Radiation pressure forces have recently been studied in the context of mechanically compliant optical microcavities for the sensing, actuation, and damping of micromechanical motion [1, 2]. A wide variety of cavity geometries have been explored, from Fabry-Perot cavities with movable internal elements or end-mirrors [3–8], to monolithic whispering-gallery glass microtoroids [9]. Nanoscale guided-wave devices have also been studied due to their strong optomechanical coupling resulting from the local intensity gradients in the guided field [10–16]. In addition to radiation pressure forces, there exists in each of these cavity geometries competing thermally driven effects [17], a result of optical absorption. Thermally induced processes include strain-optical, thermo-optical, and a variety of thermo-mechanical effects (early measurements of radiation pressure, for instance, were plagued by thermo-mechanical “gas action” effects [18]). In many cases (but not all [19, 20]), thermal effects can be neglected at the high frequencies associated with micromechanical resonances [9]; however, for static cavity tuning thermal effects may play a significant, if not dominant, role. Calibration of thermal effects is important not only in identifying the contribution of pure radiation pressure effects, but also in understanding the parasitic local heating processes, which in the realm of quantum optomechanics may limit optical cooling methods [21, 22], or for tunable photonics applications [13, 23, 24] where deleterious inter-device thermal coupling may arise.

 

Fig. 1. (a) Slot width vs. extension offset, simulation and experiment. Support extensions placed near the outside of the beam width cause outward bowing, but inward bowing when placed near the inside. (b) Scanning electron microscope images of a device with support extensions placed just inside of center, causing slight inward bowing resulting in approximately a slot-gap of s=40 nm at the cavity center. The device parameters that are common amongst all the devices tested in this work are: nominal lattice constant an=590 nm, extension width w_e=211 nm, hole width h_x=190 nm, hole height h_y=416 nm, beam width w=833 nm, beam length l_b=18.4 µm, extension length l_e=33.1 µm. In order to vary the slot-gap size from 40–560 nm, the extension offset was varied between o_e=70 to -235 nm.

Download Full Size | PDF

In this work we describe the characterization of the low frequency (static) optical and thermal effects in a nanoscale photonic crystal cavity. This so-called zipper cavity [14, 25, 26] consists of a matched pair of Si₃N₄ nanobeams, placed in the near-field of each other, and patterned with a one-dimensional (1D) array of air holes. The resonant optical modes of the zipper cavity [25] consist of manifolds of even (bonded) and odd (anti-bonded) symmetry supermodes of the dual nanobeams, localized along the long-axis of the beams to a central defect in the photonic lattice. The strength of the gradient optical force applied to the beams is exponentially dependent upon the inter-beam slot gap (s) and is described by a dispersive coupling coefficient, g _OM≡(∂ω_c/∂α), where ω_c is the gap-dependent optical cavity resonance frequency and α parametrizes the change in the inter-beam slot gap. Utilizing the connection between cavity mode dispersion and applied optical force, a pump-probe scheme is described below to accurately quantify the optomechanical, thermo-optic, and thermo-mechanical contributions to the static tuning of the bonded and anti-bonded modes versus internal cavity photon number.

The zipper cavity devices studied here were fabricated from optically thin (t=400 nm) stoichiometric silicon nitride (Si₃N₄), deposited using low-pressure chemical vapor deposition on a silicon wafer. The deposition process results in residual in-plane stress in the nitride film of approximately σ~1 GPa [27]. Electron-beam lithography is used to pattern zipper cavities with beams of length l_b=33.1 µm, width of w=833 nm, and an inter-beam spacing of s=120 nm. The beams are clamped at each end using extensions with length equal to l_e=18.4 µm and width of w_e=211 nm. The nanobeam and photonic crystal hole pattern are transferred into the Si₃N₄ film using a C₄F₈/SF₆ plasma etch. The underlying Si substrate is selectively etched using KOH, releasing the patterned beams. The devices are dried using a critical point CO2 drying process to avoid surface-tension-induced adhesion of the nanobeams.

 

Fig. 2. Experimental setup for optical testing. Two separate lasers (pump and probe) with independent power control, via variable optical attenuators (VOA), and polarization control are combined into a fiber taper waveguide placed in the near-field of the photonic crystal cavity. Cavity transmission at both the pump and probe wavelengths are multiplexed and demultiplexed using a matched set of fiber-based filters and separately monitored using calibrated photodetectors.

Download Full Size | PDF

The internal stress of the Si3N4 thin-film is used to create a range of slot-gaps. This is achieved through misalignment of the support extensions and central nanobeams (Fig. 1(b)), breaking the symmetry of the internal stress along the length of the beams. As shown by the finite-element-method (FEM) simulations and device measurements plotted in Fig. 1(a), extensions placed near the outside edge of the beams cause outward bowing of the beams, while extensions placed near the inside edge cause inward bowing. By varying the lateral offset of the extension (defined as the difference between the center of the main beam and the center of the extension beam) from o_e=70 to -235 nm, zipper cavities with slot-gaps at the cavity center ranging from 40–560 nm are created. Slot-gaps smaller than 40 nm could not be stably produced without the nanobeams sticking together at points of nanometer-scale roughness in the inner sidewall of the beams.

Optical spectroscopy of the zipper cavity modes is performed using the experimental setup shown in Fig. 2. In this setup, laser light from a bank of tunable external-cavity diode lasers covering the 1400–1625 nm wavelength band is coupled into an optical fiber taper nanoprobe. Using precision motorized stages, the tapered fiber can be controllably placed into the near-field of a zipper cavity [28], allowing for evanescent excitation and detection of resonant modes. Polarization of the laser field is adjusted using a fiber polarization controller, and optimized for coupling to the high-Q TE-like modes of the zipper cavity [25] (i.e., dominant electric field polarization in the plane of the device).

