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Exceptional point-based plasmonic metasurfaces for vortex beam generation

Ho Ming Leung, Wensheng Gao, Ranran Zhang, Qiuling Zhao, Xia Wang, C. T. Chan, Jensen Li, and Wing Yim Tam

Open Access Open Access

Abstract

An exceptional point occurring in a tailor-made lossy optical system has been recently found to alter optical properties in counter-intuitive ways. In the context of tunable plasmonic devices, exceptional points can be useful as a driving mechanism to enhance tunability. Here, we experimentally demonstrate how a plasmonic exceptional point can be incorporated in metasurface Q-plates to have the generated vortex beam tuned through a change of structural parameter. We have observed an orbital rotation in the far-field by 45 degrees in crossing the exceptional point. We expect a new generation of tunable plasmonic devices in polarization control, beam structuring and holograms, which can take advantage of the huge sensitivity from exceptional points.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Plasmonic metasurfaces, with only a thin layer of metamaterial atoms, serve as an attractive platform to develop optical devices based on their potential compact sizes and high efficiency. These include various optical components for focusing, polarization control, polarization sensing, beam structuring and generating high-resolution holograms [1–7]. From an information perspective, the subwavelength metamaterial atoms, with varying structures or orientations, store all the complexity of the device function so that a single layer of metamaterial atoms is sufficient in generating the designed effects without the need of cascading different optical components. These developments are currently leading to a revolution in downsizing optical components, ranging from lenses and waveplates in an optics laboratory to zoom lenses, 3D displays for the next generation of mobile and wearable optics [8,9].

To expand the capability of plasmonic metasurfaces, an immediate way is to spatially multiplex individual sets of metamaterial atoms onto a single metasurface to obtain different functions at separate wavelengths or polarizations, such as a series of distinct focal points, beam directions or holograms [10–14]. In this case, the tunability resides on the change of incident beam. However, due to the limited information that can be stored on a fixed area of a plasmonic metasurface, the addition of functionality unavoidably has a compromise on the resolution or accuracy in the wavefront control. In fact, a more direct way is to seek for dynamic tunability. For example, an active material, e.g. VO2, can be added on a metasurface. A refractive index change, induced by a temperature change, a pumping light or a change of applied voltage, is able to shift the resonating frequency response of the metasurface in a dynamic manner [15–18]. In this case, it is favorable to design metasurfaces to enable a large change of optical properties from only a tiny index change of the active material.

Recently, it is found that material gain and loss in optical systems, termed as non-Hermitian systems, adds a new dimension in manipulating light. In particular, when the system is driven towards a so-called exceptional point in which the response matrix becomes defective, the optical properties become very sensitive to a small change of the geometric parameters or the gain-loss contrast [19,20]. This has been leading to a range of counter-intuitive wave phenomena, such as laser-absorber, unidirectional zero reflection and single laser-mode selection [21–26]. The plasmonic metasurfaces, in our current context, may thus be highly tunable, subject to tiny change of geometric parameters if we can capture an exceptional point into their designs. In the case of metasurfaces simply constructed by a periodic set of metamaterial atoms, an exceptional point can be readily incorporated into the Jones matrix by appropriate designs. A detailed analysis from the transmission coefficients reveal the eigenvector degeneracy at the exceptional point in the Terahertz regime [27]. In this work, we demonstrate metasurfaces equipped with a varying profile of metamaterial orientations, termed as Q-plates, is able to reveal a plasmonic exceptional point through beam structuring as a directly observable phenomenon. The far-field orbitals observed by configurationally sweeping the exceptional point-enabled Q-plates show a rotation when crossing an exceptional point. Interestingly, there is a current thrust to be the first to experimentally demonstrate a plasmonic exceptional point using either waveguides or symmetry broken structures [28,29]. In Ref. [28], sensitivity in the exceptional point is translated to a large change of resonating frequency shift in sensing application. Here, the sensitivity of the exceptional point is translated to a nonlinear rotation in sweeping through plasmonic particle separation. Our work, with a demonstration of vortex beam generation, can enable a new generation of exceptional point-driven plasmonic devices.

2. Metasurface capturing an exceptional point

Figure 1(a) shows our basic unit cell of a 3-layer structure, to construct the plasmonic Q-plate. The top layer of Au has rectangular slots in orthogonal directions while the bottom layer has Au bars in the same shape, forming a Babinet complementary arrangement [30,31]. In between the two metal layers there is a dielectric spacer of PMMA. Such an arrangement eases fabrication as it does not require a lift-off process after e-beam lithography and deposition of metal while metallic structures are already embedded in both layers to enhance plasmonic resonances [32,33]. The two orthogonal slots/bars have their dimensions defined in Fig. 1(b) and its caption. They are chosen to have nearly the same shape, bar like, to respect an approximated mirror symmetry at 45 degrees while small differences in their length ${l_1}$ and ${l_2}$, and also the width w1 and w2 are to control the loss contrast between the two slots/bars. This respects the so-called passive Parity-time (PT) symmetry: a typical way to construct non-Hermitian systems in order to capture an exceptional point (for one unit cell in our case) [27,34].

 figure: Fig. 1.

