Broadband Tuning of Optomechanical Cavities

We demonstrate broadband tuning of an optomechanical microcavity optical resonance by exploring the large optomechanical coupling of a double-wheel microcavity and its uniquely low mechanical stiffness. Using a pump laser with only 13 mW at telecom wavelengths we show tuning of the silicon nitride microcavity resonances over 32 nm. This corresponds to a tuning power efficiency of only 400 $\mu$W/nm. By choosing a relatively low optical Q resonance ($\approx$18,000) we prevent the cavity from reaching the regime of regenerative optomechanical oscillations. The static mechanical displacement induced by optical gradient forces is estimated to be as large as 60 nm.


INTRODUCTION
The control of light using optical microcavities has important applications spanning from quantum aspects of light-matter interaction [1][2][3][4] to information routing in advanced photonic networks [5][6][7][8][9][10]. The key characteristic of these cavities is their resonant response, which occurs only when the wavelength of light is an integer fraction of the cavity's optical path length. Therefore in order to reconfigure these devices, a method to tune the optical path length is necessary. Electro-optic tuning has been demonstrated using LiNBO 3 [11,12] however only sub-nm tuning was achieved; thermo-optic [9,13] or free-carrier injection based [5,7] tuning has also been demonstrated with tens of nm tuning range. These methods however not only are limited to materials with high thermo-optic coefficient or strong free-carrier dispersion, but also require high temperatures (> 400 K) or suffer from free-carrier induced losses [5,7,13]. Another way to control the cavity length is to manipulate their mechanical degrees of freedom [6,[14][15][16][17][18][19][20][21][22], such manipulation can be achieved using the optical forces provided by photons circulating inside the optical microcavities [6,14,18,23,24]. For example, it has been recently shown that optical gradient forces can be used to actuate the mechanical motion in these cavities with tuning ranges exceeding 2 nm [6,14,23], and also proposed as a tuning method for various waveguide and microcavity parameters [25,26]. Here we demonstrate that the use of optical forces can provide tuning of a microcavity resonance over 30 nm using only 13 mW of laser power. This device power efficiency is 400 µW/nm and its tuning range can span the full telecom C or L-band.

OPTICAL FORCES IN DOUBLE-RING CAVITIES
To achieve a strong per-photon optical gradient force we choose to work with a coupled pair of microring cavities similar to our previous work [14]. As illustrated in the schematic of Fig. 1a, it consists of two vertically stacked microring cavities. The subwavelength air-gap between the rings allows for strong coupling between the optical modes of the two rings. For air gaps around 170 nm, the optical modes of each ring evanescently couple to each other forming symmetric and anti-symmetric optical super-modes; in Fig. 1a we show the electric field profile of the symmetric one. The coupling induces a splitting in their resonant wavelength which depends exponentially on the gap between the rings, as shown in the blue curve of Fig. 1b for the symmetric optical super-mode. Therefore one can use this gapdependence of the optical resonances to tune the microcavity's optical response. According to the simulated curve in Fig. 1b a telecom band tuning of 30 nm can be achieved with a gap change of only 60 nm. In this optical microcavity, the force to drive such a change can be derived from the optical field gradient in the cavity. The optical energy inside the cavity depends on the optical mode resonant frequency (U = Nhω where is N is the number of photons circulating in the cavity and ω is the cavity mode optical resonant frequency), therefore the cavity's optical energy also depends on the gap between the two ring resonators; an optical force between these rings should follow using a virtual work approach [26,27]. The optical potential energy change inside the cavity (Nhδω) must correspond to the mechanical work realized on the microrings (−F δy), therefore the optical force is given by F = −Nhg om , where g om = ∂ω/∂y is the optomechanical tuning efficiency. This tuning efficiency is shown as the red curve in Fig. 1b. When operating at small gaps (≈170 nm) the resonant frequency tuning efficiency g om /2π can be as high as 60 GHz/nm, this corresponds to a 40 fN/photon optical gradient force.
