Abstract
Light scattering by a twodimensional photoniccrystal slab (PCS) can result in marked interference effects associated with Fano resonances. Such devices offer appealing alternatives to distributed Bragg reflectors and filters for various applications, such as optical wavelength and polarization filters, reflectors, semiconductor lasers, photodetectors, biosensors and nonlinear optical components. Suspended PCS also have natural applications in the field of optomechanics, where the mechanical modes of a suspended slab interact via radiation pressure with the optical field of a highfinesse cavity. The reflectivity and transmission properties of a defectfree suspended PCS around normal incidence can be used to couple outofplane mechanical modes to an optical field by integrating it in a freespace cavity. Here we demonstrate the successful implementation of a PCS reflector on a hightensile stress Si_{3}N_{4} nanomembrane. We illustrate the physical process underlying the high reflectivity by measuring the photoniccrystal band diagram. Moreover, we introduce a clear theoretical description of the membrane scattering properties in the presence of optical losses. By embedding the PCS inside a highfinesse cavity, we fully characterize its optical properties. The spectrally, angular and polarizationresolved measurements demonstrate the wide tunability of the membrane’s reflectivity, from nearly 0 to 99.9470±0.0025%, and show that material absorption is not the main source of optical loss. Moreover, the cavity storage time demonstrated in this work exceeds the mechanical period of loworder mechanical drum modes. This socalled resolvedsideband condition is a prerequisite to achieve quantum control of the mechanical resonator with light.
Introduction
Fano resonances resulting from the interference between a discrete resonance and a broadband continuum of states are ubiquitous in physical science and constitute a unique resource in the field of nanophotonics^{1}. The scattering properties of subwavelength periodic structures can be tailored by engineering their asymmetric Fano lineshapes. These extraordinary and controllable optical properties hold great promise in the context of nanophotonics, where fully integrated photoniccrystal slabs (PCS) can replace traditional Bragg multilayers for various optical components. This approach has been successfully implemented in the context of optical filters^{2}, switches^{3}, sensors^{4}, lasers^{5}, detectors^{6}, slowlight and nonlinear devices^{7}. It has also been proposed as a possible route to reduce thermal noise in the mirrors of largescale gravitationalwave interferometers^{8}.
Moreover, nanophotonics is playing an increasing role in the field of optomechanics by allowing submicronscale mechanical resonators to interact via radiation pressure with highly confined optical fields. A very successful approach consists in colocalizing a photonic and a phononic mode in the small volume of a defect cavity in the plane of a PCS^{9, 10}. In this setup, the mechanical resonator consists of a GHz internal vibrational mode localized in a very small mode volume around the defect cavity. More recently, defectfree suspended PCSs have been used as lightweight mirrors of submicron thickness^{11, 12, 13}. The outofplane vibrational modes of the membrane can be coupled to an optical field by integrating the membrane inside a highfinesse cavity. Because of their very low mass and low mechanical frequency in the MHz range, such resonators are highly susceptible to radiation pressure. Moreover, they are good candidates to achieve hybrid optoelectromechanical transducers^{14, 15} because the outofplane vibrational modes of a suspended PCS can be simultaneously coupled with a nearby interdigitated capacitor. Such a device could have a decisive role in future (quantum) information networks because it allows upconversion of microwave signals onto an optical carrier for subsequent detection or transmission into optical fibers.
Suspended membranes made from thin layers of highstress materials are an advantageous platform for the development of optomechanical PCSs because they present excellent mechanical properties^{16}. For instance, recent experiments have demonstrated Qfactors exceeding 10^{8} with highstress Si_{3}N_{4} films at room^{17, 18} and cryogenic^{19} temperatures. Moreover, on the optical side, a stringent requirement is the resolvedsideband (or good cavity) limit^{20}, in which the cavity storage time exceeds the mechanical period. To date, PCS optomechanical resonators have been demonstrated with onedimensional periodic patterns (often referred to as highcontrast gratings)^{11, 21} or twodimensional (2D) photoniccrystal patterns^{12, 13, 18} that are insensitive to the incident polarization and preserve the mechanical stiffness of the membrane. However, in these experiments, the PCS structures suffered from low optical reflectivity, precluding the operation in the resolvedsideband regime, and in most cases, the PCS was fabricated in a low stress material, preventing high mechanical quality factors.
