is usually referred to as hydrodynamic theory (HT). (28,29) For a proper choice of the kinetic energy potential, it can be shown that the HT is equivalent to the Schrödinger equation for single-particle systems (or more precisely, single orbital systems). (30) Interestingly, the HT maintains its quantum structure for systems characterized by many electrons, as long as they can be considered as a collective quasiparticle, such as plasmons. With this premise, quantum size resonances, especially the intersubband transition plasmons, in quantum-confined systems (i.e., quantum wells) in traditional semiconductors can be seen from a theoretical point of view as equivalent to quantized LBP modes. Quantum well states have been indeed experimentally observed in metals. (31−33) The coupling between semiconductor quantum wells intersubband transitions and photonic systems has been experimentally exploited for linear (34,35) and nonlinear (11,32,33,36) optical modulations. However, modeling such collective intersubband plasmon oscillations in the conduction band of the quantum wells coupled with complex photonic structures typically relies on the combination of separate steps: calculate the band information with effective parameters derived and predict optical responses with separate semiclassical or phenomenological models such as coupled mode theory. (37,38)
In contrast, in this paper, we develop an ad hoc time-domain electromagnetic model integrated with semiclassical HT that can directly access the linear and nonlinear optical responses arising from the quantum nature of electrons, which is typically nontrivial for conventional semiconductor frameworks based on Schrödinger-Poisson equations. (31,32,39) With this model, we consider for the first time the plasmonic hydrodynamic contributions to Kerr nonlinearity in heavily doped semiconductors and demonstrate that LBP strongly coupled with gap surface plasmons can indeed drive a very large Kerr effect with a wide hysteresis in subwavelength volumes. As a result, an ultralow threshold (∼1.55 mW/μm2) and compact optically bistable device is proposed.

Results and Discussion

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The many-body nature of a free electron gas driven by electromagnetic fields E(r, t) and H(r, t) can be described, within the HT, through two macroscopic fields, the electron density n(r, t) and velocity v(r, t):
me(∂∂t+v·∇+γ)v=−e(E+v×μ0H)−∇δG[n]δn
(1)
with me and e being the effective electron mass and the electron charge, respectively; γ is the phenomenological damping rate; and μ0 is the vacuum permeability. The terms from left to right in eq 1 account for time evolution, convection, dissipation, Coulomb, Lorentz, and internal energy potential, i.e., quantum pressure, respectively. Since we are not interested in electron spill-out effects, here, we consider the energy functional within the Thomas–Fermi (TF) approximation G[n] ≃ TTF[n]. (28) Introducing the polarization field defined as ∂P/∂t = J = −env, expanding the nonlinear terms up to the third order and neglecting the higher-order nonlinear terms, (16) we obtain the following equation (see Supporting Information for detailed derivation):
∂2P∂t2+γ∂P∂t=n0e2meE+β2∇(∇·P)+SNL
(2)
where the first-order quantum pressure term has been written as ∇(δTTFδn)1=meen0β2∇(∇·P), where the parameter β is related to the speed of sound in the Fermi-degenerate plasma vF, β2=13vF2=109cTFmen02/3, cTF=ℏme2310(3π2)2/3. All of the higher-order terms containing the nonlinear responses to the external field and electron–electron interactions are encapsulated into the total nonlinear source SNL, which will be elaborated on later in eq 4a and 4b. Its origin should not be confused with the bulk nonlinear susceptibility χ(3) that comes from the anharmonic potential experienced by the electrons in the crystals. The former is mostly a nonlocal response depending on the gradient of the fields, and the latter is a local response that will be considered separately in the wave equation as eq 5.
Let us start with the simplest system supporting quantized LBP modes (hereafter, we directly use LBP to refer to this quantized LBP resonance for simplicity): a thin slab excited by p-polarized light at oblique incidence of θ = 60°, as depicted in Figure 1a. Note that s-polarized light cannot excite the LBP. An electric field with a nonzero normal component at the material interface is required to align with the charge oscillations necessary for driving LBP. The linear properties of LBP can be obtained in the frequency domain by solving the linear part of eq 2 (i.e., SNL = 0) coupled to the wave equation:
∇×∇×E−εrω2c2E−μ0ω2P=0
(3)
where εr is the relative permittivity accounting for the dielectric local response and c is the speed of light in vacuum.

Figure 1

Figure 1. Optical absorption of a heavily doped InGaAs thin slab. (a) Schematic. (b) Linear absorption spectra of the thin slab with a local response approximation model (LRA, cyan) and nonlocal hydrodynamic theory (HT, purple). (c) Zoom-in plot of the higher-order LBPs. (d) Induced charge ∇·P and electric field of LBPs with an order of n = 1, 3, 5.

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We consider a thin layer (g = 10 nm) of heavily doped InGaAs with equilibrium carrier density n0 = 6 × 1018 cm–3. The doped InGaAs can be described as a Drude-like material with εinf = 12, γ = 8.9 ps–1, and me = 0.041m0 (m0 being the electron mass). The parameters were taken by fitting the experimental data in ref (19) such that the screened bulk plasma wavelength lays at λp=2πc(meε0εinf/(n0e2))=9.56μm.
The linear absorption spectra with and without quantum pressure are shown in Figure 1b,c. With local response approximation (LRA), a prominent absorption peak arises due to the epsilon-near-zero (ENZ) enhancement at λp. However, as electron–electron interactions are considered with the HT, the ENZ resonance exhibits a noticeable blueshift, accompanied by the emergence of additional higher-order resonances fulfilling the LBP condition of ε(ω, k) = 0. Only LBP modes with odd orders (n = 1, 3, 5, ···) have a net dipole moment that can be excited by a plane wave. In Figure 1d, the charge density and electric field distributions clearly show the nodes and antinodes of the standing waves formed by the charge density oscillations. The energy and the intensity of these quantized modes greatly depend on the thickness of the slab due to the confinement in that dimension, similar to quantum wells.
Although appreciable, LBP modes in a thin slab display an absorption, in the best case, of only ∼1.5% (similar values are observed for the reflectance). Higher-order resonances, shown in Figure 1c, have even smaller absorption peaks (less than 0.2%). This is not surprising since the LBPs can couple only weakly to the transverse plane wave in free space. This characteristic hinders both the experimental observation and the application in linear and nonlinear devices.
To overcome this issue, we consider a hybrid system constituted by a thin layer of heavily doped InGaAs sandwiched between a silver film and an array of silver stripes (Figure 2a), which is experimentally feasible. (36,38) Such systems are usually referred to as nanopatch antennas, in analogy to their radio frequency homologue. Nanopatch antennas have an absorption cross-section that is much larger than their physical dimensions and can concentrate far-field radiation into deep subwavelength nanogaps. (40) Additionally, the strong nanogap normal fields (Ez) are perfectly aligned with the longitudinal fields of the LBP supported by the semiconductor film (lower panel in Figure 2a).

Figure 2

Figure 2. Hybrid bulk-plasmon slab-nanopatch system. (a) Schematic of the hybrid structure; height h and periodicity Px of the antenna are 50 and 200 nm, respectively. Length, L, can vary. Bottom panels: the induced charges and electric field of the LBP in the thin spacer. (b) Strong coupling between LSP and LBP with a clear Rabi splitting in reflectance spectra, where three particular cases of different L marked by dashed lines are shown in (c) for details. Gray shades in (c) represent the reflectance of the bare nanopatch without nonlocality. Reflectance spectrum of the on-resonance case (L = 124 nm) is fitted with a Rabi splitting of Ω = 33.5 meV. Green shades mark the center wavelengths of the pump for probing Kerr nonlinearities.

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The hybrid structure is designed as follows: Since the LBP is mostly determined by the thickness and doping of the InGaAs, we fix g and n0 as the same as those described in Figure 1 such that the third-order LBP resonance is within the range of the nanopatch fundamental mode, i.e., ∼2.15 μm, as shown in Figure 2b. By adjusting the length L of the nanopatch antenna, we can tune the resonances of the antenna localized surface plasmon (LSP) and its spectral overlap with the LBP. Because of the relatively large size and high equilibrium electron density of the silver nanopatches, the nonlocality and free electron nonlinearity can be negligible. In Supporting Information S3, we also apply hydrodynamic theory to silver for validation. Thus, we treat silver as a local Drude metal with εinf = 4.039, ωp = 9.172 eV, and γ = 0.0207 eV. (41)
The detuning between the LBP and the LSP is an essential factor for the coupling. In Figure 2b, we map the reflectance of the system as a function of the incident wavelength and nanopatch L, which highlights the anticrossing dispersion between the LBP and LSP. The map shows antipeaks down to virtually zero reflectance, unveiling a nearly perfect absorption behavior. (42) Even when the two modes are quite detuned (e.g., L = 140 nm), the absorption (81.7%) of the LBP is still enormously enhanced compared to the bare InGaAs slabs (0.16% in Figure 1c). This is due to the broad line width of the LSP resonances of the nanopatch, which can still provide decent field enhancements at certain detunings. This is substantial as the hybrid mode provides a practical and efficient way to access the elusive LBP in the previous experiment. (26)
When the two uncoupled states perfectly overlap, i.e., for L = 124 nm, they hybridize into two modes with identical line width. By fitting the Rabi splitting with a temporal coupled mode theory, (43) it is possible to confirm that the system is working in a strong coupling regime. We have found that the relation between the coupling strength gc (contributing to Rabi splitting Ω = 2gc = 33.5 meV) and the sum of half-width at half-maximum of each bare mode Σi(Γi/2) satisfies the well-known condition 4gc/Σi(Γi) = 1.26 > 1, which indicates by a large margin that the system is indeed working in the strong coupling regime. As a consequence, the two resulting dressed states (i.e., the two peaks in the spectra) are hybrid bulk-surface plasmons. These hybrid modes hold the potential to inherit both the strong field enhancement from the LSP and the strong free electron nonlinearity from the LBP, providing a promising direction for the development of efficient and low-threshold nonlinear devices.
To prove this, let us investigate the hydrodynamic Kerr nonlinearity supported by the dressed LBP state in the hybrid system. The free carrier Kerr-type nonlinearity in the doped semiconductor is assumed to mainly come from the third-order hydrodynamic contributions in the constitutive relation eq 2, which arise from the convection and TF pressure, i.e., SNL = Scv + STF, where (see Supporting Information for derivations):
Scv=−1e2n02[(∇·P)P˙(∇·P˙*)+(∇·P)P˙*(∇·P˙)+(∇·P*)P˙(∇·P˙)+(∇·P)P˙·∇P˙*+(∇·P)P˙*·∇P˙+(∇·P*)P˙·∇P˙+P˙·P˙∇(∇·P*)+2P˙·P˙*∇(▽·P)]
(4a)
STF=−19β2e2n02∇[(∇·P)2(∇·P*)]
(4b)
The “·” over the variables denotes the time derivative, while the “∗” denotes the complex conjugated. Note that because SNL ∝ 1/n02, these nonlinear terms, usually neglected in noble metals, become significant in semiconductors. (18) In our calculation, the equilibrium density n0 in heavily doped InGaAs is nearly 4 orders of magnitude less than that of noble metals, enhancing nonlinear sources by 8 orders of magnitude.
Equations 2 and 4a are solved with the wave equation for the magnetic vector potential A(r, t):
∇×∇×A+εrc2∂2A∂t2+μ0∂∂t(P+PdNL)=0
(5)
The vector potential A works in the Coulomb gauge with a vanishing gradient of the scalar potential due to the absence of a static field such that the electric field E = −∂A/∂t and magnetic field μ0 H = ∇ × A. The linear and nonlinear hydrodynamic contributions are coupled into the wave equation through a polarization vector P. The lattice Kerr nonlinearity from the dielectric χ(3) = 1.6 × 10–18 m2/V2 is considered through another polarization term PdNL = 3ε0χ(3)|E|2E. (1) The contributions from the free carriers (i.e., SNL) and dielectric χ(3) (i.e., PdNL) will be separated and compared. Other nonlinear contributions including higher-order and cascaded processes are beyond the scope of this letter and have been excluded from consideration.
The system of eqs 2, 4a, 4b, and 5 is solved with a finite element method in the time domain through a customized implementation carried out using COMSOL Multiphysics. The driving field is continuous, and the incident intensity is gradually increased and decreased (will be discussed as Figure 3c) to adiabatically sweep out the nonlinear reflectance and hysteresis loop. Also, see video in Supporting Information for the probing scheme. To improve the computational efficiency and reduce the cost, we have implemented a time domain pulse envelope method in which the fast time-varying components from the full time-evolving field are separated, i.e., all the fields are expressed as F(r, t) = F̃(r, t)e–ik·r+iωt, with F = E, H, P, A. Here, k and ω are the carrier wave vector and frequency, respectively, of the initial field. The tilde indicates the corresponding slowly varying envelope field. Solving for the envelopes allows for a much larger time step (e.g., ∼102 times the wave period). To take into account the transformation above, all of the equations must be modified assuming the following transformations: ∇ → ∇ – ik and ∂/∂t → ∂ /∂t + iω. In our calculations, we neglect the hydrodynamic nonlinear contribution from silver due to its high equilibrium charge density, which leads to a negligible nonlinear contribution. (16) To verify this, we also consider the hydrodynamic contributions from silver in Supporting Information S3; however, this does not result in any significant improvement. This is due to the hydrodynamic contributions in the nonlinear source from InGaAs being much stronger than those from silver (Supporting Information Figure S2c).

