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Abstract
Exceptional points (EPs) have been shown to be useful in bringing about sensitive optical properties based on non-Hermitian physics. For example, they have been applied in plasmonics to realize nano-sensing with extreme sensitivity. While the exceptional points are conventionally constructed by considering parity-time symmetric or anti-parity-time symmetric media, we theoretically demonstrate the possibility of generating a series of non-Hermitian systems by transforming a seed system with conventional parity-time symmetry within the transformation optics framework. The transformed systems do not possess PT-symmetry with a conventional parity operator after a spatial operation, i.e. hidden from conventional sense, but are equipped with exceptional points and phase transitions, hinting an alternative method to design non-Hermitian plasmonic systems with sensitive spectra or eigenmodes.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Non-Hermitian optical systems have attracted a lot of attention due to their ability to generate sensitive optical properties around the so-called exceptional points [1–5]. A range of exotic wave phenomena have been demonstrated, ranging from unidirectional invisibility [6], spontaneous PT-symmetry breaking [7,8], coherent perfect absorption and lasing [9–11], etc. In particular, the sensitive optical properties around exceptional points, also an associated phase transition, can be very useful for sensing [12–14], mode discrimination (e.g. for lasing) [15–17] and isolation (non-reciprocal transmission) [18–20]. Recently, exceptional points have been successfully demonstrated in the optical regime, allowing the realization of exceptional point-based devices, such as ultrasensitive plasmonic sensors [21,22], and vortex beam generators [23]. In this case, the gain and loss are not merely compensating each other (e.g., active plasmonic systems [24,25]). The coupling and the gain-loss contrast between two different spatial domains interplay with each other and induce an exceptional point, in which both the eigenvalues and eigenvectors coalesce.
It is not trivial to guarantee the appearance of such an exceptional point for a generally non-Hermitian system [26]. One possible way is to impose a symmetry between two spatial domains through Parity-time (PT) operation on the material parameters [27–29]. The exceptional point can then be swept across configurationally by varying a single parameter (the ratio between coupling and gain-loss contrast). There are alternative routes, such as employing bianisotropy [30,31], considering anti-PT symmetry [32–38] and pseudo-Hermiticity [39,40], to construct non-Hermitian exceptional points. It is worth to mention that employing anti-PT symmetry to construct EPs is a relatively recent approach. EPs in these anti-PT systems have been found to enhance Sagnac effect in ring laser gyroscopes [34,35], in coupled waveguides with lossy buffers for on chip applications [36] and in side-coupled cavities for robust sensing [37] or electromagnetically induced transparency [38]. The conventional P-symmetry can therefore be replaced by other kinds of symmetries. This work, in the same spirit of this route, investigates how transformation optics (TO) can help generate exceptional points by coordinate-transforming a known PT-symmetric system. We apply TO on a conventional PT-symmetric system and generate different daughter systems. Since the coordinate transformation does not preserve the original PT symmetry, the transformed daughter systems will not possess PT-symmetry with a conventionally defined parity operator, but are still equipped with exceptional points and phase transitions. In other words, the P-operation is effectively coordinate transformed while the T-operation stays the same. This approach is different from a previous scheme that employs a complex coordinate transformation to obtain a PT-symmetric system without phase transition from a lossless system [41,42]. TO has been previously used to design plasmonic systems with hidden symmetries, evolving from the earlier applications on designing invisibility cloaks to specific optical components [43–47]. For example, a broadband resonance can be hidden in a finite plasmonic structure with a singular geometry, with applications in nanofocusing, biosensing, energy harvesting, electron-energy loss spectroscopy, and strong coupling [48–57]. A third spatial dimension can be compacted and hidden into a two-dimensional plasmonic structure [58]. In the current case, the PT-symmetry is effectively hidden by the coordinate transformation without a conventionally defined P-operator (inversion or mirror) as a result of the TO approach.
