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Abstract
Active phase-control metasurfaces show outstanding capability in the active manipulation of light propagation, while the previous active phase control methods have many constraints in the cost of simulation or the phase modulation range. In this paper, we design and demonstrate a phase controlled metastructure based on two circular split ring resonators (CSRRs) composed of silicon and Au with different widths, which can continuously achieve an arbitrary Pancharatnam-Berry (PB) phase between -π and π before or after active control. The PB phase of such a metasurface before active control is determined by the rotation angle of the Au-composed CSRR, while the PB phase after active control is determined by the rotation angle of the silicon-composed CSRR. And active control of the PB phase is realized by varying conductivity of silicon under an external optical pump. Based on this metastructure, active control of light deflection, metalens with arbitrary reconfigurable focal points and achromatic metalens under selective frequencies are designed and simulated. Moreover, the experimental results demonstrate that focal spots of metalens can be actively controlled by the optical pump, in accord with the simulated ones. Our metastructure implements a plethora of metasurfaces’ active phase modulation and provides applications in active light manipulation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurfaces, the 2D counterpart of artificial metamaterials, have provided extraordinary capabilities in controlling the electromagnetic (EM) response of media at subwavelength resolution, making them promising for optical modulation devices [1–22]. Many applications based on metasurfaces have been demonstrated by designing the cell elements in metasurfaces, such as polarization converters [1,2], perfect absorbers [3], optical imaging encoding [4,5], etc. Furthermore, by properly utilizing the phase response of metastructures, metasurfaces have opened the gate to novel applications such as generalized Snell’s Law of deflection [6,7], flat metalenses [8–11], vortex-beam generation [12,13] and holograms [14,15], etc. Comparing with traditional optical modulation methods, the ultrathin nature, freedom of design and ease of fabrication of metasurfaces make it a promising way to manipulate the amplitude and phase of EM wave.
Active control of the metasurfaces’ phase profiles provides much more potential in optical devices. To date, scientists have introduced active media like liquid crystal [16], GST [17,18] or graphene [19–21] into metasurfaces and have designed the corresponding reconfigurable optical devices. Phase transition of such active media causes variance of the refractive index in the waveguide-like [22] metastructure, resulting in the active phase control. However, the phase profiles of such dielectric metastructures are not easy to predict. An applicable way is to simulate and filter for a series of metastructures to meet the discrete phase demands. But the design of these reconfigurable metasurfaces requires plenty of simulations and cannot easily realize precise phase modulation since the achieved phase profiles are discrete. Additionally, when the incident light has a large wavelength, taking terahertz wave around 0.3-0.8 THz for example [11], thickness of the designed metastructure is required to be at the magnitude of ∼500 μm, which is not convenient either for fabrication or integration. There are also some quasi-single-layer active metastructures based on active media like graphene, but the “active” phase control has limited phase modulation range and has much restricted functions. Therefore, a predictable, continuous and unlimited active phase modulation method with wavelength-independent thickness is strongly demanded. Since Pancharatnam-Berry phase (PBP) [23,24] is dispersion-free and has no constraint in geometry of the single cell of a metastructure, exploiting active controlled PBP of designed metastructure is a reasonable way to achieve this goal.
Here, we propose and demonstrate an active arbitrary PBP control metastructure with multiple phase control layers on a dielectric substrate. The relationship of an arbitrary metastructure’s tetragonal discrepancy and conversion efficiency between left-circular polarization (LCP) and right-circular polarization (RCP) is proved. Under this theory, we design a metastructure with two circular split ring resonators (CSRRs) composed of Au and silicon, and its phase profile is controlled by the rotation angles of these CSRRs. The metastructure can achieve arbitrary PBP from –π to π both before and after active control, thus can complete orthogonally phase modulation by different phase control layers. To further explore the application of EM wave modulation by such metastructure, we design several novel functional metasurfaces like metalens with arbitrary reconfigurable focal points, achromatic metalens under selective frequencies and active light deflector. The practicability of these metasurfaces are verified by numerical simulations and experiments.
2. Theory, design and discussion
2.1 Relationship of cross-polarization efficiency and tetragonal discrepancy
Figure 1(a) shows the structure of a single unit in the arbitrary active PBP control metasurface. The blue-color square is a sapphire (Al2O3) substrate, upon which two CSRRs with different geometry parameters and specified rotation angles are stacked. The smaller CSRR is composed of Au, a good conductor, while the larger CSRR is composed of Silicon, which is an active media due to that its conductivity can be easily large-range modulated by optical pump [25,26]. Upon the illumination of a specified circularly polarized terahertz beam, the cross-polarized component’s phase delay (either incident LCP to transmitted RCP or incident RCP to transmitted LCP) in comparison with the incident light is φ1. When the active media’s conductivity in our metasurface is modulated via optical pump, the cross-polarized component’s phase delay respond is φ2. It is noted that both φ1 and φ2 are arbitrary and can be orthogonally manipulated by either of the CSRRs’ rotation angles.
