Thin film sensing near exceptional point utilizing terahertz plasmonic metasurfaces

Non-Hermitian quantum systems along with engineered metasurfaces enable a versatile podium for sensor designs from industrial to medical sectors. The singularity points known as exceptional points (EPs) can be realized in such non-Hermitian systems. EP demonstrates a square root topology on minute perturbations, hence promising to be a potential candidate to sense external parameters, such as temperature, thermal fluctuations, refractive index, and biomolecules. Hence, in this work, through numerical and analytical investigations, we explore the sensing capabilities in the vicinity of EP utilizing suitably designed terahertz metasurfaces. Here, we propose a non-Hermitian metasystem comprising two orthogonally twisted square split ring resonators coupled by near-field Electromagnetic interactions that can exhibit dark-bright modes. In such a system, the presence of an active (photo-doped) material in the split gap of one of the resonators opens up an effective avenue to introduce controllable asymmetric losses, ultimately leading to the emergence of EPs in the polarization space. Hence, thin film sensing at the proximity of the emerged EP is investigated for different refractive indices by coating with an overlayer atop the metasurface. In such a configuration, the sensitivities of the eigenstates are calculated in terms of the refractive index unit, which turns out to be −0.044 THz RIU−1 and −0.063 THz RIU−1 when the system is perturbed near EP. Our proposed metasurface-inspired EP-based sensing strategy can open up novel ways to sense the refractive index of unknown materials besides other physical parameters.


Introduction
In recent times, the non-Hermitian Hamiltonians that obey Parity-Time (PT) symmetry demonstrating real domain or observable quantities picked up attention due to the associated interesting physics [1].Previously, the dominance of the Hermitian system (conservative system) largely disregarded the concepts of non-Hermitian or open systems on the grounds of unobservable quantities.Nevertheless, the behavior of some particular classes of Hermitian systems along with PT symmetry to observe quantities in real spectra has garnered attention in recent times.In the context of quantum mechanics, a non-Hermitian system is a non-conservative system (open system) whereas, a Hermitian system is a closed system obeying energy conservation.The total energy operator (Hamiltonian) of a non-Hermitian system is expressed as Ĥ ̸ = Ĥ † [2].Unlike, Hermitian Hamiltonians, non-Hermitian Hamiltonians have eigenstates in complex domains, which are generally unobservable.However, a non-Hermitian system that obeys PT symmetry can lead to eigenstates in the real domain exhibiting observable quantities.PT-symmetric Hamiltonian in the parameter space observes real eigenvalues (PT unbroken phase) in some domains, while they are complex in other domains (broken PT phase).The phase transition between the PT and BPT regimes takes place through an extraordinary transition point known as the exceptional points (EPs) [3].The EP is known as the branch point singularity of eigenstates [4].Another peculiar feature is that not only do the eigenvalues of the non-Hermitian system coalesce but also their corresponding eigenvectors [5].Generally, the notions of PT-symmetric systems are explored in the field of optics by bringing in gain components along with naturally existing loss components [6].Additionally, higher orders of EPs are found to be conceptually even richer and are promising for scientific explorations [7,8].However, the striking features of EPs have been the topic of intense studies with various futuristic applications in laser science [9], optical sensing [10], acoustics [11], photon transport engineering [12], nonlinear optics [13], and waveguides [14], just to mention a few examples.