Figure 3

Figure 3. Hysteresis loop of two-dimensional nonlinear nanopatch systems. Prominent optical bistabilities with two particular antenna sizes: (a) L = 110 nm and (b) L = 135 nm. (c) Nonlinear reflection (pulse distortion) due to the bistability with L = 135 nm.

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To understand the role played by the LBP-LSP coupling in the nonlinear process, we now study in detail three specific detuning situations, as summarized in Figure 2c. As the LBP is red-detuned from the LSP (L = 110 nm, bottom panel of Figure 2c), we use the sharp resonance at 2.142 μm for achieving optical bistability. We set the input signal carrier wavelength slightly above the resonance, at 2.165 μm, to probe the hysteresis loop. As a result, with the increase of pumping power, the Kerr nonlinearity will shift the resonance toward lower energies so that the resonance will gradually overlap with the pump frequency. As shown by the guide-to-eyes gray arrow in Figure 2b for L = 110 nm (orange line), the reflectance would see a decrease until it crosses the dip and then experiences an increase again.
By modulating the input intensity, we can generate the full hysteresis loop shown in Figure 3a, whose reflectance has bistable states with “on” and “off” states between ∼60% and 30% with a threshold starting around 2.45 mW/μm2. This situation could have a higher reflectance at the starting point, but the red-detuned LBP tends to move away from LSP which degrades the low reflectance dip. Likewise, we could switch off the hydrodynamic nonlinear sources (SNL = 0) and study the responses solely from enhanced dielectric nonlinearity PdNL. The results show no bistability, confirming that the hydrodynamic Kerr nonlinearity is dominating (Supporting Information Figure S1a). In addition, we keep increasing the incident intensity to allow dielectric PdNL to trigger the nonlinearity. As shown in Supporting Information Figure S1b, the threshold intensity is 5 orders of magnitude higher than that with hydrodynamic Kerr effect. In addition, the hysteresis window becomes extremely narrow compared with Figure 3a. It means that the combination of the narrow line width and the strong hydrodynamic Kerr effect is the key to the low-threshold wide hysteresis loop.
While the situation in which the LBP and LSP resonances are overlapped (Figure 2c, with L = 124 nm) is interesting from a linear response point of view, it does not represent a viable starting point for carrying out bistability since the LBP sharp resonance is hybridized and hence broadened. This results in a higher intensity threshold and a narrower hysteresis loop.
Let us now consider the opposite situation where the LBP is blue-detuned from the LSP, taking L = 135 nm for instance, and tuning the pumping wavelength at 2.135 μm. Instead of getting away from the LSP, the LBP is now pushed by the nonlinearity toward the LSP, as shown by the guide-to-eyes gray arrow in Figure 2b for L = 135 nm (purple line). The reflectance will decrease and cross the dip, instead of increasing as in the previous situation, due to the broadening of the branch in Rabi splitting. This causes a “flat” off-state in Figure 3b with a 20% reflectance. We can interpret this result as a self-modulated dynamic transition from detuned coupling to resonant strong coupling driven by the nonlocal Kerr nonlinearity. Moreover, the LBP is moving toward the LSP to gradually get a larger field enhancement and greater coupling due to the spectra overlap. This further lowers the intensity threshold for achieving optical bistability: 1.55 mW/μm2 for achieving 70% to 20% branches with 5.4 dB modulation depth, inferring the effectiveness. As shown in Supporting Information Figure S3, this low-intensity threshold can remain robust even when accounting for increased damping γ, which may come from fabrication imperfections or other factors in the experiment. The threshold is maintained at a level of 10 mW/μm2 as the damping grows to 30 ps–1. Even as damping becomes very large (γ = 40 ps–1) such that the LBP modes become weak and flat (Figure S3a), a sigmoid-function-like nonlinear response can still be expected at a level of 10 mW/μm2. Besides, the dielectric χ(3) contribution is again negligible compared with the hydrodynamic ones (Supporting Information Figure S1), revealing the advantage of the free carriers as a nonlinear source for the Kerr effect.
In Figure 3c, we depict the nonlinear reflection against the input light (red dash line). We can observe that the reflected signal is dramatically distorted due to the transitions at a threshold intensity. Other thermal and nonthermal nonlinearity might be considered separately especially when the system is excited by an ultrashort pulse with strong peak intensity. (5,44) These are not considered in the current work because the scope of the present work focuses on the hydrodynamic contributions in Kerr nonlinearity with a much lower excitation intensity. Although this threshold is comparable to previously considered systems, (3,4,7,45,46,47) the required intensity could be further reduced to ∼μW/μm2 if one can overcome the high computational cost and consider a full three-dimensional nanopatch antenna system, which would provide an extra enhancement due to the additional transverse confinement. More interestingly, applying a bias voltage to the semiconductors through the metal patches on top can be expected to allow for the active control of the Kerr nonlinearity due to the charge depletion. Potentially, 1–2 orders of the magnitude drop of the threshold can be expected. (17) Furthermore, the switching time, which determines the limit of the operation frequency, is another important factor. In Supporting Information Figure S4, we simulate the optical switching response of the structure. The switching-on and -off times can be approximately 1 ps. (see detailed discussion in Supporting Information S5). Overall, this low-power threshold enables an efficient switch between two states, while the decent modulation depth ensures the effectiveness and fidelity of the modulation process. Our finding of high hydrodynamic Kerr nonlinearity could enable crucial applications in optical data processing, such as memory, binary switching, and logic functions.

Conclusions

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In conclusion, we proposed an ad hoc numerical approach based on the finite element method in the time domain to self-consistently solve intensity-dependent equations at the nanoscale. By selecting appropriate kinetic energies and quantum potentials in the semiclassical HT, this approach is able to describe the behaviors of the quantized resonances in degenerate electron systems, including the intriguing nonlinearities from the intersubband transitions. More importantly, by leveraging the strong coupling between LBP and LSP, we explored the impact of the hydrodynamic Kerr nonlinearity and demonstrated large contrast bistability with a power threshold of ∼1 mW (considering a full patch and a focused beam area to the diffraction limit). Our findings open a viable avenue for studying and designing ultrafast all-optical integrated circuits.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphotonics.4c01308.

  • Derivation and implementation method of hydrodynamic nonlinear sources and contributions from dielectric χ(3) in the nonlinear reflectance, hydrodynamic contributions from silver and heavily doped InGaAs, robustness of the system with different dampings, and switching time (PDF)

  • Video of temporal evolution of the nonlinear system (MP4)

Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Authors
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  • Funding

    This work was supported by funding from the European Innovation Council through its Horizon Europe program with Grant Agreement No. 101046329.

  • Notes
    The authors declare no competing financial interest.

