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Arrays of half-symmetric Fabry-Perot micro-cavities were fabricated by controlled formation of circular delamination buckles within a-Si/SiO2 multilayers. Cavity height scales approximately linearly with diameter, in reasonable agreement with predictions based on elastic buckling theory. The measured finesse (F > 103) and quality factors (Q > 104 in the 1550 nm range) are close to reflectance limited predictions, indicating that the cavities have low roughness and few defects. Degenerate Hermite-Gaussian and Laguerre-Gaussian modes were observed, suggesting a high degree of cylindrical symmetry. Given their silicon-based fabrication, these cavities hold promise as building blocks for integrated optical sensing systems.
©2011 Optical Society of America
1. Introduction and background
MEMS-based Fabry-Perot cavities have many applications in fiber and sensing systems and for fundamental physics studies. However, the quality factor (Q) and finesse (F) of flat-mirror cavities has typically been limited by defects such as surface roughness and non-parallelism or uncontrolled curvature of the mirrors [1,2]. Furthermore, to mitigate finesse reduction arising from walk-off of non-collimated beams, flat-mirror cavities typically must operate in a low mode order with relatively large lateral dimensions [3]. Tayebati et al. [4] demonstrated half-symmetric cavities with improved stability and finesse, by using thin-film stress to control the curvature of a tethered mirror in a surface micromachining process. Similar results were obtained by Halbritter et al. [5], using a bulk (two-wafer) micromachining process. Half-symmetric cavities with F ~3×103 are reportedly used in commercial MEMS-based micro-spectrometers [6].
Curved mirror resonators have also been applied to the study of cavity quantum electrodynamics (CQED), including Bose-Einstein condensation [7], cavity optomechanics [8], and optical interrogation of single molecules [9]. For these studies, open-access air-core cavities with ultra-high finesse and low modal volume (Vm) are sought [10]. Since mirror roughness and shape deformations can ultimately limit the finesse, researchers have used novel fabrication processes such as transfer of a thin-film mirror from a lens to a fiber [7], CO2-laser ablation of glass fibers or substrates [9], and focused ion beam drilling in silicon [10]. Recently, single cavities with F > 105 [11] and arrays of micro-cavities with F ~460 [10] have been reported. For the most part, the devices mentioned were fabricated using relatively time-consuming, serial processing techniques. For applications in lab-on-chip systems, there is a need for parallel fabrication of micro-cavity arrays [12].
Here, we describe a method that employs standard silicon processing steps (film deposition, lithography) to produce variable-size micro-cavities on a single chip. Circular regions of low adhesion were embedded within Si/SiO2 multilayer stacks, and delamination buckles were subsequently induced to form in these regions. The resulting structures closely resemble half-symmetric resonators with one flat mirror and one nearly spherical mirror, and their optical properties were found to be in excellent agreement with the well-known predictions (derived from the paraxial wave equation) for macroscopic cavities of that type. Due to the nearly perfect symmetry of the cavities, modes from both the Laguerre-Gaussian and Hermite-Gaussian basis sets could be coupled and observed. The cavities exhibit F as high as 3×103 and Q as high as 4×104.
2. Fabrication and morphology of buckled dome micro-cavities
General details of the fabrication process were provided elsewhere [13] in the context of aircore waveguide channels, but a brief summary is as follows. First, a Bragg mirror (4 periods of SiO2 and a-Si) was deposited by reactive magnetron sputtering onto a double-side-polished Si wafer. Next, a low adhesion, vapor-phase deposited fluorocarbon layer (~10 nm thick) was patterned on the top (a-Si) surface of this mirror. Subsequently, a second 4-period mirror (starting with a-Si), capped by a double-thickness a-Si layer, was deposited. All layers in both the upper and lower mirror were targeted as quarter-wave layers at 1550 nm, except for the half-wavelength (latent) capping layer. The capping layer was added to increase the stiffness of the upper mirror, thereby improving the thermal stability as discussed below. The total thickness of the upper mirror with capping layer is ~1.7 μm. Sputtering parameters were as described previously [13], except that here we used a slightly lower background pressure (3 mTorr) for the a-Si layers and a slightly higher substrate temperature (170 °C). For magnetron sputtered a-Si, these conditions have been associated with higher film density, higher index, and lower loss [14]. Our process produced a-Si layers with refractive index ~3.7 and extinction coefficient ~0.001 at 1550 nm, as estimated from VASE (variable-angle spectroscopic ellipsometry) measurements. The SiO2 layers were estimated to have refractive index ~1.47 in the same range. Using well-known formulae [15] (and confirmed by transfer matrix results), these values imply a best-case reflectance R ~0.999 for the 4-period mirrors, corresponding to a best-case (reflectance-limited) finesse FR ~3140 in the absence of defects.
