Observation of an exceptional point in a non-Hermitian metasurface

2.3.1 Eigenvalue surface topology

Here, we validate the existence of an EP by examining the transmission eigenvalue surface. As mentioned previously, the eigenvalue surface of a non-Hermitian matrix forms a self-intersecting Riemann sheet in which the EP is located at the point of intersection (see Figure 2A). This type of topology is unique to non-Hermitian systems and not observed in Hermitian systems. For Hermitian systems, the eigenvalue surface forms a double-cone structure with the degeneracy located at the conical intersection. Hence, observing the self-intersecting Riemann surface structure will provide strong evidence for the observation of an EP.

Figure 2:

Eigenvalue surface topology near an exceptional point revealed through transmission simulation and experiment results of the non-Hermitian metasurface.

(A) Schematic of the complex eigenvalue surface in [ω, G_xy(S)] parameter space. At a given value of G_xy, we see that the crossing behavior as a function of frequency for the magnitude and phase of the eigenvalues is opposite. As the coupling is varied through the EP, the crossing behavior of the magnitude and phase are interchanged. Such behavior is a result of the unique topology of the self-intersecting Riemann surface formed in the vicinity of an EP. Purple lines show the evolution of the eigenvalues as the EP is encircled. (B) Simulation and experimental results of the complex eigenvalues as a function of frequency conducted at different separation S are shown. When S=5 μm, we see that the magnitude the eigenvalues do not intersect while the phase crosses. When S is changed to 6 μm, the crossing behavior is swapped, and we see crossing in the magnitude and anti-crossing in the phase. As graphically shown in (A), this indicates that an EP exists between S=5 μm and S=6 μm.

A non-Hermitian degeneracy has a codimension of 2, meaning that we need two parameters to capture the degeneracy [2]. Based on the ease and independence with which they can be controlled, parameters ω and G_xy are chosen. ω is directly controlled through incident radiation and G_xy is controlled by changing the distance between SRRs in the unit cell [9], [10]. Computing the eigenvalues and eigenvectors of the transmission matrix given in Eq. (2), we find that EPs exist at (ωEP, GxyEP)=(ω0, ±gx2γy−gy2γx2gxgy). When calculating the position of the EP, we assume that G_xx is negligible. This is because the largest contribution to G_xx comes from dipoles in neighboring unit cells, whereas G_xy has contributions from within the unit cell. The eigenstate at each EP is (1, ±i)/2, which corresponds to left and right circularly polarized light. In this paper, we focus on the EP with left circularly polarized light as its singular eigenstate.

The topology of the self-intersecting Riemann surface can be captured by observing the level crossing behavior in the magnitude and phase of the eigenvalues [13], [14]. Although it is usual to examine the real and imaginary parts of the eigenvalues, these quantities lack a direct physical interpretation for the complex transmission eigenvalues. Therefore, we instead examine the magnitude and phase of the eigenvalues. As using the magnitude and phase instead of the real and imaginary parts of the eigenvalues can be considered as a mapping from Cartesian to polar coordinates in the complex plane, the topology near EPs will be preserved. When the value of G_xy is smaller than GxyEP, the magnitude (phase) of the eigenvalues anti-cross (cross). As G_xy is increased and passes through the EP, the crossing behavior interchanges such that the magnitude (phase) now cross (anti-cross). This type of crossing behavior is unique to non-Hermitian systems and cannot be observed in the double-cone eigenvalue surface of Hermitian systems.

The transmission eigenvalues calculated from simulation and experimental results are shown in Figure 2B. Assuming that the SRR separation distance S is inversely proportional to the coupling strength G_xy, we observe that the magnitude (phase) of the transmission eigenvalues change from anti-crossing (crossing) to crossing (anti-crossing) as coupling strength is increased. Such behavior exactly follows the topology of the self-intersecting Riemann surface and provides strong evidence that an EP is located in the parameter range 5<S^EP<6.

