Suspended photonic crystal membranes in AlGaAs heterostructures for integrated multi-element optomechanics

We present high-reflectivity mechanical resonators fabricated from AlGaAs heterostructures for use in free-space optical cavities operating in the telecom wavelength regime. The mechanical resonators are fabricated in single- and doubly layer slabs of GaAs and patterned with a photonic crystal to increase their out-of-plane reflectivity. Characterization of the mechanical modes reveals residual tensile stress in the GaAs device layer. This stress results in higher mechanical frequencies than in unstressed GaAs and can be used for strain engineering of mechanical dissipation. Simultaneously, we find that the finite waist of the incident optical beam leads to a dip in the reflectance spectrum. This feature originates from coupling to a guided resonance of the photonic crystal, an effect that must be taken into account when designing slabs of finite size. The single- and sub-$\upmu$m-spaced double-layer devices demonstrated here can be directly fabricated on top of a distributed Bragg reflector mirror in the same material platform. Such a platform opens a new route for realizing integrated multi-element cavity optomechanical devices and optomechanical microcavities on chip.

We present high-reflectivity mechanical resonators fabricated from AlGaAs heterostructures for use in freespace optical cavities operating in the telecom wavelength regime. The mechanical resonators are fabricated in single-and doubly layer slabs of GaAs and patterned with a photonic crystal to increase their out-of-plane reflectivity. Characterization of the mechanical modes reveals residual tensile stress in the GaAs device layer. This stress results in higher mechanical frequencies than in unstressed GaAs and can be used for strain engineering of mechanical dissipation. Simultaneously, we find that the finite waist of the incident optical beam leads to a dip in the reflectance spectrum. This feature originates from coupling to a guided resonance of the photonic crystal, an effect that must be taken into account when designing slabs of finite size. The single-and sub-µm-spaced double-layer devices demonstrated here can be directly fabricated on top of a distributed Bragg reflector mirror in the same material platform. Such a platform opens a new route for realizing integrated multi-element cavity optomechanical devices and optomechanical microcavities on chip.
Cavity optomechanical devices explore the interaction between light and mechanical resonators in a cavity 1 and rely on strongly coupled, high-quality optical and mechanical resonators. When several independent mechanical resonators are coupled to a single cavity field, one is in the realm of multi-element optomechanics 2, 3 , which has been proposed as a route to reach the elusive single-photon strong optomechanical coupling regime 4,5 . Recent experiments along these lines [6][7][8][9] have used SiN membranes placed in free-space optical cavities and require precise alignment of their tilt angle and position and, additionally, a uniformity of the mechanical and optical properties of individual membranes.
Using III-V heterostructures such as AlGaAs would allow for the realization of a multi-element cavity optomechanical system in a fully integrated approach 10,11 . A heterostructure can integrate one of the cavity mirrors via a distributed Bragg reflector together with an array of near-uniform mechanical resonators on a single wafer [10][11][12] , and can even be combined with micromirrors on an independent chip 13 . In particular, the III-V materials system has already been used to realize (opto)mechanical systems in, e.g., (Al)GaAs [14][15][16][17][18][19][20][21][22][23][24][25] or In(Ga)P 11,26-28 . These crystalline materials have been shown to be of high optical 11,29 and mechanical quality 11,20,24 as required for cavity optomechanics. Further device functionalization based on the piezoelectricity of III-V materials or by embedding quantum emitters can lead to versatile nano-electro-optomechanical systems 30 . In this Letter, we demonstrate the fabrication and characterization of integrated single-and doublelayer, high-reflectivity mechanical resonators in AlGaAs heterostructures 31 . The mechanical resonators are fabricated in 100 nm-thin GaAs membranes, which are grown on top of sacrificial AlGaAs layers. This allows us, for example, fabrication of double-layer devices with sub-µm spacing, which is crucial for reaching high coupling strengths in multi-element optomechanics 2,3 . We engineer mechanical resonators of free-free-type geometry 11,17 and characterise the mechanical properties of single-layer devices. We demonstrate control over their out-of-plane optical reflectivity in the telecom wavelength regime by patterning a photonic crystal (PhC) into the GaAs membranes 32 , as has been demonstrated in optomechanics 27,33-37 and optical communication technologies [38][39][40] . Our devices constitute a significant step towards the realization of an array of nearuniform mechanical resonators integrated in a free-space, fully chip-based cavity optomechanical device.
