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Abstract
We numerically study metasurfaces that incorporate electro-optic materials and show that they can achieve large amplitude and phase modulations across a distance that is a fraction of the operation wavelength. We show that the metasurfaces made of dielectric discs placed on a film of lithium niobate can exhibit three main types of resonances, associated with the Fabry-Perot modes in the structure, guided modes of the film and Mie modes of the disks. We compare metasurface performance in these different regimes for achieving largest electro-optic modulation and find that in the proposed geometry the strongest amplitude modulation can be achieved through excitation and re-emission of the guided modes in the substrate. We further show that to achieve larger 70 degrees phase modulation while maintaining high transmission, we need to utilise more complex metasurfaces that have at least two resonators per unit cell.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurfaces are two-dimensional arrays of subwavelength resonant elements that interact with the electric and magnetic fields of the electromagnetic waves, showing unique properties that are not found in nature. Numerous optical components made of metasurfaces have already been demonstrated, including lenses [1], holograms [2], wave retarders [3] and other types of functional components [4]. Although these functional optical components are hundreds of times thinner than human hair, their performance often matches or even surpasses that of conventional bulky optical elements. The functionalities of most of the developed components are fixed after nanofabrication, but in many applications it would be beneficial to have the performance of the structures tunable.
Tunable metasurfaces whose properties change under external stimulus were proposed in a number of works. There are several physical tuning mechanisms employed, this includes carrier excitation in semiconductors, use of phase change materials, mechanical deformation of structures, infiltration with liquid crystals, etc. [5]. Their corresponding stimuli include thermal excitation [6], electric field, optical pump [7], mechanical force, and microfluidics [8]. However, the highest dynamic modulation speed of existing active optical metasurfaces achieved by external applied electric fields is in the kHz range, limiting applications of the dynamic metasurfaces [9].
More recently, the electro-optic properties of lithium niobate (LN), were employed for making tunable metasurfaces [10–19]. LN is used in many bulk optical devices, including electro-optic modulators, which are known to operate at GHz modulation frequencies, and this gives us hope of making fast tunable metasurfaces based on LN. Latest studies of the LN metasurfaces have verified that it is a very promising candidate for making tunable optical elements based on metasurfaces. Previous works have demonstrated LN pillars [12] or gratings [11] on silica substrates. However, their modulation efficiencies are limited, or they may sacrifice bandwidth when using high quality factor structures [20,21].
In this work, we numerically study several metasurfaces that use LN film as a substrate. Our main aim is to explore the possibility to achieve a large phase modulation while maintaining the large transmission amplitude through the structure. Such regime is typically associated with a Huygens’ condition, when two overlapping resonances ensure full transmission while changing the phase by up to $2\pi$. We consider the structures that contain a film of LN with silicon metasurface(s) attached to it. We start by studying a metasurface containing silicon disks on one side of the metasurface, and discover that in the Huygens’ regime the tunability of LN properties is not strong enough to produce a desired effect. However the amplitude modulation in this structure can be very strong in the regime of excitation of guided modes in the LN film. Then, we consider the structures containing metasurfaces on either side of the film. Our motivation is that by having such a structure we can find generalized Huygens’ regimes that appear due to interference of modes in the two metasurfaces. At the same time, when the modes of the metasurfaces hybridize, we expect that a substantial part of the electric field will penetrate into the LN layer ensuring strong tunability. The modulation of the amplitude can be achieved with nearly 100% modulation depth in all considered structures. The continuous phase modulation is more challenging, and in the proposed dual-metasurface structures the modulation of phase can reach approximately 70 degrees, while maintaining high transmission at over 90%.