After deposition of the upper mirror, samples were placed on a hot plate and subjected to an empirically optimized heating process, to induce loss of adhesion between the upper and lower Bragg mirrors in the regions of the embedded fluorocarbon. The multilayers exhibit an effective medium compressive stress ~200 MPa immediately after deposition, and this stress reduces with subsequent annealing [13]. From numerous trials, delamination buckles formed at a sample-dependent temperature, typically in the 250 to 350 °C range. Variation in the buckling temperature is likely due to uncontrolled variation in the properties (i.e. thickness, roughness) of the fluorocarbon layer. In any case, it results in uncertainty regarding the effective stress at the time of buckle formation, which is the subject of ongoing work.
As shown schematically in Fig. 1(a) , the compressive stress causes the upper Bragg mirror to buckle away from the substrate, producing a hollow cavity between a curved mirror and a flat mirror. Within a certain range of diameter for a given net stress, the circular delamination produced a dome-shaped buckle with a nearly spherical shape at its center. The cavities have diameters in the 100 to 800 μm range and peak heights in the ~2 to 25 μm range. However, larger buckles exhibited greater deviation from a spherical shape, such as partial collapse or flattening of the central region. In the following, we focus mainly on buckles with diameters of 400 μm or less (see Fig. 1(b)), which exhibited the best morphology and optical properties.
Fig. 1 Buckled dome micro-cavities: (a) schematic cross-section showing optimized coupling to the fundamental cavity mode by a nearly Gaussian beam (with waist radius ω0) from a lensed fiber. For most cavities tested, the fiber mode field diameter was actually larger than 2ω0, resulting in the excitation of multiple modes. (b) Microscope image of pairs of 150, 200, and 250 μm diameter domes. Some dust particles are also visible.
Neglecting plastic deformation, the theory for elastic buckling of a clamped circular plate can be used to predict the preconditions and height of a circular delamination buckle [16]. The critical buckling stress is σc = 1.2235[E/(1-ν 2)](h/a)2, where E is Young’s modulus, ν is Poisson’s ratio, and h and a = D/2 are the thickness and radius of the plate. For a given stress and assuming fixed E, ν, and h, this implies a minimum diameter (Dmin) for buckling to occur. For D > Dmin, the peak deflection of the plate can be approximated as:
where σ is the biaxial compressive stress, and the last approximation holds for σ >>σc (i.e. for D >> Dmin). Thus, for a given σ, ν, and E, and for D >> Dmin, Eq. (1) predicts that buckle height will increase approximately linearly with diameter. Equation (1) is an approximate closed-form solution to a nonlinear problem [16], and is expected to over-estimate the height for large σ/σc (i.e. for large D). In fact, bifurcation to a nonaxisymmetric buckling mode is predicted at high values of stress (for σ/σc > 56 when ν = 1/3 [16]).Figure 2(a) shows average peak height versus diameter, as obtained from profilometer scans (Alpha-Step IQ, KLA-Tencor) on several domes of each diameter. From previous work [13] (and the layer thicknesses described above), we estimated effective medium parameters h ~1.7 μm, E ~50 GPa and ν ~0.3 for the buckled mirror with the capping layer. Using these and a buckling stress σ = 150 MPa produced good agreement between Eq. (1) and the measured heights. Consistent with the plot, Eq. (1) is expected to be most accurate in the small diameter range, while over-estimating the height for large dome diameters.
Fig. 2 (a) The red curve shows peak height versus buckle diameter as predicted by elastic buckling theory, using the film parameters described in the main text and an effective medium compressive stress of 150 MPa. The blue symbols show average height measured using a profilometer. (b) The symbols show experimental profiles for representative 200, 250, and 300 μm diameter domes. The curves are circular sections with radius of curvature estimated by fitting the profile data from the top portion of each buckle, as described in the text.