The mechanically-compliant PhC slabs are fabricated in an AlGaAs heterostructure that is epitaxially grown on a GaAs substrate using molecular-beam epitaxy. We fabricated devices from two different wafers. The heterostructure of the first wafer is used for single-layer mechanical resonators [ Fig. 1(a)]. It consists of a GaAs device layer of 100 nm thickness grown on top of a 4 µmthick Al 0.65 Ga 0.35 As sacrificial layer. The AlGaAs layer exhibited a large peak-to-peak surface height variation of 15 nm that is partially smoothened by the top GaAs layer to 10 nm.
The second wafer is designed for fabricating subµm spaced, double-layer GaAs mechanical devices  [ Fig. 1(b)], each of 100 nm thickness on top of a 729 nm Al 0.625 Ga 0.375 As sacrificial layer, which defines the spacing between the two mechanical resonators. These Al-GaAs layers were grown with growth interruption, 41 yielding a surface height variation and roughness smaller than 1 nm and 0.2 nm, respectively. We used standard AlGaAs heterostructure 39,42 microfabrication techniques to define the patterned mechanical resonators and their subsequent release (see the Appendix for detailed information).
Our mechanical resonators are engineered with a freefree-type geometry 17 , where the suspended slab is of rectangular shape and held by four tethers at the nodes of the free-free oscillation mode 18 [see Fig. 2(b)]. We characterized first the mechanical properties of the slabs, focusing on the mode shapes and the corresponding eigenfrequencies and quality factors. To this end, we detected the outof-plane displacement of the slab via optical homodyne interferometry at room temperature in a high vacuum (∼ 5 × 10 −5 mbar)-for details of the setup we refer the reader to the Appendix. Fig. 2(a) shows a typical displacement noise power spectrum of a free-free-type PhC slab with a size of 50 × 50 × 0.1 µm 3 . The fundamental mode lies at 80 kHz and the free-free mode at 178 kHz. Mechanical mode tomography 43 enabled us to compare the measured mode shape of the device to finite element modeling (FEM) simulations 44 , see Fig. 2(b). We find good agreement between experimental and FEM data when accounting for a tensile stress of 10 MPa in the GaAs device layer.
We attribute the residual tensile stress to a mismatch between the lattice constants of the AlGaAs sacrificial and the GaAs device epilayers. The AlGaAs grown on the GaAs substrate relaxes to its native lattice constant if the layer thickness exceeds the critical thickness of 0.33 µm or 30 µm, according to Ref. 45 Figure 2. Characterization of mechanical modes of a free-freetype PhC slab. (a) Noise power spectrum (NPS) of the thermally driven mechanical motion (red) with mechanical modes labeled 1 to 6. (b) Mechanical-mode tomography of the same device along with FEM-simulated mode shapes and their frequencies. Scale bar: 10 µm. Note that the device boundary inferred from mode tomography is largely determined by the rectangular 50 × 38 µm 2 PhC area that reflects more light than the non-patterned part and, thus, leads to an apparent deviation from the square shape of the slab. (c) FEMsimulated von Mises stress distribution of the device.
tively (see also the Appendix). These predictions differ by two orders of magnitude such that the AlGaAs layer can be in a state between fully relaxed and fully strained, depending on the model used 45,46 . The GaAs device layer of 100 nm is thinner than its critical thickness and, thus, adapts to the lattice constant of the AlGaAs layer in any case. Then, the GaAs device layer can exhibit a tensile stress of between 0 and 77.5 MPa. In Fig. 3(a) we examine the effect of tensile stress in the GaAs device layer on the eigenfrequencies of the suspended slab. We observe that the frequencies increase with stress and find a match between data and FEM for a stress around 10 MPa. Upon removal of the sacrificial AlGaAs layer, an anisotropic stress distribution develops in the suspended GaAs slab, as shown in Fig. 2(c). We also observe buckling of the slabs 47 and a static deformation 48 , see Fig. 1(c). We conclude, therefore, that the GaAs layer exhibits residual tensile stress induced by the underlying AlGaAs layer, as was observed in other GaAs on AlGaAs resonators 20,47,48 .