References

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    Nonlinear optical processes rely on the intrinsically weak interactions between photons enabled by their coupling with matter. Unfortunately, many applications in nonlinear optics are severely hindered by the small response of conventional materials. Metallic nanostructures partially alleviate this situation, as the large light enhancement assocd. with their localized plasmons amplifies their nonlinear response to record high levels. Graphene hosts long-lived, elec. tunable plasmons that also interact strongly with light. Here we show that the nonlinear polarizabilities of graphene nanoislands can be elec. tuned to surpass by several orders of magnitude those of metal nanoparticles of similar size. This extraordinary behavior extends over the visible and near-IR spectrum for islands consisting of hundreds of carbon atoms doped with moderate carrier densities. Our quantum-mech. simulations of the plasmon-enhanced optical response of nanographene reveal this material as an ideal platform for the development of elec. tunable nonlinear optical nanodevices.
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    Christensen, T.; Yan, W.; Jauho, A.-P.; Wubs, M.; Mortensen, N. A. Kerr nonlinearity and plasmonic bistability in graphene nanoribbons. Phys. Rev. B 2015, 92, 121407  DOI: 10.1103/PhysRevB.92.121407
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    Cox, J. D.; Silveiro, I.; García de Abajo, F. J. Quantum effects in the nonlinear response of graphene plasmons. ACS Nano 2016, 10, 1995– 2003,  DOI: 10.1021/acsnano.5b06110
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    Quantum Effects in the Nonlinear Response of Graphene Plasmons
    Cox, Joel D.; Silveiro, Ivan; Garcia de Abajo, F. Javier
    ACS Nano (2016), 10 (2), 1995-2003CODEN: ANCAC3; ISSN:1936-0851. (American Chemical Society)
    The ability of graphene to support long-lived, elec. tunable plasmons that interact strongly with light, combined with its highly nonlinear optical response, has generated great expectations for application of the atomically thin material to nanophotonic devices. These expectations are mainly reinforced by classical analyses performed using the response derived from extended graphene, neglecting finite-size and nonlocal effects that become important when the carbon layer is structured on the nanometer scale in actual device designs. Here we show that finite-size effects produce large contributions that increase the nonlinear response of nanostructured graphene to significantly higher levels than those predicted by classical theories. We base our anal. on a quantum-mech. description of graphene using tight-binding electronic states combined with the RPA. While classical and quantum descriptions agree well for the linear response when either the plasmon energy is below the Fermi energy or the size of the structure exceeds a few tens of nanometers, this is not always the case for the nonlinear response, and in particular, third-order Kerr-type nonlinearities are generally underestimated by the classical theory. Our results reveal the complex quantum nature of the optical response in nanostructured graphene, while further supporting the exceptional potential of this material for nonlinear nanophotonic devices.
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    Ryabov, D.; Pashina, O.; Zograf, G.; Makarov, S.; Petrov, M. Nonlinear optical heating of all-dielectric super-cavity: efficient light-to-heat conversion through giant thermorefractive bistability. Nanophotonics 2022, 11, 3981– 3991,  DOI: 10.1515/nanoph-2022-0074
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    Niu, J.; Shao, H.; Feng, Y.; Gao, B.; Zhang, Y.; Li, Y.; Chen, H.; Qian, H. All-Optical Nonlinear Neuron Based on Metallic Quantum Wells. Adv. Opt. Mater. 2023, 11, 2300223  DOI: 10.1002/adom.202300223
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    Lee, J.; Tymchenko, M.; Argyropoulos, C.; Chen, P.-Y.; Lu, F.; Demmerle, F.; Boehm, G.; Amann, M.-C.; Alu, A.; Belkin, M. A. Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions. Nature 2014, 511, 65– 69,  DOI: 10.1038/nature13455
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    Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions
    Lee, Jongwon; Tymchenko, Mykhailo; Argyropoulos, Christos; Chen, Pai-Yen; Lu, Feng; Demmerle, Frederic; Boehm, Gerhard; Amann, Markus-Christian; Alu, Andrea; Belkin, Mikhail A.
    Nature (London, United Kingdom) (2014), 511 (7507), 65-69CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
    Intersubband transitions in n-doped multi-quantum-well semiconductor heterostructures make it possible to engineer one of the largest known nonlinear optical responses in condensed matter systems-but this nonlinear response is limited to light with elec. field polarized normal to the semiconductor layers. In a different context, plasmonic metasurfaces (thin conductor-dielec. composite materials) have been proposed as a way of strongly enhancing light-matter interaction and realizing ultrathin planarized devices with exotic wave properties. Here we propose and exptl. realize metasurfaces with a record-high nonlinear response based on the coupling of electromagnetic modes in plasmonic metasurfaces with quantum-engineered electronic intersubband transitions in semiconductor heterostructures. We show that it is possible to engineer almost any element of the nonlinear susceptibility tensor of these structures, and we exptl. verify this concept by realizing a 400-nm-thick metasurface with nonlinear susceptibility of greater than 5 × 104 picometres per V for second harmonic generation at a wavelength of about 8 μm under normal incidence. This susceptibility is many orders of magnitude larger than any second-order nonlinear response in optical metasurfaces measured so far. The proposed structures can act as ultrathin highly nonlinear optical elements that enable efficient frequency mixing with relaxed phase-matching conditions, ideal for realizing broadband frequency up- and down-conversions, phase conjugation and all-optical control and tunability over a surface.
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    Taliercio, T.; Biagioni, P. Semiconductor infrared plasmonics. Nanophotonics 2019, 8, 949– 990,  DOI: 10.1515/nanoph-2019-0077
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    Semiconductor infrared plasmonics
    Taliercio, Thierry; Biagioni, Paolo
    Nanophotonics (2019), 8 (6), 949-990CODEN: NANOLP; ISSN:2192-8614. (Walter de Gruyter GmbH)
    The coupling between light and collective oscillations of free carriers at metallic surfaces and nanostructures is at the origin of one of the main fields of nanophotonics: plasmonics. The potential applications offered by plasmonics range from biosensing to solar cell technologies and from nonlinear optics at the nanoscale to light harvesting and extn. in nanophotonic devices. Heavily doped semiconductors are particularly appealing for the IR spectral window due to their compatibility with microelectronic technologies, which paves the way toward their integration in low-cost, mass-fabricated devices. In addn., their plasma frequency can be tuned chem., optically, or elec. over a broad spectral range. This review covers the optical properties of the heavily doped conventional semiconductors such as Ge, Si, or III-V alloys and how they can be successfully employed in plasmonics. The modeling of their specific optical properties and the technol. processes to realize nanoantennas, slits, or metasurfaces are presented. We also provide an overview of the applications of this young field of research, mainly focusing on biosensing and active devices, among the most recent developments in semiconductor plasmonics. Finally, an outlook of further research directions and the potential technol. transfer is presented.
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    Scalora, M.; Vincenti, M. A.; de Ceglia, D.; Roppo, V.; Centini, M.; Akozbek, N.; Bloemer, M. Second-and third-harmonic generation in metal-based structures. Phys. Rev. A 2010, 82, 043828  DOI: 10.1103/PhysRevA.82.043828
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    Second- and third-harmonic generation in metal-based structures
    Scalora, M.; Vincenti, M. A.; de Ceglia, D.; Roppo, V.; Centini, M.; Akozbek, N.; Bloemer, M. J.
    Physical Review A: Atomic, Molecular, and Optical Physics (2010), 82 (4, Pt. B), 043828/1-043828/14CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    We present a theor. approach to the study of second- and third-harmonic generation from metallic structures and nanocavities filled with a nonlinear material in the ultrashort pulse regime. We model the metal as a two-component medium, using the hydrodynamic model to describe free electrons and Lorentz oscillators to account for core electron contributions to both the linear dielec. const. and harmonic generation. The active nonlinear medium that may fill a metallic nanocavity, or be positioned between metallic layers in a stack, is also modeled using Lorentz oscillators and surface phenomena due to symmetry breaking are taken into account. We study the effects of incident TE- and TM-polarized fields and show that a simple reexamn. of the basic equations reveals addnl., exploitable dynamical features of nonlinear frequency conversion in plasmonic nanostructures.
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    Ciracì, C.; Poutrina, E.; Scalora, M.; Smith, D. R. Second-harmonic generation in metallic nanoparticles: Clarification of the role of the surface. Phys. Rev. B 2012, 86, 115451  DOI: 10.1103/PhysRevB.86.115451
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    Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface
    Ciraci, Cristian; Poutrina, Ekaterina; Scalora, Michael; Smith, David R.
    Physical Review B: Condensed Matter and Materials Physics (2012), 86 (11), 115451/1-115451/10CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We present a numerical investigation of the second-order nonlinear optical properties of metal-based metamaterial nanoresonators. The nonlinear optical response of the metal is described by a hydrodynamic model, with the effects of electron pressure in the electron gas also taken into account. We show that as the pressure term tends to zero the amt. of converted second-harmonic field tends to an asymptotic value. In this limit it becomes possible to rewrite the nonlinear surface contributions as functions of the value of the polarization vector inside the bulk region. Nonlocality thus can be incorporated into numerical simulations without actually utilizing the nonlocal equation of motion or solving for the rapidly varying fields that occur near the metal surface. We use our model to investigate the second-harmonic generation process with three-dimensional gold nanoparticle arrays and show that nanocrescents can easily attain conversion efficiencies of ∼6.0 × 10-8 for pumping peak intensities of a few tens of MW/cm2.
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    Khalid, M.; Ciracì, C. Enhancing second-harmonic generation with electron spill-out at metallic surfaces. Commun. Phys. 2020, 3, 214,  DOI: 10.1038/s42005-020-00477-0
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    De Luca, F.; Ortolani, M.; Ciracì, C. Free electron nonlinearities in heavily doped semiconductors plasmonics. Phys. Rev. B 2021, 103, 115305  DOI: 10.1103/PhysRevB.103.115305
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    Free electron nonlinearities in heavily doped semiconductors plasmonics
    De Luca, Federico; Ortolani, Michele; Ciraci, Cristian
    Physical Review B (2021), 103 (11), 115305CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    Heavily doped semiconductors have emerged as tunable low-loss plasmonic materials at midinfrared frequencies. In this article we investigate nonlinear optical phenomena assocd. with high concn. of free electrons. We use a hydrodynamic description to study free electron dynamics in heavily doped semiconductors up to third-order terms, which are usually negligible for noble metals. We find that cascaded third-harmonic generation due to second-harmonic signals can be as strong as direct third-harmonic generation contributions even when the second-harmonic generation efficiency is zero. Moreover, we show that when coupled with plasmonic enhancement free electron nonlinearities could be up to two orders of magnitude larger than conventional semiconductor nonlinearities. Our work might open a new route for nonlinear optical integrated devices at midinfrared frequencies.
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    De Luca, F.; Ciracì, C. Impact of surface charge depletion on the free electron nonlinear response of heavily doped semiconductors. Phys. Rev. Lett. 2022, 129, 123902  DOI: 10.1103/PhysRevLett.129.123902
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    De Luca, F.; Ortolani, M.; Ciracì, C. Free electron harmonic generation in heavily doped semiconductors: the role of the materials properties. EPJ. Appl. Metamat. 2022, 9, 13,  DOI: 10.1051/epjam/2022011
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    Rossetti, A. Origin of optical nonlinearity in plasmonic semiconductor nanostructures. arXiv preprint 2024, arXiv:2402.15443, https://arxiv.org/abs/2402.15443 (accessed Feb 23, 2024).
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    Ruppin, R. Extinction properties of thin metallic nanowires. Opt. Commun. 2001, 190, 205– 209,  DOI: 10.1016/S0030-4018(01)01063-X
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    Extinction properties of thin metallic nanowires
    Ruppin, R.
    Optics Communications (2001), 190 (1-6), 205-209CODEN: OPCOB8; ISSN:0030-4018. (Elsevier Science B.V.)
    The classical electromagnetic scattering theory for a circular cylinder is extended so as to allow for the nonlocal response properties of metals. Using hydrodynamic dielec. functions and applying appropriate addnl. boundary conditions, the generalized scattering coeffs. are derived. For very thin nanowires, the main extinction peak, due to the surface plasmon, is shifted from its classical position towards the high frequency side and subsidiary peaks, due to bulk plasmon excitation, appear above the plasma frequency.
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    Raza, S.; Toscano, G.; Jauho, A.-P.; Wubs, M.; Mortensen, N. A. Unusual resonances in nanoplasmonic structures due to nonlocal response. Phys. Rev. B 2011, 84, 121412  DOI: 10.1103/PhysRevB.84.121412
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    Unusual resonances in nanoplasmonic structures due to nonlocal response
    Raza, Soren; Toscano, Giuseppe; Jauho, Antti-Pekka; Wubs, Martijn; Mortensen, N. Asger
    Physical Review B: Condensed Matter and Materials Physics (2011), 84 (12), 121412/1-121412/4CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We study the nonlocal response of a confined electron gas within the hydrodynamical Drude model. We address the question as to whether plasmonic nanostructures exhibit nonlocal resonances that have no counterpart in the local-response Drude model. Avoiding the usual quasistatic approxn., we find that such resonances do indeed occur, but only above the plasma frequency. Thus the recently found nonlocal resonances at optical frequencies for very small structures, obtained within quasistatic approxn., are unphys. As a specific example we consider nanosized metallic cylinders, for which extinction cross sections and field distributions can be calcd. anal.
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    Anderegg, M.; Feuerbacher, B.; Fitton, B. Optically excited longitudinal plasmons in potassium. Phys. Rev. Lett. 1971, 27, 1565,  DOI: 10.1103/PhysRevLett.27.1565
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    Experimental verifiication of optically excited longitudinal plasmons
    Lindau, I.; Nilsson, P. O.
    Physica Scripta (1971), 3 (2), 87-92CODEN: PHSTBO; ISSN:0031-8949.
    Anomalous optical absorption is obsd. in thin Ag films just above the bulk plasma frequency. The result is interpreted as excitation of longitudinal plasmons, an effect which is normally neglected when discussing optical properties. The dispersion relation of the plasma wave is obtained from the data.
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    De Ceglia, D.; Scalora, M.; Vincenti, M. A.; Campione, S.; Kelley, K.; Runnerstrom, E. L.; Maria, J.-P.; Keeler, G. A.; Luk, T. S. Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms. Sci. Rep. 2018, 8, 9335,  DOI: 10.1038/s41598-018-27655-z
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    26
    Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms
    de Ceglia Domenico; Scalora Michael; Vincenti Maria A; Campione Salvatore; Keeler Gordon A; Luk Ting S; Kelley Kyle; Runnerstrom Evan L; Maria Jon-Paul; Luk Ting S
    Scientific reports (2018), 8 (1), 9335 ISSN:.
    Optical nonlocalities are elusive and hardly observable in traditional plasmonic materials like noble and alkali metals. Here we report experimental observation of viscoelastic nonlocalities in the infrared optical response of epsilon-near-zero nanofilms made of low-loss doped cadmium-oxide. The nonlocality is detectable thanks to the low damping rate of conduction electrons and the virtual absence of interband transitions at infrared wavelengths. We describe the motion of conduction electrons using a hydrodynamic model for a viscoelastic fluid, and find excellent agreement with experimental results. The electrons' elasticity blue-shifts the infrared plasmonic resonance associated with the main epsilon-near-zero mode, and triggers the onset of higher-order resonances due to the excitation of electron-pressure modes above the bulk plasma frequency. We also provide evidence of the existence of nonlocal damping, i.e., viscosity, in the motion of optically-excited conduction electrons using a combination of spectroscopic ellipsometry data and predictions based on the viscoelastic hydrodynamic model.
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    Christensen, T.; Yan, W.; Raza, S.; Jauho, A.-P.; Mortensen, N. A.; Wubs, M. Nonlocal response of metallic nanospheres probed by light, electrons, and atoms. ACS Nano 2014, 8, 1745– 1758,  DOI: 10.1021/nn406153k
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    Nonlocal Response of Metallic Nanospheres Probed by Light, Electrons, and Atoms
    Christensen, Thomas; Yan, Wei; Raza, Soeren; Jauho, Antti-Pekka; Mortensen, N. Asger; Wubs, Martijn
    ACS Nano (2014), 8 (2), 1745-1758CODEN: ANCAC3; ISSN:1936-0851. (American Chemical Society)
    Inspired by recent measurements on individual metallic nanospheres that cannot be explained with traditional classical electrodynamics, the authors theor. study the effects of nonlocal response by metallic nanospheres in 3 distinct settings: at. spontaneous emission, EELS, and light scattering. These constitute 2 near-field and 1 far-field measurements, with zero-, 1-, and 2-dimensional excitation sources, resp. The authors search for the clearest signatures of hydrodynamic pressure waves in nanospheres. The authors employ a linearized hydrodynamic model, and Mie-Lorenz theory is applied for each case. Nonlocal response shows its mark in all 3 configurations, but for the 2 near-field measurements, the authors predict esp. pronounced nonlocal effects that are not exhibited in far-field measurements. Assocd. with every multipole order is not only a single blueshifted surface plasmon but also an infinite series of bulk plasmons that have no counterpart in a local-response approxn. These increasingly blueshifted multipole plasmons become spectrally more prominent at shorter probe-to-surface sepns. and for decreasing nanosphere radii. For selected metals, the authors predict hydrodynamic multipolar plasmons to be measurable on single nanospheres.