Fig. 1. (a) Active PBP control function of our metastructure. Phase of the transmission cross-polarized component can be arbitrarily achieved by optical pump. (b, c) Schematic illustration of a metalens with reconfigurable focal points.
In order to design the metasurface with such function, the derivation of PBP must be addressed. For an arbitrary 2D sub-wavelength metasurface, taking non-linear effect out of consideration, the transmitted electric field can be expressed by Jones matrix as:
Fig. 2. (a) Origin of Pancharatnam-Berry phase: a derivation is addressed by an inverse rotation of coordinate axes. (b) An intuitive representation of tetragonal-axis discrepancy. The geometry of a metastructure and its 90-degree rotation result is shown. (c) Schematic diagram of an Au-composed CSRR with a narrow connection gap, where yellow stands for gold and blue indicates the sapphire substrate. Geometrical parameters include r1 = 21μm, w1 = 11μm, p = 56 μm, r2 = 17 μm when w2 = 3 μm. The geometry is considered as two overlapping CSRRs with the rotation angles over x-axis are α1 and α2, respectively. (d) Simulation result about the cross-polarization efficiency of the metastructure shown in (c) as w2 varies. Gray and blue colored part in the up-right sketch represents for the tetragonal discrepancy of this metastructure. (e) Simulated electromagnetic response of the metastructure illuminated by RCP at 1.62THz when we rotate the bigger CSRR from 0 to 2π. Sketches over this figure show the geometric outline at coordinate rotation angles.
For a metastructure possessing tetragonal-axis, whose 90-degree rotation result returns to itself, thus $\mathrm{\Delta }{\boldsymbol T} = {{\boldsymbol T}_{xy}}\left( {\frac{\pi }{2}} \right) - {{\boldsymbol T}_{uv}}(0 )= 0$ and ${t_u} = {t_v}$. As a result, such metasurface has no cross-polarization effect. Moreover, it can be derived from Eqs. (5) and (6) that: the more ${{\boldsymbol T}_{xy}}(\frac{\pi} {2})$ varies from ${{\boldsymbol T}_{uv}}(0 )$, the higher cross-polarization amplitude of a metastructure. Geometrically, the cross-polarization amplitude has a positive correlation with the tetragonal discrepancy along its normal-vector.
Next, we will simulate the cross-polarization efficiency of a designed metasurface to verify this theory by employing full-wave simulations using COMSOL Multiphysics. As the schematic shown in Fig. 2(c), the metastructure is a closed circular split ring resonator (c-CSRR) which contains a CSRR with a split gap angle of 90° and a narrow connection composed of Au along its gap. The narrow connection’s width w2 varies in our simulation. The colored shadow in Fig. 2(d) shows the variance between the 90-degree rotated c-CSRR and its origin. It can be seen from the simulation result in Fig. 2(d) that the difference of this c-CSRR with its 90-degree rotation result is diminishing as w2 varies from 2μm to 10μm, resulting in the reduction of cross-polarization efficiency. In further simulation, we set w2 = 3μm in order to maintain the tetragonal-axis discrepancy of this c-CSRR which can achieve a highest cross-polarization efficiency of 24.1% at 1.62 THz. Moreover, we can consider this c-CSRR as the outline of two embedded CSRR with different radius and width but the same gap angle of 90°. When taking the rotation of these two CSRRs into consideration, we rotate the CSRR’s arc counter-clockwise, and define α1 and α2 as the angle of x-axis with the start side of bigger CSRR and smaller CSRR, respectively. Figure 2(e) shows the transmitted phase and amplitude of LCP with the incidence of RCP when α2 is set to 0 and α1 varies in this Au-composed metastructure. For α1<90° and α1>270° parts, the geometric of this metastructure is not a c-CSRR thus both the LCP amplitude and phase depend on α1. When α1 = 0°, the metastructure achieves its highest cross-polarization efficiency at 36.3%. While α1 varies from 90° to 270°, the geometric outline becomes a c-CSRR and the variation of α1 causes only the outlined c-CSRR’s rotation. Equation (4) shows that such rotation will only cause an additional cross-polarization phase delay -2Δα1 corresponding to the rotation angle Δα1, but no effect on LCP amplitude, as is presented in Fig. 2(e). Here the LCP transmission is about 24%. If ensuring the geometric outline of this metastructure to be a c-CSRR, α1 is restricted by an arbitrary value in: $\frac{\pi }{2} \le {\alpha _1} \le \frac{\pi }{2} + \pi $. Thus the LCP phase φ(α1) of this c-CSRR is:
2.2 Design of an arbitrary active PBP control metastructure
We designed an active PBP control metasurface based on the same geometric parameters of this c-CSRR, as is shown in Fig. 3(a). The bigger CSRR (Si-CSRR) is composed of epitaxial growth silicon with thickness of 600 nm on sapphire substrate, while the smaller CSRR (Au-CSRR) is composed of 1$\mathrm{\mu }$m-thick Au depositing on silicon. Similar to Fig. 2(c), the rotation angles of Si-CSRR and Au-CSRR (represented as α1 and α2, respectively) are modifiable by utilizing the discussion we already addressed. Silicon is an active media due to its carrier concentration can be easily modulated by optical pump, resulting in a variation of conductivity from 103 S/m to 106 S/m [25,26]. We simulated the electromagnetic (EM) response of this metastructure using COMSOL Multiphysics by altering the dielectric constant of silicon using equation: ${\tilde{\varepsilon }_{Si}} = {\varepsilon _{Si}} + i\frac{{{\sigma _{Si}}}}{{{\varepsilon _0}\omega }}$ where ${\varepsilon _{Si}} = 11.7$ and ${\sigma _{Si}}$ varies.
Fig. 3. (a) Schematic illustration of the active PBP control metastructure. Geometrical parameters defined similar with Fig. 2(c) include r1 = 21μm, w1 = 11μm, p = 56 μm, r2 = 17 μm and w2 = 3 μm. The rotation angles of Au-CSRR and Si-CSRR are defined as the angle of x-axis with the start side of these CSRRs, similar with the definition in Fig. 2(c). (b) Simulated cross-polarization response of this metastructure at 1.62 THz with α2 = 0 while α1 and the conductivity of silicon varies. The solid lines represent for the LCP phase response and the dot lines represent for the LCP transmission. (c) Electric field distribution of this metastructure when α1=π and α2 = 0 as ${\sigma _{Si}}$ varies. (d-i) Simulated LCP phase response when illuminated by RCP at 1.62 THz when α1 and α2 are at arbitrary value, ${\sigma _{Si}}$ varies from 103 S/m to 106 S/m in these figures. Only half area of these figures represents for the geometry with an outlined c-CSRR.
Figure 3(c) shows the electric field distributions of this metastructure with different silicon conductivity. When silicon is at low conductivity state like a dielectric media, the electric field is localized around the Au-CSRR. Thus, the PBP of this metastructure is controlled by α2, the rotation angle of Au-CSRR, which can be arbitrary. As ${\sigma _{Si}}$ increases, the electric field is gradually localized around the outline conductive c-CSRR and there is no electric field inside due to a high ${\sigma _{Si}}$. Since α2 has little effect on PBP when silicon is at high conductivity state, the PBP is only controlled by α1. The relationship of LCP response with ${\sigma _{Si}}$ under the illumination of RCP is shown in Fig. 3(b). LCP phase is fixed by α2 and the average of cross-polarization efficiency is about 30% when ${\sigma _{Si}} < {10^3}S/m$, but when ${\sigma _{Si}} > {10^6}S/m$, the LCP phase is controlled by α1 and can be arbitrary since α1 has a limitation range of π. When the conductivity of silicon is at 107 S/m, the average of cross-polarization efficiency is about 22%. Figures 3(d)–3(i) illustrate the α1, α2-controlled PBP under different ${\sigma _{Si}}$. Under the limitation of α1 and α2 represented by Eq. (8), only half area of this figure is valid since we require an outline c-CSRR. It can be observed that the gradient of LCP phase in Fig. 3(d) is parallel with α2-axis, representing that LCP phase is only controlled by α2 when silicon is at low conductivity state. While in Fig. 3(i), the LCP phase gradient in the limited parts by Eq. (8) is parallel with α1-axis. Moreover, it can be deduced from Eq. (8) or the periodicity of Figs. 3(d-i) that for an arbitrary value of α2, the LCP phase can also be arbitrary if an appropriate α1 value is chosen. It is obvious that the RCP response of this metastructure under the illumination of LCP has similar result but a different sign on PBP. As a conclusion, we achieved two arbitrary active controlled PBP in this metastructure which can be orthogonally modulated by the rotation angle of either Si-CSRR or Au-CSRR.