Here we report the implementation and full optical characterization of a 2D PCS directly etched on a hightensile stress 200 nmthick Si_{3}N_{4} nanomembrane. We reached the resolvedsideband condition for the first time with a PCS used as the end mirror of a Fabry–Perot cavity. The cavity finesse of 6390±150 is, to the best of our knowledge, an unprecedented value with freespace PCS cavities. The inferred membrane reflectivity is given by 1−R=530±25 ppm. Moreover, we measured the band diagram of the guided resonances responsible for the Fano resonances in the PCS with an angleresolved spectrometer. In contrast with previous studies conducted in the context of PCS optical filters^{22} and reflectors^{23}, we give a physical interpretation of the observed band diagram and shed new light on the physics of these structures. Finally, we introduce a theoretical description of the scattering properties of the PCS in the presence of optical losses and demonstrate that the current reflectivity is not limited by material absorption. Demonstrating highreflectivity PCS in a hightensile stress film, such as Si_{3}N_{4}, as evidenced here could also be beneficial in fields other than optomechanics, such as the generation of tunable optical filters^{24} or optomechanical lasers^{5}.
Materials and methods
Membrane and photoniccrystal fabrication
The commercial membranes^{25} are made of a 200nm stoichiometric Si_{3}N_{4} film deposited on a Si substrate and released over a 1 × 1 mm square by chemical etching of the substrate (Figure 1a). The highstress in the film ensures mechanical modes with quality factors as high as 10^{8} at low temperature^{19} and ultralow mass^{26}. The photonic crystal is realized in the center of the membrane. It is obtained by electronbeam lithography on a layer of 200 nm of polymethyl methacrylate (PMMA) resist, followed by reactive ion etching with a plasma of CHF_{3} and SF_{6}. The PMMA is then stripped off with oxygen plasma. The photonic crystal consists of a 270μm diameter disk patterned with a square lattice of circular holes (Figure 1). The lattice parameter is Λ=830 nm, and the design value for the hole radius is 293 nm.
PCS characterization
Band diagram measurement
We implemented the whitelight illumination and imaging setup shown in Figure 2a to measure the PCS band diagram. A white halogen lamp is combined with a set of three lenses (L_{1}, L_{2} and L_{3}) and two diaphragms (D_{1} and D_{2}) to obtain a Köhler illumination configuration. A field diaphragm (D_{1}) conjugated to the PCS restricts the illumination to the PCS area, thus reducing the impact of light scattered on the optical elements and mounts. The aperture diaphragm D_{2} is used to select the incident beam angles at the PCS position. It is centered on the optical axis, and its aperture width is chosen to reject high angleofincidence components that are irrelevant for this experiment. A magnified image of the PCS is obtained with a microscope objective (L_{4}) and a lens (L_{5}). A field diaphragm (D_{3}) sits at the image plane, conjugated to diaphragm D_{1}, providing additional filtering of unwanted scattered light. A fiber tip is positioned in the focal plane of lens L_{6}, that is, in the Fourier plane of this imaging system. The light collected by the fiber is guided to a nearinfrared spectrometer. By scanning the fiber in the transverse (x, y) plane, the transmission spectrum can be measured at different angles. Moreover, a polarizer (Pol) is introduced between L_{1} and L_{2} for polarization selection.
PCS reflection, transmission and loss measurements
We implemented a singleended cavity with the PCS as the end mirror (Figure 2b) to characterize its reflectivity, transmission and losses. The input mirror of the cavity has a 20mm radius of curvature, and the cavity length is 17.4 mm. This ensures a waist size w_{0}=48 μm on the PCS, smaller than the lateral extent of the photonic crystal, and an angular dispersion rad that has a negligible influence on the optical properties of the PCS. A motorized Littman–Metcalf stabilized diode laser system from Sacher Lasertechnik GmbH (Marburg, Germany) (Lion TEC520) is used to probe the cavity. The cavity transmission is monitored with an avalanche photodiode (Thorlabs, Newton, NJ, USA, APD110C) and is recorded continuously, while the laser wavelength is scanned over 4.8 nm. Simultaneously, the fringes of an imbalanced fiber interferometer are counted by a digital system (based on an Arduino microcontroller^{27}) to measure the laser frequency in real time. The freespectral range, bandwidth and visibility of each peak are extracted from Lorentzian fits.
