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Open-access optical microcavities are emerging as an original tool for light-matter studies thanks to their intrinsic tunability and the direct access to the maximum of the electric field along with their small mode volume. In this article, we present recent developments in the fabrication of such devices demonstrating topographic control of the micromirrors at the nanometer scale as well as a high degree of reproducibility. Our method takes into account the template shape as well as the effect of the dielectric mirror growth. In addition, we present the optical characterization of these microcavities with effective radii of curvature down to 4.3 µm and mode volume of . This work opens the possibility to fully engineer the photonic potential depending on the required properties.
© 2015 Optical Society of America
1. Introduction
Optical microcavities play a crucial role in technologies based on enhanced light-matter interactions [1]. Such enhancements increase our capacity to control and sense any matter placed within the microcavity and has revolutionized various domains such as quantum optics [1], lasing [2], optomechanics [3] and optical sensing [4]. In this context, epitaxial structures [5] and photonic crystals [6, 7] have already shown sufficiently high optical quality to perform cavity quantum electrodynamics (CQED) experiments. However, they are intrinsically limited in terms of cavity length tunability. Moreover, once the device is made, there is no access to the maximum of the optical field. In this respect, the open-access microcavity offers an attractive alternative. Such a device is made up of two independent mirrors facing each other separated by an empty space. Therefore, one can directly access the confined mode while maintaining a full tunability of the cavity resonance. At the micron-scale, they offer a wide range of unique applications of CQED as has been shown with nanocrystals [8], nanodiamonds [9] or even optomechanics with carbon nanotubes [10]. Furthermore, it is possible to realize hybrid devices in which a concave mirror is used in conjunction with epitaxial structures, releasing the lattice matching criterion for the top mirror while keeping the high quality of expitaxial structures [11]. In addition to these achievements, open-access microcavities can also be used as lab-on-a-chip optical sensors by integrating them into a fluidic system [12].
Significant effort has been devoted to minimizing the size of open cavities in order to maximize the light-matter interaction strength. Mode volumes of order of λ3 have been achieved with Q-factors of ≈ 104 or higher, approaching the regimes achieved by photonic crystal cavities (PCCs) and microtoroids. Miniaturization of open-access microcavities is a challenging task which requires a high degree of control over the mirror shape while maintaining ultralow surface roughness to minimize scattering. Currently, fabrication of these microcavities has relied mainly on CO2 laser machining or Focused Ion Beam (FIB) milling. The former has achieved the highest reported Q-factors for open access cavities due to the high optical quality of the processed surfaces [13–17]. However, the shape of the cavities is dictated by the ablation mechanisms and, therefore, is not directly controlled. Typically, the Radius of Curvature (RoC) of the feature fabricated by CO2 ablation is inversely linked to its depth. The FIB fabrication method is attracting increased interest and is producing smaller RoC cavities [18, 19]. As a bottom-up patterning approach with a lateral resolution down to 7 nm, this method allows more precise control over the cavity shape.
Here, we report developments in open access cavity fabrication using the FIB method to produce cavities which combine small mode volumes with high quality factors. We describe how nanometer precision in the cavity shape can be achieved, enabling precise engineering of the mode structure and systematic optimization of the cavity properties. We include a discussion of coating growth onto micromachined surfaces, and show that cracking that occurs when making smaller and steeper concave features can be removed by careful design of the substrate profile. As an example of mode engineering, we have fabricated Gaussian mode phase-matched mirrors with a minimum physical RoC of 1.5 µm and measured cavity modes with effective RoC’s down to 4.3 µm. This fabrication technique offers new ways to fully engineer the mode, depending on the goal targeted. For example, it allows the improvement of the out-coupling mode-matching, reduction of the coupling to leaky-modes or reduction of the mode volume whilst maintaining a high Q factor. As supplementary examples, it is possible to shape the concave mirror to obtain a true single mode cavity [20] or to avoid resonant coupling of different transverse modes leading to increased losses of the fundamental mode [21]. More generally, full control on the mirror topography and therefore the photonic potential opens the possibility of more complex photonic lab-on-a-chip applications.
2. Fabrication method
There are two stages in the fabrication of an open-access microcavity: the FIB milling of the mirror templates and the coating of these templates. We have chosen a plano-concave configuration in order to avoid the difficulties associated with in-plane alignment. Flat mirror substrates consist of UV fused silica slips from UQG optics (10 mm diameter and 0.5 mm thickness), whilst concave mirrors substrates are UV fused silica plates from UQG optics (50mm×50mm and 0.5 mm thickness). The concave mirrors are fabricated by FIB milling of the substrate prior to coating with Distributed Bragg Reflectors (DBR) as showed in Fig. 1(a).