Fig. 1. (a) and (b) Schematic diagram of the unit cell, highlighted by black dotted line, on a glass substrate. Thicknesses of gold, PMMA, and ITO layers are 45 nm, 180 nm, and 65 nm, respectively. s denotes the separation between the two orthogonal slots/bars with lengths ${l_1} = $ 140 nm, ${l_2} = 170$ nm and widths ${w_1} = $ 98 nm, ${w_2} = $ 110 nm. (c) SEM image of a plasmonic Q-plate constructed by these unit cells with s = 200 nm. (d) and (e) show the simulated and measured transmittance in LP basis respectively if the unit cells are arranged into a square lattice of periodicity 450 nm and separation distance $s$ = 320 nm. (f) and (g) show the corresponding measured transmittance in CP basis. (h) and (i) show the corresponding results of transmittance in CP basis when the separation distance is further reduced to $s$ = 180 nm. Insets (e) and (i) show the SEM images of the samples with white color bar denotes a length of 200 nm.

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When the same set of unit cells are assembled with orientations aligned to the radial direction on a metasurface, it forms a plasmonic Q-plate with unit topological charge, being capable to have spin-induced coupling to the orbital angular momentum for an incident beam [4,35]. Figure 1(c) shows the SEM image of such a Q-plate, in which the unit cells are placed periodically every 450 nm in the radial direction and keep a similar periodicity in the azimuthal direction. The plasmonic Q-plate has a non-patterned center (7.8 µm diameter), small comparing to the full size of the metasurface (200 µm diameter). To explore the condition in order to capture a plasmonic exceptional point, we fabricate additional metasurfaces with square lattices of the unit cells of the same periodicity. Figures 1(d) and 1(e) show the simulated (by CST Studio SuiteTM) and measured transmittance across such a plasmonic metasurface for the case with the separation distance between the two orthogonal bars set at $s = 320$ nm. The difference in the co-polarization transmittance in the linear polarization (LP) basis (polarization along either one of the two slots/bars) comes from the small anisotropy and loss contrast between the two slots/bars. It results in a small but non-zero cross-polarization transmittance ${T_{ +{-} }}$ and ${T_{ -{+} }}$ in the circular polarization (CP) basis, being shown in Figs.  1(f) and 1(g) for the simulation and measured transmittance. The first (second) subscript denotes the polarization of the outgoing (incoming) beam. Right-handed (left-handed) CP is represented by “+” (“-”). In the case of large separation $\; s$, the coupling between the two bars/slots is negligible and we have equal magnitude for ${T_{ +{-} }}$ and ${T_{ -{+} }}$. (Here, we use capital T to represent transmittance while small t to represent the corresponding complex transmission coefficient for the metamaterial unit cell, e.g. ${T_{ +{-} }} = {|{{t_{ +{-} }}} |^2}$.) On the contrary, when the separation distance s is reduced, these two cross-polarization transmittances deviate from each other due to the coupling between the two bars/slots. Figure 1(h) shows the simulated results for a special case when $s$ = 180 nm. Spectrum ${T_{ +{-} }}$ stays much lower than ${T_{ -{+} }}$. From the simulation results, ${T_{ +{-} }}$ stays nearly zero from around 600 nm to 700 nm in wavelength. In this regime, we regard the metasurface is approaching an exceptional point: an eigenvalue and eigenvector degeneracy of the Jones matrix at ${T_{ +{-} }} \cong 0$ (see Ref. [27] for details). The measured transmittance in Fig. 1(h) shows similar behavior.

3. Constructing exceptional point-based plasmonic Q-plates

After ensuring a plasmonic exceptional point can occur in our design phase space, we fabricate a series of metasurface Q-plates using these unit cells with varying s for 10 different samples. Each unit cell on one sample has the same structure except its orientation is aligned to the radial direction. When a Gaussian beam shines normally on such a plasmonic Q-plate with unit topological charge, the intensity/orbital profile $I({\rho ,\phi } )$ on a vertical screen at a distance $f\; $ from the metasurface (Fraunhofer diffraction) follows