Description and characterization of the PCS losses
In the following, we provide a complete theoretical description of the PCS optical properties in the presence of losses. The membrane can be described by its scattering matrix, transforming the fields incoming from left and right into outgoing fields
(Figure 2b):
where the symmetry of the membrane with respect to the xOy plane is used to assume that the direct (r,t) and reciprocal (r̄, t̄) complex reflection and transmission coefficients are equal. The previous assumption allows S to be diagonalized
For a lossless membrane, S is unitary (r+t=r−t=1), such that within a global phase factor, it is entirely determined by a single real parameter, for instance the transmission intensity T=t^{2}. However, in the presence of losses, the scattered fields have a total intensity smaller than the input fields, and both eigenvalues obey
Three real parameters (plus a global phase term) are required to fully describe S: in addition to the transmission intensity T, we introduce the optical losses
associated with the symmetric and antisymmetric combinations of the left and right incoming fields . To obtain a measurement of L_{±}, the membrane can be positioned in the center of a highfinesse cavity. This setup, initially proposed by the group of J. Harris^{28}, consists of two singleended cavities coupled via the transmission of the membrane. If the input and output mirrors have the same reflectivity amplitude r' and if both subcavities have the same length l, one obtains, in the absence of fields incident from outside the cavity (Figure 2b):
where is the wavenumber of the propagating field. The eigenmodes of the total optical cavity correspond to nontrivial solutions of this system for complex angular frequencies . By taking the sum and difference of the previous equations, we obtain
Nontrivial solutions of this system correspond to symmetric (+) (antisymmetric (−)) modes with angular frequency ω_{p,+} (ω_{p,−}) and damping (full width at half maximum) γ_{+} (γ_{−}) given by
where p is an integer corresponding to the longitudinal mode number. From Equation (10), the frequency splitting between the symmetric and antisymmetric modes of identical longitudinal numbers is given by:
Finally, if the membrane losses L_{±} and mirror losses are small, Equation (11) simplifies to
Hence, the damping of the symmetric and antisymmetric modes gives direct access to the two parameters L_{+} and L_{−}.
To perform this measurement, a 32mm long cavity is formed by two mirrors with transmissionlimited losses L′=455 ppm and radii of curvature 20 mm. The PCS is set in the center of the cavity on a threeaxis translation stage. To align the cavity axis with respect to the PCS, the end mirror is also mounted on a twoaxis translation stage. Finally, the PCS frame is glued on a piezoelectric stack to allow automatic scans of its z position over several free spectral ranges.
Results and discussion
Band diagram of the PCS
The physics of highcontrast gratings has previously been described in detail^{29, 30}. Here we extend it to the case of a PCS, following earlier work^{13, 31}. The highreflection properties of the PCS result from Fano resonances arising from an interference between the direct reflection from the PCS and light leaking out from internal highQ guided modes in the PCS. To unveil the band structure of the guided modes, we calculate the transmission of the PCS by rigorous coupled wave analysis (RCWA)^{32}, which is suitable for periodic structures, such as our square lattice of holes (lattice parameter Λ=830 nm, membrane thickness h=213 nm and hole radius r_{h}=276 nm). We begin the analysis with the unpatterned membrane limit, for which the dielectric constant of both the membrane material and the holes is set to an average value The field solutions are guided modes, referred to as ‘unperturbed modes’ in the following discussion. They are plane waves reflected by total internal reflection upon the two faces of the membrane. In this regime, the unperturbed modes are evanescent outside of the membrane: the z component of their wavevector fulfils inside the membrane and in the surrounding medium. TE (TM) modes have their electric (magnetic) field within the plane of the membrane and are represented in Figure 3 in red (blue). The eight modes are sorted according to their symmetry class with respect to the xOz plane. Symmetric modes (TM(±1, 0)) are not excited under sillumination, whereas antisymmetric modes (TE(±1, 0)) are not excited under pillumination. The complex coefficients describing the reflection inside the membrane for the electric field are given by Fresnel’s law
We have numerically solved the roundtrip resonance condition within the membrane
for j=TE or TM and for the lowest possible value of , (where are the diffraction orders in the x and y directions, respectively). Note that the periodicity Λ is not a property of the unpatterned membrane and is arbitrarily chosen to match that of the patterned membrane. Given the phases of r_{TE} and r_{TM}, we find that TE modes have a lower fundamental resonance frequency than TM modes. Finally, the symmetry of the mode with respect to the xOy plane is given by the sign of the half roundtrip propagation coefficient
From the expressions of r_{TE} and r_{TM}, it follows that the fundamental TE (TM) mode is antisymmetric (symmetric) with respect to xOy. The calculated mode frequencies as a function of transverse wavevector k_{x} are represented by the lines in Figure 3c and 3d. To first order, the frequency of the modes diffracted along the y direction is not affected by the transverse k_{x} component, contrary to the modes diffracted along the x direction.
