On-chip asymmetric microcavity optomechanics - OSA Publishing

Vol. 24, No. 26 | 26 Dec 2016 | OPTICS EXPRESS 29613

On-chip asymmetric microcavity optomechanics SOHEIL SOLTANI,1 ALEXA W. HUDNUT,2 AND ANDREA M. ARMANI1,2,3,* 1 Ming Hsieh Department of Electrical Engineering-Electrophysics, University of Southern California, Los Angeles, CA 90089, USA 2 Department of Biomedical Engineering, University of Southern California, Los Angeles, CA 90089, USA 3 Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089, USA * [email protected]

Abstract: High quality factor (Q) optical resonators have enabled rapid growth in the field of cavity-enhanced, radiation pressure-induced optomechanics. However, because research has focused on axisymmetric devices, the observed regenerative excited mechanical modes are similar. In the present work, a strategy for fabricating high-Q whispering gallery mode microcavities with varying degrees of asymmetry is developed and demonstrated. Due to the combination of high optical Q and asymmetric device design, two previously unobserved modes, the asymmetric cantilever and asymmetric crown mode, are demonstrated with submW thresholds for onset of oscillations. The experimental results are in good agreement with computational modeling predictions. © 2016 Optical Society of America OCIS codes: (140.3945) Microcavities; (140.4780) Optical resonators; (200.4880) Optomechanics; (220.0220) Optical design and fabrication.

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Journal © 2016

http://dx.doi.org/10.1364/OE.24.029613 Received 24 Oct 2016; revised 30 Nov 2016; accepted 1 Dec 2016; published 13 Dec 2016


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1. Introduction Optomechanical devices convert optical energy into mechanical vibrations at well-defined frequencies. Because of the predictability of the conversion process and the stability of the mechanical oscillation frequency, the devices have been used in a broad range of applications from sensing individual nanoparticles [1–3] to studying quantum mechanics [4–8]. Initial research efforts focused on understanding the fundamental science governing the optomechanical behavior [6,9–11], but with the advent of on-chip, integrated optomechanical devices, the field is now moving towards application development [2,12–14]. Thus far, a variety of symmetric devices with optomechanical properties have been studied, and their mechanical behavior has been extensively characterized [15–17].


One common feature of the devices studied to date is that they are symmetric with respect to at least one axis. This symmetry enables preferential energy transfer between the optical and mechanical modes along the symmetric axis. To study this behavior, high quality factor (Q) whispering gallery mode cavities are frequently used [18,19]. The high quality factors result in large circulating optical intensities within the device, which translates to efficient coupling between the optical and mechanical modes and low threshold for the onset of the optomechanical vibrations. Among the possible device geometries integrated ultra-high-Q microtoroids and microdisks are particularly attractive as they can be fabricated in large arrays, and the device geometry can be easily and precisely defined [20–24]. In previous work using these devices, two broad categories of mechanical modes have been identified and defined: cantilever and crown modes [25,26]. There are two approaches for experimentally studying the optomechanical behavior. In one strategy, the thermomechanical spectrum of the modes is measured [25]. While this approach allows the entire suite of mechanical modes of a device to be identified, it does not allow the modes to be regeneratively excited via optical excitation [11,27,28]. Alternatively, the optical spectrum can be analyzed for optical bistability induced by the mechanical behavior. This approach directly measures the optomechanical behavior of the structure. To date, regenerative excitation at sub mW input powers has only been observed for a distinct group of radially symmetric, cantilever modes [27,28]. While the cantilever modes can be easily excited with sub-mW thresholds, the crown modes require significantly higher input powers to be regeneratively excited due to the poor efficiency of the optomechanical energy transfer process [29]. Therefore, by improving the energy transfer process, a pathway to accessing these modes with optical excitation can be realized. One strategy is to change how the optical field is converted to mechanical energy by intelligently engineering the device architecture. In past work, the primary research focus was on lithographically fabricated symmetric devices, in which the force is uniformly transferred to the device [27,28]. Additional work in accidentally fabricated asymmetric devices demonstrated interesting chaos behavior. Unfortunately, due to the uncontrolled and random nature of the fabrication process used for the asymmetric devices, a systematic experimental and theoretical study of these devices was not possible [29]. In the present work, a repeatable and precise approach for fabricating asymmetric, high optical Q devices integrated on silicon is developed. A schematic highlighting the general principles of this device and key points of asymmetry is presented in Fig. 1. By introducing asymmetry mechanical modes are regeneratively excited at sub-mW threshold powers. 3D Finite Element Modeling (FEM) simulations show that there is strong agreement between the measured and simulated frequencies.