3. Metasurface applications
3.1 Metalens with arbitrary reconfigurable focal points
A functional metasurface such as reconfigurable metalens can be constructed by employing our metastructure, as is shown in Figs. 1(b) and 1(c). Array of such metastructure with coordinate-dependent α1 or α2 can provide designed PBP at different area of the metasurface, resulting in a distribution of in-plane wave vector. To focus the incident collimated light in a diffraction limited spot, a metalens must impart a spatial phase profile, which is determined by the Fermat’s theorem [8,19]:
Fig. 4. Simulated LCP electric field distributions for the metalens designed in Section 2.2 when illuminated by RCP at 2.52THz. The geometric parameters distribution is controlled by Eqs. (10) and (11) when ${f_{x1}} ={-} {f_{x2}} = 2mm,{\; }{f_{y1}} = {f_{y2}} = 0,\; {f_{z1}} = {f_{z2}} = 15mm$. (a)-(e) LCP electric field distributions in y = 0 plane as ${\sigma _{Si}}$ varies from 103 S/m to 106 S/m. (f)-(j) LCP electric field distributions in the z = 15 mm focal plane under different conductivity of silicon, respectively with (a)-(e). (k)(l) Spatial angular distributions of Si-CSRRs and Au-CSRRs, respectively, to achieve metalens with reconfigurable focal points.
We also designed an experiment to verify our theory, as is shown in Fig. 5. Our metasurface is fabricated on the Silicon-on-Sapphire wafer, which consists of a 600 nm-thick silicon epitaxial layer on 460 μm-thick and 10 mm × 10 mm-area R-plane (1-102) sapphire substrate. The metasurface is inversely designed for the incident light from the sapphire substrate and the angular distributions of these hybrid CSRRs, which can be calculated from Eqs. (10) and (11), are shown in Fig. 5(b). The Si-CSRRs are fabricated via a standard UV-photolithography process, and the redundant silicon is removed by reactive ion etching. After the residual photoresist is cleared, the Au-CSRRs are patterned by the second step photolithography. The Au-CSRRs are sputtered on the Si-CSRRs with a thickness of 1μm by electric beam evaporation, followed by a lift-off process. Figure 5(c) shows the optical microscopy images of our metastructure. As is shown in Fig. 5(a), a linear polarized continuous THz beam at 2.52 THz is generated by a FIR laser based on high power CO2 laser pump and low pressure of methanol molecule. The polarization state of the output laser is carefully adjusted by a THz polarizer and incident into the sapphire substrate as an ordinary ray to eliminate the birefringence effect of R-plane sapphire. An aperture is placed before our sample to cut the diameter of the THz wave into about 5 mm. The EM response of our sample is detected by a real time ultrasensitive terahertz camera. Since linear polarized light is the superposition of LCP and RCP with equal intensity, the incident RCP component is focused by our metasurface. Meanwhile, such metasurface generates an inverse PBP for the incident LCP component, causing the divergence of transmitted RCP with little power density at similar order of magnitude with the noise of our THz detector, which can be neglected in our experiment. The transmitted co-polarized components, whose propagation directions are not changed, can be considered as the background signal in our experimental data. When at focal points, the background electric field is much less than the intensity of the converged LCP. Thus the focal points are clearly observable by an ultrasensitive THz camera. The optical pump is generated by a continuous 532 nm green-light laser with a radius of about 2.5 mm and incident into our sample at an angle of about 25°. Before the THz camera, a 500 μm-thick high-resistivity silicon wafer is placed to protect the THz camera from the incident optical pump beam. Figures 5(d)-(i) shows the THz electric field distributions detected by a THz CCD at the focal plane of z = 15 mm. Similar to the simulation result, Fig. 5(d) shows the focal point F1 without optical pump at around x = 2 mm. When there is no optical pump, a small conductivity of silicon caused the slight electric field of the second focal point. As the pump power increases, the electric field of F2 increases and F1 fades. When the pump fluence is over 1.50 W/cm2, the photo-carrier concentration is saturated and silicon achieves a high conductivity, thus lead to the highest intensity of F2. According to Figs. 4 and 5, metasurface with reconfigurable focal points is verified.
Fig. 5. Experimental results of the metalens with reconfigurable focal points. (a) Schematic illustration of our THz imaging system. The continuous THz beam generated by a THz laser can pass through the sapphire substrate as an ordinary ray and can be detected by a THz CCD. (b) Angular distributions of the Au-CSRRs and Si-CSRRs, respectively in our metasurface. The metasurface is composed of a 125*125 array of the metastructures we discussed in Section 2.2. (c) Optical microscopy images of the fabricated metalens with reconfigurable focal points. The radius of Si-CSRR is 21 μm and the radius of Au-CSRR is 17 μm. (d)-(i) Normalized THz electric field distributions detected by the THz CCD under different power of optical pump. The intensity of the pump fluence is annotated above every figure.