The physical explorations of such non-Hermitian systems are generally challenging and complicated endeavours.However, the emergence of artificially structured materials (known as metamaterials/ metasurfaces) lends a helping hand to serve this purpose.These artificial structures can form an effective test bed to investigate and explore the interesting physics of non-Hermitian systems and EPs.The ability of the metasurface to manipulate electromagnetic (EM) waves by engineering the subwavelength unit cell has widened its capabilities to accomplish numerous interesting properties and applications [15][16][17].Unfortunately, loss in a system is inevitable and can only be minimized.Thus, the non-Hermitian system has an exciting new opportunity through engineering these losses, paving the way to expand the functionalities of metasurfaces as well as explore new functional dimensions for photonic devices.Hence, the scientific community has been actively researching different feasible routes encircling non-Hermitian physics and EPs in metasurfaces in connection with PT symmetry.Nevertheless, the realization of the EP phenomenon is tedious in the optical gain medium, thus limiting the explorations in photonics and optics.However, the introduction of asymmetric losses in such a system is helpful to overcome this hurdle and explore the science and technology surrounding EP.The asymmetric loss can be introduced in the system by applying mismatched material properties constituting the resonators [18,19], bi-anisotropic metasurfaces [20], different structural configurations [21,22], or the establishment of new active materials [23,24].It is learned that the appropriate balance of asymmetric losses and coupling strength can ultimately lead to the realization of EP.The signatures of attaining EP are usually characterized by a phase jump [25], change in polarization state [22,26], swapping of eigenstates [21], etc apart from degeneration of eigenvalues and corresponding eigenvectors.In such a way realized EP phenomenon in metasurfaces, can be utilized in various applications like unidirectional reflectionless light propagation [27], sensors [28], chirality [29,30], absorbers [31], and many more.Among these applications, sensors have become an integral part of human life with their utility in a vast area of practical fields.A sensor's potential capability is determined by how much little change the sensor can detect.The sensitivity of a sensor along with its figure of merit are the important factors determining the quality of the sensor [32][33][34].Typically in conventional sensors, the perturbation encountered by the system shows minute and linear changes in the sensing device [35,36].However, the EP, being ultrasensitive, qualifies as a promising yet viable candidate for the same.The square root topology of the EP and enormous effect on such a system under minute perturbations has been noticed in recent times [37][38][39][40].That is if the perturbation experienced is δ resulting in ∆ω frequency split, that leads to ∆ω ∝ √ δ [41].Hence, EP has opened up new avenues of ultrasensitive optical and photonic-based sensors.Therefore, the EP is employed in sensing refractive index [24], thermal variations [37], gas [25], and heat [42].The EP also finds significant applications in biosensors like anti-immunoglobulin G [43], single nanoparticle detection [44], optomechanical mass sensors [45], viruses [46], etc.
Based on these backgrounds, in this work, we have put forward a plasmonic metasurface design to explore the thin film sensing near the EP while operating in the Terahertz frequency regime.Here, the polarization-dependent metasurface consists of orthogonally arranged split ring resonators placed beside each other to form the unit cell.One of the resonators is embedded with an active Si patch for modulating the losses through photo doping, eventually permitting to alteration and control of asymmetric loss coefficients associated with these coupled resonators.The split ring resonators in this non-Hermitian two-level system are coupled through near-field EM interactions.The emergence of the EP is characterized in the polarization space by a sudden and abrupt overturn in the polarization state of the eigenvectors.Our further studies involve the introduction of an overlayer on top of the metasurface for reviewing the sensing characteristics near EP.The presence of an overlayer disturbs the EP paving the way for sensing unknown materials, which essentially signifies a unique route of sensing a thin film material utilizing the degenerated eigenstates.It is worth mentioning that we have adopted a generic approach in this work, which can be utilized for terahertz sensing of biomolecules, single particle detection, temperature detection, etc. utilizing EP-compatible plasmonic metasurfaces while operating in the polarization space.

Unit cell design of the metasurface