The results of the RCWA for the patterned membrane are superimposed as a color plot in Figure 3c and 3d (the refractive index of Si_{3}N_{4} is given by^{33} . In this case, the modes described in the unpatterned case are now coupled by diffraction, giving rise to avoided crossings. In addition, the reflections at the membrane/air interfaces couple the guided modes to the zerothorder modes propagating in freespace, leading to an effective loss channel. The guided resonances acquire a finite lifetime inversely proportional to the width of the Fano resonance (cut along a fixed value of k_{x} in Figure 3). The lifetime of the TM modes is ~20 times that of the TE modes. At normal incidence, the yOz plane constitutes an additional plane of symmetry (antisymmetry) in the s (p) polarization, and, hence, only the combinations TE(1, 0)+TE(−1, 0) and TM (0,1)+TM(0, −1) (respective TE(0, 1)+TE (0, −1) and TM(1, 0)+TM(−1, 0)) are effectively coupled to freespace modes: the linewidth of the other Fano resonances vanishes close to k_{x}=0. The whitelight angleresolved spectrometer described previously allowed us to measure the band structure of the guided modes for small transverse wavevectors k_{x}. The experimental transmission with s (p) illumination is presented in Figure 3e and 3f, together with three different cuts with incidence angles 3.57°, 1.23° and 0.33°. The simulation is fitted to the data with the hole radius and membrane thickness as free parameters. The former is only specified to within 10% accuracy, and the latter may differ from the design value due to imperfect pattern transfer on electronbeam lithography and etching. The fitted values r_{h}=276 nm and h=213 nm are in good agreement with the design values. The theoretical fits are represented in blue in Figure 3g and 3h. The reduced visibility of the peaks, particularly pronounced with the TM modes, is accounted for by convolving the theoretical map with the 4nm wide response of the spectrometer. In the next section, we analyze in greater detail the TE Fano resonance responsible for the largest dip in transmission close to v=c × 0.929 μm^{−1}=c/(1.076 μm) in Figure 3g and 3h.
Detailed characterization of a Fano resonance
For optical frequencies v=c/λ close to a guided mode resonance, the PCS complex reflection and transmission coefficients r and t are given in good approximation by the simple twomode model^{31}
where r_{d} and t_{d} are the offresonant coefficients , λ_{0} is the wavelength of the guided mode resonance, γ is the resonance width and the plus/minus sign in Equation (15) reflects the symmetry of the guided mode: as discussed in the previous section, the lowestorder TE mode is antisymmetric with respect to the xOy plane; thus, the minus sign applies. In the absence of optical losses, a striking feature of this Fano lineshape is the existence of a frequency λ_{1}=λ_{0}+iγr_{d}/t_{d} for which the PCS transmission vanishes: , and, accordingly, the reflectivity tends to unity: .
