1. Introduction

Many natural systems subject to dissipation or decay can be closely modeled by an open system coupled to an external bath with a non-Hermitian Hamiltonian containing non-orthogonal eigenvectors and complex eigenvalues. In such open systems, exceptional points (EPs) are singular degeneracies in the parameter space, where eigenvectors (along with their eigenvalues) coalesce [1, 2]. The existence of EPs has been theoretically and experimentally demonstrated in a great variety of physical systems, and have provided new designs with exotic functionalities [1–6]. Not limited to the level repulsion and crossing, bifurcation, chaos, as well as quantum phase transitions in the systems where EPs naturally occur, counter intuitive phenomena resulting from EPs have been widely observed in synthesized optical systems, e.g. loss-induced transparency, coherent perfect absorption, and unidirectional light reflection [7–21]. Especially, the robust asymmetric mode switching occurs when encircling an EP of the waveguide system [22]. The topological energy transfer between these two eigenmodes can be manipulated due to the presence of the EP [23]. Then the asymmetric mode switching around the EP is theoretically explained in detail [24].Taking EP induced unidirectional reflection as an example, Lin et al. theoretically predicted the coincidence between the unidirectional reflection and EP in a periodic structures guaranteed by PT-symmetry [15]. However this result is not restricted to PT-symmetric optical systems. Feng et al. experimentally demonstrated a similar phenomenon in a fullly passive synthesized structure [18]. Recently, the connection between the Dirac point and EP has been established in a photonic crystal slab without material gain, and the spawning ring of EP out of the Dirac cone has also been experimentally explored by angle-resolved reflection measurements [25]. In situations such as this, each EP in the parameter space accompanies zero reflection and vice versa. Thus, zero reflection provide a potential avenue to explore EP related physics, which is different to the approach used in coupled resonant cavities [4–6].

Among the synthesized optical systems for exploring the physics of EPs, metamaterial made from assemblies of resonant elements is a potential platform. The versatility afforded by a metamaterial is that preselected resonant elements with precise control over their structural parameters can provide predetermined smart optical properties and preset controllable coupling. By exploring the interplay between near-field coupling and resonator loss rate in the hybridized picture, similar mappings to PT symmetric Hamiltonians have been theoretically and experimentally explored including coherent perfect absorption (CPA), perfect polarization conversion (PPC), and maximally asymmetric transmission for circular polarization [26–30]. Unidirectional zero reflection responsible for the occurrence of the EP is also theoretically explored in an ultrathin metamaterial [31]. Besides the zero reflection, anomalous phase delay around the EP is also discussed, which is theoretically and experimentally verified in a synthesized resonator by a quantum weak measurement perspective [32].

Revisiting a hybridized metamaterial we previously investigated [31], here, we theoretically explore the angle-resolved reflection in such a metamaterial against different thickness of the substrate. In contrast to the normal incidence case, we find that the number of EPs along the Γ → X direction for TE mode can be transited from zero to one and then two as we tune the thickness of the substrate, where the occurrence of the EP is indicated by the unidirectional zero reflection in the angle-resolved reflection spectra. We show for the TE mode of the system considered in this paper, the single EP can be continuously deformed into a ring in the incident angle space as we change the substrate thickness. To get a deeper insight into this observed exotic phenomena, a fitted coupled-mode theory (CMT) is employed, which agrees well with the numerical results obtained by finite-difference time-domain simulations. We should note that the EP we discussed here is the EP of the scattering matrix S we defined rather than the EP of the response matrix of the investigated metamaterial. So the perfect unidirectional reflection could occur at the EP when the mteamaterial is carefully designed.

2. Analytical model

To investigate the EPs in question associated with the scattering matrix S, we start by formulating coupled mode theory (CMT) describing coupled bright and dark resonant modes of a representative system exhibiting Fano resonance [33–35]. In such Fano-type model, q⃗ = (q_b, q_d)^T denotes the complex amplitudes for the bright (b) and dark modes (d). The matrices of the system is described by the coupled matrices:

Here δ_μ (μ = b, d) is the frequency detuning for each resonator, ${\gamma }_{\mu }^s$ is the radiative scattering rate, ${\gamma }_{\mu }^d$ is the dissipation rate, and κ is the near-field coupling rate. The dark mode cannot be directly coupled to the incident and output ports, so ${\gamma }_d^s=0$. In contrast to this, the bright mode can be directly coupled to the incident and output ports, so ${\gamma }_b^s\ne 0$. All these parameters are real. a = (a₊, a₋)^T and b = (b₊, b₋)^T indicate the complex amplitudes of the incoming and outgoing light in the positive and negative directions, respectively. K_1,2 describes the coupling between the metasurface and the incident or transmitted waves, while C represents direct reflection. If the investigated structure is lack of mirror symmetry along the propagation direction, γ₁ and γ₂ could be different. The scattering matrix S, b = Sa, then takes the form

It is illuminating to explain why we selected a mirror symmetry broken structure to observe EPs via unidirectional reflection. Suppose the contrary where the investigated structure is mirror symmetric along the propagation direction; i.e. γ₁ = γ₂ and r_L = r_R = r. The result is that the scattering matrix S will have symmetric and antisymmetric eigenvectors (1, ±1)^T, which implies that we could not find an EP. Thus, mirror symmetry breaking is a necessary condition to observe EP in our system, and perfect unidirectional reflection phenomenon (r_L = 0 or r_R = 0) could indicate the presence of EP.

3. Numerical results

The theoretical results we derived using CMT in the previous section is general and can be applicable to various systems complying with our model assumptions. For example, we consider a hybridized metamaterial sysytem, where the unit cell of our investigated metamaterial is shown schematically in Fig. 1. The dielectric substrate is glass with frequency independent refractive index, n_d = 1.47. The permittivity of gold follows the Drude model, ${\epsilon }_{\text{m}}={\epsilon }_{\infty }-f_{\text{p}}^2/\left(f^2+i{\gamma }_{\text{p}}f\right)$, where f_p = 2.18 × 10¹⁵ Hz, γ_p = 1.62 × 10¹² Hz, and ɛ_∞ = 9 [36]. As shown in the Fig. 1, the hybridized metamaterial lacks the mirror symmetry along the propagation direction. Moreover owing to the C₄ rotational symmetry of the hybridized metamaterial along the z direction, this setup cannot sustain any mode conversion between TE and TM modes. The optical response of the investigated metamaterial can be formulated by a (2 × 2)-dimensional scattering matrix for each TE or TM mode, as displayed in Eq. (2). The TE mode (or S mode) described here represents a plane wave that has normal electric field component to the incident plane. Similarly, the TM mode (or P mode) described here represents a plane wave has normal magnetic field component to the incident plane. The response of the investigated metamaterial under TE mode can be described by the theoretical CMT presented above. Although the structure we investigated is the same as the one in our previous investigation [31], there are two significant differences: 1. inclusion of the substrate thickness for investigating angle resolved reflection, 2. consideration of the specific structure of spawned EPs (e.g. ring) against angle resolved spectra.

 

Fig. 1 (a) Schematic of the scattering process for the investigated metamaterial. (b) Schematic of the hybridized metamaterial design. Each unit cell contains three layers. The top layer is a crosshshaped strip antenna with thickness t_m = 20 nm, length a = 150 nm, and width w = 50 nm. The bottom layer is the complementary structure to the top antenna with the same thickness t_m = 20 nm. The middle layer is the glass substrate with thickness t_d. All layers are with a lattice period d = 400 nm.