The diagram in figure 1 comprises the design of the proposed polarization-dependent metasurface in the Terahertz region.Figure 1(a) illustrates a general depiction of a two-level asymmetric lossy non-Hermitian system mimicking the metasurface design.The asymmetric loss coefficients experienced by the two resonators are represented by γ L and γ R .These resonators are coupled to each other with a coupling strength denoted as κ.The delicate balance and interrelation of κ, γ R, and γ L can transform the coupled system from PT-symmetric unbroken (real eigenstates) to broken (complex eigenstates) phases through a transition point called an 'EP' [47].In this study, an overlayer of 3 µm is deposited on top of the proposed metasurface.The coated layer is studied under different refractive indices to study the sensing capability of the EP-compatible metasurface.Figure 1(b) gives an approximated electrical equivalent model of the proposed metasurface.The RLC circuit is depicted with two loops, one represented by 'L' representing the left resonator (here, in our case SRR) and another loop with 'R' for the right resonator.The LCR (inductance capacitance and resistance) represents a typical split ring resonator under the influence of a biased voltage.The resistor (R) is the loss experienced in the SRR, the capacitor (C) is due to the split gap of the SRR where the energy is stored in its electric field form and the inductor (L) gives the energy is stored in its magnetic field form.The parameter Lκ (effective inductance) accounts for the coupling effects between the L-and R-resonators [48]. Figure 1(c) gives an artistic view of the periodic array of the metasurface encountering Terahertz radiation at normal incidence.Here, the refractive index of the deposited overlayer varies as n = 1.0, 1.41, 1.65, and 1.87 for sensing studies.In figure 1(d), the unit cell of the bare metasurface is shown, which means, without the presence of the overlayer, for a better understanding termed as intrinsic meta structure.The proposed unit cell design is depicted with periodicities P 1 and P 2 taken as 100 µm and 57 µm.The square-shaped split ring resonators are made of gold of thickness 200 nm.The gold metal is considered a lossy metal with an electrical conductivity of 4.56 × 10 7 S m −1 .The thickness of the lossless sapphire substrate is considered as 40 µm with ϵ = 9.4.The left and right resonators marked as 'L' and 'R' are placed at a constant separation distance, 's' of 3 µm.The length of the outer square in both the L-and right split ring resonators (R-SRRs) are taken as 40 µm and represented by parameter 'b.'The split gap (u) is taken to be 2 µm for each of the resonators.The presence of an active silicon patch incorporated in the left split ring resonator (L-SRR) acts as an active material.The conductivity of Si can be modified to introduce differential losses in the coupled system.The lossy Si patch is with ϵ = 11.9 and an electrical conductivity of 0.000 25 S m −1 .The conductivity (σ) of the embedded Si patch can be actively controlled [49] to achieve the conditions of the EP under different refractive indices of the overlayer.For excitation of similar LC (inductive-capacitive) resonance modes in the L-and R-SRRs, the cell parameters are kept identical, except for the inclusion of the Si patch in the L-SRR which is essential to introduce asymmetric loss.Figure 1(e) represents the resonance frequency of the fundamental mode of a single SRR that took shape at ν 0 = 0.4882 THz.The metasurface considered in our study is polarization dependent, such that when x-polarized THz radiation is incident on the metasurface, L-SRR couples strongly with the incident radiation and behaves as a bright resonator.For the same polarization, the R-SRR behaves as a dark resonator, that is weakly coupled with the incident THz radiation [50][51][52][53].Similarly, for the y-polarization of THz radiation, R-SRR behaves as the bright resonator and L-SRR, as the dark resonator.The excitation of perpendicularly oscillating magnetic dipoles through near-field coupling in the SRRs takes care of the PT symmetry in this proposed metasurface [18].Hence, the proposed metasurface follows indispensable conditions (non-Hermitian PT-symmetry) to attain an EP.

Numerical simulation
Here, in this section, the rigorous numerical simulations of the metasurface are discussed in detail.First, we studied the bare metasurface and then metasurface coated with an overlayer.In this way, the metasurface is explored by changing the ambient surroundings, that is by changing the refractive indices of the overlayer deposited on top of the metasurface.The numerical simulations are accomplished using the software, CST Microwave Studio where the finite element frequency domain solver is applied.This software solves Maxwell's equations according to the design and complexity of the 3D metasurface.We have adopted tetrahedral meshing in this work.To generate the periodic array of the unit cells, along the x-and y-directions unit cell boundary conditions are applied.Open boundary conditions are adopted in the z-axis along with the Floquet ports.The Floquet's conditions employed are Floquet TE00 and TM00 modes for the metasurface for excitation as well as detection of x-and y-polarized THz waves, respectively.Finally, the transmission data procured for the T xx /T yy (co-polarizations), and T xy /T yx (cross-polarizations) along the xand y-directions in terms of S parameters according to the applied conditions.The active control of the Si patch embedded in the L-SRR is carried out by varying the conductivity, which can be realized by an optical pump terahertz probe (OPTP) setup [49,54].In the OPTP setup, the optical properties of the Si patch can be actively tailored by an external optical pump with an 800 nm pulsed laser having a beam spot diameter around ∼10 mm, with a repetition rate of 1 KHz and ∼120 fs pulse width [55,56].
