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Abstract
We investigate a reconfigurable dielectric metasurface merging optomechanical interaction and quasi-bound states in the continuum promising for all-optical light control light. The surface consists of a dimerized high-contrast grating with a compliant bilayer structure. The optical forces induced by a control light field lead to structural deformations changing the optical response. We discuss requirements for the geometry and optical force distribution to enable an efficient optomechanical coupling, which can be exploited to tune reflectivity, phase and polarization of a beam impinging on the metasurface. Numerical results explore some tunable devices as mirrors, saturable output couplers, phase modulators and retarder plates.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Bound states in the continuum (BIC), i.e. guided eigenmodes above the light cone within the continuum of radiation modes in photonic nanostructures [1–5], have recently attracted lots of interest for diverse applications such as lasing, filtering, sensing, biosensing, light shaping and second-harmonic generation [6–10]. These bound states theoretically have infinite quality factors (Q-factor). However, technology related structural imperfections and finite device sizes result in a weak coupling of BIC to the radiation continuum leading to leaky modes known as quasi-BICs. These quasi-BICs are no longer perfectly confined but instead exhibit sharp Fano-type resonances with ultra high Q-factors. One kind of BICs, the so-called symmetry-protected BICs [5,11], in which the radiative leakage is forbidden due to symmetry incompatibility between the excited mode and the external field, can also become radiative quasi-BICs by breaking the symmetry e.g. through off-normal incidence or with asymmetric unit cells [1,12–14]. Photonic structures in which BICs can appear include metasurfaces, photonic crystals slabs, optical waveguides and fibers, plasmonic gratings, all-dielectric gratings and integrated photonic circuits [1,5,12,15–21].
All-optical light control is essential in applications ranging from optical signal processing to beam shaping [22]. Control of light fields is commonly based on piezoelectric, electro-mechanical, electrostatic and thermal actuation mechanisms [23,24]. In contrast to these actuation mechanisms, all-optical modulation enables the manipulation of the properties of a light field through light, leading to devices with faster responses and without additional elements as heaters or electrodes [24]. Metasurfaces have proved to be optically tunable based on e.g. phase change materials or controlled absorption [25,26].
Diverse optomechanical systems [27–32] have been investigated for all-optical light control based on optomechanical actuation, i.e. actuation through optical forces on devices where the light field couples efficiently to the mechanical motion via radiation pressure, for example: integrated optomechanical systems such as as nano-beams and micro-rings [23,33], as well as optomechanical cavities [34]. Furthermore, tunable dielectric metasurfaces have proved optomechanical actuation, in which the array is supported by a free-standing elastic nano-membrane or nano-beams [24,35–37]. In these optomechanical systems, strong optical forces counteract the elastic forces in the structures with dimensions in the nanoscale, reaching a new equilibrium position and thereby changing the structural arrangement of the constituent elements, and consequently the optical response [36]. Since a strong optomechanical interaction is realized by coupling an optical mode of high optical Q-factor to a mechanical mode of high mechanical Q-factor [27], quasi-BICs are very appealing for optomechanics. The high-Q optical modes are very sensitive to the structural deformation induced by optical forces, so that even small induced deflections can enable tuning of the optical response.
In this contribution we theoretically investigate a free-space device, in contrast to integrated approaches, for optomechanical actuation based on high contrast gratings sustaining quasi-BICs. Our device is a dielectric metasurface, but it does not rely on suspended structures to realize mechanical resonators as in previous works [35–38]. The device is instead based on a bilayered grating with T-shaped building blocks, as realized for high reflectivity mirrors in high-precision metrology [39,40]. The bottom grating with high-aspect ratio provides a large mechanical susceptibility and compliance to the top grating. Furthermore, the grating includes a dimerization allowing us to work with BICs for optomechanical light control in metasurfaces. We analyze the physical requirements to achieve a large optomechanical interaction and hence a large optical tunability, including the geometry and the optical force distribution. We illustrate with numerical results that such structure is a promising platform to dynamically shape intensity, phase and polarization state of light fields, which are of interest for devices with a tunable response like saturable output couplers, phase modulators and retarder plates.