LN is transparent in the optical frequency range, and it has relatively large electro-optic coefficients [16]. In the linear regime, LN is a uniaxial crystal characterized by $n_o$ and $n_e$, and all the refractive index data in this article are retrieved from [22]. LN is stable at room temperature, which provides a platform for light manipulation at the optical frequency band. In addition, when applying an external electric field to LN, a linear electro-optic effect provides a continuous change of refractive index. An equation that expresses the relationship between external electric field and refractive index is [16]
where $\mathrm{\Delta} (1/n^2)$ characterizes the change of relative dielectric permittivity caused by the applied electric field $E_{z}$ along z light propagation direction, $\textrm{r}_{\textrm{ij}}$ denotes the linear electro-optic coefficients, and $i,j=x,y$. The matrix $\textrm{r}_{\textrm{ij}}$ for LN can be described by four independent coefficients $\textrm{r}_{51}, \textrm{r}_{22}, \textrm{r}_{13}$ and $\textrm{r}_{33}$. The external electric field is usually applied in such a way, so that the largest coefficient $\textrm{r}_{33} = 28.8 \times 10^{-12} \textrm{m/V}$ is responsible for the index change in order to maximize the electro-optic effect [16,23].2. Results and discussion
2.1 Single-disk metasurfaces
In our work, we use commercial software CST Microwave Studio, and start with the study of a simple structure, where silicon disks are placed on a 450 nm thick Z-cut LN substrate, and consider that external DC voltage can be applied to the LN, without considering the actual possible experimental implementation, see Fig. 1(a). The height of the disks and lattice constant of this metasurface are fixed at 200 nm and 620 nm, respectively. By varying the voltage, we can change the tensor of the refractive index of the LN substrate, aiming to tune the amplitude and/or phase of the transmitted light. As shown in Fig. 1(b), this disk-based metasurface exhibits four resonances in the frequency range between 950 and 1350 nm. To identify these four resonances, we choose a disk radius of 125 nm (white dashed line in Fig. 2(b)) and mark the four intersections with resonances as A, B, C, and D. By analysing their electric field distribution, these four resonances can be identified as the magnetic dipole (MD), electric dipole (ED), guided mode (GM) of the LN substrate, and Fabry-Perot resonance (FP) in the substrate, as shown in Figs. 1(c-f). The MD and ED have most of their electric field located in high-index silicon disks and resemble Mie-type modes [24]. When disk radius is 146 nm, MD and ED overlap and this corresponds to the Huygens condition [25]. The guided mode and the Fabry-Perot mode have a substantial part of the energy located within the substrate (see Fig. 1(e,f)) and we expect that these modes will experience stronger tunability when the parameters of the LN change with an external voltage.
Fig. 1. Transmission properties of a metasurface made of silicon disks placed on an LN substrate. (a) Schematics of an LN metasurfaces whose properties are controlled by external electric field. Our first example is a metasurface made of silicon disks placed on a surface of a z-cut LN substrate. (b) The dependence of the transmission spectrum on wavelength and the disk radius. MD - magnetic dipole; ED - electric dipole; GM - guided mode; FP - Fabry-Perot resonance. The parameters of the metasurfaces are: disk height = 200 nm, substrate height = 450 nm, lattice constant = 620 nm. (c-f) Electric field distribution of four resonances corresponds to the (A-D) points in (b). The white dash line in (b) indicates the nanodisk radius of 125 nm for which the corresponding modes are plotted in (c-f). The black arrows in (c-f) denote the electric field.
Fig. 2. Resonance tuning of an LN metasurface made of silicon disks. (a) Change of the transmission amplitude ($|\Delta {T}|$ in log scale) of the metasurface caused by the applied external electric field. (b-e) Simulated tunability of the amplitude and phase of the metasurface transmission for different modes. (b,c) Shows the transmission amplitude (blue) and phase (red) at negative (solid) and positive (dashed) applied electric field for the Mie mode in the case of the disk radius of 146 nm (near the point A on the black dashed line in (a)). The induced difference in the amplitude of the transmission coefficient is shown by purple line in (b); and the difference of the phase of the transmission coefficient is shown by green line in (c). (d,e) Same as (b,c) but for the guided-mode case with the disk radius of 94 nm (white dashed line near point B in (a)).
