Tuning the Effective $\mathcal{PT}$-Phase of Plasmonic Eigenmodes

Hermitian systems and their operators are used to describe a wide range of physical phenomena to predict the evolution of Eigenstates via unitary operations containing purely real Eigenvalues Dunford and Schwartz (1957); Shankar (2012). Recently, increasing attention has been dedicated to non-Hermitian systems, which are non-conservative and generally yield complex Eigenvalue spectra Heiss (2012). Non-Hermitian systems are typically created by opening a Hermitian system to the environment by including dissipation and/or gain. Interestingly, coupled non-Hermitian systems can yield purely real Eigenvalues which respect parity-time ($\mathcal{PT}$) symmetry via an appropriate balance of gain and loss Bender and Boettcher (1998); Feng et al. (2017). In an unbalanced scenario, a $\mathcal{PT}$-broken system shows complex conjugate Eigenvalue pairs. The $\mathcal{PT}$-symmetric (PTS) and $\mathcal{PT}$-broken (PTB) regimes are separated by the exceptional point (EP), where the Eigenvalues coalesce, which is associated with several interesting physical phenomena, such as level repulsion Heiss (2000), reflectionless propagation Feng et al. (2013), and topological phase transitions Rudner and Levitov (2009).

Photonics has been identified as the ideal landscape for investigating the subtle properties of non-Hermitian systems Feng et al. (2017); El-Ganainy et al. (2018); Miri and Alù (2019); Özdemir et al. (2019), since tailored amounts of optical gain and loss can be introduced by appropriately engineered materials and structures. Many important technological advances have recently been provided by EP-photonics, such as sensitive modal manipulation Hodaei et al. (2014), topological energy transfer Xu et al. (2016), unidirectional propagation Wang et al. (2009), and polarization conversion Hassan et al. (2017). These advances rely on the unique characteristics of the Eigenstates close to the EP – for example, even nanoscale events can be detected by measuring their impact on macroscopic Eigenstates Chen et al. (2017).

For many applications, balancing gain and loss is not necessary for accessing the underlying physics. Most investigations Guo et al. (2009); Feng et al. (2013); Zhen et al. (2015); Feng et al. (2017) harness effective $\mathcal{PT}$-symmetric systems with a global loss offset, where PTS and PTB states, separated by an EP, are achieved with Eigenvalues that are shifted along the imaginary axis with respect to the perfectly balanced case. Such effective $\mathcal{PT}$-symmetric structures open up many opportunities in the design of photonic systems that harness $\mathcal{PT}$-symmetries without requiring gain, simplifying designs.

One state that can be used for non-Hermitian photonics is the surface plasmon polariton (SPP) Novotny and Hecht (2012), which is a propagating surface wave at metal/dielectric interface. SPPs continue to attract attention due to their ability to confine light down to a fraction of the wavelength, and their high sensitivity to RI changes of the environment Novotny and Hecht (2012); Wieduwilt et al. (2015). Owing to their large wave-vectors, coupling energy to SPPs is often challenging. One approach relies on using (lossless) dielectric waveguides running parallel to (lossy) plasmonic waveguide Tuniz and Schmidt (2016), leading to a system with hybrid Eigenmodes (EMs) and complex Eigenvalues. Such dielectric/plasmonic systems are inherently non-Hermitian, and can be designed to possess an effective-$\mathcal{PT}$ phase transition – full $\mathcal{PT}$-symmetric properties can then be recovered by re-introducing gain Alaeian and Dionne (2014); Salgueiro and Kivshar (2016). Despite its great potential in a multitude of areas Miri and Alù (2019), and a growing number of studies Alaeian and Dionne (2014); Salgueiro and Kivshar (2016); Kodigala et al. (2016); Turitsyna et al. (2017); Lourenço-Martins et al. (2018), to the best of our knowledge effective $\mathcal{PT}$-phase transitions, tuned across the EP, have yet to be experimentally reported in plasmonic waveguides. Here, we experimentally observe the effective $\mathcal{PT}$-phase transition across the plasmonic EP of a tuneable hybrid dielectric/plasmonic waveguide system. Transmission experiments reveal fundamental changes in the properties of the underlying system by tailoring the refractive index of the surrounding environment and unambiguously showing a crossing through the plasmonic EP.