Fig. 1. (a) Rendering of a fiber taper-coupled asymmetric cavity. (b) Schematic where the parameters of minimum minor radius (rmin), maximum minor radius (rmax), maximum major radius (Rmax), minimum major radius (Rmin), total diameter (D), vertical offset (Δz) and maximum pillar shift (ΔLmax) are labeled. (c) SEM of a cross-section of the smaller side of an asymmetric device with large vertical offset.

2. Theory A mass in motion can be described by the damped harmonic oscillator equation:


 x(t ) +

γ0 2

x (t ) + Ω 2 x(t ) =

f (t ) FL + meff meff

(1)

where x is the displacement of the device in radial direction, γ0 is the damping coefficient of the structure, Ω is the mechanical frequency, meff is the effective vibrating mass of the cavity, and f(t) is the optical force. For the specific case of a circular whispering gallery mode cavity, the optical force is:

(

f ( t ) = 2π n a

2

) / ( cT ) r

(2)

where n is the effective refractive index of the mode, |a|2 is the optical energy amplitude inside the cavity, c is the speed of light and Tr is the cavity round trip time. The intrinsic damping coefficient is defined as γ0 = Ω/(Qm) where Qm denotes the mechanical quality factor [17]. To satisfy Fluctuation Dissipation Theorem, FL is included to account for the Langevin Force. For macroscopic particles over long time scales, the Langevin force has no correlation with itself at any other time and hence its correlation function takes the form of a Delta function: FL (t ), FL (t ') = γ 0 k BTmeff δ (t − t ')

(3)

where kB and T are Boltzman's constant and resonator temperature, respectively [27]: The equation relating the mechanical motion to circulating light in the cavity is: da 1 1 1 1 s = i Δ ( x)a − ( + )a + i dt 2 τ 0 τ ex τ ex

(4)

where τ0 is the intrinsic cavity life-time, τex is the extrinsic cavity life-time and |s|2 is the launched input power [17]. These equations are coupled through the dependence of the resonant frequency on variations of x: Δ ( x) = ω − ω0 ( x) = Δ 0 − g om x

(5)

where, Δ0 is the initial detuning of the laser frequency with respect to the resonant wavelength and gom is the optomechanical coupling coefficient. When the above equations are applied to a circular whispering gallery mode cavity, the major radius of the cavity (R) is constant over the periphery of the resonator and represents x [27]. In the adiabatic regime, and with the assumptions given above, one can solve Eqs. (1-4) in steady state to solve for the threshold for the onset of self-oscillations: Pth = Cmech

Qc (1 + f 2 )3 P , γ = γ 0 (1 − ) 4 Qloaded Qmech f Pth

(6)

where Pth is the threshold and γ is the damping factor. Qc, Qloaded, and Qmech are the coupling quality factor, loaded quality factor, and mechanical quality factor. The constant Cmech = R 2ω02 meff Ω / 64 is a measure of the strength of the coupling between the optical and mechanical modes. Additionally, f = Qloaded

