Chiral TeraHertz surface plasmonics
Ian Aupiais Romain Grasset Dmitri Daineka Javier Briatico Luca Perfetti Jean-Paul Hugonin Jean-Jacques Greffet Yannis Laplace



	Estimation Network Design framework for efficient distributed optimization

    Mattia Bianchi and  Sergio Grammatico
     
	M. Bianchi is with the Automatic Control Laboratory (IfA), ETH Zürich, Switzerland (). S. Grammatico is with the Delft Center for Systems and Control (DCSC), TU Delft, The Netherlands (). This work is supported by NWO under research project OMEGA (613.001.702), by the ERC under research project COSMOS (802348) and by ETH Zürich funds.

    Received 28 February 2024 / Accepted 23 April 2024
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 Chiral engineering of TeraHertz (THz) light fields and the use of the handedness of light in THz light-matter interactions promise many novel opportunities for advanced sensing and control of matter in this frequency range. Unlike previously explored methods, this is achieved here by leveraging the chiral properties of highly confined THz surface plasmon modes. More specifically, we design ultrasmall surface plasmonic-based THz cavities and THz metasurfaces that display significant and adjustable chiral behavior under modest magnetic fields. For such a prototypical example of non-hermitian and dispersive photonic system, we demonstrate the capacity to magnetic field-tune both the poles and zeros of cavity resonances, the two fundamental parameters governing their resonance properties.
 Alongside the observed handedness-dependent cavity frequencies, this highlights the remarkable ability to engineer chiral and tunable radiative couplings for THz resonators and metasurfaces. The extensive tunability offered by the surface plasmonic approach paves the way for the development of agile and multifunctional THz metasurfaces as well as the realization of ultrastrong chiral light-matter interactions at low energy in matter with potential far-reaching applications for the design of material properties.




§ INTRODUCTION

Despite its long history, the role of chirality in light-matter interactions remains intriguing, drawing intense focus both from a fundamental perspective <cit.> and for its technological applications in sensing and controlling matter across diverse length scales, energy scales and scientific disciplines <cit.>. In this context, the design of chiral optical fields and optically-active devices at THz frequencies holds particular significance for sensing applications across a large class of systems. This stems from the wide range of excitations exhibiting chirality within this frequency band, spanning from chemically and biologically relevant excitations in molecules <cit.> to quantum collective modes such as magnons <cit.> and phonons <cit.> in condensed matter systems. Alongside sensing capabilities, the ultrastrong interaction between matter and chiral optical fields at low energies has recently emerged as a new paradigm for manipulating material properties through Floquet engineering <cit.> and has been proposed to realize novel phases of matter through hybridization with the vacuum field of chiral cavities <cit.>. Notably, the potential to induce magnetism and non-trivial topologies in otherwise non-magnetic and topologically trivial materials <cit.>, or photon condensation in the ground state <cit.>, represent some of the most far-reaching prospects enabled by chiral vacuum fields. For these reasons, the ability to engineer ultrastrong and chiral light-matter interactions at THz frequencies, achievable through the design of cavities with ultrasmall mode volumes, appears as a new frontier in the THz range <cit.> that would open up many avenues for chiral sensing and matter manipulation with applications in chemistry, biology and physics.

In the past years, large efforts have been devoted to the design of chiral and optically-active devices for polarization control of THz light <cit.>. 
These have been obtained following mostly two distinct routes. The first one is a synthetic or metamaterial-like approach, in which the geometric chirality of metallic resonators is tailored and transferred onto the resonating electromagnetic (EM) fields <cit.>. The second approach relies on the natural magneto-optical activity of the bulk of semiconductors <cit.> and is enabled by the propagation of THz magneto-plasma waves <cit.>. The latter originate from the microscopic Lorentz force acting on carriers with low effective mass m^*, giving rise to a cyclotron frequency ν_c=eB/2π m^* that falls within the THz frequency range for moderate magnetic fields below 1T <cit.>. Yet, achieving high EM-field confinement sufficient enough to reach the ultrastrong coupling regime of light-matter interaction represents to date a challenge for both of these approaches. On the one hand, shrinking the size of metallic resonators while preserving operating frequencies in the THz range is a difficult task that often requires the design of complex and technologically challenging resonator architectures <cit.>. On the other hand, utilization of magneto-plasma waves necessitates light propagation through the bulk of the semiconductor and hence only offer sizeable chiral effects at the expense of large interaction volumes <cit.>. 