2. Optical forces in high-contrast gratings induced by quasi-bound states in the continuum
The time-averaged optical forces acting on a nanostructured surface are determined by ponderomotive light pressure $p_i$ acting on the boundaries of the structure [41,42]:
where $n_{i(j)}$ are the unit vectors and the quantity $\Delta \sigma _{i,j}$ is the difference of the time-averaged Maxwell stress tensor components $\sigma _{i,j}$ acting on the inner and the outer sides of the structure boundaries given by:High-index-contrast subwavelength gratings (HCGs) [43,44] are promising to induce large pressures, since they are made of a material with large refractive index and can sustain BIC modes associated to optical resonances with high Q-factors in the range of $10^8$ and, hence, enabling large field enhancements inside and around the ridges [4]. To use these field enhancement for large effective displacement forces, an asymmetry of the spatial field distribution is necessary. Otherwise the ridges experience either no deformation or just a position shift leading to no geometrical changes. One possibility to achieve this spatial asymmetry is a structural change such as the dimerization of the HCG by introducing a perturbation doubling the periodicity of the HCG [12]. The structure we propose is shown in Fig. 1, whose physical background will be explained in the following. These dimerized HCG sustain symmetry-protected BICs that can become leaky high-Q resonant modes excitable by free-space illumination by introducing a symmetry breaking condition, such as off-normal incidence illumination [12,17]. In combination with the structural asymmetry, these resonant modes enables an effective net force that leads to a deformation of the structure.
Fig. 1. Geometry of the dimerized T-shaped HCG. The unit cell consists of two top-ridges (width $w_{\mathrm {1}}$, height $h_{\mathrm {1}}$, separation $d$ and period $\Lambda$) on top of two bottom-ridges (width $w_{\mathrm {2}}$ and height $h_{\mathrm {2}}$).
For the dimerized HCG to be mechanically compliant to the optical forces, they need to be free to move to some extent. This has been achieved in previous works by using free-standing approaches as nano-beams and nano-membranes to support the building blocks [35,36]. We use a non-free-standing design by introducing a narrow grating as a support structure between the dimerized HCG and the substrate, forming T-shaped ridges. Larger and narrower support gratings, i.e. gratings with high-aspect ratio, reduce the mechanical stiffness enabling larger bending moments and optimizing the deformation for a given optical net force. The bilayered grating and the substrate are made of silicon ($n_{Si}=3.48$ at 1550 nm [45]). The dimerized silicon T-shaped HCG is shown in Fig. 1. The dimerization of the T-shaped HCG is realized with a gap perturbation [12] leading to a supercell with period $\Lambda$ that contains two non-equidistant ridges. The symmetry-protected BICs related to the gap-dimerization become leaky, i.e. quasi-BICs, for off-normal incidence illumination [12,17]. We used computations with rigorous coupled wave analysis (RCWA) [46] to optimize the geometry to get resonant modes in the 1550 nm wavelength regime fullfilling the requirements exposed above. Furthermore RCWA computations for the parameters listed in Table 1 also reveal the angular dependence of the resonance as is illustrated in the reflectivity spectrum in Fig. 2 for the lowest-frequency symmetry-protected mode for TE-polarized light, the $\mathrm {TE}_{\mathrm {1,1}}$ mode. The modes are denoted as $\mathrm {TE}_{\mathrm {m,n}}$, where m and n represent the number of lobes in the $x$- and $z$-direction in the top-ridge. The spectral line width and thus the optical Q-factor of the resonance increases with the angle of incidence (see inset in Fig. 2). For normal incidence, the coupling to free space is forbidden due to symmetry-incompatibility and the high-Q resonance becomes an optical BIC mode [4].
Fig. 2. Reflectivity spectrum of a quasi-BIC in dependence of the incidence angle for transverse-electric (TE) polarization. The inset shows the dependence of the Q-factor on the incidence angle.
Table 1. Geometry parameters of dimerized HCG grating, illumination conditions and material parameters of silicon.