Spatial variations of the resonator geometry 47 or defect-driven material anisotropy 28 also influence the mechanical properties. While analysis of the latter is beyond the scope of our work, the former can be caused by growth-related thickness variation or microfabricationinduced changes. Geometry variations influence the mode-dependent oscillating mass of the resonator and, thus, its eigenfrequencies. We account for the geometry of the devices in FEM with the simplifying assumption of a constant GaAs layer thickness. Fig. 3(b) shows frequencies for devices of various slab length and width. For a GaAs layer thickness of 105 nm, we find good agreement between measured and FEM-simulated frequencies (see the Appendix for detailed simulation results). Thus, residual tensile stress in the GaAs layer and the simplifying assumption of its constant thickness yields a reason- able explanation for the observed mechanical frequencies of the suspended PhC slabs. The mechanical quality factor, Q, is an important figure of merit for (opto)mechanical devices that we determined using ringdown measurements. We find that devices fabricated from the first wafer have quality factors just below 10 5 and similar devices from the second wafer reach 5 × 10 5 (see the Appendix for all data), which is by a factor of 10 larger (4 smaller) than Refs. 18,48 (Ref. 20 ). We do not observe a systematic discrepancy between the Q of patterned and unpatterned devices. We expect an increase in Q by at least an order of magnitude when using samples with smoother surfaces 29 , operating at lower temperatures 24,48 , and using strain engineering 34,49,50 .
The unpatterned membrane has an out-of-plane optical reflectance of 69% at a free-space wavelength of 1550 nm, which is too low for reaching singlephoton strong coupling in a multi-element optomechanical device 2,3 . By patterning the membranes as a PhC with air holes arranged in a square lattice, 27,[32][33][34]36 we can engineer a reflectance between 0 % and 100 % ( Fig. 4), which we calculated using rigorous coupled wave analysis, e.g., via the S4 package 51 . To demonstrate this capability, we fabricated devices aiming at a reflectance of (i) 99 %, (ii) 75 % and (iii) 50 % at 1550 nm.
We focus on device (i) in Fig. 5 and discuss devices (ii) and (iii) in the Appendix along with a description of the optical setup used to measure reflectance 52 . In Fig. 5(a), we observe a maximum of the reflectance around 1510 nm, away from the designed maximum at 1550 nm. We attribute this shift to a local slab thickness of 87.5 nm (see below) deviating from the assumed 100 nm in the PhC simulation and the 105 nm extracted from the mechanical properties. We attribute this difference to the growth-related thickness variation of the slab and the fact that optical reflectance is probed by a focused light beam that senses only part of the slab, whereas mechanical properties are determined by the entire slab. Notably, the reflectance spectrum in Fig. 5(a) shows a pronounced dip at 1581 nm. This dip can only be reproduced when taking into account the finite waist of the incident beam. To this end, we model the incident Gaussian beam as a weighted sum of plane waves incident at polar angle θ and azimuthal angle φ [ Fig. 4(a) and inset Fig. 5(b)], see Refs. 37,53 and the Appendix. The dip results from coupling of plane waves at oblique incidence to a guided resonance of the PhC. This can be best illustrated with the reflectance map of the PhC slab shown in Fig. 5(b). The dispersion relation of the guided resonance at 1581 nm at wave vector β = 0 shows a decrease in frequency with an increase in β. Hence, the guided resonance appears at longer wavelengths for light impinging under oblique incidence. As a Gaussian beam is formed by the weighted sum of many plane waves, a reflectance dip of finite spectral width is formed.