The radius of curvature of the buckled mirror is a key parameter, since it determines the modal volume and the spacing of the high-order transverse modes. However, there is no closed-form expression for the profile of a circular buckled plate, even if the buckling is assumed to be an elastic deformation [16]. Moreover, thin-film buckling often deviates from the predictions of elastic theory, due to plastic deformation of the layers near the buckle boundaries [17]. Assuming for simplicity a perfectly spherical dome shape, the expected radius of curvature is given by:
where D is the buckle diameter (i.e. chord length of the dome) and δ is the peak height of the buckle (i.e. the sagittal length of the spherical section).The experimental radius of curvature was estimated by fitting to data from the scanning profilometer. Typical scans are shown in Fig. 2(b), along with circular fits to the top portions of the buckled mirrors. Somewhat arbitrarily, but with the intent of capturing the curvature for the portion of the mirror sampled by the low-order cavity modes, the fit in each case was based on the profile data within +/− 20 μm (along the lateral direction) of the peak. Smaller domes (diameters of 200 μm or less) exhibited an approximately spherical shape, whereas larger domes appear somewhat flattened at the top. This flattening is exacerbated by the force of the profilometer needle, which might also explain the slight asymmetry for the 200 μm dome. However, the same basic trends were also observed using a non-contact optical profilometer (Zygo). For the 200, 250, and 300 μm diameter domes shown in Fig. 2(b), Eq. (2) predicts RCD ~0.9, 1.1, and 1.3 mm, respectively. Rather than being flattened, RC (at the peak) of the smallest domes was smaller than the value predicted by Eq. (2). This is reminiscent of the profile for a straight-sided (Euler) buckle [16], w(x) = (δ/2)[1 + cos(πx/a)], where w is vertical deflection, x is the distance across the buckle, and a is the half-width of the buckle, which exhibits minimum radius of curvature at its center.
3. Optical properties and characterization
Assuming a spherical shape for the buckled upper mirror, the domes form half-symmetric Fabry-Perot cavities [4]. Depending on the degree of cylindrical symmetry, mode-fields for such cavities are traditionally described using one of two alternative sets of orthogonal basis functions [15,18]. In a rectangular coordinate system, the solutions are Hermite-Gaussian (HG) functions Hm,n(x,y,z), where m and n are integer mode indices for the x and y transverse coordinates. In a cylindrical coordinate system, the solutions are Laguerre-Gaussian (LG) functions Lp,l(r,ϕ,z), where p and l are integer mode indices for the radial and azimuthal coordinate directions, respectively. In most macroscopic cavities, deviation from cylindrical symmetry is significant so that it is predominately HG modes that are observed experimentally [18]. However, a predominance of LG modes has been reported for some micro-cavities [19].
Each family of solutions forms a complete set of orthogonal basis functions, so that a given HG mode can be expressed as a linear weighted sum of degenerate LG modes, or vice-versa [20]. The degeneracy condition requires that the modes have equivalent Gouy phase shift, and is expressed as follows:
For example, the HG1,1, LG1,0, and LG0,2 modes form a nominally degenerate set. Degenerate modes are expected to share the same resonance frequency, although slight imperfections (such as deviations from spherical mirror symmetry) will perturb this degeneracy [10]. For RC >> δ, the nominal wavelength spacing between non-degenerate transverse spatial modes can be approximated [15] as:where λ is the resonant wavelength, and z0 is the Rayleigh range, of the fundamental cavity mode. Table 1 shows predicted and measured mode properties for 4 representative cavities, where the predictions are based on the RC fit described above (+/− 20 μm from the peak).
Table 1. Predicted and Measured Optical Properties for Representative Microcavitiesa
Optical properties of the cavities were tested using the experimental setup illustrated in Fig. 1(a). Light from a tunable laser was coupled into the cavities via lensed optical fiber with focal spot diameter ~20 μm, somewhat larger than the fundamental mode field diameter for most of the cavities tested. This resulted in significant coupling to higher-order spatial modes. However, lower-order modes could be isolated and imaged by tuning the laser to the corresponding resonant wavelength of a given mode. Transmitted light was captured by either an infrared camera or a cooled photodetector.