Here, we overcome these challenges by exploiting THz surface plasmons-based cavities in semiconductors. As demonstrated recently <cit.>, this approach allows to downscale cavity dimensions at fixed operating frequencies and provide an efficient way to confine THz EM-fields down to the ultimate limit allowed by plasmonics. In this work, we show that surface plasmons inherit the chiral activity of the bulk and use this as a means to create chiral THz cavities with ultrasmall mode volumes. When assembled together, arrays of such cavities lead to a novel type of chiral THz metasurfaces, fully tunable by geometry, magnetic-field and even temperature. This work opens the door to exciting perspectives for the exploration of ultrastrong chiral light-matter interactions at low energies in matter.



§ WORKING PRINCIPLE OF THE CHIRAL THZ SURFACE PLASMONIC CAVITIES

In the Faraday configuration, where a homogeneous magnetic field B=Bz is perpendicular to the plasmonic medium's interface and a plane wave is incident normally, the magnetic field acts exclusively onto the spin-sector of the EM-field (i.e. its polarization state) and normal modes correspond to left- and right-circularly polarized (LCP/RCP) photons
E_σ_±=e^ik_zze^-iω t.|σ_±.⟩  <cit.>. Interaction between light and the plasma involves two distinct magneto-plasmonic permittivities ϵ_σ±(ν,B) and results in reflection coefficients denoted as r̃_σ_±(ν,B) (see supplementary information).

Normal incidence experimental power reflectivity spectra R_σ_±(ν)=r̃_σ_±(ν,B)^2 as well as the phase ϕ_σ_±(ν,B)=Arg(r̃_σ_±(ν,B)) accumulated by the electric field of the light upon reflection onto the THz plasma of bulk semiconductor InSb are shown in Fig. 1a for σ_± polarizations. As the cyclotron frequency ν_c varies, the THz plasma edge initially located at ν≈ν_p splits into two different plasma edges. Their frequencies ν_p,σ_±≈1/2(√(4ν^2_p+ν^2_c)±ν_c) depend on the polarization handedness (see Fig. 1a-b) and highligths the bulk chirality of the plasmonic medium <cit.>.
 
In contrast to plane waves, where orbital and spin components of the EM-field are independent, the normal modes of the proposed ultrasmall chiral THz cavity illustrated in Fig. 1c are sustained by surface plasmons and their orbital and spin degrees of freedom are intertwined within the highly subwavelength volume of the resonator, leading to a more complex chiral behavior. This can be seen in Fig. 1d where we present the characteristics of the normal modes of circular patch cavities, which, owing to their cylindrical symmetry, represent a natural choice of plasmonic resonators in the present context. In the absence of magnetic field, their normal modes have been studied previously <cit.> and are characterized by an orbital angular moment L (L= … ,-2,-1,0,1,2,… ) and a radial number n (n=1,2, … ). For lossless cavities, modes of opposite orbital momentum are time-reversal symmetric of one another (E_n,-L=E^*_n,L) and hence have degenerate frequencies ν_n,L=ν_n,-L. In Fig 1d, we display the EM-fields sustained by the surface plasmon inside the resonator for a set of representative (n,L) modes: 1) the normal component of the electric field  (Re(Ẽ_z))  and 2) the relative density Δσ_n,L(r)=σ_+,n,L(r)-σ_-,n,L(r) of the σ_± polarized photons (or spin density of the mode), where σ_±,n,L(r)∝|Ẽ_x ± iẼ_y | ^2 depends only on the radial coordinate r. 