$$I \propto {|{{F_{2q}}[{a(\rho )} ]({e^{i4q\phi }}{t_{ -{+} \; }} - {t_{ +{-} }}) + {F_0}[{a(\rho )} ]{e^{i2q\phi }}\; ({{t_{ +{+} }} - {t_{ -{-} }}} )} |^2}$$
where $({\rho ,\phi } )$ is the cylindrical coordinate on the screen and q is the topological charge of the Q-plate ($q = 1$ for the current case). The auxiliary function ${F_m}[{\cdot} ]$ is responsible for integrating the contribution from each structure on the plasmonic metasurface with aperture function $a(\rho )$ and is governed by
$${F_m}[{a(\rho )} ]= \frac{1}{{2\pi }}\mathop \smallint \nolimits_0^\infty \mathop \smallint \nolimits_0^{2\pi } {e^{ - \frac{{ik\rho ^{\prime}\rho }}{{{f^2}}}\cos \phi ^{\prime}}}{e^{im\phi ^{\prime}}}a({\rho^{\prime}} )d\phi ^{\prime}\rho ^{\prime}d\rho ^{\prime}.$$
The first term inside $|\cdot |^2\; $ in Eq. (1) dominates and is responsible for a four-fold rotational symmetry of the orbital while the second term, being responsible for a two-fold rotational symmetry, is much smaller as ${t_{ +{+} }} \cong {t_{ -{-} }}\; $ in our case. The change of the rotational symmetry of the orbital from a Gaussian beam is due to the fact that the Q-plate couples the spin-angular momentum of the incident beam into the orbital angular momentum of the transmitted beam [4]. Furthermore, when either ${t_{ -{+} }}$ or ${t_{ +{-} }}$ has amplitude approaching zero at an exceptional point, the phase of it will change sharply. Suppose the separation s is varied as an external parameter in crossing an exceptional point. We can approximate ${t_{ +{-} }}/{t_{ -{+} }} \cong {\tau _0}({s - {s_{ex}} - i\delta } )$ and ${s_{ex}}$ is where the exceptional point occurs. ${\tau _0}$ is a complex constant slope and $\delta $ is a tiny real number, ideally zero and indicates how accurate we can approach an exceptional point. When s sweeps across ${s_{ex}}$, the phase changes sharply from $\arg ({ - {\tau_0}} )$ to $\arg ({{\tau_0}} )$ by a $\pi $ phase change. As Eq. (1) (by neglecting the second term) indicates the orbital intensity profile, the phase change thus translates itself into a change of orbital profile I from ${({\cos 4\phi + Re({{\tau_0}} )|{s - {s_{ex}}} |} )^2} + {({\sin 4\phi + Im({{\tau_0}} )|{s - {s_{ex}}} |} )^2}$ to ${({\cos 4\phi - Re({{\tau_0}} )|{s - {s_{ex}}} |} )^2} + {({\sin 4\phi - Im({{\tau_0}} )|{s - {s_{ex}}} |} )^2}$ from small to large s. We note that the change of the sign can be compensated by replacing $\phi \to \phi + \pi /4$. As the transition is continuous when we vary s, the orbital profile thus appears to be rotated by 45 degrees. A nonzero $\delta $ comes from fabrication deviation from the design while the magnitude of $\delta $ controls how sharply the phase change occurs. In our case, 2$|\delta |\cong 40\; \textrm{nm}$ is approximately the transition distance needed in s that the phase changes from one to another phase, or the orbital rotation occurs. Figure 2 shows the measured far-field diffraction pattern (at a wavelength of 600 nm) when s is varied from 140 nm to 320 nm, in steps of 20 nm using the 10 different Q-plates. The observed orbitals have 4 outer lobes, showing approximately a four-fold rotational symmetry induced by the first term in Eq. (1). The bright dot at the center reveals the appearance of the residual term (second term in Eq. (1)). We have also put a white cross on the figure to indicate the orientation of the orbital (the 4 outer lobes). The orbital rotates in the anticlockwise direction as s increases. In crossing the exceptional point by these samples, the orbital is rotated by 45 degrees, in agreement to the theory prediction. The overall intensity dimming at larger s is due to the smaller cross-polarization transmittance. The orbital rotation accompanies with a $\pi $ phase change in ${t_{ + \; - }}/{t_{ - \; + }}$, as mentioned, which is plotted in Fig. 2(k) for the simulated results.

 figure: Fig. 2.

Fig. 2. (a)-(j) Experimental cross-polarization diffraction patterns measured at wavelength λ=600 nm for s ranging from 140 nm to 320 nm. The white crosses added to show the orientation of the diffraction orbitals. (k) Corresponding (simulated) phase change $\arg ({{t_{ +{-} \; }}/{t_{ - \; + }}} )$ in degrees when we vary s at the specified wavelength.