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    Ciracì, C.; Della Sala, F. Quantum hydrodynamic theory for plasmonics: Impact of the electron density tail. Phys. Rev. B 2016, 93, 205405  DOI: 10.1103/PhysRevB.93.205405
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    Quantum hydrodynamic theory for plasmonics: impact of the electron density tail
    Ciraci, Cristian; Della Sala, Fabio
    Physical Review B (2016), 93 (20), 205405/1-205405/14CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
    Multiscale plasmonic systems (e.g., extended metallic nanostructures with subnanometer inter-distances) play a key role in the development of next-generation nanophotonic devices. An accurate modeling of the optical interactions in these systems requires an accurate description of both quantum effects and far-field properties. Classical electromagnetism can only describe the latter, while time-dependent d. functional theory (TD-DFT) can provide a full first-principles quantum treatment. However, TD-DFT becomes computationally prohibitive for sizes that exceed few nanometers, which are instead very important for most applications. In this article, we introduce a method based on the quantum hydrodynamic theory (QHT) that includes nonlocal contributions of the kinetic energy and the correct asymptotic description of the electron d. We show that our QHT method can predict both plasmon energy and spill-out effects in metal nanoparticles in excellent agreement with TD-DFT predictions, thus allowing reliable and efficient calcns. of both quantum and far-field properties in multiscale plasmonic systems.
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    Toscano, G.; Straubel, J.; Kwiatkowski, A.; Rockstuhl, C.; Evers, F.; Xu, H.; Asger Mortensen, N.; Wubs, M. Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics. Nat. Commun. 2015, 6, 7132,  DOI: 10.1038/ncomms8132
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    Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics
    Toscano Giuseppe; Straubel Jakob; Kwiatkowski Alexander; Rockstuhl Carsten; Evers Ferdinand; Xu Hongxing; Mortensen N Asger; Wubs Martijn
    Nature communications (2015), 6 (), 7132 ISSN:.
    The standard hydrodynamic Drude model with hard-wall boundary conditions can give accurate quantitative predictions for the optical response of noble-metal nanoparticles. However, it is less accurate for other metallic nanosystems, where surface effects due to electron density spill-out in free space cannot be neglected. Here we address the fundamental question whether the description of surface effects in plasmonics necessarily requires a fully quantum-mechanical ab initio approach. We present a self-consistent hydrodynamic model (SC-HDM), where both the ground state and the excited state properties of an inhomogeneous electron gas can be determined. With this method we are able to explain the size-dependent surface resonance shifts of Na and Ag nanowires and nanospheres. The results we obtain are in good agreement with experiments and more advanced quantum methods. The SC-HDM gives accurate results with modest computational effort, and can be applied to arbitrary nanoplasmonic systems of much larger sizes than accessible with ab initio methods.
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    Ciracì, C. Current-dependent potential for nonlocal absorption in quantum hydrodynamic theory. Phys. Rev. B 2017, 95, 245434  DOI: 10.1103/PhysRevB.95.245434
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    Current-dependent potential for nonlocal absorption in quantum hydrodynamic theory
    Ciraci, Cristian
    Physical Review B (2017), 95 (24), 245434/1-245434/10CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    The quantum hydrodynamic theory is a promising method for describing microscopic details of macroscopic systems. The hydrodynamic equation can be partially obtained from a single-particle Kohn-Sham equation and improved by adding a viscoelastic kinetic-exchange-correlation tensor term, so that broadening of collective excitation can be taken into account, as well as a correction to the plasmon dispersion. The result is an accurate self-consistent and computationally efficient hydrodynamic description of the free-electron gas. A very accurate agreement with full quantum calcns. is shown.
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    Silberberg, Y.; Sands, T. Optical properties of metallic quantum wells. IEEE J. Quantum Electron. 1992, 28, 1663– 1669,  DOI: 10.1109/3.142553
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    Qian, H.; Xiao, Y.; Liu, Z. Giant Kerr response of ultrathin gold films from quantum size effect. Nat. Commun. 2016, 7, 13153  DOI: 10.1038/ncomms13153
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    Giant Kerr response of ultrathin gold films from quantum size effect
    Qian, Haoliang; Xiao, Yuzhe; Liu, Zhaowei
    Nature Communications (2016), 7 (), 13153CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
    With the size of plasmonic devices entering into the nanoscale region, the impact of quantum physics needs to be considered. In the past, the quantum size effect on linear material properties has been studied extensively. However, the nonlinear aspects have not been explored much so far. On the other hand, much effort has been put into the field of integrated nonlinear optics and a medium with large nonlinearity is desirable. Here we study the optical nonlinear properties of a nanometer scale gold quantum well by using the z-scan method and nonlinear spectrum broadening technique. The quantum size effect results in a giant optical Kerr susceptibility, which is four orders of magnitude higher than the intrinsic value of bulk gold and several orders larger than traditional nonlinear media. Such high nonlinearity enables efficient nonlinear interaction within a microscopic footprint, making quantum metallic films a promising candidate for integrated nonlinear optical applications.
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    Qian, H.; Li, S.; Chen, C.-F.; Hsu, S.-W.; Bopp, S. E.; Ma, Q.; Tao, A. R.; Liu, Z. Large optical nonlinearity enabled by coupled metallic quantum wells. Light Sci. Appl. 2019, 8, 13,  DOI: 10.1038/s41377-019-0123-4
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    Large optical nonlinearity enabled by coupled metallic quantum wells
    Qian Haoliang; Li Shilong; Chen Ching-Fu; Ma Qian; Liu Zhaowei; Hsu Su-Wen; Tao Andrea R; Bopp Steven Edward; Tao Andrea R; Liu Zhaowei; Liu Zhaowei
    Light, science & applications (2019), 8 (), 13 ISSN:.
    New materials that exhibit strong second-order optical nonlinearities at a desired operational frequency are of paramount importance for nonlinear optics. Giant second-order susceptibility χ((2)) has been obtained in semiconductor quantum wells (QWs). Unfortunately, the limited confining potential in semiconductor QWs causes formidable challenges in scaling such a scheme to the visible/near-infrared (NIR) frequencies for more vital nonlinear-optic applications. Here, we introduce a metal/dielectric heterostructured platform, i.e., TiN/Al2O3 epitaxial multilayers, to overcome that limitation. This platform has an extremely high χ((2)) of approximately 1500 pm/V at NIR frequencies. By combining the aforementioned heterostructure with the large electric field enhancement afforded by a nanostructured metasurface, the power efficiency of second harmonic generation (SHG) achieved 10(-4) at an incident pulse intensity of 10 GW/cm(2), which is an improvement of several orders of magnitude compared to that of previous demonstrations from nonlinear surfaces at similar frequencies. The proposed quantum-engineered heterostructures enable efficient wave mixing at visible/NIR frequencies into ultracompact nonlinear optical devices.
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    Pirotta, S.; Tran, N.-L.; Jollivet, A.; Biasiol, G.; Crozat, P.; Manceau, J.-M.; Bousseksou, A.; Colombelli, R. Fast amplitude modulation up to 1.5 GHz of mid-IR free-space beams at room-temperature. Nat. Commun. 2021, 12, 799,  DOI: 10.1038/s41467-020-20710-2
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    34
    Fast amplitude modulation up to 1.5 GHz of mid-IR free-space beams at room-temperature
    Pirotta, Stefano; Tran, Ngoc-Linh; Jollivet, Arnaud; Biasiol, Giorgio; Crozat, Paul; Manceau, Jean-Michel; Bousseksou, Adel; Colombelli, Raffaele
    Nature Communications (2021), 12 (1), 799CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)
    Applications relying on mid-IR radiation (λ ∼ 3-30μm) have progressed at a very rapid pace in recent years, stimulated by scientific and technol. breakthroughs like mid-IR cameras and quantum cascade lasers. On the other side, standalone and broadband devices allowing control of the beam amplitude and/or phase at ultra-fast rates (GHz or more) are still missing. Here we show a free-space amplitude modulator for mid-IR radiation (λ ∼ 10μm) that can operate at room temp. up to at least 1.5 GHz (-3dB cutoff at ∼750 MHz). The device relies on a semiconductor heterostructure enclosed in a judiciously designed metal-metal optical resonator. At zero bias, it operates in the strong light-matter coupling regime up to 300 K. By applying an appropriate bias, the device transitions toward the weak-coupling regime. The large change in reflectance is exploited to modulate the intensity of a mid-IR continuous-wave laser up to 1.5 GHz.
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    Malerba, M.; Pirotta, S.; Aubin, G.; Lucia, L.; Jeannin, M.; Manceau, J.-M.; Bousseksou, A.; Lin, Q.; Lampin, J.-F.; Peytavit, E.; Barbieri, S.; Li, L.; Davies, G.; Linfield, E. H.; Colombelli, R. Ultrafast (≈ 10 GHz) mid-IR modulator based on ultrafast electrical switching of the light–matter coupling. Appl. Phys. Lett. 2024, 125, 041101  DOI: 10.1063/5.0213965
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    Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption
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    Nature Physics (2014), 10 (11), 830-834CODEN: NPAHAX; ISSN:1745-2473. (Nature Publishing Group)
    The ability to drive a system with an external input is a fundamental aspect of light-matter interaction. The key concept in many photonic applications is the 'crit. coupling' condition: at criticality, all the energy fed to the system is dissipated within the system itself. Although this idea was crucial to enhance the efficiency of many devices, it was never considered in the context of systems operating in a non-perturbative regime. In this so-called strong-coupling regime, the matter and light degrees of freedom are mixed into dressed states, leading to new eigenstates called polaritons. Here we demonstrate that the strong-coupling regime and the crit. coupling condition can coexist; in such a strong crit. coupling situation, all the incoming energy is converted into polaritons. A general semiclassical theory reveals that such a situation corresponds to a special curve in the phase diagram of the coupled light-matter oscillators. In the case of a system with two radiating ports, the phenomenol. shown is that of coherent perfect absorption (CPA; refs , ), which is then naturally understood in the framework of crit. coupling. Most importantly, we exptl. verify polaritonic CPA in a semiconductor-based intersubband-polariton photonic crystal resonator. This result opens new avenues in polariton physics, making it possible to control the pumping efficiency of a system independent of the energy exchange rate between the electromagnetic field and the material transition.
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    Jeannin, M.; Manceau, J.-M.; Colombelli, R. Unified description of saturation and bistability of intersubband transitions in the weak and strong light-matter coupling regimes. Phys. Rev. Lett. 2021, 127, 187401  DOI: 10.1103/PhysRevLett.127.187401
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    Unified Description of Saturation and Bistability of Intersubband Transitions in the Weak and Strong Light-Matter Coupling Regimes
    Jeannin, Mathieu; Manceau, Jean-Michel; Colombelli, Raffaele
    Physical Review Letters (2021), 127 (18), 187401CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
    We propose a unified description of intersubband absorption satn. for quantum wells inserted in a resonator, both in the weak and strong light-matter coupling regimes. We demonstrate how absorption satn. can be engineered. In particular, we show that the satn. intensity increases linearly with the doping in the strong coupling regime, while it remains doping independent in weak coupling. Hence, countering intuition, the most suitable region to exploit low satn. intensities is not the ultrastrong coupling regime, but is instead at the onset of the strong light-matter coupling. We further derive explicit conditions for the emergence of bistability. This Letter sets the path toward, as yet, nonexistent ultrafast midinfrared semiconductor saturable absorption mirrors (SESAMs) and bistable systems. As an example, we show how to design a midinfrared SESAM with a 3 orders of magnitude redn. in satn. intensity, down to ≈5 kW cm-2.
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    Cominotti, R.; Leymann, H.; Nespolo, J.; Manceau, J.-M.; Jeannin, M.; Colombelli, R.; Carusotto, I. Theory of coherent optical nonlinearities of intersubband transitions in semiconductor quantum wells. Phys. Rev. B 2023, 107, 115431  DOI: 10.1103/PhysRevB.107.115431
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    Nature Materials (2019), 18 (7), 668-678CODEN: NMAACR; ISSN:1476-1122. (Nature Research)
    A review. Ultrathin dielec. gaps between metals can trap plasmonic optical modes with surprisingly low loss and with vols. below 1 nm3. We review the origin and subtle properties of these modes, and show how they can be well accounted for by simple models. Particularly important is the mixing between radiating antennas and confined nanogap modes, which is extremely sensitive to precise nanogeometry, right down to the single-atom level. Coupling nanogap plasmons to electronic and vibronic transitions yields a host of phenomena including single-mol. strong coupling and mol. optomechanics, opening access to at.-scale chem. and materials science, as well as quantum metamaterials. Ultimate low-energy devices such as robust bottom-up assembled single-atom switches are thus in prospect.
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    The optical consts. n and k were obtained for Cu, Ag, and Au from reflection and transmission measurements on vacuum-evapd. thin films at room temp., in the spectral range 0.5-6.5 eV. The film-thickness range was 185-500 Å. Three optical measurements were inverted to obtain the film thickness d as well as n and k. The estd. error in d was ±2 Å, and that in n, k was <0.02 over most of the spectral range. The results in the film-thickness range 250-500 Å were independent of thickness, and were unchanged after vacuum annealing or aging in air. The free-electron optical effective masses and relaxation times derived from the results in the near ir agreed satisfactorily with previous values. The interband contribution to the imaginary part of the dielec. const. was obtained by subtracting the free-electron contribution. Some recent theor. calcns. were compared with the results for Cu and Au. In addn., some other recent expts. are crit. compared with the present results.
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    Controlled-reflectance surfaces with film-coupled colloidal nanoantennas
    Moreau, Antoine; Ciraci, Cristian; Mock, Jack J.; Hill, Ryan T.; Wang, Qiang; Wiley, Benjamin J.; Chilkoti, Ashutosh; Smith, David R.
    Nature (London, United Kingdom) (2012), 492 (7427), 86-89CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
    Efficient and tunable absorption is essential for a variety of applications, such as designing controlled-emissivity surfaces for thermophotovoltaic devices, tailoring an IR spectrum for controlled thermal dissipation and producing detector elements for imaging. Metamaterials based on metallic elements are particularly efficient as absorbing media, because both the elec. and the magnetic properties of a metamaterial can be tuned by structured design. So far, metamaterial absorbers in the IR or visible range were fabricated using lithog. patterned metallic structures, making them inherently difficult to produce over large areas and hence reducing their applicability. Here the authors demonstrate a simple method to create a metamaterial absorber by randomly adsorbing chem. synthesized Ag nanocubes onto a nanoscale-thick polymer spacer layer on a Au film, making no effort to control the spatial arrangement of the cubes on the film. The film-coupled nanocubes provide a reflectance spectrum that can be tailored by varying the geometry (the size of the cubes and/or the thickness of the spacer). Each nanocube is the optical analog of a grounded patch antenna, with a nearly identical local field structure that is modified by the plasmonic response of the metal's dielec. function, and with an anomalously large absorption efficiency that can be partly attributed to an interferometric effect. The absorptivity of large surface areas can be controlled using this method, at scales out of reach of lithog. approaches (such as electron-beam lithog.) that are otherwise required to manipulate matter on the nanoscale.
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    Metasurfaces comprising arrays of film-coupled, nanopatch antennas are a promising platform for low-energy, all-optical switches. The large field enhancements that can be achieved in the dielec. spacer region between the nanopatch and the metallic substrate can substantially enhance optical nonlinear processes. Here we consider a dielec. material that exhibits an optical Kerr effect as the spacer layer and numerically calc. the optical bistability of a metasurface using the finite element method (FEM). We expect the proposed method to be highly accurate compared with other numerical approaches, such as those based on graphical post-processing techniques, because it self-consistently solves for both the spatial field distribution and the intensity-dependent refractive index distribution of the spacer layer. This method offers an alternative approach to finite-difference time-domain (FDTD) modeling. We use this numerical tool to design a metasurface optical switch and our optimized design exhibits exceptionally low switching intensity of 33 kW/cm2, corresponding to switching energy on the order of tens of attojoules per resonator, a value much smaller than those found for most devices reported in the literature. We propose our method as a tool for designing all-optical switches and modulators.
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    Hu, H.; Álvarez-Pérez, G.; Otomalo, T. O.; Ciracì, C. Low-power threshold optical bistability enabled by hydrodynamic Kerr nonlinearity of free-carriers in heavily doped semiconductors. arXiv preprint 2024, arXiv:2406.07750, https://arxiv.org/abs/2406.07750 (accessed Jun 11, 2024).
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  • Abstract