Notably, normal modes are seen to consist in a spatially dependent superposition of σ_± polarized photons which entails the mixing of orbital and spin degrees of freedom of the resonator. It is possible to estimate the shifts δν_n,L of the resonance frequencies ν_n,L by applying a modal perturbation theory, which, to first order in ν_c, gives (see supplementary material):


    δν_n,L/ν_n,L∝ν_c(∫^s/2_0r Δσ_n,L(r)dr)
 

indicating that the frequency shift is proportional to the spatially integrated spin density Δσ_n,L(r) of the σ_± polarized photons (s being the diameter of the patch). As depicted in Fig. 1e, the L=0 mode doesn't experience any shift as it is linearly polarized, while modes with finite orbital angular momentum (L≠ 0) split symmetrically according to δν_n,L=-δν_n,-L as they have opposite spin densities of equal magnitude. Because they carry the largest spin polarization among all, the (n=1, L=± 1) modes are expected to provide the most pronounced mode splitting and hence the largest chiral effect. 



§ SAMPLE ARCHITECTURE AND PLASMONIC RESONANCES WITHOUT MAGNETIC FIELD

Rather than an isolated cavity, our samples consist in two-dimensional and subwavelength periodic arrays (period s+a) of circular patch cavities of diameter s(Fig. 2a). Over a bulk wafer of semiconducting InSb, we deposit an insulating layer of highly subwavelength thickness d (Si_3N_4 with d≪λ_0, λ_0 being the wavelength in vacuum) followed by circular patches of metallic gold on top. In the absence of magnetic field and due to the four-fold symmetry of the structure, reflectivity spectra are independent of the incoming THz light polarization.  In Fig. 2b, we present the experimental and simulated power reflectivity spectra for three samples having circular patches of varying diameters. The major absorption observed at low frequency corresponds to the resonance frequency of the two degenerate (n,L)=(1,± 1) cavity modes, as confirmed by inspection of the mode profile obtained from numerical simulations (Fig.  2c). A second pronounced absorption at high frequency lies slightly below the THz plasma edge. As seen in Fig. 2c, it corresponds to the resonance of a pair of degenerate plasmonic modes that are fully expelled from the volume encompassed by the cavities and that are located between them. These modes are the counterpart in these structures of the inter-cavity mode observed in striped lattices of plasmonic cavities <cit.>. For completeness, we also report the presence of: 1) a small absorption for the s=43μ m sample (triangle in Fig. 2b) which corresponds to the excitation of the higher order (n,L)=(3,± 1) cavity modes and whose mode profile is shown in Fig.2c and 2) a resonance corresponding to the excitation of a continuum of propagating bulk plasma modes within the transparency region of the plasmonic medium arising the first-order diffraction of the array (s=43μ m sample, star in Fig. 2b). In the remaining of this paper, we will not further discuss the latter resonances and will focus our analysis on the main cavity resonance. 



§ SURFACE PLASMONIC CHIRALITY AND ITS MAGNETIC FIELD DEPENDENCE



Upon application of the magnetic field, we observe the lifting of the degeneracy of the (n,L)=(1,+1) and (1,-1) cavity resonance. This is featured in the polarization resolved THz spectra presented in Fig. 3a-b-c as two frequency-displaced resonant absorptions for opposite σ_± polarizations, hence demonstrating the chiral effect that is looked for. 
 
Besides the polarization handedeness-dependent frequency-shifts, we notice that the contrast of the cavity resonance varies strongly and evolves oppositely for opposite σ_± polarizations. The large variation of its contrast can be inferred from the value of the reflectivity minimum at resonance R_min which changes as much as ∼50% within the range of parameters investigated (see top and bottom panels in Fig. 3a-b-c). 