To compare the performance of different resonances for the purpose of tuning amplitude and phase of the transmission coefficient, we assume that we can apply a maximum electric field of $E=65$ kV/mm [26] to the LN substrate, and this corresponds to the change $\Delta n_e = 9.3\times 10^{-3}$. The amplitude tunability ($\Delta T = T(+E) - T(-E)$) caused by this external field change is presented in Fig.2 (a) in a logarithmic scale. As expected, the stronger tunability of the transmission amplitude is observed near the resonances. First, we look closer at the crossing of the two Mie resonances, where the interaction of the modes leads to the Huygens’ condition, characterised by high transmission and large phase change of the transmitted waves. This condition occurs when the radius of the disks is 146 nm, and it is marked by the point A on a black dashed line in Fig. 2(a). The corresponding spectrum of the transmission amplitude and phase are shown in Figs. 2(b,c). The blue solid line and the blue dashed line in Fig. 2(b) denote the amplitude of the transmission coefficient ($T$), when we apply +E and -E tuning fields, respectively. The difference between the two quantities, $\Delta T$, is shown by the purple line in Fig. 2(b). Figure 2(c) shows the phase tuning properties for the Mie resonance, with the coinciding red solid line and red dashed line denoting the transmitted phase $\varphi$ for negative and positive electric fields applied. The vanishing phase tunability $\Delta \varphi$, which is the phase difference caused by the external electrical stimuli, is denoted by the green line in (c). From Fig. 2(b-c), we see that $T$ reaches a unity transmission, and $\varphi$ changes by $2\pi$ across the frequency range, being the signature of the Huygens condition. Due to the fact that the fields are mostly concentrated inside the resonators, the $\Delta T$ and the $\Delta \varphi$ are close to zero, showing a weak tuning of transmission properties that can be achieved by these Mie modes.
It is important to note that in Fig. 2(a), the guided mode shows a stronger sensitivity to the applied stimuli. The quality factor of this GM resonance is 370. It is smaller than the Q-factors in the state-of the art metasurface structures [27,28], however the guided mode has most of its field localized inside the LN layer, thus providing the strongest tunability. We choose a disk radius of 94 nm, corresponding to the largest tunability of the amplitude in Fig. 2(a), with this parameter shown by the white dashed line, and study the response near the guided mode regime marked by the point B. As shown in Figs. 2(d-e), GM resonance shifts by several nanometers after applying stimuli, resulting in $\Delta T$ at around 1500 nm being close to one. In this case, the phase curve of the GM also shifts in frequency, and $\Delta \varphi$ reaches the values of 160 degrees. Thus, we conclude that the single-disk structure can be used for efficient amplitude modulation using GM excitation regime. While the phase modulation is relatively high, the continuous tuning will show a strong modulation of the amplitude, limiting the usefulness of this regime.
2.2 Dual-disks metasurfaces
We now look for structures that can exhibit several resonances at close frequencies and at the same time be highly tunable by the properties of LN. To do this, we propose to make metasurfaces that have two resonators per unit cell, and they are placed on either side of the LN film. In such structures, we expect a hybridisation of the modes, so that we can have modes that overlap in the frequency domain, thus providing large phase gradient, and at the same time have the fields extending over the LN film, so that they are more sensitive to the changes of the properties of the LN. The first structure is shown in Fig. 3(a), it has two similar silicon disks (shown in blue) on either side of the 450 nm thick LN film (red colour). The lattice constant is set at 620 nm, and the height of both disks is 200 nm. This dual-disks metasurface structure also supports Mie-like modes, GM, and FP resonances in the frequency range of interest.
Fig. 3. Resonance tuning of an LN metasurface made of dual silicon disks. (a) Schematic of the unit cell of the dual-disks LN metasurfaces. The design parameters of the metasurfaces are: substrate height = 450 nm, lattice constant = 620 nm, disk height = 200 nm, the top and the bottom disk have the same radius. (b,c) The amplitude and phase at negative and positive maximum electric field, and their separate tunability of Mie mode at a radius of 148 nm. (d,e) The amplitude and phase at negative and positive maximum electric field, and their separate tunability of guided-mode at a radius of 110 nm.
We performed a parameter scan over the disc radii to identify the regimes of either large amplitude modulation, or large phase modulation while maintaining high amplitude of transmission. We found that this occurs when the disks on either side have the same sizes. Figure 3(b-c) summarises our results for this case. In the spectra from 1030 nm to 1050 nm with a disk radius of 148 nm, the transmitted amplitude is near unity, and the $2\pi$ phase change across the spectrum can be seen. Additionally, $\Delta T$ is close to zero while $\Delta \varphi$ reaches values of up to 70 degrees. This means the dual-disks metasurface can modulate the transmitted phase in the range of [0, $70^{\circ }$] while still maintaining high transmission amplitude.