The fabricated microcavity structure is optimized to achieve maximum optical frequency tuning and per-photon optical gradient force. Such optomechanical frequency tuning can be expressed as a function of the optical power in the cavity. When a pump laser excites the symmetric super-mode resonance, the flexible spokes bend in response to the optical gradient force. The gap between the rings reduces and the optical resonant frequency of this mode decreases as shown in Fig. 1b. The cavity optical frequency shift due to mechanical displacement is simply given by ∆ω = g om ∆y, where ∆y is the gap change between the rings. These relations can be combined with Hooke's law (F = k∆y/2, k is the stiffness of the rings in response to the gradient optical force [28]) to yield an expression for the cavity frequency shift and the optical power coupled to the cavity, where Q i is the intrinsic optical quality, ω 0 is the optical resonant frequency, and P d = (1 − T )P in is the power dropped to the cavity resonance with normalized transmission T . According to Eq. 1, the optomechanical tuning g om and the beam stiffness k are the two cavity parameters that can be engineered in order to achieve large optical frequency tuning. There are however practical limits to them; to increase g om , as shown in Fig. 1b, it is necessary to have cavities with small gaps. When the gaps are smaller than 100 nm however, the fabrication yield is considerably smaller since the structures tend to collapse because of the short-range Van der Wals interaction [29]. This also limits the smallest spring constant that can be practically achieved. The double-ring optical cavity used here can have optical and mechanical parameters, such as g om , Q i and k that increases the threshold for regenerative mechanical in the cavity [3,30,31] and enables large static tuning. When the regenerative oscillation threshold is reached, one or more mechanical modes of the structure will oscillate with a large amplitude leading to a strong modulation of the light transmitted by the cavity. In Fig. 2a we show the mechanical displacement profile of two floppy mechanical modes that will be driven by the optical gradient force. The threshold optical power at which regenerative mechanical oscillations begins will define the maximum static frequency tuning, i.e., an upper limit for the static operation of our device. As we illustrate in Fig. 2b, when the pump laser is bluedetuned with respect to the cavity frequency (∆ ≡ ω p − ω 0 > 0), it provides optical gain for the thermally excited mechanical modes and effectively reduces the mechanical damping, narrowing the mechanical resonance linewidth (Γ m ). At the threshold power P th , the optomechanical gain exceeds the intrinsic losses of the mechanical modes (Γ m < 0, dashed black line in Fig. 2b) and they enter into regenerative oscillations [3,14,30,32]. As a result, any optical signal going through the cavity is strongly modulated at the mechanical frequency Ω m of these modes. The floppy mechanical modes shown in Fig. 2a are the first two mechanical modes that are strongly driven by optical field due to their mostly vertical and opposing (or bright) motion of the two rings. The threshold input power to om , and mechanical quality factor Q m = Ω m /Γ m is given by P T min )/2 is the ideality coupling factor for an undercoupled (−) or overcoupled (+) cavity, T min is the transmission value exactly on resonance, and Q = Q i (1 − η c ) is the loaded optical quality factor [3,30,[32][33][34][35]. Here we assumed that the cavity is excited close to the optimal cavity frequency detuning ∆ ≈ −Γ/2 (where Γ = ω 0 /Q) and that the cavity parameters are within the unresolved sideband limit, Ω m Γ. At this detuning point, the cavity transmission is given by T (∆ = −Γ/2) = 1 + 2(η c − 1)η c . Using this transmission value and the power threshold expression above together with Eq. 1, an expression can be derived for the maximum static frequency shift for an optomechanical cavity as limited by optomechanical oscillations of the m th mechanical mode, In the case of the fundamental anti-symmetric (bright) mechanical mode (m = 1), this expression does not depend on the optomechanical tuning efficiency g om since g om ≈ g om , a high value of g om however ensures that large tuning can be achieved using low optical powers (see Eq. 1). For double-ring cavities however many mechanical modes will have similar g (m) om , for example g (2,3) om /g (1) om ≈ (60%, 64%), where (2,3) stands for the second and third order bright mechanical modes, therefore Eq. 2 can still predict the maximum frequency shift as limited by optomechanical oscillations of the higher order mechanical modes. In Fig. 2c we show the maximum wavelength tuning predicted by Eq. 2 for a doublering cavity with a loaded optical Q = 18 × 10 3 (vertical dashed line), each curve represent the maximum wavelength tuning as limited by regenerative oscillations from the two mechanical modes shown in Fig. 2a. The red and blue lines corresponds to the first and second mechanical modes shown in Fig. 2a with parameters Ω m /2π = (0.646, 8.041) MHz, m ef f = (87, 26) pg, k = 1.44 N/m, and Q m = 4. Since these modes may have distinct mechanical quality factors, we represent in the wide blue and red regions the tuning range spanned when the mechanical quality factor vary between 1 < Q m < 10. The overlap of these region show that depending on their mechanical quality factor, the second order mode may reach oscillation threshold before the first mode. These parameters should allow for a maximum static tunability in the few hundred nanometers range as shown in Fig. 2c.