2.3.2 Eigenstate swapping

The eigenvalue surface structure shown schematically in Figure 2A also reveals that the EP is a square-root branch point for the complex eigenvalues. This implies that the eigenvalues of the system will not return to their original values under one full loop around the EP but will rather be swapped. They will return to their original values only under two full loops. For the eigenstates, the EP is in fact a fourth-root branch point, meaning that four full loops are required to return the eigenstates to their initial states [2]. A single loop around the EP will interchange the two eigenstates and additionally accumulate a geometric phase of π on one of the states.

To show the swapping of eigenstates under encirclement, we follow the evolution of the transmission eigenstates by observing closely spaced stationary eigenstates of the metasurface. The eigenstates of the transmission matrix in Eq. (2) are polarization states that can be represented as polarization ellipses with ellipticity angle χ and orientation angle ψ. Polarization states are plotted on a Poincaré sphere for easy visualization.

Simulation and experimental data in Figure 3A show how the polarization eigenstates of the metasurface evolve as the parameters [ω, G_xy(S)] are encircled around the EP. On the Poincaré sphere, we plot the eigenstates of the system found from simulations for SRR separations 5 and 6 μm and frequency range 0.42–0.44 THz. Each parameter gives two eigenstates located on opposite sides of the sphere with respect to the north pole. Following the states along the parameter loop (ω, S)→(0.42, 5)→(0.44, 5)→(0.44, 6)→(0.42, 6)→(0.42, 5), we find that each state moves halfway around the north pole, resulting in a swapping of eigenstates. The polarization ellipses for the eigenstates along the encircling path are explicitly shown. States from experimental data are also plotted for SRR separations 5 and 6 μm and for a frequency range 0.45–0.47 THz. We find that the states from experimental data also result in a swapping of eigenstates under encircling of the EP. Although we should be able to observe an accumulation of geometric phases on one of the states, for our current setup, it cannot be measured. This is because measurements for different coupling values are conducted separately, thus making the phase relation between measurements ill-defined.

Figure 3:

Eigenstate swapping and transmission phase singularity observed by changing coupling between resonator components.

(A) Transmission eigenstates of the metasurface are plotted on a Poincaré sphere as the EP is encircled in the [ω, G_xy(S)] parameter space. Under one full loop around the EP, the states move halfway around the north pole of the Poincaré sphere, resulting in a swapping of eigenstates [i.e. (ψ₁,ψ₂)→(ψ₂,-ψ₁)]. To the right of the Poincaré sphere, we explicitly show the evolution of polarization states. The states in the dashed box show the two eigenstates at the beginning of the loop. Thereafter, we follow how state ψ₁ evolves and find that it becomes ψ₂ at the end of a single loop. (B) The transmission matrix components in the circularly polarized basis are shown when S=6 μm. As we have anticipated from Eq. (5), T_RR and T_LL are identical, whereas T_RL is suppressed with respect to T_LR. The asymmetric transmission of circularly polarized light is a direct consequence of having a single eigenstate at the EP. (C) An abrupt phase flip in T_RL is observed as the parameter S is modulated through the EP.

2.3.3 Asymmetric transmission of circularly polarized light

Due to the existence of a single transmission eigenstate at the EP, we expect that transmission should be asymmetric for right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) states when near the EP, which are associated with an abrupt change in phase. In general, the eigenstates of the system are given by |ψ₁⟩=α|L⟩+β|R⟩ and |ψ₂⟩=α|L⟩–β|R⟩. When the system is near the EP, both eigenstates are close to the |L〉 state. Therefore, we can assume that |α|≫|β|. As T|ψ_i⟩=λ_i|ψ_i⟩, we find the transmission behavior for the RCP and LCP states to be

From Eqs. (5a) and (5b), we can expect that the direct transmission of RCP and LCP are identical. However, for cross-polarization conversion, the conversion from RCP to LCP is larger than the conversion from LCP to RCP by a factor of (α/β)². As we have |α|≫|β| near the EP, we expect to see an asymmetry in the transmission components T_RL and T_LR as the EP is approached.