3.2 Metalens with achromatic focal point under selective frequencies
Our proposed metastructure has much potential in designing other functional metasurfaces. In order to design metalens with achromatic focal point under specified frequencies, instead of filtering large amount of metastructures to achieve phase compensation for high order Taylor-expansion series in Eq. (9) [10,11], we present a much easier way to achieve by actively modulating PBP with our metasurface. Since PBP illustrated in Eq. (4) has no relationship with frequency, PBP in our metastructure can be orthogonally controlled by CSRRs in arbitrary frequencies. If it is designed that frequencies ν1, ν2 have an achromatic focal point $({{f_x},{f_y},{f_z}} )$, a practicable rotation angles distribution of Si-CSRR and Au-CSRR can be described as:
Fig. 6. Electromagnetic response for the metalens designed in Section 2.2 when illuminated by RCP wave. (a)-(d), (f)-(i) Simulation results of the metalens’ LCP electric response at y = 0 plane when the incident frequency is 1.62THz or 2.52THz, respectively as ${\sigma _{Si}}$ varies. (e), (j) Normalized LCP electric field along z-axis under different incident frequency at 1.62THz or 2.52THz, respectively.
3.3 Active control of light deflection
Another application on the basis of the metastructure shown in Fig. 3(a) is the active wavefront control of a normal incident light. When the transmitted phase distribution of one metasurface forms a non-zero gradient, the wavefront is deflected from the metasurface normal with an angle of θ following the generalized Snell’s law equation [6, 8]:
where dφ and dx represents the EM response phase difference and the geometry distance of adjacent metastructure cells, respectively, and ${\lambda _0}$ represents the incident wavelength. According to Section 2, arbitrary PBP can be achieved by either the rotation angles of Au-CSRR or Si-CSRR. In order to achieve a tilted wavefront at ${\theta _1}$, we carefully adjust the rotation angles of Au-CSRRs, which is represented as α2(x), to form gradient transmitted PBP :$- \frac{{6\pi }}{7},\; - \frac{{4\pi }}{7},{\; } - \frac{{2\pi }}{7},{\; }0,{\; }\frac{{2\pi }}{7},\; \frac{{4\pi }}{7},{\; }\frac{{6\pi }}{7}$ in every 7 cells following the rule of PBP in Eq. (4). Meanwhile, to generate another active-controlled deflection angle θ2, α2(x), the rotation angles of Si-CSRRs are also adjusted meeting an active-controlled PBP : $\frac{{4\mathrm{\pi }}}{5},{\; }\frac{{3\mathrm{\pi }}}{5},{\; } \ldots ,{\; } - \frac{{3\mathrm{\pi }}}{5},{\; } - \frac{{4\mathrm{\pi }}}{5},{\; } - \mathrm{\pi }$ in every 10 cells. Thus a 1D-supercell composed of 70 meta-atoms is formed and the simulation result of an incident RCP wave at 1.62 THz is shown in Fig. 7. Theoretically deflection angles of transmitted LCP can be predicted by Eq. (14) as:Fig. 7. EM response of the supercell composed of 70 metastructures under a normal incident RCP. (a)-(e) Real part of the transmitted LCP electric field component under different conductivity of silicon, respectively. (f)-(j) Normalized propagation angle distribution of transmitted LCP at 1.62 THz. The conductivity of silicon is corresponding to (a)-(e), respectively.
4. Conclusion
In summary, we propose a subwavelength metastructure with two arbitrary cross-polarization phase profiles which can be actively and orthogonally modulated by two stacking CSRRs as the conductivity of Si-CSRR varies from 103 S/m to 106 S/m. With suitable design parameters, we utilize the tetragonal discrepancy to maintain the cross-polarization efficiency in such metastructure. Functional metalenses with arbitrary reconfigurable focal points and with achromatic focal point in specified frequencies and active control of light deflection based on such metastructure are addressed and demonstrated both by simulation and experiment. Due to the principle of PBP, when comparing with other active phase control metasurfaces, phase profile of our metastructure is predictable and can achieve arbitrary value over a large frequency range. By adjusting the geometry parameters of our metastructure, we may achieve reconfigurable PBP in other EM frequencies apart from terahertz range. Moreover, such metastructure has realized a feasible and convenient way in modulating phase profile of a metasurface and has great potential in electromagnetic wave modulation.
Funding
National Key Research and Development Program of China (2020YFA0710100); National Natural Science Foundation of China (51627901); Anhui Initiative in Quantum Information Technologies (AHY100000).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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