EXPERIMENTAL RESULTS
We demonstrate experimentally optomechanical tuning exceeding 30 nm using only 13 mW of laser power, well below the regenerative oscillations threshold. The fabricated cavity, shown in Fig. 1c,d, has a 30 µm diameter and a 3 µm wide ring. Each ring is made of 190 nm thick stoichiometric LPCVD (low-pressure chemical vapor deposition) Si 3 N 4 . The details of the fabrication process can be found elsewhere [14]. The spokes have cross-section dimensions of 190 x 500 nm. The air gap between the rings is about 170 nm. A top view of the device under test is shown in Fig.  3a and the experimental setup schematic is shown in Fig. 3b. A typical low power transmission spectrum obtained using a tapered optical fiber coupled evanescently to the cavity is shown Fig. 3c. The loaded optical quality factor of the pump resonance was Q = 18, 000 whereas the mechanical quality factors was Q m = 2 for the mechanical mode at 8.05 MHz, for the first order mode at 646 KHz it was too low and was not measured since it was below the noise level in our direct detection setup. Such low mechanical quality factor is typical in double-ring cavities due to the strong damping caused by gas trapped between the rings [6,14,30,36]. To induce the optical force we use a 13 mW tunable external cavity pump laser centered on an optical resonance at 1580 nm (highlighted in Fig. 3c), the polarization of the pump is adjusted to maximize its coupling to the cavity TE mode. As the pump laser wavelength scans this resonance (0.5 nm steps) from shorter towards longer wavelengths, more power is dropped to the cavity as the laser approaches the resonance. At each pump wavelength step, we record the probe transmission over the range indicated by the horizontal arrow in Fig. 3c. We summarize our experimental data demonstrating 32 nm optomechanical tuning in Fig. 4. The most dramatic effect of the optomechanical tuning can be seen by simply observing under a microscope the colour change of the light reflected off the top of the cavity. As the air-gap between the rings is reduced, the thin-film interference effect allow us to see in real time such a change. We show this effect in the sequence of images on right-side of Fig. 4a, and also on the attached multimedia file on Fig. 4. A more quantitative evidence is revealed by the pump laser transmission shown in Fig. 4b. The offset curves correspond to different power levels as the laser scans from shorter to longer wavelengths. As the pump power increases (from bottom to top) the optical resonances show the typical triangular shape of a bistable cavity [6,14,37]. Such bistable transmission curves results from the red-shift induced by the change in the air-gap due to optical forces. At the highest pump power of 13 mW, the bistability extend over 32 nm, which spans twice the optical free spectral range. This indicates that a very large optomechanical shift is induced. The probe transmission curves at each pump wavelength are shown as offset curves in Fig. 4a. Since the pump power actually dropped to the cavity is related to the pump transmission curve and input power as P d = (1 − T )P in , the minimum pump transmission (around 1617 nm) is T =0.4 and therefore the maximum power dropped to the cavity is P d =7.8 mW. Using Eq. 1 with this power value and a spring constant of k=1.44 N/m, as obtained from static finite element simulations [28], we calculate the expected wavelength shift to be ∆λ=33 nm, in good agreement with the measured value.