ω − ω0 is the detuning factor of the laser 2ω0

wavelength for the resonant frequency (ω0) [17]. Although Eqs. (1-5) are generally valid for any type of optomechanical system, these equations are significantly simplified for the case of a radially symmetric device. Specifically, the changes in the major diameter of the device induced by the mechanical vibrations that


give rise to the optical bistability behavior are directly related to the mechanical degrees of freedom of the system. Therefore, the optomechanical coupling coefficient is simply gom = ωc/R [17]. On the other hand, for an asymmetric device, the major diameter varies with azimuthal angle (φ); therefore, gom is a function of R and φ. This dependence limits the appropriateness of using symmetric weighting functions to perform the mapping of the 3D mechanical displacement field to a 1D harmonic oscillator, as done in previous work [30]. Therefore, while the motion of asymmetric devices can still be described by Eqs. (1) through 5, the non-uniformity of the geometry must be included explicitly in order to accurately model the structure's mechanical frequency. To account for this change, the orthogonal eigenfrequencies of the structure can be calculated using the spatial equation of Hooke's law [3]:          (λ + μ )∇.(∇.Φ n (r )) + μ∇ 2 Φ n (r ) = − ρωn2 Φ n (r ) (7) where λ =

E σE and μ = are Lamé constants. σ and E are Poisson's ratio (1 + σ )(1 − 2σ ) 2(1 + σ )

and Young's modulus, and ρ is the density of the material. Also in Eq. (7), ωn is the eigenfrequency of nth mode, and Φn (r) is the position dependent mode profile function. This equation enables all of the eigenfrequencies of the structure independent of their time dependencies to be determined and, therefore, the system can be described in terms of a set of damped harmonic oscillators. Equation (7) is used to determine the eigenfrequencies of the asymmetric devices numerically in COMSOL Multiphysics. 3. Modeling 3.1 Finite element method modeling Due to the complexity of the asymmetric structures, it is not possible to solve for the mechanical modes analytically. To overcome this limitation, COMSOL Multiphysics, Finite Element Method (FEM) software, is used to numerically calculate the eigenfrequencies of the asymmetric structures. By extracting the exact dimensions of each device from optical microscope images, accurate models of each asymmetric device are drawn in SolidWorks and imported into COMSOL to solve for the mechanical eigenfrequencies. The mesh size for mechanical simulations is set to 0.7 µm. In addition, a second set of simulations were performed to better understand how the asymmetry changes the optical to mechanical energy conversion. In these models, a known radially symmetric force simulates the radiation pressure excitation from the whispering gallery mode, and the mechanical oscillation is determined. 3.2 Mechanical modes in asymmetric resonators The first fourteen modes and optomechanical frequencies from a single asymmetric device with an aspect ratio of 0.68 and a pillar offset of 2.1 μm are shown in Fig. 2. A cursory analysis of the modes reveals that the behavior of the asymmetric device is completely unique. While many of the conventional cantilever and crown modes are present in the simulation spectrum of this device, several previously unobserved mechanical modes appear, because the optical force is not uniformly distributed along the device periphery. This effect results in the mechanical modes becoming degenerate and aligning along the maximum and minimum axis of asymmetry. To understand the force transduction, the potential energy stored in the structure from a radial force is calculated by running a second set of FEM simulations. To simulate a whispering gallery mode optical field, a 10mN circulating force is applied perpendicular to the sidewalls. This force mimics the pressure that the circulating optical whispering gallery mode field applies to the device. Both an asymmetric and a symmetric device are modeled.


Fig. 2. FEM simulation results of the first 14 modes of an asymmetric cavity along with their corresponding frequencies. The red lines indicate the lowest threshold modes for symmetric devices: the first and third cantilever modes. The blue lines indicate the lowest threshold modes for the asymmetric devices: the second asymmetric crown mode and the second asymmetric cantilever mode. Degenerate modes are represented with α and β and linked with a bracket.