As expected, the guided mode of the dual-disks metasurfaces still exhibits a strong response to the electric stimulus, since most of the field of guided mode is located within the LN film. To illustrate the tunability of GM of the dual-disks metasurfaces, we identified in our parameter sweep the radii of the disks that provide the largest amplitude modulation. We plot the transmission spectra for a disk radius of 110 nm in Fig. 3(d,e). At the central wavelength of 1165 nm, the amplitude contrast can reach unity, resulting in a strong tunability of amplitude. However, the phase also shifts along with the amplitude, and the $\Delta \varphi$ reaches 155 degrees. Similar to the single-disk design, GM in the dual-disks metasurface can efficiently modulate the amplitude of the transmitted waves, however the phase modulation without affecting the amplitude is not possible.
2.3 Hole-disk metasurfaces
Another type of metasurfaces that was recently proposed is based on a perforated dielectric membrane [29]. In the membrane metasurface the behaviour of the electric and magnetic Mie modes changes, and we are hoping to find regimes that provide the large modulation of phase. To extend the design thought of our previous section, we substitute one disk metasurface with a perforated structure, whose unit cell is the same as that of the disk-based metasurface. As shown in Fig. 4(a), we put membrane metasurface and disk metasurfaces on either sides of the LN substrate that we make 1000 nm thick, so that all resonances appear in the frequency range of interest.
Fig. 4. Resonance tuning of a LN metasurface made of silicon disks and perforated membrane. (a) Schematic of the unit cell of the disk-hole LN metasurfaces. The design parameters of the metasurfaces are: substrate height = 1000 nm, lattice constant = 620 nm, disk height = 170 nm, hole height = 245 nm, disk radius = 210 nm, hole radius = 245 nm. (b) The amplitude of the transmission coefficient at negative and positive maximum electric field, and amplitude tunability. (c) The phase at negative and positive maximum electric field and phase tunability.
For this geometry, we performed a parameter sweep over the radii and heights of holes in a membrane and disks, and identified the parameters providing large phase modulation of the transmitted signal while maintaining high amplitude. Specifically, the optimised radii of the disks and the holes are found to be 210 nm and 245 nm; the height of the disk and the membrane are 170 nm and 245 nm, respectively. As shown in Fig. 4(c), the phase tunability of this hole-disk metasurfaces can reach values of 61 degrees, whereas transmission amplitude has a strong dependence on the wavelength, as shown in Fig. 4(b). We see that the phase modulating performance of this structure is not as good as that of the dual-disk structure. The transmission amplitude, however, can be efficiently modulated in a narrow frequency range due to the presence of sharp Fano-like line shapes in the spectrum.
3. Conclusion and outlook
In summary, we have studied the tunability of electro-optic metasurfaces based on silicon metasurfaces placed on one or two sides of lithium niobate films. We have designed three types of LN metasurfaces, including disk metasurface on one side of LN film, dual-disk, and membrane-disk metasurfaces. We explored the parameter space in order to identify the strongest possible modulation of the transmission properties of metasurfaces. We have found that the LN metasurfaces can support four main types of resonances and studied the amplitude and phase tunability of each case. We discovered that while each of these structures can provide large modulation of the amplitude of the transmission, the largest transmission phase tunability with relatively high amplitude can be achieved in a dual-disk structure, where the phase can be modulated in a 70 degrees range, while maintaining a near-unity transmission in a structure with a total thickness of less than 1 micron.
Due to the relatively small material parameter change of LN with applied voltage, in order to achieve large modulation with a full $2\pi$ phase shift we would need to search for a higher-quality Huygens’ type resonance that does not degrade as we apply voltage, and this is a subject of future studies. Alternatively, we can try to employ other materials that exhibit larger tunability of the index of refraction.
Funding
Australian Research Council (CE200100010).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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