Fig. 1 a) Process to fabricate the mirrors. Step 1: A Gallium ion beam is used to sputter the surface of a silica substrate. Step 2: DBR coating is grown on top of the template to create the concave part of the microcavity. b) RMS roughness of a feature as a function of its depth. Transparent blue area corresponds to the resolution of our AFM. Dashed line is the initial roughness of the silica substrate.
We employ a FEI FIB200 focused ion beam with a 30 keV Ga+ beam to pattern the surface of the substrate (step one on Fig. 1(a)). Prior to machining, the silica substrates are coated with a thin layer (≈ 50 nm) of gold to prevent localized charging of the surface which causes drift and distortion of the ion beam. The ion beam current is selected from a set of fixed values, ranging from low currents used for imaging (≈ 50 pA) to a maximum current of 20 nA. The beam diameter scales with the current, and ranges from 7 nm to 1 ≈µm. To provide good control over the shape, the beam diameter has to be, at least, one order of magnitude smaller than the size of the feature to be produced. Once the current is selected the field of view for the feature is digitized into a grid of 4096×4096 locations by a 12-bit digital to analogue converter (DAC) controlling the beam deflection. Dwell times as a function of position typically range between 0.1 µs and 4.7 ms, and are programmed using a pre-prepared “stream” file generated in Matlab [24]. This file provides dwell time for each pixel location in agreement with the cavity topography. The ion beam follows its order. Accurate measurement of the sputtering rate is essential in order to produce the desired shape: for the work presented here the Ga+ beam sputters the fused silica substrate at a dose rate of 4.16×10−6pC/nm3 similar to that reported in [30]. The beam current of the FIB changes over time with use. To counter this effect, we adjust the dwell times in the ”stream” files by doing a calibration of the cavity depth with an AFM before every run of template production. Several important considerations are required to achieve the high degree of control over the mirror shape and smooth surfaces using the FIB process. The beam position order is randomized in order to average out the effects of material redeposition and residual charging effects that can create systematic deviation from the desired shape for raster and serpentine sequences. It is also important to account for changes in the beam properties (current, focus, astigmatism) by regular calibration. The focus and the astigmatism are tuned by the operator immediately prior to producing a series of templates with the same beam current. This milling process is typically used to produce each individual concave feature. Further scaling to produce large arrays of features (with different shapes if desired) can be achieved using Runscript software. Therefore, it is possible to easily explore the parameter space in order to optimize the fabrication.
It is important to note that achieving sub-nanometer surface roughness from the milling process does not require super-polished surfaces on the pre-milled substrates. Indeed, it has been shown that the FIB provides a natural polishing effect because of the strong dependence of the sputtering rate with angle of beam incidence [22, 23]. Figure 1(b) shows the roughness of a feature as a function of its depth obtained by stopping the FIB process at different steps and imaging the surface with an AFM for each step. The roughness is obtained by removing the smooth component of the shape from the AFM image. The polishing effect is clearly observed for feature depths in between 20 and 60 nm. Above 60 nm, the roughness is limited by the resolution of our AFM represented by the transparent blue area on Fig. 1(b). The silica substrates we use have an initial roughness of 1.5 nm (given by the dashed line on Fig. 1(b)). Note that the roughness is much larger than the substrate roughness at the beginning of the patterning. This effect is caused by the inhomogeneous layer of gold we use to cover our substrates. At the beginning of the milling process, this layer is inhomogeneously sputtered from the surface meaning the ion beam interacts unevenly with the silica surface.
In this report the substrates were coated with sputtered DBRs in-house at the Thin Film Facility of the Department of Physics at the University of Oxford (step two on Fig. 1(a)). The coating are 1.9 µm in thickness and consists of 10 pairs of SiO2/TiO2 ( and ) for both mirrors designed to give the highest reflectivity at 640 nm above 99.7%.