Given the estimated mirror reflectance (~0.999), the circulating power at resonance is predicted to be greater than 1000 times the incident power [15]. Even for low input power (< 0.1 mW), significant drift (on the time scale of seconds) in the transmitted power was observed when the laser was tuned to a fundamental resonance frequency. We speculated that this was caused mainly by thermal expansion [21] related to residual absorption in the mirrors. This instability was reduced significantly by addition of the a-Si capping layer, as described in Section 2. To minimize error in estimating experimental line-widths, scans were performed at low laser power. Also, measurements were repeated at various scan rates, and for both decreasing and increasing laser wavelength. Experimental Q and F were estimated from the FWHM line-width of the fundamental resonance. The predicted FSR takes mirror penetration into account [22], and was confirmed using a white-light source and spectrum analyzer setup (not shown). As listed in Table 1, the experimental finesse for each cavity was in good agreement with the reflectance limited prediction (FR ~3140).
Representative results for a 250 μm diameter dome are shown in Fig. 3 . Transverse modes exhibited fixed spacing with values in good agreement with the predictions from paraxial theory, as summarized in Table 1. As mentioned, associated with each higher-order resonance line is a set of nominally degenerate HG and LG modes, whose degeneracy is perturbed by any deviation from cylindrical/spherical symmetry, such as spherical aberration or astigmatism [10,19]. Such perturbations are apparent from the multiple sub-peaks within the g = 2 and g = 3 resonant lines in Fig. 3. By making fine adjustments (typically on the order of a few picometers) to the laser wavelength, it was possible to isolate individual HG and LG modes within a given resonant line. As an example, the H3,0, H2,1, L0,3, and L1,1 modes shown represent the complete set of HG and LG modes possessing g = 3 degeneracy. To our knowledge, such direct experimental evidence for the intrinsic relationship between LG and HG modes is rarely reported. We believe it is made possible, in part, by the near-cylindrical symmetry and geometrical perfection of the self-assembled microcavities.
Fig. 3 The plot shows the transmission spectrum for a 250 μm diameter cavity, with peak height (mirror spacing) ~7.5 μm. The broad peak near 1522 nm is due to a transmission resonance outside the buckled areas. The inset plot shows the fundamental resonance line in greater detail. Mode-field images were captured with the laser tuned near one of the resonance lines, as indicated. Images for the four nominally degenerate modes associated with the third-order resonance were captured by making fine adjustments to the laser wavelength.
For the smallest cavities, the LG modes were dominant and it was in fact more difficult to isolate HG modes. Moreover, there was less evidence for the existence of multiple peaks within the high-order resonance lines, suggesting a higher degree of symmetry. As an example, Fig. 4 shows the transmission spectrum for a 200 μm diameter cavity. Also shown are representative images of Lp ,1 modes, which were found to be dominant in this case. As predicted by (3), this family of modes occupies the odd-order (g = 1,3,5 …) resonance lines. The mismatch between the incident beam and the fundamental cavity mode is particularly large here, resulting in significant coupling to higher order modes. As above, these modes exhibit a fixed spacing in good agreement with paraxial theory. Finally, the effective fundamental mode volume is on the order of Vm ~100λ3 for the smallest cavities, comparable to values reported for similar cavities [10,11].
Fig. 4 The plot shows the transmission spectrum for a 200 μm diameter dome, with peak height (mirror spacing) ~5.7 μm. Representative LP ,1 mode images are shown; they were captured by tuning the laser source to one of the resonance lines as indicated.
4. Summary and conclusions
The reflectance-limited optical properties of the cavity modes can be taken as evidence that the ‘defect finesse’ [1] of the cavities is high. This suggests that cavities formed by buckling can have very low roughness and a highly regular geometric shape. In principle, the finesse could be improved by using higher reflectance mirrors. For example, reflectance of the present mirrors is mainly limited by absorption in the a-Si layers, which could be reduced by the use of hydrogenated a-Si. Furthermore, operation at shorter wavelengths might be possible by replacing a-Si with TiO2 or similar, provided compressively stressed layers are possible.
The shape of the cavities, and especially the minimum radius of curvature and thus cavity modal volume, will ultimately be limited by the restricted combination of film thicknesses, cavity size, and stress that produces dome-shaped buckles. A full exploration of these details is left for future work. Nevertheless, the Q, F, and Vm demonstrated are already competitive with values reported in the literature [10–12]. Another major issue, particularly for sensing and atomic physics studies, is the possibility of implementing ‘open-access’ versions of the dome cavities. A possible solution might involve intersecting air channels, which we have demonstrated previously using the buckling process [13].
Acknowledgments
The work was supported by the National Sciences and Engineering Research Council of Canada.
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