This observation suggests that the magnetic field is acting to tune another crucial cavity parameter: the overall coupling efficiency of external radiation to the cavity. It can be understood qualitatively from temporal coupled mode theory <cit.> which predicts that R_min=(Γ_r-Γ_)^2/(Γ_r+Γ_)^2 for single-port cavities <cit.>, where Γ_r and Γ_ are the radiative and non-radiative decay rates of the cavity describing respectively the coupling of the resonator to external radiation and the decay of the resonance due to ohmic losses. In other words, R_min measures how close the cavity is from critical coupling at which Γ_i=Γ_r (R_min=0) and all incident radiation is coupled and dissipated into the cavity. In order to further assess quantitatively and reliably all cavity parameters for this highly non-Hermitian and dispersive photonic system <cit.>, we fitted the two quadratures of the reflection coefficient r̃_σ_±(ν,B) based on an analytical continuation of the frequency in the complex plane <cit.> (ν→ν̃ and r̃_σ_±(ν,B)→r̃_σ_±(ν̃,B), see supplementary material). This leads to a pole-zero representation of the cavity resonance shown in Fig. <ref>a. We define the complex zero ν̃_z0 by the condition  r̃_σ_±(ν̃_z0)=0.  Following this zero in the complex plane enables to characterize the undercoupled (Im(ν̃_z0)<0)), critically coupled (Im(ν̃_z0)=0) and overcoupled (Im(ν̃_z0)>0) regimes. Note that the critically coupled regime is closely linked to the coherent perfect absorption <cit.>. We observe that as the magnetic field is raised, the zero either moves towards (σ_-) the real axis (critical coupling) or departs from it (σ_+), signaling not only the tunability of the coupling but also its chiral behavior. The major results of the analysis of the cavity parameters are further presented in Fig. <ref>b.

The relative frequency-shifts of the cavity resonance are defined as δν_0/ν_0 = ν_0,σ_±(B)-ν_0(B=0)/ν_0(B=0) where ν_0,σ_±(B)=Re(ν̃_0,σ_±(B)) corresponds to the real part of the cavity pole ν̃_0,σ_±(B), i.e. the cavity resonance frequency. As anticipated, we observe that they demonstrate opposite variations for σ_± polarizations as a function of magnetic field B. While linear in B at small fields, they acquire a slight quadratic dependence at larger fields which is reminiscent of the dependences of bulk magneto-plasma modes <cit.>. At B=500 mT, the relative shifts amount to 14%, 20% and 24% for the 21, 32 and 43μ m cavities respectively in between the two σ_± polarizations. The two decay rates, determined from the knowledge of the pole and the zero as Γ_=-Im(ν̃_0+ν̃_z0)/2 and Γ_r=-Im(ν̃_0-ν̃_z0)/2 (see supplementary material), also evolve as a function of magnetic-field and exhibit chiral behaviour: Γ_r/ ,σ_+(B)≠Γ_r/ ,σ_-(B). While this is somewhat expected for the ohmic decay rate as the cyclotron resonance modulates material dissipation in the vicinity of ν=±ν_c, the most striking feature is the chiral and large variations observed for the radiative decay rate by as much as 300% in the range of parameters investigated. Such phenomenon is largely unexpected as this quantity is generally believed to depend solely on geometrical parameters of the arrays <cit.>. We attribute it to the magnetic field-tuning of the phase ϕ_σ_±(ν,B) accumulated by the THz fields upon reflection onto the underlying plasmonic medium. More specifically, denoting r̃_σ_±,0(ν,B) the background reflectivity of the structure in the absence of metallic patches (i.e. involving only the vacuum/Si_3 N_4/InSb planar multilayer), we show that the variation of the radiative decay rate can be well reproduced and explained by a functionnal dependence of the form Γ_r,σ_±(B)=α| 1+r̃_σ_±,0(ν,B) | ^2 where r̃_σ_±,0(ν,B) is evaluated at the vacuum/Si_3 N_4 interface, at the cavity resonance frequency ν=ν_0,σ_±(B), and α is a geometry-dependent but magnetic field-independent scaling factor (black solid lines in Fig. 4b). Such a relation between background reflectivity and radiative decay rates has been derived analytically for metal/insulator/metal cavities in a context where the phase did not play any significant role <cit.>. In the present case however, the phase ϕ_σ_±(ν,B)  of the reflected field is the central parameter controlling such behavior, as within a good approximation |r̃_σ_±,0(ν,B) | ≈ 1, so that Γ_r,σ_±(B) ∝cos(ϕ_σ_±(ν,B)/2)^2. The basic idea behind the tunability and chirality of the radiative decay rates that we observe is sketched in Fig. 4c. In the absence of metallic patches, interference of the incident E_i,σ_± and reflected fields E_r,σ_±=r̃_σ_±,0(ν,B)E_i,σ_± result in a total background field of amplitude | E_tot,σ_±|=| 1+r̃_σ_±,0(ν,B)| | E_i,σ_±| that takes the form of a standing wave whose locations of nodes and anti-nodes are crucially depending the phase ϕ_σ_±(ν,B) accumulated upon reflection from the plasmonic medium. Immersed into this background field, the metallic patches in- and out- couple it to the cavity with coupling rates √(2Γ_r,σ_±(B)) <cit.> being determined by the field amplitude at their location and hence scaling as | 1+r̃_σ_±,0(ν,B) |. This leads to an overall radiation decay rate given by: Γ_r,σ_±(B) ∝| 1+r̃_σ_±,0(ν,B) | ^2. This phenomenon is analogous to a resonant nanoparticle located at distance L above a metallic mirror and whose coupling to external radiation Γ_r∝sin(2π L/λ_0)^2 is adjustable by varying the location of the nanoparticle with respect to the standing wave (see Fig. 4d). Here, despite the fixed geometry, the sizeable variations of the phase provided by the underlying plasmonic medium close to the plasma frequency as well as their tunability with magnetic field play the same role as the height for the nanoparticle. 