By fitting the RCWA results with Equations (15) and (16) around the largest TE Fano resonance, we determine λ_{1}=1076 nm and γ=12 nm. To study the optical behavior of the PCS around λ_{1}, a singleended optical cavity of length l_{0} is achieved, using the PCS as the end mirror (Figure 2). The transmission of the cavity is monitored, whereas a tunable diode laser is swept across the Fano resonance. The recorded signal, shown in Figure 4a, displays a comb of peaks spaced by the freespectral range of the cavity given by
The linewidth Δλ_{n}(λ) of each longitudinal cavity mode n is then obtained by fitting a Lorentzian profile to each of the individual peaks. To minimize the impact of scanspeed fluctuations, we generate 50 MHz sidebands in the laser spectrum with an electrooptic modulator for frequency calibration of the instantaneous scanning speed. Denoting the finesse of the cavity by F(λ), the total roundtrip loss of the cavity is given by. The results are plotted in blue in Figure 4b as a function of the laser wavelength. The lowest level of Γ_{RTL}=(985±25) ppm, corresponding to a finesse of 6385±25 was found at λ_{1}=1070.9 nm. The 5.1 nm discrepancy with respect to the results of Figure 3 is quite small considering that two different membranes from the same microfabrication run were used in these experiments. In the simulations, it is accounted for by changing the hole radius from r_{h}=276 nm to r_{h}=285 nm. Taking into account the 455 ppm transmission of the input mirror, we infer a membrane reflectivity as good as 1−R(λ_{1})=530 ppm.
To discriminate between the contributions of the membrane transmission and the optical losses L(λ)=1−R(λ)−T(λ) in Γ_{RTL}(λ), we use the extra information embedded in the height of the transmitted peaks. The transmission of the cavity at resonance with the nth mode is given by
where T_{c} is the transmission of the input mirror, assumed to be constant over the wavelength range scanned here. In the experiment, a signal proportional to is measured, so that we can deduce from Equation (18) within a multiplicative constant α. By assuming that the losses L(λ) are within the interval at the highest accessible wavelength, we obtain a confidence interval for α and hence for T(λ)=Γ_{RTL}(λ)−L(λ) (in green in Figure 4c and 4d) and L(λ) (in red) over the whole wavelength range. For comparison, the transmission calculated by RCWA is displayed by a green solid line in Figure 4c and 4d. The agreement between RCWA and the experimental results is excellent. At the Fano resonance λ_{1}=1070.9 nm, the roundtrip damping is vastly dominated by membrane losses. The lowest cavity bandwidth κ/2=c/4l_{0}F(λ_{1})=675 kHz measured in this singleended cavity setup approaches the resolvedsideband condition for the firstorder flexure modes of the suspended membrane.
To understand the exact form of the scattering matrix and to gain insight into the origin of the losses observed in the previous experiment, a PCS is positioned in the middle of a highfinesse cavity. The PCS position Δl is scanned along the cavity axis with a piezoelectric actuator. In this configuration, the PCS couples two cavities of length l_{1}+Δl/2 and l_{2}−Δl/2, where l_{1}≈l_{2}≈l are the initial lengths of the first and second subcavities, which are assumed to be simultaneously resonant with the central laser wavelength λ_{c} for Δl=0. The cavity transmission, recorded as a function of Δl and laser detuning Δv ∝ λ_{c}−λ, is represented in the color plots of Figure 5a–5c. The avoided crossing of the symmetric and antisymmetric modes is visible in the center of the figure. From a Lorentzian fit of the peaks at the avoided crossing and along the vertical axis, we extract the frequency spacing Δv, and the linewidth γ_{±} of the symmetric and antisymmetric modes. The experiment is then reproduced for several laser wavelengths λ_{c} close to the Fano resonance, and the corresponding frequency spacings and linewidths are shown in Figure 5d and 5e, respectively.