The frequency of the applied force is varied over each mechanical resonance and the maximum potential energy is calculated for each mechanical mode. In order to clearly understand the efficiency of the process, the first cantilever mode is used as the reference potential energy for each device geometry. Therefore, a value greater than 1 indicates that it is a more efficient mechanical oscillator than the first cantilever mode. Given the number of mechanical modes present in the device (Fig. 2), the following discussion is focused on the two highest energy modes for both device geometries. These modes are underlined in Fig. 2, and the relative potential energy values are shown in Table 1. Table 1. Energy ratio and mechanical modes. Summary for the two primary Symmetric and Asymmetric modes calculated based on simulations. (*)Denotes the corresponding mode number in Fig. 2. Mechanical mode (symmetric) First Cantilever mode

Maximum Energy Ratio (symmetric) 1

Second Crown mode

0.00297

Second Cantilever mode

0.17376

Radial Breathing mode

47.1629

Mechanical mode (asymmetric) First Cantilever mode (2*) Second Crown mode (4-α*) Second Cantilever mode (8*) Radial Breathing mode (14*)

Maximum Energy Ratio (asymmetric) 1 30.075 8.75 1.12

In symmetric devices, the radial breathing mode has the highest energy ratio, indicating it is able to efficiently convert optical to mechanical energy. This finding agrees with previous experimental findings that demonstrated that the radial breathing mode had the lowest threshold [20]. In contrast, in asymmetric devices, the threshold power for the radial breathing mode was only slightly different from the first cantilever mode. However, notably, it was still greater than 1, indicating that these modes can still be excited. This change in behavior can be directly tied to the presence of the asymmetry. For the case of the cantilever mode, it is particularly straightforward. As mentioned previously, an rmin and rmax can be defined. For the cantilever mode, these points act as pivot points. However, unlike in the case of a symmetric or balanced device, in an asymmetric device, it takes more energy to move the rmax side than the rmin side. Moreover, given the dependence of the overhang length on r, the rmin side


experiences a larger torque, further increasing the disparity between rmin and rmax. As a result of this imbalance, this mode takes more power to excite. In the asymmetric devices, the second crown mode (mode 4α) had the highest energy ratio, followed by the second cantilever mode (mode 8). Interestingly, the second crown mode is a degenerate mode. Neither of these modes has been observed in symmetric devices. A simple analysis of the energy ratios in Table 1 provides key insight. While the radial breathing mode experiences a greater than 47x increase in energy ratio, the energy ratio for the second crown mode decreases by nearly three orders of magnitude, and for the case of the second cantilever mode, the energy ratio decreases by almost an order of magnitude. As a result, both modes are very energetically unfavorable. 3.3 Dependence of frequency on geometry To develop a more quantitative analysis for the asymmetric modes observed, we compare the frequencies of the second asymmetric crown mode (fcrn) and the second asymmetric cantilever mode (fcntl) of a device with varying average parameters. These modes are selected because their threshold power drops drastically in devices where the major and minor radii vary as previously described. A series of simulations demonstrate the effect of major and minor radii as well as the overhanging length on the frequency of the crown and cantilever modes, Fig. 4. In each simulation only one variable is swept. Finally, we fit curves to the resulting frequencies to estimate the trend of variations. The analysis reveals that the dependence on the different values of the geometrical parameters is:

f crn ∝

f cntl ∝

((L − α

(( r − α ) 5

2

(r − α1 ) 2

2

(8)

) 2 − α 3 ) *( R + α 4 )

)

1

− α 6 * ( L − α 7 ) * ( R − α8 ) 2

2

(9)

where r, R, and L are the average minor radius, major radius and overhanging length, respectively. All αn terms are offset constants. The results are plotted in Fig. 4. To have a clear comparison only one parameter is changed in each graph. From Fig. 3, it is apparent that the frequency of the cantilever mode decreases for all parameter variations. In contrast, for the crown mode, the frequency decreases for the major radius and the overhanging length, but it exhibits a significantly different trend for the minor radius.

Fig. 3. Theoretically calculated dependence of second asymmetric crown mode (black squares, left axis) and cantilever mode (blue circles, right axis) on the average (a) minor radius, (b) major radius, and (c) overhang length. Each is fit to a quadratic (red line).

The asymmetric crown mode has a shifted parabolic dependence on the minor radius. Therefore, for smaller minor radii, increasing r results in a frequency decrease due to a larger torus mass. However, for larger values of r (with respect to α1), the frequency increases as a result of an increase in mechanical stiffness. It is possible to define a threshold overhanging length, above which the impact to the frequency is negligible, which could enable the design of optimized device architectures.