Hence, the transmission curves are procured for varying refractive indices of the overlayer as n = 1.0, 1.41, 1.65, and 1.87.The representative transmission plots of co-and cross-transmission results obtained for n = 1.0 are shown in figure 2. Further, in the proposed metasurface the L-SRR and R-SRR interact through near field coupling mechanism for both the probe polarizations.Figure 2(a) represents the T yy co-polarization transmission results for varied conductivities for the y-polarized THz beam (n = 1.0).The Si patch's conductivity in the L-SRR is varied which ultimately alters the asymmetric losses associated with the SRRs.Moreover, it is observed that as the conductivity of the Si patch is increased from σ = 0 S m −1 to 5600 S m −1 , two resonance dips are transformed into a single dip (figure 2(a)).The change in the line shape of the resonance dip is remarked as a diminishing coupling strength amongst the resonators.The enhanced conductivity of the Si patch diminished the LC mode excitation in the L-SRR and effectively weakened the coupling effect between the LC modes [57].Figure 2(b) represents the plot of T xx transmission spectra when L-SRR behaves as a bright resonator while the R-SRR behaves as a dark resonator.The bright resonator (here L-SRR) directly excited by the incident THz interacts with the neighboring R-SRR unfolding resonance mode hybridization [58].At σ = 0 S m −1 , the split in the resonance is prominent as a result of hybridization between the dark and bright modes.The LC mode is excited in the bright resonator and interacts with the closely placed dark resonator through a magnetic coupling mechanism.With the increment of the Si patch's conductivity, the L-SRR is incapable of supporting the LC mode and hence unable to drive the R-SRR anymore through inductive coupling, hence split modes lose strength at higher conductivities and ultimately disappear (figure 2(b)).Further, the cross-polarization transmission data is plotted in figures 2(c) and (d).It is noted that the transmission amplitude T xy is equal to T yx .These undifferentiated curves are due to the optical reciprocity of the metasurface, which arises because of time reversal symmetry [59,60], a necessary condition for exhibiting EPs.
The T xx , T yy , T xy , and T yx transmission results acquired so far are further utilized to realize the eigenstates of the non-Hermitian system in polarization space.The transmission matrix T is represented by [59] T = The transmission (2 × 2) matrix is solved for eigenvalues as ( Here, the components of the matrix T xx and T yy represent the amplitudes of co-polarization transmissions.The component T xx is procured when the THz transmission detection is carried out along the x-axis for the x-polarized applied electric field of the THz probe.In a similar manner for the component T yy along the y-axis.Whereas, the components T xy and T yx denote the amplitude of cross-polarization transmission that is detected along the y-(x)axis when the applied electric field of the THz probe is polarized along the x(y) direction.
The (2 × 2) transmission complex matrix yields two eigenstates denoted by λ 1 and λ 2 .The eigenstates derived from the matrix take up the form of the Ae iϕ , where A is the magnitude part that provides the info regarding the transmission amplitude.The ϕ signifies the eigenphase part of the λ 1 and λ 2 eigenstates.The (2 × 2) transmission complex matrix yields two eigenstates denoted by λ 1 and λ 2 .Thus, extracted eigen amplitudes, phases, and vectors for each of the refractive indices are plotted in figure 3.
Figures 3(a)-(i) depicts the eigentransmission amplitude, phase, and the respective eigenvectors for n = 1.0 for eigenstate 1 and eigenstate 2 in orange and red color respectively.The active Si patch's conductivity is swept from σ = 0-5600 S m −1 , to realize the existence of EP.The EP is remarked as an appearance of crossing point frequency in eigen amplitude and phase (figures 3(b) and (e)) concurrently.Additionally, the associated eigen polarization states can confirm the existence of the EP in the metasurface.That is, the evolution of skewed eigenpolarization states is perceived in such a condition.Figures 3(g)-(i) denote the vertically oriented elliptical eigen vectors noticed at the EP in the polarization space.The eigenpolarization states are transformed from PT symmetry unbroken (ellipses aligning along y = ±x) to broken (ellipses orienting along the x-and y-axis) through the EP [23].As representative plots of the transition of PT symmetry unbroken to broken phase through the EP, Si conductivity of σ = 2000 S m −1 , 3800 S m −1 , and 4000 S m −1 are presented in figures 3(g)-(i) respectively.The exact feature of EP emerged at ν = 0.4599 THz for n = 1.0 and is characterized by the evolution of merged circular polarization states of the degenerate two eigenvectors.