In Fig. 6(a) we examine the effect of varying beam waist on reflectance. For larger waists, the dip in the spectrum narrows. The reason for this behavior is that larger waists are represented by plane waves with weighting factors that favor less oblique contributions and, thus, less dispersion of the guided resonance is collected. Furthermore, a larger waist reaches a larger maximal reflectance 37 , as seen in the inset of Fig. 6(a). In our measurements in Fig. 6(b), we observe that the dip width indeed decreases with increasing waist. However, in contrast to our prediction, we observe an overall drop in reflectance with larger waists. We attribute this drop to clipping loss due to the finite size of the slab and to diffraction loss of the guided resonance at the boundaries of the slab 54 .
Finally, we study the dependence of reflectance on parameters of the PhC device. Fig. 6(c) shows that the reflectance dip shifts to shorter wavelengths upon increasing the PhC holes. This shift is expected as the patterning determines the mode structure of the PhC. We illustrate this behavior with three devices in Fig. 6(d).  We observe that the position of the dip remains constant, as expected since the mode structure of the PhC membrane is not influenced by the air-gap. However, the reflectance at this wavelength depends strongly on the air-gap. This is the result of a spectral shift of the Fabry-Pérot resonance formed by the slab and substrate through the dip. Fig. 6(f) shows the dependence of the reflectance on slab thickness h. We observe that the dip shifts to longer wavelengths with increasing h, with a strong shift of 4.5 nm in wavelength per nm change in thickness. Hence, a precise knowledge of the slab thickness is required to engineer the position of the dip accurately.
To conclude, we have demonstrated the engineering of suspended PhC slabs in GaAs with mechanical resonance frequencies above 50 kHz, quality factors as high as 5 × 10 5 at room temperature and a maximal Q×f product of 2.3 × 10 11 Hz, and a controllable out-of-plane reflectance at telecom wavelengths 32,34,35 . The GaAs device layer exhibited residual tensile stress, which can be favorably used for strain engineering to reduce mechanical dissipation as demonstrated, e.g., with SiN 34,49,50,55,56 or III-V-based resonators 11,15,28,57 . A dip in the reflectance spectrum 35,37,53 originating from coupling to a guided resonance in the PhC was observed. Hence, PhC devices of finite size must be carefully engineered to have this dip outside of a desired high-reflectivity region.
The single-and double-layer mechanical resonator slabs in GaAs presented in this Letter can be engineered into arrays of high-reflectivity mechanical resonators of precise, epitaxially defined thickness and spacing using AlGaAs heterostructures integrated on top of a distributed Bragg reflector 10,48 . Such an integrated system presents novel perspectives for realizing freespace and fully chip-based multi-element cavity optomechanical systems 2,6-9 , optomechanical microcavities 10 , or frequency-dependent mirrors in optical cavities 58,59 .
We acknowledge fruitful discussions with Claus Gärtner, Garrett D. Cole  The mechanically compliant PhC slabs were fabricated in an AlGaAs heterostructure epitaxially grown using molecular-beam epitaxy (MBE) on a 100 oriented GaAs substrate, where the first grown layer is a 100 nm GaAs buffer layer. The AlGaAs layer of the first wafer was not grown with growth interruption. As a result, the top surface layer had a large surface height variation, as seen in Fig. 7. The AlGaAs layers of the second wafer were grown with a growth interruption 41 of 60 s after 150 nm of deposited AlGaAs material, which lead to a marked improvement of height variation (less than 1 nm) and root-mean-square surface roughness (less than 0.2 nm).