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    Figure 1

    Figure 1. Optical absorption of a heavily doped InGaAs thin slab. (a) Schematic. (b) Linear absorption spectra of the thin slab with a local response approximation model (LRA, cyan) and nonlocal hydrodynamic theory (HT, purple). (c) Zoom-in plot of the higher-order LBPs. (d) Induced charge ∇·P and electric field of LBPs with an order of n = 1, 3, 5.

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    Figure 2

    Figure 2. Hybrid bulk-plasmon slab-nanopatch system. (a) Schematic of the hybrid structure; height h and periodicity Px of the antenna are 50 and 200 nm, respectively. Length, L, can vary. Bottom panels: the induced charges and electric field of the LBP in the thin spacer. (b) Strong coupling between LSP and LBP with a clear Rabi splitting in reflectance spectra, where three particular cases of different L marked by dashed lines are shown in (c) for details. Gray shades in (c) represent the reflectance of the bare nanopatch without nonlocality. Reflectance spectrum of the on-resonance case (L = 124 nm) is fitted with a Rabi splitting of Ω = 33.5 meV. Green shades mark the center wavelengths of the pump for probing Kerr nonlinearities.

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    Figure 3

    Figure 3. Hysteresis loop of two-dimensional nonlinear nanopatch systems. Prominent optical bistabilities with two particular antenna sizes: (a) L = 110 nm and (b) L = 135 nm. (c) Nonlinear reflection (pulse distortion) due to the bistability with L = 135 nm.