3. Coating topography effect
A detailed understanding of coating topography is vital for producing high quality concave mirrors of a desired shape. Figure 2(b) presents cross-sectional SEM images of mirrors grown on spherical concave surfaces with RoC’s of 3 and 5 µm. Note that the mirrors are tilted with an angle of 52° in order to access the DBR stacks arrangement. The top of this image corresponds to the top of the mirrors (covered with gold). The SiO2 substrate is demarcated in blue on the two left hand images. One can see that the surface of the template is significantly different from the last surface of the DBR on the air side. Moreover, these micrographs reveal that for small RoC microcavities, cracks start to appear during the coating growth close to the edges. They correspond to a sharp demarcation highlighted by the red dots on Fig. 2(b). We find that such cracking is catastrophic for the optical properties of the mirrors. In order to gain some insight about the coating effect on the cavity shape, we develop a model based on a simplified version of a level set approach for film deposition [25]. Figure 2(a) presents the two main mechanisms of growth for the coating we consider. The first one is mono-directional growth upwards towards the sputtering source, and the second is isotropic growth. While the first does not change the surface topography, the isotropic growth does, creating a strain point in the coating. As an example, on the middle scheme of Fig. 2(a) the middle of the surface will result in a crack since at least two different initial surfaces will enter in contact. In reality, the growth of a coating has a mixed behavior between isotropic and mono-directional growth. The evolution of a surface defined by its position (r,z) in this model (see bottom scheme on Fig. 2(a)) is given by:
where (r0,z0) is the position of the initial surface and is an input of this model obtained from the SEM images. Δh is the surface displacement (Δh=182 nm for the Figs. in this paper). θ is the angle between the normal of the surface with the axis z. It is, as well, the angle between the surface tangent and the axis r (see bottom scheme of Fig. 2(a)). On the right images of Fig. 2(b), Δh is taken to be the thickness of one DBR pair and repeated for each pair. The only fitting parameter here is f, the mixing parameter between the two mechanisms of growth. The results of this model for f = 65% are given by the dashed lines on the SEM images on Fig. 2(b) and gives an excellent agreement with the shape evolution and the crack position. Notice that this mixing parameter may be unique to each deposition process and depends on factors such as the technique used, the position of the sources, and the rotation of the samples. However, this single parameter reproduces all our cavities coated with this technique and makes it a useful tool for our fabrication method. We are able to design mirrors in which the terminating layer takes the desired mode shape and work backwards to establish the required substrate shape. Moreover, we can predict the appearance of cracks in the DBR coating depending on the shape we are targeting.Fig. 2 a) The top and middle schemes are represent the mono-directional and isotropic growth models respectively. Bottom scheme gives the definition of the position (r,z) as well as the angle θ used in equation 1. b) SEM cross-section images of hemispherical microcavities with RoCs of 5 µm (top) and 3 µm (bottom). Left graphics show the raw images with the cracks displayed with red dots. Right graphics show the sames images with the growth model in dashed lines for f=65%.
4. Example of application: Gaussian mode phase matched microcavities with ultra-small RoCs
To demonstrate the degree of control we have over the cavity shape, we now choose the cavity surface to follow the analytic expression for an isophase of a Gaussian cavity mode. These modes are approximate solutions of Maxwell’s equation for small divergence. Exact solutions are given by the Helmholtz solutions [26–28]. However, the isophase surfaces are oblate elliptoids for such solutions and present 90° sharp corners on the edges. Consequently, the coating will be highly strained in these region, probably causing crack formation. To avoid this obstacle, our approach here is to use Gaussian isophase (GI) surfaces for radial displacements that extend well beyond the paraxial approximation. We note that the shape is not expected to be greatly detrimental to the cavity finesse since the confined mode will always adapt to the geometry of the mirrors as long as it is smooth and crack-free.
The phase of a Gaussian cavity mode at a position (r,z) is given by:
where R(z) is the RoC for r = 0 and is given by , is the Rayleigh length and k is the mode wavevector given by k = 2π/λ. The waist of a confined Gaussian mode is given by where R is the RoC and L is the cavity length. To obtain the position of the isophase, the following equation is solved for (r,z):Notice that for Gaussian phase-matched microcavities, the shape of the GI depends on the RoC on-axis and the targeted length L. For cavities with large RoCs, the GI matches with a hemispherical surface. Therefore, the targeted length L is not a relevant parameter. However, for smaller RoC cavities, ie. below 7 µm, the targeted length L specifies the length of operation of the microcavity. The left plot of Fig. 3 presents the phase map of a Gaussian cavity mode for a RoC of 4 µm on axis. The dashed line corresponds to a hemispherical shape and the difference between these two shapes is presented on the right plot. The deviation reaches 100 nm within one waist of the mode with this configuration. Using our fabrication method, we can take into account the complete GI surface and correct for these differences.
Fig. 3 Left graphic: Phase map of a Gaussian cavity mode for a RoC of 4 µm and a cavity length of 327 nm. The isophase surface associated is displayed with the blue dashed line. The equivalent hemispherical surface is displayed with the black dashed line. The dash dot line gives the extension of the waist of the beam as a function of the longitudinal position in the cavity. Right graphic: Difference between the Gaussian mode isophase and the hemispherical equivalent.