We defined the geometry of the mechanical resonator and the PhC pattern using electron beam lithography with UV-60 resist. The pattern is transferred onto the device layer by inductively coupled plasma reactive ion etching (ICP-RIE) using SiCl 4 /Ar chemistry 39 . The sacrificial AlGaAs layers are removed by HF wet etching with an approximate etch rate of 1 µm/min followed by removal of etch remnants using KOH 42 . The openings of the chosen geometry of the mechanical resonator allow the etch-products of wet etching to be flushed out, which is crucial for fabricating multi-layer devices. Finally, the devices are dried using CO 2 -based critical point drying to prevent stiction of the released slabs. We fabricated single-layer devices from the second wafer by stripping the top GaAs layer with SiCl 4 /Ar ICP-RIE followed by HF wet etch to remove the top AlGaAs layer. . False-coloured transmission electron microscopy image of the MBE-grown AlGaAs heterostructure. The top layer (green) is palladium deposition for ion beam milling, followed by the GaAs layer (yellow) and the AlGaAs layer (gray) on top of the GaAs substrate (yellow).

Appendix B: Experimental setup
The characterization of the mechanical properties of the suspended PhC slabs is carried out with an optical homodyne detection setup operating in the telecom wavelength regime, shown in Fig. 8. A diode laser tunable between 1530 nm and 1620 nm is split into a signal and a local oscillator (LO) beam path using a fiber beam splitter. The signal light is reflected off the device, which is placed inside a vacuum chamber on an xyz transla- tion stage. The displacements of the mechanical modes of the device imprint a phase shift on the reflected light beam. The reflected signal is then mixed with the local oscillator beam in a tunable fiber beam splitter, whose outputs are directed to a balanced photo receiver. The optical interferometer is locked on the phase quadrature by sending a feed back signal to a fiber-based phase modulator in the LO beam path. The electronic signal of the photo receiver is then analyzed with an electronic spectrum analyzer.
The setup for measuring the optical reflectance of the PhC slabs is shown in Fig. 9. The laser light is first passed through a polarizer. A half-wave plate is used to adjust the ratio of the light reflected off a polarizing beam splitter and going to the reference arm and the light transmitted and going to the sample. A quarterwave plate rotates the transmitted light to circular polarization. We use an aspheric lens to focus the light onto the PhC slab. By using aspheric lenses of different focal length, we can focus the beam to different beam waists on the sample. The light reflected off the sample collects a π-phase shift upon reflection and, after passing through the quarter-wave plate, is vertically polarized and, thus, reflected by the PBS into the detection arm. This type of reflectance measurement assumes that the reflection of light from the sample behaves the same for s-and ppolarized light. This is the case for our samples that use non-patterned devices or devices patterned with a PhC of C 4 symmetry.
The reflectance of the PhC slab, R PhC , is then given by is the reflected signal intensity of the PhC slab (a mirror of known reflectivity) measured by photodetector 1 (PD1) and I PhC PD2 I Mirror   PD2 is the reference signal intensity measured simultaneously by photodetector 2 (PD2). This means that we normalize the signal in PD1 by the one in PD2 to account for laser intensity fluctuations. In order to account for any undesired wavelength dependence of the utilized optical components, we independently measure the reflectivity of a mirror of known reflectance, R mirror , in our setup, i.e., I Mirror PD1 /I Mirror PD2 and normalize by this measurement.   This strain leads to a maximal tensile stress σ in the GaAs layer of where E GaAs = 85.9 GPa is the Young's modulus of GaAs. We note that a more detailed calculation of the residual stress in the layer could also consider the strain imprinted in the layer during growth at elevated temperatures and combine these two strains, see Refs. 60,61 . As we want to give here only an approximate upper bound on the maximal observable stress in the GaAs layer, this level of detail is not required in our case.  Fig. 11 shows the noise power spectrum of a free-freetype PhC slab and the mode tomography of the first six mechanical modes. In addition to the modes shown in the main text, we show here also the tilt, tilt-prime and skew 2nd-order mode data and FEM simulation results. Fig. 12 shows FEM simulation results for the mechanical frequencies of different modes when varying the thickness and residual tensile stress in the GaAs slab. We observe that tensile stress leads to an increase of the eigenfrequencies in most cases. However, for the fundamental [ Fig. 12(a)] and tilt-prime modes [ Fig. 12(c)] we observe that for thinner slabs the eigenfrequencies reach a maximum at around 15 MPa.