As the dropped power in the cavity increases we observe tuning of mechanical resonant frequency (i.e., optical spring effect) as well as a reduction in the mechanical resonance linewidth (i.e., optomechanical amplification). These two effects can be seen in Fig. 4c and 4d where we show a density plot of the transmitted pump RF spectrum measured using a 125 MHz photodetector (PD2 in Fig. 4a). The observed shift in the mechanical resonant frequency, from 8 MHz to 32 MHz, corresponds to a stiffening of the mechanical resonator of k /k = (Ω /Ω) 2 ≈ 16. The stiffening of the resonator is so large that the mechanical mode exhibit anti-crossings [38] with higher frequency modes, two of such anti-crossings at 17 and 31 MHz are highlighted in Fig. 4d. Each of these anti-crossings are actual doublets formed by a symmetric (dark) lower frequency mechanical mode Ω d /2π = (16, 31) MHz, shown in Fig. 4e,f, and an anti-symmetric (bright) higher frequency mechanical mode Ω b /2π = (17, 32) MHz, shown in Fig. 4g,h; the mechanical frequencies shown in these figures are obtained from finite element simulations and agree with the measurements within < 10% difference. This identification is justified since after the anti-crossing with the dark modes, the mechanical mode preserves its frequency and appears as a straight vertical trace in density plot of Fig. 4d, whereas after the anti-crossing with the bright mode, the mechanical mode keeps increasing its frequency due to the spring effect. The major benefit of this stiffening for static tuning applications is the reduction of thermal driven vibrations, which scales as k B T /k according to the equipartition of energy [6,14,37]. The reduction of the mechanical linewidth leads to an increase in the mechanical quality factor, as shown in Fig. 4c which was measured to increase from Q m ≈ 2 to Q m ≈ 30. This confirms that our device is indeed well below the optomechanical oscillations threshold. The optical absorption inside the cavity raises its temperature and also contributes to the measured shift. There is a thermo-optic contribution arising from the refractive index change (δn = α∆T , where α = 4 × 10 −5 K −1 for Si 3 N 4 ) and a thermo-mechanical contribution due to thermal expansion of the cavity. Although such thermal expansion could also cause a change in the gap between the rings, it has been shown that the major thermo-mechanical shift is caused by radial expansion [6,14]. We verified both numerically and experimentally that these thermal contributions are negligible. Considering both thermal contributions we may write ∆ω th = g T δT + g R δR where g T = ωα/n g and g R = ω/2πR, one can estimate the radial expansion of the ring using the relation δR/R = α L δT , where α L ≈ 10 −6 K −1 is the SiN thermal expansion coefficient. To estimate the temperature change we assume an absorption loss of 0.06 dB/cm [39], which corresponds to an optical absorption quality factor of Q abs = 5.2 × 10 6 . The total thermal resistance of the cavity, as calculated through the finite element method [40], is R th = 7 × 10 5 K/W. Using these parameters, the estimated temperature change in the ring is δT = R th P heat =23 K, where the heating power is calculated from the intra-cavity energy U as P heat = ω 0 U/Q abs [14]. Using the above relation we calculate the thermal contribution to the frequency shift, the thermal expansion term gives g T δT /2π = −4.5 GHz, whereas the thermo-optic effect gives g T δT /2π = −99 GHz. The total thermal shift is therefore ∆ω th /2π = −103 GHz, or equivalently ∆λ th =0.86 nm, which corresponds to 3% of the measured shift. To verify that the thermal contribution is indeed small we tested a device in which the two rings collapsed and were stuck to each other and therefore do not experience the usual optomechanical tuning. Such a small contribution from thermal shift was verified experimentally is in agreement with previous results on double-ring and spider-web cavities where the optomechanical tuning is the dominant effect. Some contribution could also arise from the nonlinear Kerr effect, however due to low finesse of our cavity (F ≈ 180) and relatively large effective mode area (A ef f ≈ 8 × 10 −13 m 2 ), we estimated the Kerr contribution ∆λ Kerr = λn 2 FP d /(πn g A ef f ), where n 2 = 2.5 × 10 −15 cm 2 W −1 is the nonlinear refractive index of Si 3 N 4 and n g = 1.8 is the cavity mode group index, to be below 1 pm and therefore negligible in our device.

CONCLUSIONS
In conclusion, we show efficient (400 µW/nm), broadband (across C &L bands) tuning of optical resonances using gradient force actuation of optical devices. We also show that competing effects such as thermo-optic effect and Kerr effect contribute only to a small extent to the overall optical frequency shift. This optomechanical tuning approach is not only competitive with other known tuning methods, but also advantageous since it simplifies the fabrication process by avoiding metal contacts etc. Static tuning beyond the one achieved here should be possible using such devices, although further optimization of the resonator parameter, such as spokes thickness, inter-ring gap, ring width, and perhaps higher laser powers may be necessary. While we focused on the static effects in this work, using appropriate experimental conditions, high tuning efficiency gradient force optomechanical devices may offer a potential for studying dynamic effects of radiation and near field forces.