Furthermore, the examination of the presence of EP due to the presence of an overlayer of varied refractive indices on the metasurface is also explored.Thus, figures 3(j)-(r) maps out the transmission amplitudes of eigenstates λ 1 and λ 2 at different refractive indices (n = 1.41, 1.65, and 1.87) of the overlayer.Note that, in the proposed system when a sensing material is deposited on the metasurface the EP is disturbed and eigenstates are no longer in coalescence at EP but degenerate.So, to re-establish the EP conditions in the perturbed system, the Si patch conductivity is tuned accordingly and EP is evoked back.Ultimately, adjusting the Si conductivity results in tuning the asymmetric loss parameters of the two-resonator system.It is noted that the amplitude dips of the eigenstates 1 and 2 tend to have an overlapping nature in all four cases (n = 1.0, 1.47, 1.65, and 1.87).The plots in figures 3(j)-(o) illustrate the transmission phases corresponding to the amplitudes.Similar to the trend exhibited by the amplitudes of n = 1.0, the λ 1 and λ 2 , and their respective eigenphases also come closer and cross each other.Therefore, the EP is evoked at this simultaneous crossing of both transmission amplitude and phase.It is also noted that in all four different refractive indices, the eigenstates 1 and 2 crossing takes place at different frequencies.For n = 1.0, the crossing point frequency is observed at ν = 0.4599 THz at σ = 3800 S m −1 whereas for n = 1.47 around ν = 0.4450 THz at σ = 3810 S m −1 .Subsequently, for refractive index n = 1.65, the EP is encircled at ν = 0.4310 THz for σ = 4100 S m −1 , and finally, for n = 1.87, it is perceived at ν = 0.4270 THz at σ = 4125 S m −1 .
Moreover, it is noticed that as the refractive index of the coated material is increased, the EP frequency is redshifted.The resonant frequencies of the metasurfaces are influenced by the refractive indices surrounding the resonators and dielectric substrate.Such resonance modes of the metasurfaces can simplistically be represented as f eff = f0 n eff where f 0 is the resonant frequency of the metasurface in air, n eff is the effective refractive index experienced by the metasurface in the presence of the overlayer [61].Notably, the non-orthogonality of the eigenvectors apart from frequency crossing is observed at EP for all the different refractive indices of the overlayer.This feature is intensified after re-establishing the EP through conductivity variations (figure 3).

Analytical modeling
Next, the results of the numerical simulations are further substantiated using analytical modeling.The coupled mode theory (CMT) is employed on the proposed perturbed non-Hermitian two-level system.The model helps to understand the principles of coupled orthogonally oriented split ring resonators experiencing asymmetric loss coefficients [29].CMT solves time-dependent second-order differential equations for two closely placed resonators mimicking bright and dark resonators.The orthogonally positioned SRRs act differently in the presence of an external electric field.The bright mode of the SRR is excited through strong coupling with the incident THz radiation.In contrast, the other SRR is termed as a dark mode which gets excited through the nearby bright resonator.The separation distance between the closely placed SRRs determines the coupling strength experienced by the resonators.Thus, the Hamiltonian of the system can be written as [62] H = The L-and R-SRRs are assumed to have closely identical resonance modes (ω L ∼ ω R ) considering stronger mode hybridization and better resonance coupling between the resonators.The coupling coefficient is determined by the parameter 'κ' , and the effective losses (radiation and dissipative) are expressed as γ 1 , and γ 2 .The Hamiltonian Hψ = Eψ is further solved to obtain the eigenstates of the system in terms of κ, γ 1 , and Here, since ω L ∼ ω R is considered as ω 0 for calculation of λ 1 and λ 2 Thus, different conditions related to the term under the square root (4κ 2 − (γ 1 − γ 2 ) 2 ) defines the eigenstates λ 1 and λ 2 .The PT symmetry unbroken phase is given by 2κ ≫ |γ 1 − γ 2 | and broken phase is defined as 2κ ≪ |γ 1 − γ 2 |.The transition between the PT unbroken and broken phase is a critical singularity point known as an EP.Thus, the distinguished eigenstates λ 1 and λ 2 become identical at EP.The CMT model conveys an analytical/theoretical way to understand the coupled bright and dark modes of the non-Hermitian PT-symmetric system.The physical parameters (γ 1 , γ 2 and κ) are chosen in such a way as to reproduce the metasurface responses derived from the numerical study, see figure 4. While the transmission plots of T yy are shown as a representative plot to demonstrate the agreement of our numerical and theoretical calculations.