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      Low-electron-d. Drude (LEDD) materials such as indium tin oxide (ITO) are receiving considerable attention for their combination of CMOS compatibility, unique epsilon-near-zero (ENZ) behavior, and giant ultrafast nonlinear thermo-optic response. However, current understanding of the electronic and optical response of LEDD materials is limited due to the simplistic modeling that extends only the known models of noble metals without considering the interplay among the lower electron d., relatively high Debye energy, and the nonparabolic band structure. We bridge this knowledge gap and provide a complete understanding of the nonlinear electronic-thermal-optical response of LEDD materials. In particular, we rely on state-of-the-art electron dynamics modeling, as well as a time-dependent permittivity model for LEDD materials under optical pumping within the adiabatic approxn. We find the electron temp. may reach values much higher than realized before, even exceeding the Fermi temp., in which case the effective chem. potential dramatically decreases and even becomes neg., thus, transient giving the material some characteristics of a semiconductor. We further show that the nonlinear optical response of LEDD materials originating from the changes to the real part of the permittivity is assocd. with changes of the population. This resolves the argument about the rise time of the permittivity, showing that it is instantaneous. In this vein, we show that referring to the LEDD permittivity as having a Kerr or "saturable" nonlinearity is unsuitable since its permittivity dynamics is absorptive rather than nonresonant and does not originate from population inversion. Finally, we analyze the probe-pulse dynamics and unlike previous work, we obtain a quant. agreement with the results of recent expts.
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      Nature Communications (2014), 5 (), 5725CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
      Nonlinear optical processes rely on the intrinsically weak interactions between photons enabled by their coupling with matter. Unfortunately, many applications in nonlinear optics are severely hindered by the small response of conventional materials. Metallic nanostructures partially alleviate this situation, as the large light enhancement assocd. with their localized plasmons amplifies their nonlinear response to record high levels. Graphene hosts long-lived, elec. tunable plasmons that also interact strongly with light. Here we show that the nonlinear polarizabilities of graphene nanoislands can be elec. tuned to surpass by several orders of magnitude those of metal nanoparticles of similar size. This extraordinary behavior extends over the visible and near-IR spectrum for islands consisting of hundreds of carbon atoms doped with moderate carrier densities. Our quantum-mech. simulations of the plasmon-enhanced optical response of nanographene reveal this material as an ideal platform for the development of elec. tunable nonlinear optical nanodevices.
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      Quantum Effects in the Nonlinear Response of Graphene Plasmons
      Cox, Joel D.; Silveiro, Ivan; Garcia de Abajo, F. Javier
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      The ability of graphene to support long-lived, elec. tunable plasmons that interact strongly with light, combined with its highly nonlinear optical response, has generated great expectations for application of the atomically thin material to nanophotonic devices. These expectations are mainly reinforced by classical analyses performed using the response derived from extended graphene, neglecting finite-size and nonlocal effects that become important when the carbon layer is structured on the nanometer scale in actual device designs. Here we show that finite-size effects produce large contributions that increase the nonlinear response of nanostructured graphene to significantly higher levels than those predicted by classical theories. We base our anal. on a quantum-mech. description of graphene using tight-binding electronic states combined with the RPA. While classical and quantum descriptions agree well for the linear response when either the plasmon energy is below the Fermi energy or the size of the structure exceeds a few tens of nanometers, this is not always the case for the nonlinear response, and in particular, third-order Kerr-type nonlinearities are generally underestimated by the classical theory. Our results reveal the complex quantum nature of the optical response in nanostructured graphene, while further supporting the exceptional potential of this material for nonlinear nanophotonic devices.
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      There is no corresponding record for this reference.
    11. 11
      Lee, J.; Tymchenko, M.; Argyropoulos, C.; Chen, P.-Y.; Lu, F.; Demmerle, F.; Boehm, G.; Amann, M.-C.; Alu, A.; Belkin, M. A. Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions. Nature 2014, 511, 65– 69,  DOI: 10.1038/nature13455
      11
      Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions
      Lee, Jongwon; Tymchenko, Mykhailo; Argyropoulos, Christos; Chen, Pai-Yen; Lu, Feng; Demmerle, Frederic; Boehm, Gerhard; Amann, Markus-Christian; Alu, Andrea; Belkin, Mikhail A.
      Nature (London, United Kingdom) (2014), 511 (7507), 65-69CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
      Intersubband transitions in n-doped multi-quantum-well semiconductor heterostructures make it possible to engineer one of the largest known nonlinear optical responses in condensed matter systems-but this nonlinear response is limited to light with elec. field polarized normal to the semiconductor layers. In a different context, plasmonic metasurfaces (thin conductor-dielec. composite materials) have been proposed as a way of strongly enhancing light-matter interaction and realizing ultrathin planarized devices with exotic wave properties. Here we propose and exptl. realize metasurfaces with a record-high nonlinear response based on the coupling of electromagnetic modes in plasmonic metasurfaces with quantum-engineered electronic intersubband transitions in semiconductor heterostructures. We show that it is possible to engineer almost any element of the nonlinear susceptibility tensor of these structures, and we exptl. verify this concept by realizing a 400-nm-thick metasurface with nonlinear susceptibility of greater than 5 × 104 picometres per V for second harmonic generation at a wavelength of about 8 μm under normal incidence. This susceptibility is many orders of magnitude larger than any second-order nonlinear response in optical metasurfaces measured so far. The proposed structures can act as ultrathin highly nonlinear optical elements that enable efficient frequency mixing with relaxed phase-matching conditions, ideal for realizing broadband frequency up- and down-conversions, phase conjugation and all-optical control and tunability over a surface.
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    12. 12
      Taliercio, T.; Biagioni, P. Semiconductor infrared plasmonics. Nanophotonics 2019, 8, 949– 990,  DOI: 10.1515/nanoph-2019-0077
      12
      Semiconductor infrared plasmonics
      Taliercio, Thierry; Biagioni, Paolo
      Nanophotonics (2019), 8 (6), 949-990CODEN: NANOLP; ISSN:2192-8614. (Walter de Gruyter GmbH)
      The coupling between light and collective oscillations of free carriers at metallic surfaces and nanostructures is at the origin of one of the main fields of nanophotonics: plasmonics. The potential applications offered by plasmonics range from biosensing to solar cell technologies and from nonlinear optics at the nanoscale to light harvesting and extn. in nanophotonic devices. Heavily doped semiconductors are particularly appealing for the IR spectral window due to their compatibility with microelectronic technologies, which paves the way toward their integration in low-cost, mass-fabricated devices. In addn., their plasma frequency can be tuned chem., optically, or elec. over a broad spectral range. This review covers the optical properties of the heavily doped conventional semiconductors such as Ge, Si, or III-V alloys and how they can be successfully employed in plasmonics. The modeling of their specific optical properties and the technol. processes to realize nanoantennas, slits, or metasurfaces are presented. We also provide an overview of the applications of this young field of research, mainly focusing on biosensing and active devices, among the most recent developments in semiconductor plasmonics. Finally, an outlook of further research directions and the potential technol. transfer is presented.
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    13. 13
      Scalora, M.; Vincenti, M. A.; de Ceglia, D.; Roppo, V.; Centini, M.; Akozbek, N.; Bloemer, M. Second-and third-harmonic generation in metal-based structures. Phys. Rev. A 2010, 82, 043828  DOI: 10.1103/PhysRevA.82.043828
      13
      Second- and third-harmonic generation in metal-based structures
      Scalora, M.; Vincenti, M. A.; de Ceglia, D.; Roppo, V.; Centini, M.; Akozbek, N.; Bloemer, M. J.
      Physical Review A: Atomic, Molecular, and Optical Physics (2010), 82 (4, Pt. B), 043828/1-043828/14CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      We present a theor. approach to the study of second- and third-harmonic generation from metallic structures and nanocavities filled with a nonlinear material in the ultrashort pulse regime. We model the metal as a two-component medium, using the hydrodynamic model to describe free electrons and Lorentz oscillators to account for core electron contributions to both the linear dielec. const. and harmonic generation. The active nonlinear medium that may fill a metallic nanocavity, or be positioned between metallic layers in a stack, is also modeled using Lorentz oscillators and surface phenomena due to symmetry breaking are taken into account. We study the effects of incident TE- and TM-polarized fields and show that a simple reexamn. of the basic equations reveals addnl., exploitable dynamical features of nonlinear frequency conversion in plasmonic nanostructures.
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    14. 14
      Ciracì, C.; Poutrina, E.; Scalora, M.; Smith, D. R. Second-harmonic generation in metallic nanoparticles: Clarification of the role of the surface. Phys. Rev. B 2012, 86, 115451  DOI: 10.1103/PhysRevB.86.115451
      14
      Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface
      Ciraci, Cristian; Poutrina, Ekaterina; Scalora, Michael; Smith, David R.
      Physical Review B: Condensed Matter and Materials Physics (2012), 86 (11), 115451/1-115451/10CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We present a numerical investigation of the second-order nonlinear optical properties of metal-based metamaterial nanoresonators. The nonlinear optical response of the metal is described by a hydrodynamic model, with the effects of electron pressure in the electron gas also taken into account. We show that as the pressure term tends to zero the amt. of converted second-harmonic field tends to an asymptotic value. In this limit it becomes possible to rewrite the nonlinear surface contributions as functions of the value of the polarization vector inside the bulk region. Nonlocality thus can be incorporated into numerical simulations without actually utilizing the nonlocal equation of motion or solving for the rapidly varying fields that occur near the metal surface. We use our model to investigate the second-harmonic generation process with three-dimensional gold nanoparticle arrays and show that nanocrescents can easily attain conversion efficiencies of ∼6.0 × 10-8 for pumping peak intensities of a few tens of MW/cm2.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xhs1Srt7nL&md5=78946117b3c46a91611bfd730f9454a7
    15. 15
      Khalid, M.; Ciracì, C. Enhancing second-harmonic generation with electron spill-out at metallic surfaces. Commun. Phys. 2020, 3, 214,  DOI: 10.1038/s42005-020-00477-0
      There is no corresponding record for this reference.
    16. 16
      De Luca, F.; Ortolani, M.; Ciracì, C. Free electron nonlinearities in heavily doped semiconductors plasmonics. Phys. Rev. B 2021, 103, 115305  DOI: 10.1103/PhysRevB.103.115305
      16
      Free electron nonlinearities in heavily doped semiconductors plasmonics
      De Luca, Federico; Ortolani, Michele; Ciraci, Cristian
      Physical Review B (2021), 103 (11), 115305CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      Heavily doped semiconductors have emerged as tunable low-loss plasmonic materials at midinfrared frequencies. In this article we investigate nonlinear optical phenomena assocd. with high concn. of free electrons. We use a hydrodynamic description to study free electron dynamics in heavily doped semiconductors up to third-order terms, which are usually negligible for noble metals. We find that cascaded third-harmonic generation due to second-harmonic signals can be as strong as direct third-harmonic generation contributions even when the second-harmonic generation efficiency is zero. Moreover, we show that when coupled with plasmonic enhancement free electron nonlinearities could be up to two orders of magnitude larger than conventional semiconductor nonlinearities. Our work might open a new route for nonlinear optical integrated devices at midinfrared frequencies.
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    17. 17
      De Luca, F.; Ciracì, C. Impact of surface charge depletion on the free electron nonlinear response of heavily doped semiconductors. Phys. Rev. Lett. 2022, 129, 123902  DOI: 10.1103/PhysRevLett.129.123902
      There is no corresponding record for this reference.
    18. 18
      De Luca, F.; Ortolani, M.; Ciracì, C. Free electron harmonic generation in heavily doped semiconductors: the role of the materials properties. EPJ. Appl. Metamat. 2022, 9, 13,  DOI: 10.1051/epjam/2022011
      There is no corresponding record for this reference.
    19. 19
      Rossetti, A. Origin of optical nonlinearity in plasmonic semiconductor nanostructures. arXiv preprint 2024, arXiv:2402.15443, https://arxiv.org/abs/2402.15443 (accessed Feb 23, 2024).
      There is no corresponding record for this reference.
    20. 20
      Ruppin, R. Optical properties of a plasma sphere. Phys. Rev. Lett. 1973, 31, 1434,  DOI: 10.1103/PhysRevLett.31.1434
      There is no corresponding record for this reference.
    21. 21
      Ruppin, R. Extinction properties of thin metallic nanowires. Opt. Commun. 2001, 190, 205– 209,  DOI: 10.1016/S0030-4018(01)01063-X