In Figure 5a and 5c, the upper and lower branches of the avoided crossing are strongly asymmetric. This is a direct consequence of the difference between L_{+} and L_{−}. For λ_{c}<λ_{1} (Figure 5a), the phase of (r−t)/(r+t) is positive, such that from Equation (12), the symmetric mode (angular frequency ω_{+}) constitutes the upper branch of the crossing, whereas the antisymmetric mode (angular frequency ω_{−}) is the lower branch. On the other hand, for λ_{c}>λ_{1} (Figure 5c), Δv=(ω_{+}−ω_{−})/2π changes its sign and the upper and lower branches are exchanged. Finally, at the Fano resonance λ_{c}=λ_{1}, r and t are in phase, so the symmetric and antisymmetric modes are perfectly degenerate. Combined with the inequalities of Equation (3), an upper bound for the transmission of the membrane can be derived: at the Fano resonance. For 1−R=985 ppm, this corresponds to a transmission as low as T≤0.2 ppm. Hence, the residual transmission of 100 ppm observed in Figure 4 is in contradiction with the twoport description developed in this work. A first explanation is the possible scattering into nonoverlapping spatial modes due to surface roughness. Moreover, given our experimental parameters, we expect from RCWA that the wavevector components of the input Gaussian beam not aligned with a symmetry plane of the membrane should give rise to ≈20 ppm transmission into the polarization orthogonal to the input beam (with a spatial distribution that is antisymmetric with respect to the xOz and yOz planes). Nevertheless, as shown in Figure 4, the residual transmission only accounts for a small fraction of the total cavity losses close to the Fano resonance, and the nature of these losses remains to be elucidated.
The difference between the losses L_{±} experienced by the symmetric and antisymmetric modes is captured by a simple phenomenological model; because the Fano resonance results from the interference between direct PCS transmission and a highQ guided resonance, it can be safely assumed that the losses mostly affect the former. Hence, Equations (15) and (16) are modified to include a phenomenological loss rate in the highQ response as
Using this model, it follows that
thus, only the antisymmetric mode of the twocavity setup is affected by the optical losses, with a value proportional to the Lorentzian response of the guided resonance. This is an intuitive result because the antisymmetric guided resonance under study is not driven by a symmetric incoming field (and vice versa for the symmetric TM resonance). To quantitatively evaluate the model, Figure 5e shows the total expected losses , where nm is the only adjustable parameter because λ_{1}=1079.1 nm, γ=12 nm and are independently determined. Therefore, optical losses are found to account for <1‰ of the total decay rate of the guided mode resonance. The previous model shows that the asymmetry between L_{+} and L_{−} is a direct consequence of the physics of Fano resonances and is not specific to a particular loss mechanism, such as absorption or scattering, because both would preferentially affect the highQ guided mode. This finding is in stark contrast with a model recently proposed in Ref. 21, where such an asymmetry was attributed to absorption in the material. Moreover, we show in the following that absorption should have a negligible influence on the optical loss of our PCS. If the losses observed in our experiment were entirely due to optical absorption, the value Im(n)=(3.5±1.5) × 10^{−5} could be derived for the absorption of the Si_{3}N_{4} layer. The values of obtained by RCWA simulation, including an absorption of 2 × 10^{−5}≤Im(n)≤5 × 10^{−5} are located within the shaded areas in Figure 5e. Because previous experiments^{34} have found values for the absorption of a similar layer that are an order of magnitude smaller (Im(n)=2.0 × 10^{−6}), we conclude that scattering is the main source of losses. In the singleended cavity case, where the field is essentially incident on one side of the membrane, the relevant optical losses are given by L=(L_{+}+L_{−})/2. Thus, the decrease of optical losses with optical wavelength visible in Figure 4b is consistent with Equations (19) and (20). With the previous smaller value of Si_{3}N_{4} absorption, RCWA simulation shows that a reflectivity as high as R≈1–80 ppm, corresponding to a finesse close to 80 000, could be achieved by improving the PCS fabrication process.
Conclusions
Highstress Si_{3}N_{4} films have been successfully patterned to achieve a 2D PCS and have provided a detailed study of the rich mode structure underlying the high reflectivity of the crystal. We have fully characterized the scattering and loss properties of the device. We have also experimentally demonstrated the wide tunability of the membrane reflectivity via adjustment of the laser wavelength close to a Fano resonance. Such tunability is highly desirable in the context of membraneinthemiddle optomechanics experiments. In particular, the ability to reach a very narrow energy gap between the eigenmodes of the coupled cavities can be used to enter the regime where the photon exchange time becomes comparable to the mechanical period. This would enable the observation of mechanically induced coherent photon dynamics, such as Autler–Townes splitting or Landau–Zener–Stueckelberg dynamics^{35}. Finally, the bandwidth of the singleended cavity described in this work, as low as 675 kHz, is in the resolved sideband regime for the firstorder harmonics of the drum mode, ranging from 596 kHz for the (0, 1) mode to 1.2 MHz for the (2, 2) mode used in previous quantum optomechanics experiments^{16}. Hence, this technique results in devices suitable for the development of optomechanical transducers.