Figure 4 represents the analytically derived and numerically simulated T yy at n = 1.0, 1.4, 1.65, and 1.87 of the overlayer.Figures 4(a)-(d) gives the transmission spectra at silicon conductivity of 0 S m −1 for varied The physical parameters (γ 1 , γ 2 and κ) derived from the CMT model are plotted in figure 5 for a proper understanding of their behaviour.Figures 5(a)-(d) give the plot of the γ 1 , the loss parameter of the bright resonator, and figures 5(e)-(h) denotes the loss parameter (γ 2 ) of the dark resonator.The conductivity of the Si active material in the L-SRR is extended from σ = 0 S m −1 to the corresponding conductivity value where the EP is achieved from the numerical simulations.In general, similar trends of the fitted parameters (γ 1 , γ 2 and κ) are observed as the refractive indices are varied (from n = 1.0-1.87).It can be noted that γ 1 /γ 2 increases gradually with the increase in conductivity of the Si patch starting from σ = 0 S m −1 .The loss coefficient of the bright SRR (γ 1 ) and dark SRR (γ 2 ) showed an akin tendency to increase with increasing conductivities.However, the relative change in γ 2 is comparatively larger than γ 1 .This is because the dark resonator (γ 2 ) experiences a more dramatic change in loss parameter because of external laser-induced photo doping inside the split gap.
Similarly, in the case of the parameter κ, as represented by figures 5(i)-(l), it is noticed that the coupling parameter reduces with an increase in conductivity.In general, the parameter κ has a range varying from 0.188 THz to 0.09 THz.The κ parameter ultimately defines the coupling strength between the dark and bright resonators.As the Si patch conductivity is increased the dark resonator fails to support the LC resonance and eventually becomes a closed-loop resonator.This effectively reduces the inter-resonator coupling between the bright and dark resonators at LC resonance.Figures 5(m)-(p) depict the plots of the eigenstates 1 and 2 calculated by equation (3).The main feature of the coalescence of eigenstates at the EP is observed in figures 5(m)-(p).The eigenstates λ 1 and λ 2 , are found to show a converging nature as the conductivity is increased which is a remarkable feature of EP.The EP is analytically estimated at ν = 0.54 THz in the case of n = 1.0 and ν = 0.534 THz for n = 1.41.The EP frequency further decreased to ν = 0.5275 THz and ν = 0.5235 THz on a further escalation of refractive index to 1.65 and 1.87 respectively.This implies that an increase in refractive index or dielectric constant, redshifts EP's position in the frequency spectrum.
Furthermore, to understand the capability of the designed terahertz metasurface to detect small perturbations in the vicinity of the EP, refractive index sensing is explored.The overlayer deposited on the metasurface with a constant thickness of 3 µm with differing refractive indices is studied further for thin film sensing near EP (ν = 0.4599 THz). Figure 6(a) gives a comparative plot of, the EP frequencies calculated from analytical and numerical methods.It can be noted that our theoretical calculations are reasonably consistent with the numerical calculations.The EP frequencies red shift with increasing refractive indices for a fixed overlayer thickness (figure 6(a)).Further to study the capability of the metasurface to operate as a refractive index sensor, sensitivity (S) is calculated for the degenerate eigenstates originated as soon as the system is perturbed from EP.The sensitivity (S) is extracted by invoking the relation ∆f ∆n , where ∆f is the variation in resonance frequency and ∆n is the fractional change in refractive indices.The change in ∆n is caused by modifying the surrounding environment, represented by the unit, refractive index unit (RIU).The slope of the frequency v/s refractive index plot gives the sensitivity (S).Therefore, to quantify the refractive index sensing, eigen frequencies procured from the numerical simulations are plotted (figure 6(b)).In this way, the sensitivity of the eigenstates is calculated at σ = 3800 S m −1 for varying refractive indices, i.e. n = 1.0, 1.41, 1.65, and 1.87 (tabulated in table 1).It is observed that the value of S = −0.044THz RIU −1 for eigenstate 1 and in the case of eigenstate 2, it is calculated as −0.063THz RIU −1 .It is noted that as soon as the system is disturbed at EP by a change in its environment (here, the introduction of different refractive indices), the eigenstates which otherwise have identical values at EP move away from each other.The eigenstates start to  develop a diverging nature as the refractive indices increase from n = 1.0 to n = 1.87.The system shifts in a direct proposition from EP as the physical environment changes and loses its dimensionality.Since the EP condition is satisfied at a higher conductivity of Si, it is worth noting that the system lesser sensitivity at σ = 0 S m −1 and then the sensitivity increases drastically till it reaches EP.The frequency shifts of the eigenstates from its EP degeneracy under variable refractive indices is an effective parameter check to study the capability of the EP sensor.The enhanced performance of the sensor under external perturbations can be further determined by a sudden and abrupt change in the eigen polarization states from circular to elliptically polarized states.Theoretical and numerical studies conducted in our study reveal that it approaches the singular point (EP) and further external parameters like refractive index sensing can be carried out.Thus, the proposed actively tunable metasurface in planar configuration encircling EP is eligible as an effective platform for sensing materials in thin film forms.