      21
      Extinction properties of thin metallic nanowires
      Ruppin, R.
      Optics Communications (2001), 190 (1-6), 205-209CODEN: OPCOB8; ISSN:0030-4018. (Elsevier Science B.V.)
      The classical electromagnetic scattering theory for a circular cylinder is extended so as to allow for the nonlocal response properties of metals. Using hydrodynamic dielec. functions and applying appropriate addnl. boundary conditions, the generalized scattering coeffs. are derived. For very thin nanowires, the main extinction peak, due to the surface plasmon, is shifted from its classical position towards the high frequency side and subsidiary peaks, due to bulk plasmon excitation, appear above the plasma frequency.
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    22. 22
      Raza, S.; Toscano, G.; Jauho, A.-P.; Wubs, M.; Mortensen, N. A. Unusual resonances in nanoplasmonic structures due to nonlocal response. Phys. Rev. B 2011, 84, 121412  DOI: 10.1103/PhysRevB.84.121412
      22
      Unusual resonances in nanoplasmonic structures due to nonlocal response
      Raza, Soren; Toscano, Giuseppe; Jauho, Antti-Pekka; Wubs, Martijn; Mortensen, N. Asger
      Physical Review B: Condensed Matter and Materials Physics (2011), 84 (12), 121412/1-121412/4CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We study the nonlocal response of a confined electron gas within the hydrodynamical Drude model. We address the question as to whether plasmonic nanostructures exhibit nonlocal resonances that have no counterpart in the local-response Drude model. Avoiding the usual quasistatic approxn., we find that such resonances do indeed occur, but only above the plasma frequency. Thus the recently found nonlocal resonances at optical frequencies for very small structures, obtained within quasistatic approxn., are unphys. As a specific example we consider nanosized metallic cylinders, for which extinction cross sections and field distributions can be calcd. anal.
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    23. 23
      Benisty, H.; Greffet, J.-J.; Lalanne, P. Introduction to nanophotonics; Oxford university press, 2022.
      There is no corresponding record for this reference.
    24. 24
      Anderegg, M.; Feuerbacher, B.; Fitton, B. Optically excited longitudinal plasmons in potassium. Phys. Rev. Lett. 1971, 27, 1565,  DOI: 10.1103/PhysRevLett.27.1565
      There is no corresponding record for this reference.
    25. 25
      Lindau, I.; Nilsson, P. Experimental verification of optically excited longitudinal plasmons. Phys. Scr. 1971, 3, 87,  DOI: 10.1088/0031-8949/3/2/007
      25
      Experimental verifiication of optically excited longitudinal plasmons
      Lindau, I.; Nilsson, P. O.
      Physica Scripta (1971), 3 (2), 87-92CODEN: PHSTBO; ISSN:0031-8949.
      Anomalous optical absorption is obsd. in thin Ag films just above the bulk plasma frequency. The result is interpreted as excitation of longitudinal plasmons, an effect which is normally neglected when discussing optical properties. The dispersion relation of the plasma wave is obtained from the data.
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    26. 26
      De Ceglia, D.; Scalora, M.; Vincenti, M. A.; Campione, S.; Kelley, K.; Runnerstrom, E. L.; Maria, J.-P.; Keeler, G. A.; Luk, T. S. Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms. Sci. Rep. 2018, 8, 9335,  DOI: 10.1038/s41598-018-27655-z
      26
      Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms
      de Ceglia Domenico; Scalora Michael; Vincenti Maria A; Campione Salvatore; Keeler Gordon A; Luk Ting S; Kelley Kyle; Runnerstrom Evan L; Maria Jon-Paul; Luk Ting S
      Scientific reports (2018), 8 (1), 9335 ISSN:.
      Optical nonlocalities are elusive and hardly observable in traditional plasmonic materials like noble and alkali metals. Here we report experimental observation of viscoelastic nonlocalities in the infrared optical response of epsilon-near-zero nanofilms made of low-loss doped cadmium-oxide. The nonlocality is detectable thanks to the low damping rate of conduction electrons and the virtual absence of interband transitions at infrared wavelengths. We describe the motion of conduction electrons using a hydrodynamic model for a viscoelastic fluid, and find excellent agreement with experimental results. The electrons' elasticity blue-shifts the infrared plasmonic resonance associated with the main epsilon-near-zero mode, and triggers the onset of higher-order resonances due to the excitation of electron-pressure modes above the bulk plasma frequency. We also provide evidence of the existence of nonlocal damping, i.e., viscosity, in the motion of optically-excited conduction electrons using a combination of spectroscopic ellipsometry data and predictions based on the viscoelastic hydrodynamic model.
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    27. 27
      Christensen, T.; Yan, W.; Raza, S.; Jauho, A.-P.; Mortensen, N. A.; Wubs, M. Nonlocal response of metallic nanospheres probed by light, electrons, and atoms. ACS Nano 2014, 8, 1745– 1758,  DOI: 10.1021/nn406153k
      27
      Nonlocal Response of Metallic Nanospheres Probed by Light, Electrons, and Atoms
      Christensen, Thomas; Yan, Wei; Raza, Soeren; Jauho, Antti-Pekka; Mortensen, N. Asger; Wubs, Martijn
      ACS Nano (2014), 8 (2), 1745-1758CODEN: ANCAC3; ISSN:1936-0851. (American Chemical Society)
      Inspired by recent measurements on individual metallic nanospheres that cannot be explained with traditional classical electrodynamics, the authors theor. study the effects of nonlocal response by metallic nanospheres in 3 distinct settings: at. spontaneous emission, EELS, and light scattering. These constitute 2 near-field and 1 far-field measurements, with zero-, 1-, and 2-dimensional excitation sources, resp. The authors search for the clearest signatures of hydrodynamic pressure waves in nanospheres. The authors employ a linearized hydrodynamic model, and Mie-Lorenz theory is applied for each case. Nonlocal response shows its mark in all 3 configurations, but for the 2 near-field measurements, the authors predict esp. pronounced nonlocal effects that are not exhibited in far-field measurements. Assocd. with every multipole order is not only a single blueshifted surface plasmon but also an infinite series of bulk plasmons that have no counterpart in a local-response approxn. These increasingly blueshifted multipole plasmons become spectrally more prominent at shorter probe-to-surface sepns. and for decreasing nanosphere radii. For selected metals, the authors predict hydrodynamic multipolar plasmons to be measurable on single nanospheres.
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    28. 28
      Ciracì, C.; Della Sala, F. Quantum hydrodynamic theory for plasmonics: Impact of the electron density tail. Phys. Rev. B 2016, 93, 205405  DOI: 10.1103/PhysRevB.93.205405
      28
      Quantum hydrodynamic theory for plasmonics: impact of the electron density tail
      Ciraci, Cristian; Della Sala, Fabio
      Physical Review B (2016), 93 (20), 205405/1-205405/14CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
      Multiscale plasmonic systems (e.g., extended metallic nanostructures with subnanometer inter-distances) play a key role in the development of next-generation nanophotonic devices. An accurate modeling of the optical interactions in these systems requires an accurate description of both quantum effects and far-field properties. Classical electromagnetism can only describe the latter, while time-dependent d. functional theory (TD-DFT) can provide a full first-principles quantum treatment. However, TD-DFT becomes computationally prohibitive for sizes that exceed few nanometers, which are instead very important for most applications. In this article, we introduce a method based on the quantum hydrodynamic theory (QHT) that includes nonlocal contributions of the kinetic energy and the correct asymptotic description of the electron d. We show that our QHT method can predict both plasmon energy and spill-out effects in metal nanoparticles in excellent agreement with TD-DFT predictions, thus allowing reliable and efficient calcns. of both quantum and far-field properties in multiscale plasmonic systems.
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    29. 29
      Toscano, G.; Straubel, J.; Kwiatkowski, A.; Rockstuhl, C.; Evers, F.; Xu, H.; Asger Mortensen, N.; Wubs, M. Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics. Nat. Commun. 2015, 6, 7132,  DOI: 10.1038/ncomms8132
      29
      Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics
      Toscano Giuseppe; Straubel Jakob; Kwiatkowski Alexander; Rockstuhl Carsten; Evers Ferdinand; Xu Hongxing; Mortensen N Asger; Wubs Martijn
      Nature communications (2015), 6 (), 7132 ISSN:.
      The standard hydrodynamic Drude model with hard-wall boundary conditions can give accurate quantitative predictions for the optical response of noble-metal nanoparticles. However, it is less accurate for other metallic nanosystems, where surface effects due to electron density spill-out in free space cannot be neglected. Here we address the fundamental question whether the description of surface effects in plasmonics necessarily requires a fully quantum-mechanical ab initio approach. We present a self-consistent hydrodynamic model (SC-HDM), where both the ground state and the excited state properties of an inhomogeneous electron gas can be determined. With this method we are able to explain the size-dependent surface resonance shifts of Na and Ag nanowires and nanospheres. The results we obtain are in good agreement with experiments and more advanced quantum methods. The SC-HDM gives accurate results with modest computational effort, and can be applied to arbitrary nanoplasmonic systems of much larger sizes than accessible with ab initio methods.
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    30. 30
      Ciracì, C. Current-dependent potential for nonlocal absorption in quantum hydrodynamic theory. Phys. Rev. B 2017, 95, 245434  DOI: 10.1103/PhysRevB.95.245434
      30
      Current-dependent potential for nonlocal absorption in quantum hydrodynamic theory
      Ciraci, Cristian
      Physical Review B (2017), 95 (24), 245434/1-245434/10CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      The quantum hydrodynamic theory is a promising method for describing microscopic details of macroscopic systems. The hydrodynamic equation can be partially obtained from a single-particle Kohn-Sham equation and improved by adding a viscoelastic kinetic-exchange-correlation tensor term, so that broadening of collective excitation can be taken into account, as well as a correction to the plasmon dispersion. The result is an accurate self-consistent and computationally efficient hydrodynamic description of the free-electron gas. A very accurate agreement with full quantum calcns. is shown.
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    31. 31
      Silberberg, Y.; Sands, T. Optical properties of metallic quantum wells. IEEE J. Quantum Electron. 1992, 28, 1663– 1669,  DOI: 10.1109/3.142553
      There is no corresponding record for this reference.
    32. 32
      Qian, H.; Xiao, Y.; Liu, Z. Giant Kerr response of ultrathin gold films from quantum size effect. Nat. Commun. 2016, 7, 13153  DOI: 10.1038/ncomms13153
      32
      Giant Kerr response of ultrathin gold films from quantum size effect
      Qian, Haoliang; Xiao, Yuzhe; Liu, Zhaowei
      Nature Communications (2016), 7 (), 13153CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
      With the size of plasmonic devices entering into the nanoscale region, the impact of quantum physics needs to be considered. In the past, the quantum size effect on linear material properties has been studied extensively. However, the nonlinear aspects have not been explored much so far. On the other hand, much effort has been put into the field of integrated nonlinear optics and a medium with large nonlinearity is desirable. Here we study the optical nonlinear properties of a nanometer scale gold quantum well by using the z-scan method and nonlinear spectrum broadening technique. The quantum size effect results in a giant optical Kerr susceptibility, which is four orders of magnitude higher than the intrinsic value of bulk gold and several orders larger than traditional nonlinear media. Such high nonlinearity enables efficient nonlinear interaction within a microscopic footprint, making quantum metallic films a promising candidate for integrated nonlinear optical applications.
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    33. 33
      Qian, H.; Li, S.; Chen, C.-F.; Hsu, S.-W.; Bopp, S. E.; Ma, Q.; Tao, A. R.; Liu, Z. Large optical nonlinearity enabled by coupled metallic quantum wells. Light Sci. Appl. 2019, 8, 13,  DOI: 10.1038/s41377-019-0123-4
      33
      Large optical nonlinearity enabled by coupled metallic quantum wells
      Qian Haoliang; Li Shilong; Chen Ching-Fu; Ma Qian; Liu Zhaowei; Hsu Su-Wen; Tao Andrea R; Bopp Steven Edward; Tao Andrea R; Liu Zhaowei; Liu Zhaowei
      Light, science & applications (2019), 8 (), 13 ISSN:.
      New materials that exhibit strong second-order optical nonlinearities at a desired operational frequency are of paramount importance for nonlinear optics. Giant second-order susceptibility χ((2)) has been obtained in semiconductor quantum wells (QWs). Unfortunately, the limited confining potential in semiconductor QWs causes formidable challenges in scaling such a scheme to the visible/near-infrared (NIR) frequencies for more vital nonlinear-optic applications. Here, we introduce a metal/dielectric heterostructured platform, i.e., TiN/Al2O3 epitaxial multilayers, to overcome that limitation. This platform has an extremely high χ((2)) of approximately 1500 pm/V at NIR frequencies. By combining the aforementioned heterostructure with the large electric field enhancement afforded by a nanostructured metasurface, the power efficiency of second harmonic generation (SHG) achieved 10(-4) at an incident pulse intensity of 10 GW/cm(2), which is an improvement of several orders of magnitude compared to that of previous demonstrations from nonlinear surfaces at similar frequencies. The proposed quantum-engineered heterostructures enable efficient wave mixing at visible/NIR frequencies into ultracompact nonlinear optical devices.
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    34. 34
      Pirotta, S.; Tran, N.-L.; Jollivet, A.; Biasiol, G.; Crozat, P.; Manceau, J.-M.; Bousseksou, A.; Colombelli, R. Fast amplitude modulation up to 1.5 GHz of mid-IR free-space beams at room-temperature. Nat. Commun. 2021, 12, 799,  DOI: 10.1038/s41467-020-20710-2
      34