References
Zhou WD, Zhao DY, Shuai YC, Yang HJ, Chuwongin S et al. Progress in 2D photonic crystal Fano resonance photonics. Prog Quantum Electron 2014; 38: 1–74.
Lin CC, Lu ZL, Shi SY, Jin G, Prather DW . Experimentally demonstrated filters based on guided resonance of photoniccrystal films. Appl Phys Lett 2005; 87: 091102.
Yacomotti AM, Raineri F, Vecchi G, Monnier P, Raj R et al. Alloptical bistable bandedge Bloch modes in a twodimensional photonic crystal. Appl Phys Lett 2006; 88: 231107.
Magnusson R, Wawro D, Zimmerman S, Ding YW . Resonant photonic biosensors with polarizationbased multiparametric discrimination in each channel. Sensors 2011; 11: 1476–1488.
Yang WJ, Adair Gerke S, Wei Ng K, Rao Y, Chase C et al. Laser optomechanics. Sci Rep 2015; 5: 13700.
Yanik MF, Fan SH . Stopping and storing light coherently. Phys Rev A 2005; 71: 013803.
Yacomotti AM, Monnier P, Raineri F, Ben Bakir B, Seassal C et al. Fast thermooptical excitability in a twodimensional photonic crystal. Phys Rev Lett 2006; 97: 143904.
Friedrich D, Barr BW, Brückner F, Hild S, Nelson J et al. Waveguide grating mirror in a fully suspended 10 meter FabryPerot cavity. Opt Express 2011; 19: 14955–14963.
Gavartin E, Braive R, Sagnes I, Arcizet O, Beveratos A et al. Optomechanical coupling in a twodimensional photonic crystal defect cavity. Phys Rev Lett 2011; 106: 203902.
Alegre TPM, SafaviNaeini A, Winger M, Painter O . Quasitwodimensional optomechanical crystals with a complete phononic bandgap. Opt Express 2011; 19: 5658–5669.
Kemiktarak U, Metcalfe M, Durand M, Lawall J . Mechanically compliant grating reflectors for optomechanics. Appl Phys Lett 2012; 100: 061124.
Bui CH, Zheng JJ, Hoch SW, Lee LYT, Harris JGE et al. Highreflectivity, highQ micromechanical membranes via guided resonances for enhanced optomechanical coupling. Appl Phys Lett 2012; 100: 021110.
Makles K, Antoni T, Kuhn AG, Deléglise S, Briant T et al. 2D photoniccrystal optomechanical nanoresonator. Opt Lett 2015; 40: 174–177.
Andrews RW, Peterson RW, Purdy TP, Cicak K, Simmonds RW et al. Bidirectional and efficient conversion between microwave and optical light. Nat Phys 2014; 10: 321–326.
Bagci T, Simonsen A, Schmid S, Villanueva LG, Zeuthen E et al. Optical detection of radio waves through a nanomechanical transducer. Nature 2014; 507: 81–85.
Purdy TP, Peterson RW, Regal CA . Observation of radiation pressure shot noise on a macroscopic object. Science 2013; 339: 801–804.
Reinhardt C, Müller T, Bourassa A, Sankey JC . Ultralownoise SiN trampoline MEMS for sensing and optomechanics. Phys Rev X 2016; 6: 021001.
Norte RA, Moura JP, Gröblacher S . Mechanical resonators for quantum optomechanics experiments at room temperature. Phys Rev Lett 2016; 116: 147202.
Yuan MY, Cohen MA, Steele GA . Silicon nitride membrane resonators at millikelvin temperatures with quality factors exceeding 10^{8}. Appl Phys Lett 2015; 107: 263501.
Schliesser A, Rivière R, Anetsberger G, Arcizet O, Kippenberg TJ . Resolvedsideband cooling of a micromechanical oscillator. Nat Phys 2008; 4: 415–419.