2. Transformation optics applied on a PT-symmetric system
To illustrate the idea of TO in generating non-Hermitian systems with exceptional points, we start from a seed PT-symmetric plasmonic metal-insulator-metal (MIM) structure [59]. Figure 1(a) shows the MIM structure in complex coordinate $w = u + iv$. It comprises a thin slab of insulator of thickness d with permittivity ${\epsilon _0} = 1$ (white, chosen as air) sandwiched by a lossy semi-infinite metal slab (blue) with permittivity ${\epsilon _L} = \epsilon + i\gamma $ ($\epsilon < 0$, $\gamma \ge 0$) for $u < - d/2$ and a semi-infinite gain material (orange) with permittivity${\; }{\epsilon _R} = \epsilon - i\gamma $ for $u > d/2$. The system is parity-time symmetric, with parity being the mirror operation in u-direction, satisfying ${\epsilon _L} = \epsilon _R^\ast $ [60]. Now, we apply an exponential conformal map [61]
which transforms the MIM system in coordinate w to concentric shells in coordinate $z = x + iy$, as shown in Fig. 1(b). We note that complex coordinates w and z are introduced as a compact notation for u, v and x, y which are the actual Cartesian coordinates, for simplicity in expressing conformal maps.
Fig. 1. Coordinate transformation from a mother metal-insulator-metal (MIM) structure in complex coordinate w (a) to concentric shells in complex coordinate z (b). In (a) the MIM structure is a well-defined PT-symmetric system. The blue (Orange) slab has a permittivity $\epsilon + i\gamma $ ($\epsilon - i\gamma $) in loss (gain), and the white area represents an insulator slab with unit permittivity. In (b), the daughter system consists of the same materials (represented with the same colours) in cylindrical shells.
According to TO [43,48], the permittivity and permeability of the transformed structure $\varepsilon ^{\prime},\mu ^{\prime}$are given by $\varepsilon ^{\prime} = \mathrm{\Lambda \varepsilon }{\mathrm{\Lambda }^\textrm{T}}/({\det \mathrm{\Lambda }} ),\mu ^{\prime} = \mathrm{\Lambda }\mathrm{\mu }{\mathrm{\Lambda }^\textrm{T}}/({\det \mathrm{\Lambda }} )$, where $\mathrm{\Lambda }$ is the Jacobian transformation matrix:
A local plasmon mode on the $w$-plane, is the so-called gap plasmon [59,63] with transverse-magnetic (TM) polarization, satisfying in the near-field limit:
Fig. 2. Local plasmon condition in terms of $\epsilon $ and $\gamma $ (the real and imaginary parts of the permittivities) for the MIM system in $w$-plane (a) and for the cylindrical shells in the z-plane (b). The colour map shows the average of ${|{{E_x}} |^2}$ over the metal/dielectric interface (in log scale). The red colour shows the locations with highest E-field intensity, i.e. the local plasmon condition. The dashed line in both (a) and (b) shows the local plasmon condition by solving $\epsilon $ from Eq. (2) with varying $\gamma .$ $m = 1$ (the larger parabola) and $m = 2$ (the smaller parabola) are the two plasmon conditions in the near-field limit. The geometric parameters are $d = 20.0$ nm, ${R_1} = 3.3$ nm, and ${R_2} = 13.3$ nm. The conformal map in Eq. (1) has parameters $g = 6.6$ nm and $H = 90.0$ nm. (c) gap plasmon mode (showing only ${E_x}$ distribution) at $\gamma = 0.02,\epsilon ={-} 0.6$ in the w plane and its coordinate transformation result (d) (showing only in ${E_x}$ distribution). (e) ${E_x}$ distribution obtained using a dipole excitation directly in the cylindrical z plane at $\gamma = 0.02,\epsilon ={-} 0.6$. (f) gap plasmon mode (showing only ${E_x}$ distribution) at $\gamma = 0.02,\epsilon ={-} 1.73$ in the w plane and its coordinate transformation result (g) (showing only in ${E_x}$ distribution). (h) ${E_x}$ distribution obtained using a dipole excitation directly in the cylindrical z plane at $\gamma = 0.02,\epsilon ={-} 1.73$.