      Fast amplitude modulation up to 1.5 GHz of mid-IR free-space beams at room-temperature
      Pirotta, Stefano; Tran, Ngoc-Linh; Jollivet, Arnaud; Biasiol, Giorgio; Crozat, Paul; Manceau, Jean-Michel; Bousseksou, Adel; Colombelli, Raffaele
      Nature Communications (2021), 12 (1), 799CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)
      Applications relying on mid-IR radiation (λ ∼ 3-30μm) have progressed at a very rapid pace in recent years, stimulated by scientific and technol. breakthroughs like mid-IR cameras and quantum cascade lasers. On the other side, standalone and broadband devices allowing control of the beam amplitude and/or phase at ultra-fast rates (GHz or more) are still missing. Here we show a free-space amplitude modulator for mid-IR radiation (λ ∼ 10μm) that can operate at room temp. up to at least 1.5 GHz (-3dB cutoff at ∼750 MHz). The device relies on a semiconductor heterostructure enclosed in a judiciously designed metal-metal optical resonator. At zero bias, it operates in the strong light-matter coupling regime up to 300 K. By applying an appropriate bias, the device transitions toward the weak-coupling regime. The large change in reflectance is exploited to modulate the intensity of a mid-IR continuous-wave laser up to 1.5 GHz.
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    35. 35
      Malerba, M.; Pirotta, S.; Aubin, G.; Lucia, L.; Jeannin, M.; Manceau, J.-M.; Bousseksou, A.; Lin, Q.; Lampin, J.-F.; Peytavit, E.; Barbieri, S.; Li, L.; Davies, G.; Linfield, E. H.; Colombelli, R. Ultrafast (≈ 10 GHz) mid-IR modulator based on ultrafast electrical switching of the light–matter coupling. Appl. Phys. Lett. 2024, 125, 041101  DOI: 10.1063/5.0213965
      There is no corresponding record for this reference.
    36. 36
      Jeannin, M.; Cosentino, E.; Pirotta, S.; Malerba, M.; Biasiol, G.; Manceau, J.-M.; Colombelli, R. Low intensity saturation of an ISB transition by a mid-IR quantum cascade laser. Appl. Phys. Lett. 2023, 122, 241107,  DOI: 10.1063/5.0153891
      There is no corresponding record for this reference.
    37. 37
      Zanotto, S.; Mezzapesa, F. P.; Bianco, F.; Biasiol, G.; Baldacci, L.; Vitiello, M. S.; Sorba, L.; Colombelli, R.; Tredicucci, A. Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption. Nat. Phys. 2014, 10, 830– 834,  DOI: 10.1038/nphys3106
      37
      Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption
      Zanotto, Simone; Mezzapesa, Francesco P.; Bianco, Federica; Biasiol, Giorgio; Baldacci, Lorenzo; Vitiello, Miriam Serena; Sorba, Lucia; Colombelli, Raffaele; Tredicucci, Alessandro
      Nature Physics (2014), 10 (11), 830-834CODEN: NPAHAX; ISSN:1745-2473. (Nature Publishing Group)
      The ability to drive a system with an external input is a fundamental aspect of light-matter interaction. The key concept in many photonic applications is the 'crit. coupling' condition: at criticality, all the energy fed to the system is dissipated within the system itself. Although this idea was crucial to enhance the efficiency of many devices, it was never considered in the context of systems operating in a non-perturbative regime. In this so-called strong-coupling regime, the matter and light degrees of freedom are mixed into dressed states, leading to new eigenstates called polaritons. Here we demonstrate that the strong-coupling regime and the crit. coupling condition can coexist; in such a strong crit. coupling situation, all the incoming energy is converted into polaritons. A general semiclassical theory reveals that such a situation corresponds to a special curve in the phase diagram of the coupled light-matter oscillators. In the case of a system with two radiating ports, the phenomenol. shown is that of coherent perfect absorption (CPA; refs , ), which is then naturally understood in the framework of crit. coupling. Most importantly, we exptl. verify polaritonic CPA in a semiconductor-based intersubband-polariton photonic crystal resonator. This result opens new avenues in polariton physics, making it possible to control the pumping efficiency of a system independent of the energy exchange rate between the electromagnetic field and the material transition.
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    38. 38
      Jeannin, M.; Manceau, J.-M.; Colombelli, R. Unified description of saturation and bistability of intersubband transitions in the weak and strong light-matter coupling regimes. Phys. Rev. Lett. 2021, 127, 187401  DOI: 10.1103/PhysRevLett.127.187401
      38
      Unified Description of Saturation and Bistability of Intersubband Transitions in the Weak and Strong Light-Matter Coupling Regimes
      Jeannin, Mathieu; Manceau, Jean-Michel; Colombelli, Raffaele
      Physical Review Letters (2021), 127 (18), 187401CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      We propose a unified description of intersubband absorption satn. for quantum wells inserted in a resonator, both in the weak and strong light-matter coupling regimes. We demonstrate how absorption satn. can be engineered. In particular, we show that the satn. intensity increases linearly with the doping in the strong coupling regime, while it remains doping independent in weak coupling. Hence, countering intuition, the most suitable region to exploit low satn. intensities is not the ultrastrong coupling regime, but is instead at the onset of the strong light-matter coupling. We further derive explicit conditions for the emergence of bistability. This Letter sets the path toward, as yet, nonexistent ultrafast midinfrared semiconductor saturable absorption mirrors (SESAMs) and bistable systems. As an example, we show how to design a midinfrared SESAM with a 3 orders of magnitude redn. in satn. intensity, down to ≈5 kW cm-2.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXis1Smt7nN&md5=7553f32da759e6009091b30bc42e29ce
    39. 39
      Cominotti, R.; Leymann, H.; Nespolo, J.; Manceau, J.-M.; Jeannin, M.; Colombelli, R.; Carusotto, I. Theory of coherent optical nonlinearities of intersubband transitions in semiconductor quantum wells. Phys. Rev. B 2023, 107, 115431  DOI: 10.1103/PhysRevB.107.115431
      There is no corresponding record for this reference.
    40. 40
      Baumberg, J. J.; Aizpurua, J.; Mikkelsen, M. H.; Smith, D. R. Extreme nanophotonics from ultrathin metallic gaps. Nat. Mater. 2019, 18, 668– 678,  DOI: 10.1038/s41563-019-0290-y
      40
      Extreme nanophotonics from ultrathin metallic gaps
      Baumberg, Jeremy J.; Aizpurua, Javier; Mikkelsen, Maiken H.; Smith, David R.
      Nature Materials (2019), 18 (7), 668-678CODEN: NMAACR; ISSN:1476-1122. (Nature Research)
      A review. Ultrathin dielec. gaps between metals can trap plasmonic optical modes with surprisingly low loss and with vols. below 1 nm3. We review the origin and subtle properties of these modes, and show how they can be well accounted for by simple models. Particularly important is the mixing between radiating antennas and confined nanogap modes, which is extremely sensitive to precise nanogeometry, right down to the single-atom level. Coupling nanogap plasmons to electronic and vibronic transitions yields a host of phenomena including single-mol. strong coupling and mol. optomechanics, opening access to at.-scale chem. and materials science, as well as quantum metamaterials. Ultimate low-energy devices such as robust bottom-up assembled single-atom switches are thus in prospect.
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    41. 41
      Johnson, P. B.; Christy, R.-W. Optical constants of the noble metals. Phys. Rev. B 1972, 6, 4370,  DOI: 10.1103/PhysRevB.6.4370
      41
      Optical constants of the noble metals
      Johnson, P. B.; Christy, R. W.
      Physical Review B: Solid State (1972), 6 (12), 4370-9CODEN: PLRBAQ; ISSN:0556-2805.
      The optical consts. n and k were obtained for Cu, Ag, and Au from reflection and transmission measurements on vacuum-evapd. thin films at room temp., in the spectral range 0.5-6.5 eV. The film-thickness range was 185-500 Å. Three optical measurements were inverted to obtain the film thickness d as well as n and k. The estd. error in d was ±2 Å, and that in n, k was <0.02 over most of the spectral range. The results in the film-thickness range 250-500 Å were independent of thickness, and were unchanged after vacuum annealing or aging in air. The free-electron optical effective masses and relaxation times derived from the results in the near ir agreed satisfactorily with previous values. The interband contribution to the imaginary part of the dielec. const. was obtained by subtracting the free-electron contribution. Some recent theor. calcns. were compared with the results for Cu and Au. In addn., some other recent expts. are crit. compared with the present results.
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    42. 42
      Moreau, A.; Ciracì, C.; Mock, J. J.; Hill, R. T.; Wang, Q.; Wiley, B. J.; Chilkoti, A.; Smith, D. R. Controlled-reflectance surfaces with film-coupled colloidal nanoantennas. Nature 2012, 492, 86– 89,  DOI: 10.1038/nature11615
      42
      Controlled-reflectance surfaces with film-coupled colloidal nanoantennas
      Moreau, Antoine; Ciraci, Cristian; Mock, Jack J.; Hill, Ryan T.; Wang, Qiang; Wiley, Benjamin J.; Chilkoti, Ashutosh; Smith, David R.
      Nature (London, United Kingdom) (2012), 492 (7427), 86-89CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
      Efficient and tunable absorption is essential for a variety of applications, such as designing controlled-emissivity surfaces for thermophotovoltaic devices, tailoring an IR spectrum for controlled thermal dissipation and producing detector elements for imaging. Metamaterials based on metallic elements are particularly efficient as absorbing media, because both the elec. and the magnetic properties of a metamaterial can be tuned by structured design. So far, metamaterial absorbers in the IR or visible range were fabricated using lithog. patterned metallic structures, making them inherently difficult to produce over large areas and hence reducing their applicability. Here the authors demonstrate a simple method to create a metamaterial absorber by randomly adsorbing chem. synthesized Ag nanocubes onto a nanoscale-thick polymer spacer layer on a Au film, making no effort to control the spatial arrangement of the cubes on the film. The film-coupled nanocubes provide a reflectance spectrum that can be tailored by varying the geometry (the size of the cubes and/or the thickness of the spacer). Each nanocube is the optical analog of a grounded patch antenna, with a nearly identical local field structure that is modified by the plasmonic response of the metal's dielec. function, and with an anomalously large absorption efficiency that can be partly attributed to an interferometric effect. The absorptivity of large surface areas can be controlled using this method, at scales out of reach of lithog. approaches (such as electron-beam lithog.) that are otherwise required to manipulate matter on the nanoscale.
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    43. 43
      Hu, H.; Shi, Z.; Zhang, S.; Xu, H. Unified treatment of scattering, absorption, and luminescence spectra from a plasmon–exciton hybrid by temporal coupled-mode theory. J. Chem. Phys. 2021, 155, 074104  DOI: 10.1063/5.0059816
      There is no corresponding record for this reference.
    44. 44
      Silvestri, M.; Sahoo, A.; Assogna, L.; Benassi, P.; Ferrante, C.; Ciattoni, A.; Marini, A. Resonant third-harmonic generation driven by out-of-equilibrium electron dynamics in sodium-based near-zero index thin films. Nanophotonics 2024, 13, 2003– 2013,  DOI: 10.1515/nanoph-2023-0743
      There is no corresponding record for this reference.
    45. 45
      Huang, Z.; Baron, A.; Larouche, S.; Argyropoulos, C.; Smith, D. R. Optical bistability with film-coupled metasurfaces. Opt. Lett. 2015, 40, 5638– 5641,  DOI: 10.1364/OL.40.005638
      45
      Optical bistability with film-coupled metasurfaces
      Huang, Zhiqin; Baron, Alexandre; Larouche, Stephane; Argyropoulos, Christos; Smith, David R.
      Optics Letters (2015), 40 (23), 5638-5641CODEN: OPLEDP; ISSN:0146-9592. (Optical Society of America)
      Metasurfaces comprising arrays of film-coupled, nanopatch antennas are a promising platform for low-energy, all-optical switches. The large field enhancements that can be achieved in the dielec. spacer region between the nanopatch and the metallic substrate can substantially enhance optical nonlinear processes. Here we consider a dielec. material that exhibits an optical Kerr effect as the spacer layer and numerically calc. the optical bistability of a metasurface using the finite element method (FEM). We expect the proposed method to be highly accurate compared with other numerical approaches, such as those based on graphical post-processing techniques, because it self-consistently solves for both the spatial field distribution and the intensity-dependent refractive index distribution of the spacer layer. This method offers an alternative approach to finite-difference time-domain (FDTD) modeling. We use this numerical tool to design a metasurface optical switch and our optimized design exhibits exceptionally low switching intensity of 33 kW/cm2, corresponding to switching energy on the order of tens of attojoules per resonator, a value much smaller than those found for most devices reported in the literature. We propose our method as a tool for designing all-optical switches and modulators.
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    46. 46
      Argyropoulos, C.; Chen, P.-Y.; D’Aguanno, G.; Engheta, N.; Alu, A. Boosting optical nonlinearities in ε-near-zero plasmonic channels. Phys. Rev. B 2012, 85, 045129  DOI: 10.1103/PhysRevB.85.045129
      There is no corresponding record for this reference.
    47. 47
      Hu, H.; Álvarez-Pérez, G.; Otomalo, T. O.; Ciracì, C. Low-power threshold optical bistability enabled by hydrodynamic Kerr nonlinearity of free-carriers in heavily doped semiconductors. arXiv preprint 2024, arXiv:2406.07750, https://arxiv.org/abs/2406.07750 (accessed Jun 11, 2024).
      There is no corresponding record for this reference.

  • Supporting Information

    Supporting Information

    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphotonics.4c01308.

    • Derivation and implementation method of hydrodynamic nonlinear sources and contributions from dielectric χ(3) in the nonlinear reflectance, hydrodynamic contributions from silver and heavily doped InGaAs, robustness of the system with different dampings, and switching time (PDF)

    • Video of temporal evolution of the nonlinear system (MP4)


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