Download Full Size | PDF

To demonstrate key features of our proposed structure, we carry out finite-difference time-domain simulations which takes into account the incident angle, the substrate thickness, and the frequency. First, we numerically show that the number of the EPs can exhibit a transition at different substrate thicknesses, which can be resolved by finding the zero reflection occurring in one side of the metamaterial. We would like to point out that such a transition is not related to any geometry symmetry breaking process. As illustrated in Fig. 2, we present the reflection and transmission spectra for TE mode illumination along the Γ→ X direction at three thicknesses, t_d = 20 nm, 30 nm, 40 nm. When the substrate thickness t_d is 20 nm, the logarithmic reflection intensities in both sides increase monochromatically as the incident angle increases, where the dip with −1.93 is found at (θ = 0, f = 248.9 THz). There is no zero reflection occurring in any sides of the metamaterial. The transmission clearly shows a Fano type profile at each incident angle, which implies we can understand and describe it by the physics of Fano resonance. At t_d = 30 nm, the logarithmic reflection intensity from left side is almost zero up to −5.15 at (θ = 0, 264.8 THz), and monochromatically increases as the incident angle increases, which implies zero reflection occurring around this position, i.e. an EP. In contrast with t_d = 20 nm, the increase of the thickness leads to the decrease of the near-field coupling between the top and bottom layers, and results in a blue shift of the reflection dip. Especially, at t_d = 40 nm, there are two almost zero dips up to −5.09 at (θ = ±42°, f = 273.2 THz) for the left side reflection, which implies two EPs. The reflection at the right side still shows monochromatic increase as the incident angle increases, which indicates no zero reflection occurring in this side. Without any geometry symmetry breaking of the investigated metamaterial, the number of the EPs cam be dramatically changed as we increase the thickness of the substrate. To further indicate the appearance of EP, we plot the phase spectra of the reflection coefficients at t_d = 40 nm in Fig. 3. Two topological phase change can be seen around (θ = ±42°, f = 273.2 THz) for r_L, which is an alternative way to indicate the existence of an EP. The obvious unidirectional reflection in our system leads to different absorption when excited by the left and right sides, as shown in Fig. 3 (c) and (d). As we change the thickness of the substrate, the positions of the EPs do change. For example, at the thickness of t_d = 35 nm, the appearance of the EPs are around (θ = ±30°, f = 269.1 THz), while at the thickness of t_d = 55 nm, we find the appearance of the EPs are around (θ = ±56°, f = 279.4 THz).

 

Fig. 2 Numerical reflection and transmission for the investigated metamaterial at different thickness t_d = 20, 30, 40 nm along the Γ → X direction. A logarithmic intensity expression is developed to describe the appearance of EP, where 0 implies the unit intensity of the reflection or transmission and −∞ indicates the zero intensity of the reflection or transmission. The first row shows reflection from left side log₁₀ |r_L|², the second row indicates the reflection from right side log₁₀ |r_R|², and the third row illustrates the transmission log₁₀ |t|².

Download Full Size | PDF

 

Fig. 3 Numerical reflection phase and absorption for the investigated metamaterial with the thickness t_d = 40 nm. The first row shows reflection phase from left and right sides, the second row indicates the absorption of the system when excited from left and right sides.

Download Full Size | PDF

According to the equations from CMT, we could fit the angle-resolved numerical reflections and transmissions along the Γ → X and Γ → M directions at three different substrate thicknesses: t_d = 20 nm, t_d = 30 nm, and t_d = 40 nm. These microscopic parameters are complicated functions of the incident angle, leading to macroscopic complex reflection and transmission coefficients. Consider the symmetry of the unit cell, we fit all these parameters with the “cos“ function as we change the incident angle. When we change the thickness of the substrate, the near field coupling strength is also changed. This means thicker substrates lead to weaker near field coupling and smaller frequency splitting between the bright and dark resonant modes. The weaker coupling also leads to the frequency blue shift of the minimum reflection according to the Eq. 6. Since the occurrence of the EP is manily determined by the reflection coefficients from both sides, the numerical and fitted angle-resolved log₁₀ |r_L/r_R| are plotted in Fig. 4. Table 1 shows the detailed fitted function for each microscopic parameter along the Γ → X direction. Along the Γ → M direction, only the microscopic parameter f_d is changed, and the other parameters are the same as the parameters along the Γ → X direction for each substrate thickness. At t_d = 20 nm, f_d is changed to be f_d = 284.3 + 11.5(cos θ − 1). At t_d = 30 nm, f_d is changed to be f_d = 301.5 + 13.5(cos θ − 1). At t_d = 40 nm, f_d is changed to be f_d = 314.6 + 21(cos θ − 1). To coincide with numerical configuration to obtain the reflection and transmission, r_L,R and t need to multiply by a accumulated phase factor $e^{-i{k}_0{l}_{t_d}^{L,R,T}\mathit{cos}\theta }$, where k₀ is the wave vector in vacuum, and the accumulated lengths at different thickness are $l_{t_{20}}^L=3060\hspace{0.17em}\text{nm}$, $l_{t_{20}}^R=3030\hspace{0.17em}\text{nm}$, $l_{t_{20}}^T=3080\hspace{0.17em}\text{nm}$, $l_{t_{30}}^L=3080\hspace{0.17em}\text{nm}$, $l_{t_{30}}^R=3040\hspace{0.17em}\text{nm}$, $l_{t_{30}}^T=3090\hspace{0.17em}\text{nm}$, $l_{t_{40}}^L=3100\hspace{0.17em}\text{nm}$, $l_{t_{40}}^R=3040\hspace{0.17em}\text{nm}$, and $l_{t_{40}}^T=3100\hspace{0.17em}\text{nm}$. The superscripts L,R and T indicate left side, right side, and transmission, respectively. The fitted spectra agree well with the numerical spectra. Not only the spectra, these EP positions predicted by the fitted CMT find agreement with the numerical results. At t_d = 20 nm, no EP occurs at the parameter position. At t_d = 30 nm, single EP occurs at the parameter position, (θ = 0°, f = 264.8 THz), both along the Γ → X and Γ → M directions. The single EP appearance implies only one EP at (θ = 0°, f = 264.8 THz) in the parameters space including incident angle and frequency. At t_d = 40 nm, the EP positions occur in different parameters positions, (θ = 42°, f = 273.2 THz) along the Γ → X direction, while (θ = 39°, f = 270.7 THz) along the Γ → M direction. To clearly show the EP positions in the parameter space, we numerically show the EP positions as we change the incident azimuthal angle, as shown in Fig. 5. The slight fluctuation (3°, 2.7 THz) may be due to the finite cell size of the investigated metamaterials, and an approximate EP ring can be generated. In the parameters space including incident angle and frequency, the EP can be deformed into an exceptional ring structure with slight fluctuation. By only tuning the thickness of the substrate, the hybridized metamaterials show a spontaneous transition from single EP to an exceptional ring, free of any geometry breaking of the investigated metamaterials.