FEM-simulations for varying slab thickness and stress
The fundamental [ Fig. 12(a)], tilt [ Fig. 12(b)] and skew modes [ Fig. 12(e)] show a strong dependence on stress, while they are much less dependent on thickness. From this behavior we can extract a residual tensile stress in the slab of between 7.5 MPa and 12.5 MPa when assuming a realistic slab thickness between 75 nm and 125 nm in order to match to the measured mechanical frequencies. The opposite behavior is the case for the free-free mode [ Fig. 12(d)], which is strongly dependent on thickness, but much less on stress. The latter is due to stress relaxation within the pad itself as seen in Fig. 2(c) and the fact that the free-free oscillation mode is barely affected by the tethers. However, the free-free mode eigenfrequency is directly proportional to the thickness of the PhC slab. Hence, we can use this fact to estimate the slab thickness and get a good match between the simulated and measured frequency at a thickness of 105 nm, when searching in the stress region between 7.5 MPa and 12.5 MPa.
The tilt-prime [ Fig. 12(c)] and skew 2nd-order modes [ Fig. 12(f)] also depend strongly on thickness and are consistent with the stress and thickness estimates made from the previous four modes.
We, thus, conclude that the slab thickness is around 105 nm and the residual tensile stress in the GaAs layer between 7.5 MPa and 12.5 MPa. Fig. 13 shows the mechanical quality factors that we measured for 151 single-layer devices fabricated from the first and second wafer, both of patterned and unpatterned GaAs slabs.  Fig. 14 shows the reflectance of a GaAs PhC slab of lattice constant a = 1081 nm and hole radius r = 418 nm for varying slab thickness. We observe three distinct operating regions indicated by the blue dashed lines. In region (i) where λ < a, diffraction into higher order modes is dominant. In region (ii), where a < λ < a · n eff with n eff = (1 − η) · n GaAs + η · n Air and η = πr 2 /a 2 , the zero-order mode interferes with higher order modes leading to high reflectance regions. In region (iii), where λ > a · n eff , a Fabry-Pérot effect is seen between the two interfaces of the PhC slab 27 . The black dashed vertical line represents the operating wavelength λ = 1550 nm and the black dashed horizontal line represents the slab thickness h = 87.5 nm indicating that we operate in the near-wavelength regime.

Reflectance map of a PhC slab on a substrate
In our system, the high-reflectance PhC slab is on top of a GaAs substrate forming a low-finesse Fabry-Pérot cavity. Fig. 15 shows the reflectance of a PhC slab on top of a GaAs substrate for varying the spacing between slab and substrate. We observe that around λ = 1510 nm, i.e., the designed wavelength of the PhC, high reflectance is achieved due to the guided resonance. At wavelengths away from the guided resonance, we observe the Fabry-Pérot resonances of the low-finesse cavity, visible as tilted black lines.  Figure 15. Reflectance map of a low-finesse Fabry-Pérot cavity formed by a PhC slab and GaAs substrate, where the spacing between slab and substrate is changed. This map is calculated for a plane wave at normal incidence. The vertical dashed line represents the spacing used and the horizontal dashed lines represent the wavelength region for simulations shown in the main text.

Finite beam waist incident on a PhC slab
Using the angular spectrum representation 62 , an impinging Gaussian beam can be considered as a sum of plane wave components incident at various angles and weighted by a Gaussian distribution with a standard deviation given by the beam divergence. Rather than using an explicit Gaussian source, we run a series of rigorous coupled wave analysis (RCWA) simulations, each with a plane wave source, for various angles of incidence and then we construct a weighted superposition of the results. This has the advantage that once we have the simulation results, we need only to change the weights to explore the role of the beam waist.