Conclusions
This study introduces a method for refractive index sensing of thin film near an EP using a terahertz metasurface that imitates a non-Hermitian quantum system while operating in polarization space.Here, the unit cell of the non-Hermitian metasurface is realized by employing asymmetric loss-based two-coupled resonators.The coupled system is comprised of a L-SRR with an embedded Si active material in the split gap and a R-SRR without any active material.The asymmetric losses are introduced into the system by selectively modifying the conductivity of the Si patch incorporated in the L-SRR.Such selective conductivity variation in terahertz metasurfaces can be achievable through photo doping [63].Further, the numerical and theoretical calculations are systematically carried out to attain the EP in the system.Next, thin film sensing is carried out by varying the refractive index (n) from 1.0 to 1.87 with the help of an overlayer.The EP positions are numerically observed at ν = 0.4599 THz, ν = 0.4450 THz, ν = 0.4310 THz, and ν = 0.4270 THz for n = 1.0, 1.41, 1.65, and 1.87 respectively at Si conductivities of σ = 3800 S m −1 , 3810 S m −1 , 4100 S m −1 , and 4125 S m −1 correspondingly.To validate the numerical outcomes, the theoretical/analytical calculations are performed by employing CMT.Our analytical/theoretical calculations reveal EPs at ν = 0.54 THz, ν = 0.534 THz, ν = 0.527 THz, and ν = 0.5235 THz for n = 1.0, 1.41, 1.65, and 1.87 respectively.It can be observed that numerically and analytically calculated EP positions are within ∼18% deviations with similar monotonically decreasing trends with increasing refractive indices.Further to understand the sensing capabilities of the proposed EP metasurface, the sensitivities of the eigenstates are derived from the resonance frequency v/s refractive index plot.The sensitivities of the degenerated eigenstates are found to be around −0.044 THz RIU −1 and −0.063 THz RIU −1 respectively, when the system is disturbed at EP. Therefore, this work demonstrates a way to sense the refractive index of unknown materials by utilizing PT-symmetric, non-Hermitian THz metasurfaces operating near the EP.These results suggest a wide range of potential applications for EP-based sensors operating in the THz regime, spanning from refractive index sensing to single-particle detection in the near future.

Figure 1 .
Figure 1.(a) A general representation of a non-Hermitian system coupled with asymmetric loss coefficients represented by γL and γR.The two resonators are coupled with a coupling strength of parameter κ.(b) An approximated electrical equivalent model of the proposed unit cell.(c) An artistic view of the periodic array of the studied metasurface.An overlayer of thickness of 3 µm is placed on top of the proposed metasurface for sensing.(d) The unit cell of the metasurface with active material embedded in the gap of one of the split ring resonators.(e) The fundamental signature of the single SRR at ν0 = 0.4882 THz when the electric field is incident along the gap of the resonator (y-axis).The inset represents the corresponding single SRR.

Figure 2 .
Figure 2. The transmission plots of the active metasurface where the refractive index of the overlayer is 1.0.The conductivity of the active Si patch is varied from σ = 0 S m −1 to σ = 5600 S m −1 .The inset depicts the corresponding unit cell with the electric field direction applied along the gap of the SRR.The co-polarization transmission results from electric field direction along the y-axis (a) Tyy and along the x-axis (b) Txx.The cross-polarization results from (c) Tyx and (d) Txy.