5. Nanometer control of open-access microcavity shapes
In order to test our fabrication method, six types of cavities with different nominal RoCs ranging from 20 µm to 1.5 µm were produced. Their characteristics are summarized in table 1. The diameter of each feature is at least four times larger than the expected waist on the concave mirror. For a Gaussian cavity mode, it means that 99.97% of the mode is located within the feature. For large RoCs (20 µm and 12 µm), the targeted length will be taken at the middle of the stability region while for smaller RoC (7 µm, 4 µm, 2.5 µm and 1.5 µm), it corresponds to an integer number of half wavelengths.
Table 1. Characteristics of the six different Gaussian phase-matched cavities type fabricated using the method developed in this work.
Figure 4(a) presents the AFM data of the templates for the cavity with a RoC of 4 µm. The black circles correspond to the initial targeted shape, without any fitting parameter except the in-plane position. This targeted shape includes the GI surface and the effect of the coating. Therefore, the last layer after coating the DBRs, will follow the targeted isophase surface. The fit is performed over 85% of the feature diameter. The fitted RoC and targeted length as well as the RMS deviation from the fit are reported in table 1 for the six different cavities. They all agree with the nominal parameters and demonstrate the control over the targeted shape that our fabrication method provides. The RMS deviation from the fit includes the intrinsic roughness of the surface and the deviation from the fitted shape. The roughness can be estimated by performing a 6th order polynomial fit and gives a value 0.8 nm ± 0.1nm for all the cavities. This value is close to the resolution of our AFM (≈ 0.7 nm). The last column of table 1 reports the RMS deviation from the targeted shape, ie. without any free parameters. The agreement, within a few nanometers, emphasizes the accuracy this fabrication method has over the shape. The residual deviation from the targeted shape for the cavity with RoC 4 µm is given in Fig. 4(b). It shows that the feature is slightly elliptical with respect to the targeted shape. This effect comes from a slight anisotropy in the FIB beam positioning and can be further optimized. Figure 4(c) presents the correlation function of the residual deviation. It is interesting to note that most of the roughness occurs on a length scale well below the wavelength of light. Consequently, this roughness will not efficiently scatter light out of the cavity mode. The cross-section of the coated cavity with a RoC of 4 µm is presented on Fig. 4(d). The DBR growth model is superimposed with the white dashed lines and reproduces the general behavior of the DBR growth. The black line is at four times the extension of the mode waist to give an indication of the region in which the mirror needs to be of highest quality in order to give a finesse ∼10,000. The cavity is crack-free within this range, and even beyond, in contrast with to the hemispherical surfaces.
Fig. 4 a) AFM image of a template for a cavity of RoC of 4 µm. The initial targeted shape is given by the black line b) Residual deviation between AFM data and the targeted shape presented in a). c) Correlation function of the residual deviation between the AFM data and a smooth polynomial function. Most of the roughness correlation is below λ d) SEM cross-section of the same cavity after coating. Dashed white lines correspond to the DBR growth model. Black dashed line gives the extension of four times the waist expected for the nominal cavity parameters.
6. Optical characterization
To complete our study, we performed an optical characterization of the cavities presented in table 1 after coating. DBR properties being significantly wavelength dependent, we decided to perform a transmission experiment at a fixed wavelength (637 nm) and varying cavity length. We use a TOPTICA laser DL100 in a transmission configuration. The laser is focused using an aspheric lens and collected with a standard ×10 objective. A ring piezo-actuator from Piezomechanik is used to sweep the cavity length. We record the transmission spectrum with an APD from Thorlabs (ref: APD120A2/M). The absolute cavity length is initialized using a white light transmission experiment. The length is then calibrated using the mode resonances appearing every half wavelength (318.5 nm).
Figure 5(a) displays one of the transmission spectra for the cavity with nominal RoC of 4 µm. The broad peaks at 1450 nm and 1760 nm are the bidimensional modes in the flat region of the mirrors, ie. out of the cavity. The cavity modes for q = 5 are indicated with black arrows up to l + m = 4.
Fig. 5 a) Transmission spectrum for the cavity with nominal RoC of 4 µm as a function of the optical length. The cavity modes for q=5 are indicated with black arrows up to l+m=4. b) Histogram of the spread of the optical length mode positions for 142 cavities. Red curve is a Gaussian fit giving a standard deviation of 1.1 nm.