Here, we set $d = 20.0$ nm, ${R_1} = 3.3$ nm, and ${R_2} = 13.3$ nm as the geometric parameters of the current example. With ${\epsilon _L} = \epsilon + i\gamma $ and ${\; }{\epsilon _R} = \epsilon - i\gamma $, we can obtain the local plasmon dispersion (over $\epsilon $ and $\gamma $, as shown in Fig. 2(a)) for the MIM system using full-wave simulations (COMSOL Multiphysics). In the simulation, the system is excited by an array of dipoles pointing in the u-direction at a wavelength of 600 nm, periodically placed in the v-direction with period H. The colour profile plots the averaged field intensity over the metal/dielectric surfaces (where the local plasmon modes localize) in log scale with red colour (high intensity) indicates the local plasmon mode dispersion obtained numerically. The two $m = 1$ modes start with $\epsilon ={-} 1.73$ and $\epsilon ={-} 0.6$ at $\gamma = 0$. Then the splitting between the two modes becomes smaller when $\gamma $ increases until a certain point at $\gamma \cong 0.55$ at which the two $m = 1$ modes become degenerate. The $m = 2$ modes show up as a smaller “parabola,” while the $m = 3$ modes appear as an even smaller one around the point $\epsilon ={-} 1$ and $\gamma = 0$ (the higher-order modes are not clearly resolved around the same point). In Fig. 2(b), the dispersion diagram is solved by full-wave simulations in the x-y plane (Fig. 1(b)) using a dipole excitation pointing along the x-direction at $({g,0} )$ [48]. The two dispersions exhibit a similar permittivity dependence, confirming the equivalence between the two systems for the mode spectra. It should be noted that different points in the color map represent different systems with fixed wavelength excitation in the full wave simulation in principle. However, one can substitute a frequency dispersion, e.g., the Drude model of permittivity into Eq. (2) and solve for ${k_0}$ to get distinct modes of a single system at any specific $\gamma $ (see later Fig. 3). In order to have an intuitive sense of the equivalence, Fig. 2(c) and (f) shows the ${E_x}$-field profile for the gap plasmon mode $\gamma = 0.02,\; \; \epsilon ={-} 0.6$(odd mode for ${E_x}$) and $\gamma = 0.02,\; \; \epsilon ={-} 1.73\; $(even mode${\; }$for${\; }{E_x})$ in the w plane and the field fits one wavelength in a period H in the vertical direction. We transform this field to the z plane by the coordinate transformation according to TO [50] and the TO-transformed field (again only showing ${E_x}$ profile) is plot in Fig. 2(d) and (g). The transformed field shows a very similar quality to the result in Fig. 2(e) and (h), which is directly excited in the cylindrical z plane. We note that the equivalence is not exact as there are approximations here. The permeability $\mu $ is kept at value 1 for the whole domain in both w-plane and z-plane in the near-field limit. For TO theory, the permeability $\mu $ becomes inhomogeneous in the transformed $z$-plane, as discussed above. The near-field approximation becomes more accurate only when the wavelength is much larger than the sizes of the cylinders [64]. In the present case, the wavelength is sufficiently large to ensure the equivalence between the two domains. On the other hand, the retardation effect shows up by plotting the analytic mode dispersions obtained from Eq. (2) (solving for $\epsilon $ by varying $\gamma $). The analytic mode dispersions are plotted in the two panels of Fig. 2 as dashed lines for both $m = 1$ and $m = 2$ modes. The size of the $m = 1$ “parabola” has a small mismatch with the colour map in Fig. 2(a) and (b) due to the retardation effect. Nonetheless, we can conclude that the two systems have very similar mode dispersions (confirmed by the full-wave simulations). The system in the $z$-plane, although lacking a well-defined PT-symmetry, has an origin from the system in the $w$-plane, which is purposely designed to be conventional PT-symmetric. In the dispersion relation of $m = 1$, when the two branches become degenerate at around $\gamma \cong 0.55$, such a non-Hermitian system exhibits an exceptional point, which will be investigated in the next section. We note that the mode in the broken phase (with $\gamma $ beyond the exceptional point) is not showing up in Fig. 2 as the full-wave simulations (color map) are performed with a point source excitation with only a real value of $\epsilon $, while the mode in the broken phase has an imaginary $\epsilon $ in solving the mode equation.
Fig. 3. Exceptional point dynamics. The complex wavenumber ${k_0}$ satisfying the mode equation with retardation effect is plotted against the gain-loss contrast $\gamma $ (according to Drude model (Eq. (9)) in (a) $Re({{k_0}} )$ against $\gamma $ and in (b) trajectory of complex ${k_0}$ when $\gamma $ increases from zero to $5.5 \times {10^6}{\textrm{m}^{ - 1}}$. The solid black lines are the theoretical results by plugging in Drude model into Eq. (2) without considering retardation effect. The sizes of the cylinders are scaled down by 50% with parameters listed in the caption of Fig. 2.