The most effective structural deformation to tune the optical response is the variation of the gap between the ridges within the dimer. Forces in $x$-direction either decrease or increase the gap and thereby shift the optical resonance’s wavelength. To study this effect we first compute the respective optical forces of the $TE_{1,1}$ mode with COMSOL Multiphysics [47] using the parameters listed in Table 1. For the finite-elements simulations in frequency domain on Comsol, we take a single 2D unit cell consisting of a dimer surrounded by air on a substrate. We implement plane wave excitation, S-parameter ports at the top and bottom of the unit cells and periodic boundary conditions on the right and left side, so that the grating is considered as infinitely periodic in the x-direction and invariant in the y-direction. Figure 3 shows the field distribution and the optical forces for the optical mode as a function of the structural gap perturbation $\delta /\Lambda$, i.e. the relative difference of gap widths, being $\delta =\lvert d_1 - d_2 \rvert$. The force distribution results in $F_1>F_2$, i.e. a net repulsive force in $x$-direction between the ridges of the dimer. By increasing the structural gap perturbation and hence the asymmetry, the resulting net force $\Delta F=F_1-F_2$ decreases as a result of the interplay of two effects: On the one hand the optical Q-factor scales inversely proportional to $\propto 1/\delta ^2$ [12], and so do the forces $F_1$ and $F_2$. On the other hand increasing the asymmetry the light intensity maximum moves towards the outer parts of the dimer (see contour plots of Fig. 3). Even though this increases the ratio $F_{\mathrm {1}}/F_{\mathrm {2}}$ it does not compensate the effect of the decreasing Q-factor. Thus, the net force is larger for dimers with small gap perturbations.
Fig. 3. Upper part: Computed normalized $E_{\mathrm {y}}$ field component and schematic of the optical force distribution. Lower part: Horizontal forces $F_1$, $F_2$ and net force $\Delta F$ per unit length versus structural gap perturbation $\delta /\Lambda$ as a measure for the supercell asymmetry. $\delta =\lvert d_1 - d_2 \rvert$ is the difference in gap widths where $\delta /\Lambda =0$ represents a symmetric grating with equal gaps and $\delta /\Lambda =0.45$ a grating with the two ridges at zero distance. The remaining geometrical and illumination parameters are those listed in Table 1 for the $\mathrm {TE}_{\mathrm {1,1}}$ mode. $F_1$ and $F_2$ correspond to the left axis, $\Delta F$ to the right one.
3. Light control by optical forces
To compute the structural deformation induced by the optical forces and the resulting back-action on the optical response, we implement a COMSOL simulation coupling two interfaces: one for the calculation of the electromagnetic field and the other one for the calculation of the structural deformation. Additionally, a third interface modeling the deformed mesh is necessary to account for the structural deformation due to the optical forces. Since the structural deformation changes the electromagnetic field distribution, the computation cycle is repeated until a self-consistent solution is achieved. This is the case when the optical forces equal the mechanical restoring forces of the ridge. We compute the back-action for a structure with parameters as in Table 1 and with a gap $d_1=0.18\Lambda$. Figure 4(a) shows the ridge displacement in dependence of a static control beam intensity at the resonance wavelength $\lambda _{1,1}$ of the $TE_{\mathrm {1,1}}$ mode, whose optical resonance has a Q-factor of $6\times 10^4$. The displacement value rises and then starts saturating for increasing beam intensities. For example, for the largest beam intensity in Fig. 4(a) (corresponding to an input power of $160$ mW for a beam radius of $50\,\mathrm {\mu m}$) the optically induced net line force reaches $4.9 \times 10^{-5}$ N/m and each ridge is displaced by $32$ pm, i.e. the gap is increased by $64$ pm.
Fig. 4. (a) Displacement of the ridges and reflectivity for the control beam $\lambda _{1,1}$ and signal beam $\lambda _{1,2}$ as a function of the control beam’s intensity. The displacement curve correspond to the left axis, the reflectivity curves to the right one. (b) and (c) Resonance for the $\mathrm {TE_{1,1}}$ and $\mathrm {TE_{1,2}}$ modes and corresponding $E_{\mathrm {y}}$-components.