To see how this works in practice, we note that an arbi-trary beam can be split into s (transverse electric) and p (transverse magnetic) components with respect to planes of constant z (we take z as the propagation direction of the beam) 63 . This division is achieved most succinctly by introducing two unit vectors for a given plane wave:n, which is normal to the plane of incidence, andt =k ×n defined asn where k 2 = k 2 x + k 2 y and the explicit forms are derived by the condition of orthogonality with one another and with the wave vector k. They can be used, along with the angular spectrum representation, to express the electric field at an arbitrary position along z: whereẼ s andẼ p are the projections of the spectral distribution Ẽ (defined as the spatial Fourier transform of the electric field on the plane z = 0, we use a tilde to denote Fourier-transformed functions in this appendix) on the unit vectorsn andt, respectively. We assume a material interface that is located at the focus of the beam and in the xy-plane. The reflection of the plane wave components of the Gaussian beam is described by the reflection coefficients r s and r p , which correspond to polarization perpendicular and parallel to the plane of incidence, respectively. The reflected field at position z can therefore be written as 62 From here we take the projection along the propagation direction of the time-averaged Poynting vector and integrate over the transverse direction to get the power flux of the reflected beam. Because of the symmetry of the PhC slab we are concerned with, we may take the beam to be x-or y-polarized for simplicity, e.g. Ẽ =Ẽŷ. After some lengthy calculations and taking the paraxial beam approximation, which is appropriate for the beam waist and wavelength range we consider, we obtain which is expressed in terms of the polar angle θ and azimuthal angle φ.Ẽ(θ) is the electric field distribution of a Gaussian beam at the waist position, which is given bỹ where w 0 is the beam waist. Note that we use the opposite spherical coordinate convention to the equivalent expressions given in Ref. 37 . An illustration of the weighting factor of the reflection coefficients, i.e., |Ẽ(θ)| 2 sin (θ)dφ for varying polar angles θ is shown in Fig. 16(a). Fig. 16(b) shows the simulated reflectance spectra of an infinite PhC slab on top of a substrate with a Gaussian beam of varying waist size impinging on it. We observe the narrowing of the linewidth of the dip with increasing waist size. As we see in Fig. 16(a), as the waist increases the weighting factor for plane waves incident at larger angles curtails their contribution and leads to less dispersion into the guided resonance resulting in a narrowing of the dip. Wavelength ( Figure 17. (a, b, c) Simulation and measurement of reflectance spectra for devices (ii) and (iii) and an unpatterned device [for the simulations we used h = 87.5 nm, l = 4300 nm, a beam waist of 4.2 µm and for device (ii) a = 1452.8 nm and for device (iii) a = 1162.8 nm]. The measured data (black) are compared to simulated spectra for a plane wave/Gaussian beam of waist 4.2 µm incident on the PhC slab (orange)/(-) or on the slab on top of a GaAs substrate (purple)/(blue). Reflectance spectra for varying (d, e, f) PhC device layer thickness h, (g, h, i) air-gap thickness l and (j, k) PhC hole radius r.
terned device (ii) used a PhC pattern with a = 1452.8 nm and r = 318.8 nm, aiming at a reflectance of R = 50% , and device (iii) used a = 1162.8 nm and r = 159.18 nm, aiming at a reflectance of R = 75% .
The deviation of the dip position in Fig. 17 (a) is caused by a different thickness assumed in the simulation than the one the actual device has. Recall that the position of the dip depends strongly on the thickness of the PhC, see also Fig. 17(d).
In the reflectance spectrum of an unpatterned device Fig. 17(c,f,i), the Fabry-Pérot cavity resonance is the dominating feature. We observe that slightly changing the thickness of the slab [ Fig. 17(f)] has a minute influence on the spectrum, whereas increasing the air-gap shows the expected spectral shift of the Fabry-Pérot reso-nance [ Fig. 17(i)]. Interestingly, the reflectance spectrum of the unpatterned device [ Fig. 17(f)] is close to the reflectance spectrum of patterned device (iii) [Fig. 17(e)], apart from the sharp feature in the spectrum occurring for the patterned device. This feature is explained by coupling to a guided resonance of the PhC slab, which is not existent in the unpatterned one. Similar behaviour is seen in the reflectance spectrum of the two devices when varying the air-gap, see Fig. 17(h) and Fig. 17(i).