As such, for the proposed plasmonic cavities, we demonstrate that chirality proceeds not only by frequency splitting of degenerate modes but also by a chiral and tunable coupling of the resonators to external radiation. This constitutes an important result of this study.


§ CHIRAL THZ SURFACE PLASMONIC METASURFACES

We discuss here an application where the proposed structures can be exploited as resonant and tunable reflective THz chiral metasurfaces for polarization control of the THz light. In order to demonstrate the advantages of structuring a metasurface over that of using a bulk semiconductor for this purpose, we compare in Fig. 5 the optical activity of the unstructured bulk material and that of a structured metasurface. Optical activity is commonly characterized in terms of: 1) circular dichroism (CD) via the ellipticity η=|r_σ_+(ν)|-|r_σ_-(ν)|/|r_σ_+(ν)|+|r_σ_-(ν)| and 2) optical rotary dispersion (OR) via the rotation angle θ=ϕ_σ_+(ν)-ϕ_σ_-(ν)/2<cit.>.  For linearly polarized incident light, CD and OR result in the present context in reflected light that is elliptically polarized (CD phenomenon) and rotated by an angle θ compared to the incident polarization (OR phenomenon), see Fig. 5a.
In Fig. 5b, we compare the ellipticity and optical rotation angle of the bulk and structured metasurface at B=500mT.

We notice profound differences between the two responses. Firstly, we observe that CD and OR of the bulk material display rather complex shapes in an otherwise large frequency window centered around the plasma edge. This results in overall poor spectral control of the optical activity of the bulk material. In contrast, the ellipticity of the metasurface presents two sizeable peaks of opposite polarity at around 1THz and 2THz, which correspond to the frequencies of the cavity and intercavity resonances respectively for this s=32 μ m cavity metasurface. Ellipticity around the cavity resonance demonstrates in particular a well-defined unipolar behaviour.

Strikingly, at the cavity resonance, the ellipticity  reaches a sizeable value η≈ 0.5 that is at a frequency where the bulk ellipticity is almost null within noise level: this corresponds to a giant enhancement of the ellipticity provided by the metasurface. The ability to induce such a large dichroism in a frequency band well below the plasma frequency where the bulk material only provides a minute effect demonstrates the superiority of metasurface over the bulk material for the chiral engineering of THz light fields. The origin of such a metasurface-enabled ellipticity can be traced back to the difference in the contrast between  σ_+ and σ_- resonances. In this respect, chirality of the radiative decay rates of the cavities discussed in the previous section is crucial in providing this effect. 

In the supplementary material, we further show the wide tunability of the optical activity of the metasurfaces both spectrally via the geometrical tuning of the cavity dimensions and in its magnitude via the tuning of the magnetic field strenght. Such metasurfaces achieve ellipticities as high as η≈ 0.7 at a moderate field B=500mT relevant for applications, and we expect that full ellipticity (η=1) is reachable around 600-700mT. 