3.1 Reflectivity tuning
The mechanical deformation induced by a beam exciting the $\mathrm {TE_{1,1}}$ mode can be used as a control beam to tune the reflectivity of a signal beam. In such control-signal configuration, the control beam has high optical power needed to deform the grating, while the signal with much lower optical power impinges on a deformed grating probing its optical response. Exemplary, we choose a signal beam at the mode $\mathrm {TE_{1,2}}$ with a resonance wavelength of $\lambda =1325.522$ nm for an angle of incidence $\theta =3^\circ$. Figure 4(a) shows the reflectivity for both wavelengths as a function of the control beam intensity and Fig. 4(b) and 4(c) illustrate the corresponding resonance curves and field distributions. The signal beam’s reflectivity is tuned between $85.3\%$ and $4.5\%$ resulting in a tunability range of $80.8\%$ for control beam’s intensities up to $4000\,\mathrm {W/cm^2}$. The background reflectivity of the resonances is a result of the residual reflection from the bilayer grating. It can be minimized around the resonances by tuning the thicknesses $h_1$ and $h_2$ to get a half-wavelength top slab and a quarter-wavelength bottom slab, as was done here for the $\mathrm {TE_{1,2}}$ resonance, and by setting the effective refractive index of the bottom grating to fulfill the matching condition $n^2_{\mathrm {2,eff}}=n_{\mathrm {air}} n_{\mathrm {sub}}$, as much as other other constraints allow it.
The optomechanical interaction can be further enhanced by modulating the optical power of the control beam at a frequency close to one of the mechanical frequencies of the dimerized HCG. The displacement is then proportional to the mechanical quality factor $Q_{\mathrm {mech}}$. With a $Q_{\mathrm {mech}}$ in the range of $10^5$ for small scale structures [48,49], silicon is a suitable material for such devices. For the structure given in Table 1, the resonance frequency of the fundamental mode is 15.3 MHz. This frequency is proportional to the ratio of the effective width to the squared of the total height of single T-shaped ridge, as for a cantilever beam, and can be varied in a frequency range from a few MHz up to hundreds of MHz by changing the geometrical parameters, mainly those of the bottom-grating. The coupling of the optical and the mechanical modes enlarges the tunablility range and also allows optomechanical tuning with even smaller optical Q-factors and lower beam intensities.
Beside a control-beam configuration, the dimerized HCG can also be employed with a single beam as a nonlinear mirror with intensity-dependent reflectivity. With increasing beam intensities, the induced optical forces deform the geometry changing the reflected intensity at the same time. Such a mirror can act as a saturable output coupler (SOC) for generation of short laser pulses [50], where the mirror acts as an output coupler introducing intensity dependent losses in the laser cavity. Since the SOC’s reflectivity should increase with increasing intracavity energy, the dimerized HCG has to provide a band-pass filter transmittance spectrum.
By setting the operation point in the transmittance peak for low intensities, an increasing beam power decreases the transmittance, thus making the mirror reflect more light at higher intensities. To realize a spectral band-pass filter, the dimerized HCG (the geometry given in Table 1) is combined with a distributed Bragg mirror (DBR) at the backside of the structure (see Fig. 5(a)). This configuration forms a Fabry-Perot etalon [51]. The top-grating of the HCG acts as front mirror with abrupt changes in amplitude $R_1$ and phase $\phi _1$ at the BIC resonance (see Fig. 5(b)). The supporting structure acts as a spacer with thickness $h_2$ and a low effective refractive index $n_{\mathrm {2,neff}}$, and the distributed Bragg mirror at the back side provides a moderate reflectivity of $R_{\mathrm {DBR}}=98.8\%$. The reflected intensity $I_{\mathrm {R}}$ of the etalon is given by the Airy function [51]:
Fig. 5. (a) Sketch of saturable output coupler based on dimerized HCG with DBR mirror. (b) Reflection coefficient and phase of the HCG mirror at resonance. (c) Reflectivity as a function of the beam intensity for the etalon with a spacer height $h_2=1055$ nm and wavelength $\lambda _{1,1}=1552.38$ nm of the $\mathrm {TE}_{1,1}$ mode.
3.2 Phase tuning
At resonance, the dimerized HCG shows not only abrupt changes in the reflected intensity but also in the phase (see Fig. 5(b)). The latter effect can be used for phase modulation [52] and thus for some kinds of manipulation of polarization states. A possible realization is the Fabry-Perot configuration from the previous section (see Fig. 5(a)) now operating out of the etalon resonance (e.g. by changing the spacer height to $h_{\mathrm {s}}=1250$ nm), so that the reflectivity remains high around the BIC resonance, while the phase varies between 0 and $2\pi$. To illustrate this, we again use a beam at the $\mathrm {TE}_{1,1}$ mode at $\lambda =1552.389$ nm as control beam and a beam at the $\mathrm {TE}_{1,2}$ mode at $\lambda =1325.55$ nm as signal beam. The signal beam is slightly detuned from resonance and located close to the steepest change of phase to cover a large phase variation. We calculate the signal beam response as a function of the intensity of the control beam. Figure 6 shows how the phase varies up to $1.5\pi$ rad for control beam intensities up to $6000\,\mathrm {W/cm^2}$, while the reflectivity is almost unitary, varying only in the range between $99.52\%$ and $100\%$. Power modulation of the control beam, as explained in Sec. 3.1, enables to expand the tunability range even to $2\pi$ rad.