 

Fig. 4 Numerical(the first row) and fitted(the second row) angle-resolved log₁₀ |r_L/r_R| along the Γ → X and Γ → M directions at different thickness t_d = 20, 30, 40 nm. A logarithmic expression is developed to describe the appearance of EP, where 0 implies the unit ratio of |r_L/r_R| and −∞ indicates the zero amplitude of the reflection r_L.

Download Full Size | PDF

Table 1. Fitted microscopic parameters along the Γ → X direction

View Table

 

Fig. 5 The EP position at t_d = 40 nm for different incident azimuthal angle φ. (a) the incident polar angle θ, (b) frequency.

Download Full Size | PDF

To further illustrate the complex evolution around each EP, we present the evolution of $\pm \sqrt{r_L/r_R}$ via the incident polar angle at different frequencies $f_{t_d}^d$ for each substrate thickness in Fig. 6, including $f_{20}^d=248.9\hspace{0.17em}\text{THz}$, $f_{30}^d=264.8\hspace{0.17em}\text{THz}$, and $f_{40}^d=273.2\hspace{0.17em}\text{THz}$ for substrate thicknesses of 20 nm, 30 nm, and 40 nm, respectively. The symbol selection of ± presented here is consistent with the symbol indication of the scattering matrix S eigenvalues. At t_d = 20 nm, the obvious splitting exists. At t_d = 30 nm, accidental degeneracy occurs at Γ point, and almost linear evolution exhibits around Γ point. Meanwhile, the imaginary part is very small, and close to zero around Γ point. However, at t_d = 40 nm, both real and imaginary parts are degenerated to zero around θ = 42° along Γ → X direction. Bellow θ = 42°, splitting mainly occurs in the imaginary part, while the real parts are close to zero. Above θ = 42°, splitting mainly occurs in the real part, while the imaginary parts are very small. There is no degenerated point along Γ → M direction, implying no EP at this frequency, which is consistent with the previous results.

 

Fig. 6 The real (Wine color) and imaginary (Royal color) parts of $\pm \sqrt{r_L/r_R}$ along the → X and Γ → M directions at different thickness t_d = 20 nm,30 nm,40 nm.

Download Full Size | PDF

4. Conclusion

In conclusion, we have theoretically shown that EPs indicated by zero reflection can be realized in a mirror symmetry broken resonant metamaterial. The number of EPs can be manipulated by tuning the thickness of the substrate, where single EP point can be deformed into an ring of EPs. Such exotic phenomena can be well explained and reconstructed by our presented CMT model.