In order to describe the optical properties of these microcavities, we use the standard Gaussian cavity mode resonance equation. This equation is valid within the paraxial approximation, ie. if the divergence of the beam is smaller than 30°. For hemispherical cavity modes, this is a valid assumption for RoC’s down to 1.5 µm. Since the cavities studied in this article satisfy this criterion, we can find the resonant length L of a Hermite-Gaussian cavity mode with by the equation:
where R is the effective RoC, q is the main longitudinal mode number and l and m the transverse mode numbers. The effective RoC is the actual RoC that the mode experiences which for GI surfaces depend on the area probed by the mode. The approximate character of equation 5 for non-spherical and finite size microcavities is included in this effective RoC.Figure 5(b) presents the optical length spread for 142 cavities in a fixed array at the smallest cavity length using the equation 5. The cavities feature a standard deviation of 1.1 nm for their optical length, demonstrating the reproducibility and the level of control of our fabrication method.
The two key parameters to characterize these microcavities are the effective RoC and the Finesse of each mode which can be directly extracted from transmission spectra such as the one on Fig. 5(a). The effective RoC can be obtained from the spectrum using to the equation:
where ΔtL is the length difference between the longitudinal and the first transverse mode.The Finesse is obtained by:
where ΔL is the length difference between two consecutive modes, ie. 318.5 nm, and δL is the length linewidth of the mode.Figure 6(a) presents the effective RoC as a function of the optical cavity length. Measured and nominal values are displayed side by side. We observed excellent agreement between the nominal and the measured effective RoC’s at the design cavity length for 20 µm, 12 µm and 7 µm RoC mirrors. For smaller RoC’s, there is a substantial discrepancy between these two quantities which can be attributed to two phenomena. Firstly, the optical penetration depth into the DBR has not been taken into account in the targeted length. It has been measured to be 625 nm for the flat mirror [12] and it is predicted to be 230 nm for the concave mirror [29] corresponding to a total of 855 nm. Choosing small targeted length leads to mirror shapes not steep enough for the achieved optical lengths, resulting in higher effective RoC’s. Secondly, for small RoC, the cavity mode hybridizes with the bidimensional continuum surrounding it, resulting in renormalized energies and a smaller effective RoC. This last point will be described in a forthcoming communication.
Fig. 6 a) Effective RoC as a function of the optical cavity length. Left part is measured data and right part is the nominal values for the six types of cavities presented on table 1. b) Finesse as a function of length for the same cavities. The transparent blue area corresponds to the Finesse expected from the intrinsic reflectivity of the coating.
The physical RoC of 1.5 µm and effective RoC of 4.3 µm are to our knowledge one of the smallest values yet reported for open-access microcavities. For the minimal length achieved, the transverse splitting is measured to be 21% of the free spectral range which converts into a 24 nm splitting in terms of wavelengths. It leads to a mode volume of . As a matter of comparison, the theoretical minimal mode volume achievable with open-access microcavities, including the penetration depth, is .
Figure 6(b) presents the measured Finesse as a function of length for the same set of cavities. The blue transparent area corresponds to the Finesse deduced from a macro-reflectivity experiment on the DBR coating. The drop at long cavity length is due to the increase in diffraction losses for increasing mode waist. Indeed, at large waist, the mode starts to experience edge losses. This effect is more pronounced for small RoCs in agreement with this explanation. However, once the mode waist is small enough, the Finesse is only limited by the coating.
7. Conclusion
In this article, we have presented a fabrication technique of open-access microcavities based on the use of a FIB. We produce concave mirrors that follow precisely the GI surface. The development presented here provides nanometric control on the cavities shapes, taking into account the coating topography, and a high degree of reproducibility. Using this method, we have successfully produced ultra-small mode volume microcavities with mode volume down to with finesses mainly limited by the coating itself.
This approach provides a method with which cavity parameters such as Q factor, mode volume and structure, and beam characteristics can be engineered to a high degree of precision, and with which novel mirror geometries can be explored for advanced lab-on-a-chip applications. New cavity designs at the single cavity level such as coupled cavities or polarization split cavities can be easily developed. Moreover, the FIB fabrication paves the way for designing more complex photonic structures such as in-plane waveguides, arrays of coupled cavities using the flexibility and the control offered by our fabrication method bringing open-access microcavities into the realm of on-chip photonic circuits.
Acknowledgments
This work was supported by the Leverhulme Trust, the Royal Society and the European Commission Seventh Framework Programme ( FP7/2007-2013) under grant agreement n°618078.
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