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Alternatively, we can also plot the orbitals only along the angular $\phi $ direction at a fixed $\rho $, which is selected to capture the highest intensity of the 4 outer lobes of the orbital. Then, the intensity of the orbitals can be plotted together with the variation of separation parameter s, in Fig. 3(a). By plotting the orbital intensity with s, it is clear that each bright band in either Figs.  3(a) or 3(b) represents one of the 4 outer lobes and its orientation is added by 45 degrees in the whole increasing range of s. The exceptional point can then be identified by where the orbital rotates most rapidly. It is found at around $s = 270$ nm. For the experimental results in Fig. 3(a), although they suffer from insufficient resolution in sampling a 2D CCD-discretized image in the angular direction, there is still good agreement with the theory results in Fig. 3(b) in terms of rotation angle and the position of the exceptional point. One subtle observation is that the first and the third bright bands in the experimental results (Fig. 3(a)) looks brighter than the other two bands, especially at smaller s. This can be explained (results not shown) by performing numerical calculations of the orbital profiles from Eq. (1) without neglecting the second term. In our current experiment, we have demonstrated the embedding of an exceptional point to a metasurface with unit topological charge. As the exceptional point belongs to individual unit cells, it is possible to extend our current results to other types of Q-plates with general q to reveal the existence of the exceptional point. In this case, the orbital will rotate by $\pi /4q$. Such an exception point-induced change of orbital profile can also be extended to general inhomogeneous orientation profile of the metasurface in future exploration.

 figure: Fig. 3.

Fig. 3. Angular normalized intensity contour plots of (a) experimental and (b) simulated results of the orbitals in increasing separation parameter s. The scale chart is normalized intensity.

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4. Conclusions

In conclusion, we have demonstrated a series of metasurface Q-plates which can configurationally pass through an exceptional point for the constituting plasmonic structures. A plasmonic exceptional point is revealed through a sharp orbital rotation, being directly observable without the need of performing eigen-decomposition of the transmission coefficients. Our investigations have demonstrated a plasmonic exceptional point as an effective mean to enhance the sensitivity of optical properties to a change of geometric parameters, which will be useful to design a new generation of tunable optical components, e.g. enabled by MEMS elements in a dynamically tunable fashion.

Funding

Research Grants Council, University Grants Committee of Hong Kong (AoE P-02/12, C6013-18G); National Natural Science Foundation of China (11874232); Key Research and Development Program of Shandong (2018GGX101008).

Acknowledgments

We acknowledge K.S. Wong for providing equipment for the diffraction measurement.

Disclosures

The authors declare no conflicts of interest.

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Figures (3)
Fig. 1.
Fig. 1. (a) and (b) Schematic diagram of the unit cell, highlighted by black dotted line, on a glass substrate. Thicknesses of gold, PMMA, and ITO layers are 45 nm, 180 nm, and 65 nm, respectively. s denotes the separation between the two orthogonal slots/bars with lengths ${l_1} = $ 140 nm, ${l_2} = 170$ nm and widths ${w_1} = $ 98 nm, ${w_2} = $ 110 nm. (c) SEM image of a plasmonic Q-plate constructed by these unit cells with s = 200 nm. (d) and (e) show the simulated and measured transmittance in LP basis respectively if the unit cells are arranged into a square lattice of periodicity 450 nm and separation distance $s$  = 320 nm. (f) and (g) show the corresponding measured transmittance in CP basis. (h) and (i) show the corresponding results of transmittance in CP basis when the separation distance is further reduced to $s$  = 180 nm. Insets (e) and (i) show the SEM images of the samples with white color bar denotes a length of 200 nm.

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Fig. 2.
Fig. 2. (a)-(j) Experimental cross-polarization diffraction patterns measured at wavelength λ=600 nm for s ranging from 140 nm to 320 nm. The white crosses added to show the orientation of the diffraction orbitals. (k) Corresponding (simulated) phase change $\arg ({{t_{ +{-} \; }}/{t_{ - \; + }}} )$ in degrees when we vary s at the specified wavelength.

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Fig. 3.
Fig. 3. Angular normalized intensity contour plots of (a) experimental and (b) simulated results of the orbitals in increasing separation parameter s. The scale chart is normalized intensity.

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Equations (2)

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I ∝ | F 2 q [ a ( ρ ) ] ( e i 4 q ϕ t − + − t + − ) + F 0 [ a ( ρ ) ] e i 2 q ϕ ( t + + − t − − ) | 2
F m [ a ( ρ ) ] = 1 2 π ∫ 0 ∞ ⁡ ∫ 0 2 π ⁡ e − i k ρ ′ ρ f 2 cos ⁡ ϕ ′ e i m ϕ ′ a ( ρ ′ ) d ϕ ′ ρ ′ d ρ ′ .
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