4. Results and discussion 4.1 Fabrication

To fabricate asymmetric high-Q devices, a combination of photolithography, buffered oxide etching, and XeF2 etching are used to create an array of silica microdisk cavities integrated on silicon. To induce asymmetry in a controlled and systematic manner, the microdisk is reflowed using an intentionally misaligned CO2 laser. Specifically, the laser beam and the silicon pillar are slightly offset, creating a non-uniform thermal gradient around the periphery of the device during reflow. In the final asymmetric device, the minor and major radii vary as a function of azimuthal angle Fig. 1(b). Asymmetry is defined in terms of the location of the pillar relative to the device center. To quantify the role of major and minor diameter, we introduce the parameter M as: M =

Rmax − Rmin Rmax

(10)

where Rmax and Rmin along with other relevant structural parameters are defined in Fig. 1(b). A key aspect of this device is the non-uniformity of both the overhang length and the cavity radius. Both values change gradually as one progresses radially around the circumference of the microcavity, with the overhang length increasing as the cavity minor radius (r) decreases. As such, both the overhang length and the device radius are dependent on angle (φ). For the values of Rmax≈Rmin, the device is almost symmetric and behaves similar to symmetric toroidal microcavities. As M increases, the asymmetry in the structure increases. To achieve the range of asymmetries studied in the present work, offsets ranging from 60 to 80 µm are introduced between the center of the CO2 laser beam and the silica microdisk to control the degree of asymmetry. The CO2 laser intensity is increased at a constant rate of 5 W/s to 25 W. Specifically, devices with major diameters of 44-48 μm, AR (AR = rmin/rmax) of 0.67-0.9 and pillar offsets (Δz) of 1-3.5 μm, are studied. Symmetric devices are fabricated using the conventional method and serve as a control measurement with zero degrees of asymmetry [31, 32]. 4.2 Device characterization

Using a tapered optical fiber waveguide, light from a 1550 nm tunable laser (Newport Velocity series) is coupled into the optical cavities. While continuously scanning the laser wavelength across the resonance frequency, the transmission spectrum is recorded and fit to a Lorentzian to calculate the optical quality factor. It is important to note that the Q is measured in the under-coupled regime with minimal power to reduce artifacts that arise from thermal broadening [32]. The quality factors for the asymmetric devices range from 3x106 to 3x107. Therefore, the measured quality factors prove that by using the described method, it is possible to fabricate an asymmetric device while maintaining the requisite high quality factor to achieve optomechanical behavior. The primary optical loss mechanism is radiation loss due to the varying optical mode profile [33–35]. An Electrical Signal Analyzer (ESA) is used to measure the mechanical frequency spectrum of the transmitted light [36]. The threshold power for the onset of mechanical oscillations is determined by varying the input power of the laser from sub-threshold power to the saturation region, while monitoring the change of the output measured by the ESA. 4.3 Asymmetric crown and cantilever mode

To characterize the mechanical behavior of our device, the thresholds and frequency spectrum for mechanical oscillations are determined. Figure 4 shows the spectra of the two lowest threshold mechanical modes of an asymmetric device. The mode at 15.96 MHz has a 38 μW threshold, and the mode at 47.3 MHz has a 148 μW threshold. When these values are


compared to the previously discussed FEM results, they correspond to the second asymmetric crown and the second asymmetric cantilever modes, as predicted. Additionally, the first and third cantilever modes are excited at 9.58 MHz and 72.8 MHZ; however, the power thresholds are greater than 1.4 mW which is an order of magnitude larger than the asymmetric modes. In a symmetric device, these modes would be excited with threshold powers on the order of 20-40 µW [20].