We verify these predictions through simulation and experimental results for the transmission spectrum of the metasurface. Figure 3B shows that the conversion from LCP to RCP (T_RL) is significantly lower than the conversion from RCP to LCP (T_LR) when the system is near the EP (S=6 μm). Physically, this can be understood from the fact that, near the EP, the amplitude of the RCP state is much smaller than the amplitude of the LCP state. Hence, it is relatively improbable for conversion from a high amplitude state to a low amplitude state to occur compared to conversion in the opposite direction. As the LCP state dominates more as the system approaches the EP, we expect the transmission asymmetry to also get larger. We notice that there is a mismatch of about 0.03 THz in the resonance peak between the simulation and experimental data. This discrepancy most likely stems from a difference in the refractive index of the silicon wafer substrate. Although we do not dismiss the discrepancy as trivial, we would like to point out that it is not a major issue in observing an EP. The shift in resonance frequency will simply result in the EP also being shifted in the parameter space. One can see that as the system approaches the EP, T_RL reaches almost zero and an abrupt jump in the phase of T_RL is observed despite a small change of coupling strength between two SRRs (Figure 3C). The sensitivity in phase dispersion for T_RL originates from the sensitivity of eigenvalues and eigenstates near the EP [15], [16], [17], [18]. From Eq. (5), we find that the phase of T_RL depends on β/α and λ₁ – λ₂. As β/α gives the eigenstate position on the Poincaré sphere and λ₁ – λ₂ is the eigenvalue splitting, we see that sensitivity of the eigenvalues and eigenstates directly translate into sensitivity of the T_RL phase. This rapid phase dispersion can be used for extremely sensitive biochemical detection [24], [25], [26].

2.3.4 Modulation of the incident angle of radiation

Up to this point, we have shown that an EP can be observed in our metasurface design through variation of the incident radiation frequency and separation between SRRs. Although this provides a straightforward and intuitive path to observing the EP, having to repeatedly fabricate new samples to change the SRR separation is a definite drawback. Repeated fabrication will inadvertently add fabrication errors, making it difficult to fine-tune parameters. When parameters are near the EP, errors are further amplified because the eigenvalues exhibit an enhanced sensitivity to perturbations [15], [16]. Moreover, we are also unable to perform dynamic encircling of the EP, which gives rise to phenomena such as time-asymmetric loops around the EP [27].

To overcome these limitations, we modulate the parameter g_y instead of G_xy by controlling the angle of incident radiation. By increasing the incident angle, we leave the in-plane x-component of the field unchanged while weakening the in-plane y-component and hence decreasing g_y. Using this method, we are capable of probing the parameter space with a single metasurface sample, freeing us from the limitations pointed out above.

We now show that the EP is observable in the [ω, g_y(θ)] parameter space. In Figure 4A, the simulation and experimental data for the complex eigenvalues are shown. Just as we have seen by modulating the coupling strength, the magnitude (phase) of the eigenvalues change from crossing (anti-crossing) to anti-crossing (crossing) as the incident angle is modulated through the EP. This implies that the self-intersecting Riemann surface topology is also captured in the [ω, g_y(θ)] parameter space. Figure 4B shows the magnitude and phase of the T_RL transmission matrix component. Once again, as expected from the properties of an EP, we see that the magnitude of T_RL decreases as the EP is approached accompanied with an abrupt phase jump.

Figure 4:

Eigenvalue surface topology and transmission phase singularity demonstrated by varying the incident angle of radiation.

(A) The topology of the self-intersecting Riemann surface is observed in the [ω, g_y(θ)] parameter space. Just as we have observed by varying G_xy, we see that the crossing behavior of the eigenvalue magnitude and phase are interchanged as g_y passes through the EP. From simulation and experimental results, we find that the EP located between θ=41° and 42°. (B) We observe an abrupt phase flip in T_RL as θ is varied through the EP.