The main features of such system can be understood on the basis of mode hybridization using coupled mode theory (CMT). We consider the specific case of a lossless waveguide 1 (WG1) and lossy WG2, supporting modes with respective propagation constants $\beta_{1}=\beta_{1}^{R}$ and $\beta_{2}=\beta_{2}^{R}+i\beta_{2}^{I}$, and coupling constant $\kappa$, which for this particular case is assumed to be real Miri and Alù (2019). The coupled mode equations can be written as

where $a_{1,2}$ are the uncoupled modal field amplitudes in each waveguide. Solutions of the form $a_{j}=A_{j}\,\exp(i\beta^{\rm EM}_{j}z)$ lead to hybrid EMs with propagation constants

where $\overline{\beta_{R}}=\left(\beta_{1}^{R}+\beta_{2}^{R}\right)/2$, $\Delta\beta_{R}=\left(\beta_{1}^{R}-\beta_{2}^{R}\right)/2$ is the dephasing, and $j=1,2$. We find that the CMT model is in excellent agreement with simulations of a lossy 1D multilayer system Tuniz and Schmidt (2016) (see SM) The model predicts different transmission characteristics near the EP, which occurs when the Eigenvalues coalesce at a certain frequency $\omega_{R}$ such that $\sqrt{\kappa^{2}+\left(\Delta\beta_{R}+i\beta_{2}^{I}/2\right)^{2}}=0$, so that $\Delta{\beta_{R}}=0$ (i.e., the phase-matching condition) and $\kappa=\beta_{2}^{I}/2$. Figure 1(b)-(c) shows finite element (FE) calculations which highlight the transmission characteristics close to the EP: the PTS regime is characterized by a periodic exchange of energy between the waveguides due to modal beating between EMs with different propagation constants – analagously to what occurs in Hermitian systems. The PTB regime is unique to non-Hermitian systems and presents no periodic energy exchange between waveguides at $\omega_{R}$, since the real parts of the coupled Eigenmodes are equal, resulting in an infinite beat length. These different regimes can be accessed at $\omega_{R}$ by changing $\kappa/\beta_{2}^{I}$.

To experimentally reveal the phase transition between PTB and PTS regimes across the EP, we consider a dielectric/plasmonic hybrid waveguide system with properties that can be tailored via the environmental refractive index (RI) (Fig. 2(a)). The system is formed by a cylindrical SiO₂ waveguide (diameter: $d=20\,{\mu{\rm m}}$) coated with a gold nano-film on one side (thickness: 30 nm). The EM dispersions are tuned by immersing the waveguide in liquids with different RI, with the interaction length determined by the length of the liquid column $\ell$ Wieduwilt et al. (2015). The uncoupled modes satisfy the phase matching condition at a wavelength $\lambda_{R}$ only when the outer RI has a specific value – a scheme which has frequently been used for plasmonic sensing Novotny and Hecht (2012); Wieduwilt et al. (2015). Here, we use commercially available analytes, identified by the refractive index $n_{\rm ID}$ at $\lambda=589\,{\rm nm}$ (see Supplemental Material (SM) for the measured analyte and gold RI dispersions $n_{o}$).

We now consider the Eigenstates of this system when changing the RI environment, and how this impacts coupling. Since $d\gg\lambda$, a qualitative understanding of the modal behaviour can be obtained using a 1D multilayer system that includes the RI distribution along the connection line between the centre of the silica core and the azimuthal location of maximum gold film thickness Wieduwilt et al. (2015). The evolution of the Eigenstate dispersions for this hybrid system (Fig. 2(b), where $n_{{\rm eff},j}=\beta_{j}^{\rm EM}/k_{0}$, $k_{0}=2\pi/\lambda$) show a clear transition from PTB to PTS regimes. The EP is predicted close to $n_{\rm ID}=1.42$, but its exact location will be highly sensitive to environmental conditions Chen et al. (2017). Note that the detailed properties of hybrid multimode dielectric/plasmonic waveguides are more complex than what the CMT model predicts, also because SPP modes are close to cut-off, with obvious implications on modal dispersion.

Calculations show that the transition across the plasmonic EP can be inferred from the transmission properties of this waveguide system by using different $\ell$ and RI environments. To illustrate this, we calculate the axial intensity distribution (Fig. 2(c)), and retrieve the transmitted power (Fig. 2(d)) using the modal evolution and interference of the two hybrid EMs excited by the fundamental mode of the corresponding dielectric waveguide Tuniz and Schmidt (2016). The complex modal amplitudes at input and output are determined by overlap integrals between the fields at $\ell$ and the fundamental mode of the dielectric waveguide, allowing to calculate the transmission through the hybrid system. The plots in Fig. 2(c) are calculated at $\lambda_{R}$, which in the PTB or PTS regime respectively occurs where the imaginary- or real- parts of $n_{\rm eff}$ cross (see SM) As expected, the PTS regime ($n_{\rm ID}=1.440$) shows an oscillating damped energy distribution between dielectric core and plasmonic layer (Fig. 2(c), right), due to modal beating of lossy EMs (Fig. 2(d), right). In the PTB regime ($n_{\rm ID}=1.400$), no such oscillations are observed (Fig. 2(c), left), and the transmitted power decays monotonically at smaller loss values (Fig. 2(d), left). For $n_{\rm ID}=1.420$ we find an intermediate case, where the transmission monotonically decays with the higest overall loss. Therefore, the two regimes can be distinguished by the axial power distribution, with the PTS case yielding an oscillatory behaviour, while the PTB case shows a monotonic power decay, both of which have lower overall loss with respect to the loss at the EP (dashed lines). The fact that the EP possesses the highest overall loss in this kind of hybrid plasmonic system has been verified by a detailed numerical analysis (see SM) Since the loss of this system is maximum at the EP, the presence of a local loss maximum provides an additional property that can be used to confirm that a $\mathcal{PT}$-phase transition through the EP has occurred. Note that the simulations presented (Fig. 2(d)) enable a qualitative comparison to the experimental data and have been conducted in order to justify the choice of the hybrid waveguide used; the strong susceptibility on environmental influences, however, makes a quantitative comparison to experiments challenging.