Fig. 4. (a) Bright field image of the asymmetric device with the major and minor radii illustrated. Dashed lines are circles drawn as guides to the eye to aid in visualizing the asymmetry present in the device. ESA spectra data for the two lowest threshold modes from the asymmetric device: (b) the asymmetric crown mode and (c) the asymmetric cantilever mode. Threshold curves for (d) the asymmetric crown mode and (e) the asymmetric cantilever mode.

While the mechanical modes with the lowest threshold in the asymmetric device are different from those in the symmetric device, the absolute threshold values for the two lowest threshold modes are of the same order of magnitude [17,20]. This similarity indicates that the energy conversion process in symmetric and in asymmetric devices is equally efficient for the different mode types as predicted by the FEM simulations and the results in Table 1. Therefore, the introduction of the asymmetry provides access to previously unobserved modes at low threshold powers without completely suppressing previously characterized mechanical modes. 4.4 Dependence of frequency on asymmetry (M)

In order to fully understand the dependence of oscillation frequency on device asymmetry, a suite of devices with varying degrees of asymmetry are characterized. The experimentally determined frequencies of crown and second cantilever modes are plotted versus M in Fig. 5. Complementary FEM modeling is performed at specific M values and is included for comparison. The simulated frequency is in agreement with the measured frequency for all the modes discussed here with a maximum error of 5%. The error is calculated by subtracting the simulated values from the real value, dividing by the real value, and then multiplying by 100%. For the asymmetric crown mode and the second cantilever mode, the mechanical frequency initially increases and then plateaus as M increases. This behavior originates from


dependence of the device stiffness (spring constant) on the membrane overhang length and the AR. Specifically, as M increases, the torus minor diameter increases and the overhanging length decreases. Therefore, for small values of M, the mechanical frequency scales with M. But as M increases beyond 0.06, the impact of structural changes on the spring constant diminishes and the mechanical frequency reaches an upper limit.

Fig. 5. Overhang length normalized frequency dependence of the (a) second asymmetric crown mode and (b) second asymmetric cantilever mode on the eccentricity parameter, M. Black squares are results from FEM simulations, and red circles correspond to experimental results. The solid black line is a fit to theoretical data for illustration purposes. Maximum Error is around 5%. Note: There are error bars in both the x and y axes, but the error bars are usually smaller than the symbols. The error in the M ratio is calculated by multiple measurements of Rmin and Rmax in optical images, similar to the one shown in Fig. 4(a). The error in the normalized frequency is calculated by projecting the errors occurred in M to the values of measured frequency.

4.5 Dependence of threshold on asymmetry

To show the effect of eccentricity on the threshold for the onset of mechanical vibrations, the threshold powers of different devices with different asymmetries are measured and plotted in Fig. 6 as a function of the normalized threshold (Pth*Q3/Vrm), where Vrm is the average mode volume. There is a clear dependence of the threshold power on asymmetry. Specifically, the threshold power drops drastically in both modes, which is in agreement with the previous predictions based on energy transfer.

Fig. 6. (a) Normalized threshold graph for asymmetric crown mode vs. pillar shift. (b) Normalized threshold for asymmetric cantilever mode vs. pillar shift. Note that the threshold values have been normalized (Pth*Q3/Vrm) to desensitize them to other factors that affect threshold. Vrm is the average mode volume.


5. Conclusion

In conclusion, asymmetric whispering gallery mode optical microcavities integrated on silicon are fabricated and characterized. Quality factors range from 3x106 to 3x107. Due to the combination of high-Q and device asymmetry, optomechanical modes are regeneratively excited with sub-mW thresholds. These modes have previously only been observed indirectly using thermomechanical excitation. 3D FEM models confirm the experimental findings and support the hypothesis that the asymmetric crown and cantilever modes are enabled by spatially varying the optical force. In additional to enabling regenerative excitation of these unique optomechanical modes, high Q asymmetric resonators have many applications in quantum optics [7, 15], chaos studies and directional coupling [18,37–41]. Funding

Office of Naval Research [N00014-11-1-0910] Acknowledgments

The authors would like to thank A. Kovach, M. Reynolds, I. Kim and E. Moen.