3. Hidden PT-symmetry with exceptional point dynamics
The above discussion assumes the near-field limit and solves for $\epsilon $ from the mode equation (Eq. (2)) in the w-plane by substituting ${\epsilon _L} = \epsilon + i\gamma $ and ${\epsilon _R} = \epsilon - i\gamma $ for the analytic solution. The same mode equation can be mathematically recast as an eigenvalue problem for the MIM structure:
After making the PT-symmetry origin clear in both materials profile (PT-symmetric in w-plane) and the abstract mathematical equivalence (Eq. (5)), we now incorporate the retardation effect. The consideration of the retardation effect is necessary if we want to solve for eigen-frequency (equivalently finite wavenumber ${k_0} = \omega /c$) directly and observe the dynamics of exceptional point in the complex frequency plane. By adding $\delta A$, a function of ${k_0}$, ${\epsilon _L}$ and ${\epsilon _R}$ and can be seen as a perturbation term for a small ${k_0}$, the mode equation is then the solution of $\det ({A + \delta A} )= 0.$ The mode equation ($m = 1$) with retardation effect can be written as
We note that Eq. (8) (derived from the perturbation theory) is only quantitatively accurate for small frequencies, or equivalently at small sizes of the cylinders compared to the free-space wavelength. As an example, we scale down the physical sizes of the cylinders by half. The ${\epsilon _L}$ and $\epsilon _L^\ast $ in Eqs. (6) and (8) are now replaced by a frequency-dispersive Drude model:
4. Generating other hidden PT-symmetric systems
From the above discussion, TO serves as a guidance in obtaining non-Hermitian systems that can capture exceptional points, with an origin from a conventional PT-symmetric system. In fact, we can adopt other maps from the virtual space (w-plane) to obtain and generate other non-Hermitian systems with hidden PT-symmetries. In the current case, we further cascade a Mobius map $z = g({a + w} )/({b - w} )$ so that the metal cylinder with gain can be shifted from its current position (shown in Fig. 4(a)). There is a shift in distance d from the center of the lossy shell in the z-plane. As an example, we set ${R_1} = 3.1\; \textrm{nm}$, ${R_2} = 14.6\; \textrm{nm}$ and $d = 5.5\; \textrm{nm}$. The resultant dispersion diagram (full-wave simulations shown as the colour map) is shown in Fig. 4(b). For the concentric structure, the dispersion relation with retardation effect in Fig. 2(b) is plotted as the dashed lines for the fundamental and second modes. It is found to have a good agreement with the colour map. Therefore, the non-concentric system shown in Fig. 4(a) is also having a hidden PT-symmetry.
Fig. 4. (a). A mobius transformation $z = g({\; a + w} )({b - w} )$ converts concentric shell structure with loss (blue region $\epsilon + i\gamma $) and gain (orange region $\epsilon - i\gamma $) into a non-concentric structure with lossy cylinder shell enclosing a gain cylinder. The mapping has parameters $a = 6.6\; \textrm{nm},\; b = 33.4\; \textrm{nm},$ $g = 25.8\; \textrm{nm}$. The geometric parameters in the $w$-plane are the same as those in Fig. 1(b). The radius of the gain cylinder and lossy shell in the z plane are ${R_1} = 3.1\; \textrm{nm}$, ${R_2} = 14.6\; \textrm{nm}$. The centre of the gain cylinder is displaced by $d = 5.5\; \textrm{nm}$ from the centre of the lossy shell. (b) Local plasmon dispersions of a non-concentric structure in terms of $\epsilon\;\textrm{and}\;\gamma $ (the real and imaginary parts of the permittivities). The dashed lines are simulated fundamental and second modes of concentric structure with retardation effect in Fig. 2(b). The colour map shows the average of ${|{{E_x}} |^2}$ in logscale over the metal/dielectric interface.
5. Conclusion
We have established a generic approach to construct various non-Hermitian systems with exceptional points through coordinate transformations. We have applied this approach to transform a conventional PT-symmetric system, a plasmonic MIM structure, to generate systems with hidden PT-symmetry. The same approach can also be equivalently applied to systems (other than the MIM structure) with other origins of exceptional points, such as anti-PT symmetric or bianisotropic systems. The investigations will be useful for exceptional point-based sensing, lasing, and generally for non-Hermitian plasmonics.
Funding
Research Grants Council, University Grants Committee (C6013-18G); Gordon and Betty Moore Foundation.
Acknowledgments
J. B. P. acknowledges funding from the Gordon and Betty Moore Foundation.
Disclosures
The authors declare no conflicts of interest.
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