We perform a comprehensive series of transmission experiments on the above system – details of sample preparation and experimental setup can be found in Ref. [21]. Here, different interaction lengths were implemented using Teflon stubs of different length, which support defined droplets of analyte (spatial resolution: $\pm$ 0.05 mm.) We first measured the spectral distribution of the transmission $T$ for various analytes at constant interaction length ($\ell=2\,{\rm mm}$, Fig. 3(a)). We observe a transmission dip that shifts towards longer wavelengths and increases in contrast as $n_{\rm ID}$ is increased. The minimum transmission ($T_{\rm R}$) for each $n_{\rm ID}$ occurs at resonant wavelengths $\lambda_{R}$. Calculations show that the transmission minimum is due to the interactions between the input mode and the excited hybrid EMs excited (see SM) and depends on both $\ell$ and the effective $\mathcal{PT}$-phase accessed. By analysing $\lambda_{R}$ as function of RI at $\lambda_{R}$, we determined the experimental RI sensitivity given by $S=\partial\lambda_{R}/\partial n_{o}$, which exceeds $50\rm\mu m/RIU$ for $n_{o}>1.43$ (see SM) This represents one of the highest values ever measured for any plasmonics device to date, and is due to operation near the cutoff Wu et al. (2009) of the plasmonic mode.

Based on our previous analysis, we identify the transition between the two regimes by the different spatial power distribution (Figs. 2) by measuring the on-resonance power $T_{R}$ vs. $\ell$ at $\lambda_{R}$ under three different conditions. A high analyte RI ($n_{\rm ID}=1.440$, Fig. 3(b)) shows a strong dip in the spatial power transmission, indicating periodic energy exchange between the waveguides, i.e., directional coupling, confirming that this configuration is in the PTS regime. In contrast, a lower analyte RI (here $n_{\rm ID}\leq 1.438$, Fig. 3(c),(d)), yield an exponentially decreasing spatial power distribution with no dips, indicating no directional coupling, i.e., the PTB regime.

Note that due to power oscillations in the PTS case (Fig. 3(b)), it is difficult to estimate the overall loss; however, from peak of the envelope we can estimate a lower bound of $\sim 4.91\,{\rm dB/mm}$. This is higher than what is measured in the PTB regime (Fig. 3(d), 2.91 dB/mm), but lower than what is measured close to the EP (Fig. 3(c), 8.46 dB/mm), in qualitative agreement with the model (Fig. 2(b),(c)) – a further confirmation that the EP was crossed.

This study reveals that the effective $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phases can be tuned through the EP in a non-Hermitian plasmonic system by adjusting the environmental refractive index. The PTB regime is particularly suitable for bioanalytic sensing, as the power transmission only weakly depends on propagation distance at moderate loss level. This allows to probe thin layers with large lateral dimensions (e.g., molecular layers or 2D materials), since the evanescent field is homogeneously probing the material of interest along the entire waveguide, in contrast to all-dielectric systems. The PTS regime instead allows efficient and rapid energy transfer into plasmonic layers, leading to the extraordinary high RI sensitivity measured, which can be employed for highly-demanding RI sensing applications, e.g., detecting single DNA-binding events or protein folding.

We have found that modes at the EP possess the highest overall loss for this particular system, a property which could be harnessed for plasmonic sensing applications near the EP. A recent theoretical report presented a similar result in a different context Ke et al. (2018), warranting further study to verify its generality. Additionally, the phase indices of the EMs show a strong dependence of the effective mode index on the environmental RI, with additional implications for sensing. A modification of our structure could yield passive devices with extreme sensitivity at desired RIs (e.g., at the water index) to detect individual molecular events Chemnitz et al. (2016).

Our platform extends the capabilities of tuneable non-Hermitian photonics near the EP Miri and Alù (2019). The results and concepts presented here apply to any non-Hermitian waveguide, and immediately provide design tools for future EP-based photonic devices. Due to the unique properties of the two regimes and the unique dispersion characteristics near the EP, our study widens the scope of applications areas of plasmonics, with applications in bioanalytics, nonlinear light generation, signal processing, quantum technologies, and topological physics.

A.T. acknowledges support from the University of Sydney Postdoctoral Fellowship scheme.