Stambaugh C, Xu HT, Kemiktarak U, Taylor J, Lawall J . From membraneinthemiddle to mirrorinthemiddle with a highreflectivity subwavelength grating. Annal Phys 2015; 527: 81–88.
Qiang ZX, Chen L, Yang HJ, Pang HQ, Ma ZQ et al. Fano filter modal analysis based on transferred silicon nanomembranes on plastic substrates. Appl Phys Lett 2008; 93: 061106.
Singh Chadha A, Zhao DY, Chuwongin S, Ma ZQ, Zhou WD . Polarization and angledependent characteristics in two dimensional photonic crystal membrane reflectors. Appl Phys Lett 2013; 103: 211107.
Stomeo T, Grande M, Rainò G, Passaseo A, D’Orazio A et al. Optical filter based on two coupled PhC GaAsmembranes. Opt Lett 2010; 35: 411–413.
Norcada Inc., Edmonton, AB, Canada. Available at http://www.norcada.com.
Purdy TP, Peterson RW, Yu PL, Regal CA . Cavity optomechanics with Si3N4 membranes at cryogenic temperatures. New J Phys 2012; 14: 115021.
Arduino. Available at http://www.arduino.cc.
Thompson JD, Zwickl BM, Jayich AM, Marquardt F, Girvin SM et al. Strong dispersive coupling of a highfinesse cavity to a micromechanical membrane. Nature 2008; 452: 72–75.
Karagodsky V, Sedgwick FG, ChangHasnain CJ . Theoretical analysis of subwavelength high contrast grating reflectors. Opt Express 2010; 18: 16973–16988.
Karagodsky V, ChangHasnain CJ . Physics of nearwavelength high contrast gratings. Opt Express 2012; 20: 10888–10895.
Fan SH, Joannopoulos JD . Analysis of guided resonances in photonic crystal slabs. Phys Rev B 2002; 65: 235112.
Liu V, Fan SH . S4: A free electromagnetic solver for layered periodic structures. Comp Phys Commun 2012; 183: 2233–2244.
Philipp HR . Optical properties of silicon nitride. J Electrochem Soc 1973; 120: 295–300.
Serra E, Bawaj M, Borrielli A, Di Giuseppe G, Forte S et al. Microfabrication of large area high stress silicon nitride membranes for optomechanical applications. AIP Adv 2016; 6: 065004.
Heinrich G, Harris JGE, Marquardt F . Photon shuttle: LandauZenerStückelberg dynamics in an optomechanical system. Phys Rev A 2010; 81: 011801.
Acknowledgements
We acknowledge M Corcos, I Krasnokutska, X Pennehouat and L Pinol for preliminary experimental developments. We thank J Palomo and M Rosticher for developing the first steps of the microfabrication process. Finally, we thank J Lawall and R Guérout for insightful scientific discussions. This research has been partially funded by the Agence Nationale de la Recherche programs ‘ANR2011BS04029 MiNOToRe’ and ‘ANR14CE260002 QuNaT’, the Marie Curie Initial Training Network ‘cQOM’, and the DIM nanoK IledeFrance program ‘NanoMecAtom’. SC is supported by the Marie SklodowskaCurie Individual Fellowship program, XC is supported by a fellowship ‘Research in Paris’ from the city of Paris.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/byncsa/4.0/
About this article
Cite this article
Chen, X., Chardin, C., Makles, K. et al. Highfinesse Fabry–Perot cavities with bidimensional Si_{3}N_{4} photoniccrystal slabs. Light Sci Appl 6, e16190 (2017). https://doi.org/10.1038/lsa.2016.190
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/lsa.2016.190
Keywords
 Fano resonance
 highreflectivity membrane
 lowloss optical reflector
 nanomembrane resonator
 photoniccrystal slab
 quantum optomechanics
This article is cited by

Multibeam ultrafast laser processing of freestanding nanofilms
Applied Physics A (2023)

Spinpreserving chiral photonic crystal mirror
Light: Science & Applications (2020)

New design for highly durable infraredreflective coatings
Light: Science & Applications (2017)