§ DISCUSSION

Our approach offers a major and unique asset with respect to other alternatives for the chiral engineering of THz fields based on either synthetic <cit.> or natural <cit.> optical activity. In comparison, the large magnitude of optical activity obtained here is achieved over much smaller, i.e. deep-subwavelength, scales thanks to the remarkable confinement properties of surface plasmons. All the reported chiral effects originate from beneath the surface of the plasmonic medium within the penetration depth of the surface plasmons that is ∼ 0.01 λ_0 (where λ_0 is the free space wavelength) and within cavities with mode volumes of the order of 10^-6λ_0^3. This represents at least two orders of magnitude enhancement over previous approaches for typical figure of merits that compare the magnitude of CD/OR and the lenght or volume over which it is realized. Straightforward improvements can be readily achieved by reducing cavity mode volumes to 10^-8λ_0^3 <cit.> or by increasing the ratio ν_c/ν_p through the magnetic field or the temperature <cit.>, as this ratio governs the overall magnitude of the chiral effects. In summary, along with the numerous opportunities it allows for harnessing ultrastrong chiral light-matter interactions for the sensing and control of matter at THz frequencies, this system stands out as a singular platform within the electromagnetic spectrum for unravelling novel phenomena in chiral plasmonics thanks to the extended parameter space in ν_c/ν_p which remains beyond reach at mid-infrared and optical frequencies.



§ ACKNOWLEDGMENTS


Y.L. acknowledges support from Agence Nationale de la Recherche (grant n^∘ ANR- 17-CE30-0011-11) and Labex PALM (grant n^∘ ANR-10-LABX-0039-PALM). 



§ METHODS




 §.§ Sample manufacturing


Samples were manufactured starting from a 500  μ m thick, <100> oriented, bulk wafer of InSb commercially available (MTI Corporation). The InSb wafer was nominally N-type undoped. Each sample consisted of a ∼ 5  mm*5  mm*0.5  mm InSb substrate dice-cut from the wafer. An insulating layer of Si_3N_4 was deposited via plasma enhanced chemical vapor deposition and the top metallic patches were realized via photolithography followed by metal deposition of Ti(15 nm)/Au(200 nm). The geometric and optical parameters of the samples are provided in the supplementary information.



 §.§ THz spectroscopy


Spectroscopy of the samples was performed with a time-domain THz spectrometer driven by an ultrafast Ti:Sapphire oscillator. THz generation and detection were achieved via a photoconductive emitter (Tera-SED) and a 1 mm thick <110> ZnTe crystal, respectively, allowing for spectroscopic coverage from 0.2 THz to 2.5 THz. Delayed THz pulses originating from the optical components of the setup were eliminated by windowing the main THz pulse reflected off the samples. The time domain signals were Fourier transformed and analyzed in both their amplitude and phase. All reflectivity measurements were conducted at normal incidence and room temperature.  For absolute measurements of the reflectivity, we used the THz pulse reflected off a gold-coated sample as a reference. In all measurements, the THz spot size (∼ 2  mm in diameter) is well below the size of the manufactured samples, so that the samples can be considered infinite in the plane containing the interface (x-y plane in Fig. <ref>.a).
Polarization-resolved THz measurements were obtained by initially decomposing the linearly-polarized incident THz pulse into the σ_± circular basis. The reflection coefficient of each σ_± component was obtained thanks to a polarization-resolved detection. A description of the experimental setup and of the polarization-resolved THz measurements is reported in the supplementary information.
The magnetic field was applied by a large static magnet located behind the sample and was calibrated with a Hall probe. Homogeneity of the magnetic-field and the uncertainty on its absolute magnitude were better than 3% across the THz spot size.




 §.§ Numerical simulations


Numerical simulations of the cavities presented in Fig. 2b were performed with the Rigorous Coupled Wave Analysis method <cit.>. The optical and geometrical parameters used in the simulations are determined from experimental measurements performed on the samples (see supplementary information). The permittivity of Gold at THz frequencies was taken from <cit.> and the permittivity of Si_3N_4 from <cit.>.













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