Fig. 6. Signal beam’s phase tuning as a function of the control beam intensity. The reflectivity’s amplitude is almost unitary.
For a signal beam with linear polarization at $45^\circ$, i.e. with transverse-electric (TE) and transverse-magnetic (TM) components of the same amplitude and same phase, the dimerized grating can also be used as a tunable retarder plate. For gratings with one-dimensional periodicity, TE and TM BIC resonance spectra usually do not overlap. For instance, there is no resonance for TM-polarized light near the $\mathrm {TE}_{1,1}$ mode. Hence both reflectance $R_{\mathrm {TM}}$ and phase $\phi _{\mathrm {TM}}$ are nearly constant near the TE resonance. The etalon tuned out of resonance provides almost unitary reflectivity for both TE- and TM-components and at the same time variable phase at either TE- or TM-resonances. The phase difference of the retarder device is tunable for wavelengths close to the BIC resonance realizing e.g. an optomechanically induced quarter-wave function.
4. Conclusion
We have numerically investigated dimerized bilayer grating structures with high refractive index contrast and high aspect ratio as a promising platform for optomechanically reconfigurable metasurfaces. The dimerized grating exhibits high optical Q-factors related to bound states in the continuum (BICs) providing a large field enhancement and thereby large optical forces, which enable significant mechanical deformations. Deflections in the pm-range are sufficient to systematically manipulate amplitude, phase and polarization of reflected or transmitted light fields. We discuss the application of this structure design for the realization of tunable devices as mirrors, reflective phase modulators and retarder plates, as well as saturable output couplers. By operating the devices close to a mechanical resonance the optomechanical tuning can be further enhanced by a factor proportional to the mechanical Q-factor of the structures. As a particular benefit the bilayer configuration allows us to design the optical and mechanical response nearly independently from each other. The mechanical response depends mostly on the bottom-grating, while the optical response is given mainly by the top-grating, but further optical optimization such as the minimization of the background reflectivity involves the bottom-grating.
Our discussion here focuses on silicon T-shaped structures and two different strategies are feasible for the fabrication of these building blocks. One approach is based on a bilayer fabrication process, first the lower part of the structure is etched and embedded in a polymer, then another silicon layer is deposited and structured, finally the polymer is removed. T-shape building blocks have already been demonstrated in [39]. Another approach uses deep reactive ion etching processes, where gases are used in alternate etching and passivation cycles [53]. This process enables the realization of T-like structural shapes [40]. The dimerization is realized within the lithography process. Beside these strategies for silicon structures the combination of suitable etching processes also allows for other materials, for example, silicon-silica structures by realizing the undercut of the silicon structures with wet chemical etching [39]. Similar techniques may also be feasible in III-V semiconductors, provided they have low absorption and a sufficiently high refractive index of the upper grating to support high-Q resonances. Very recently T-shape structures have also been successfully manufactured in diamond [54] opening a wide range of application wavelengths. Our approach could also be used in the study of collective phenomena in arrays of coupled mechanical resonators,which has attracted great interest in the recent past [55,56]. We believe that our bilayer optomechanical metasurface bringing additionally optical modes into such systems is a promising platform to enhance the variability of dynamics that can be studied.
Funding
Deutsche Forschungsgemeinschaft (EXC-2123 QuantumFrontiers - 390837967).
Acknowledgments
The authors thank Frank Fuchs (Gitterwerk GmbH, Jena, Germany) for providing the RCWA code. C.B.R.H., J.D., F.F.B., S.K. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germanys Excellence Strategy - EXC-2123 QuantumFrontiers - 390837967.
Disclosures
The authors declare no conflicts of interest.
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