Figure 3(a) shows a series of wavelength scans from an array of nominally identical zipper cavities, each with slightly differing slot-gap due to variation in the lateral extension alignment. Scanning electron microscope images of the central region of each zipper cavity, from which the slot-gap is measured, are shown to the right of each wavelength scan. Even (bonded) and odd (anti-bonded) parity supermodes are identified by stepping the taper across the width of the zipper cavity and noting the lateral spatial symmetry of the mode coupling [14]. Three distinct pairs of modes are identified in the wavelength scans, with the fundamental longtitudinal cavity modes (TE_±,0) occurring at shorter wavelengths and the higher-order modes shifted to longer wavelengths (this is a result of the negative curvature of the photonic band-edge from which the modes originate [25]). Radio frequency (RF) spectra of the optical transmission intensity (see Fig. 3(b)) were performed to verify the presence of micromechanical oscillation and free movement of the nanobeams even for the smallest of slot-gaps.

Overlayed on the wavelength scan plots of 3(a) are theoretical dispersion curves generated using FEM simulations of the optical properties of the zipper cavity (a single in-plane scaling factor of 5% was used to match the TE_+,0 resonant wavelength for the largest (560 nm) slot-gap, which is within the accuracy of our SEM calibration). Figure 3(c) shows a plot of the measured and simulated splitting of the fundamental bonded and anti-bonded modes versus the SEM-measured slot gap. Good correspondence is found for all but the smallest (40 nm) slot gap, in which the theoretical curve shows significantly more dispersion for the bonded modes. The source of this discrepancy is not fully understood, but may be due to other effects such as the dispersive nature of the refractive index of the Si₃N₄ material itself (the frequency separation of the even and odd supermodes is more than 10 THz for the smallest slot-gap). As discussed below, the local sensitivity of each mode to beam displacement is consistent with the measured 40 nm slot gap.

In order to understand the various actuation methods of the zipper cavity, it is useful to consider the dispersive nature of the two supermode mode types, and their relation to the magnitude and direction of the optical force. In an adiabatic limit [10,29], the radiation pressure force can be related to the gradient of the internal optical cavity energy, F _OM=-∂(Nh̄ω_c)/∂α=-Nh̄_gOM, where N is the stored photon number and α is a displacement factor related to the movement of the nanobeams (here we choose α to be equal to one half the slot-gap size to be consistent with previous work in Refs. [14, 25]). As can be seen in the measured and simulated dispersion curves of Figure 3(c), the symmetric bonded modes with large field strength in between the beams, tunes to the red with shrinking slot-gap (i.e., g_OM for the bonded mode is positive for α=s/2). The direction of the optical force for photons stored in a bonded mode thus tends inwards, pulling the beams together (this is a result of the fact that in order to perform mechanical work on the beams, the stored photons must lose energy through a reduction in their frequency). In the case of the odd parity anti-bonded cavity modes, the resonance frequency decreases with increasing slot-gap, resulting in a negative g _OM and an optical force that pushes the beams apart.

Owing to the different dispersive character of the bonded and anti-bonded cavity modes, and the different directions in which cavity photons of each mode type apply forces to the two beams, pure radiation pressure effects actuate and tune the cavity modes in a unique and distinctive manner. This should be contrasted with thermo-optic and thermo-mechanical effects. The thermo-optic tuning of the cavity modes is related to the change in refractive index of the cavity material with temperature. For the Si₃N₄ zipper cavity studied here, the thermo-optic coefficient is known to be positive (and of magnitude ∂n/∂T≈1.9×10^-5 K^-1 [14]). The manifold of bonded and anti-bonded cavity modes thus tend to uniformly red shift with increasing temperature due to the thermo-optic effect (differences between modes in resonance frequency and modal overlap with the Si₃N₄ beams are insignificant). Increased temperature in the local cavity region of the beams, due to optical absorption, not only increases the refractive index of the clamped beams, but may also produce non-uniform thermal expansion and significant strain in the structure. Unlike the radiation pressure force, the resulting thermo-mechanical force is only dependent upon the temperature rise, and thus actuates the zipper cavity beams in the same manner independent of which cavity mode is being driven. Owing to the different dispersive nature of the cavity modes, however, the sign of the thermo-mechanical tuning will depend upon the type of supermode being considered.

 

Fig. 3. (a) Experimentally measured optical transmission as a function of wavelength in an array of six devices (SEM images of the central cavity region on right), each with a different slot-gap. Overlayed are FEM simulations of the cavity mode dispersion versus gap size, where solid curves are for the bonded modes, dashed curves for the anti-bonded modes, and the color of the curve matches the highlighting applied to the different mode orders (red=TE_±,0, green=TE_±,1, blue=TE_±,2). (b) Measured RF spectrum of the TE_+,0 mode for the largest gap (s=560 nm) zipper cavity. Inset shows the optical transmission as a function of detuning as the pump laser is swept across the cavity resonance (dashed vertical line indicates detuning for RF spectrum measurement). (c) Plot of the FEM-simulated and experimentally measured bonded and anti-bonded mode splitting versus slot-gap. The inset shows the effective optomechanical coupling length (L _OM≡λ/(dλ/dα) in the small slot-gap region (solid curve for bonded mode, dashed curve for anti-bonded mode).

Download Full Size | PDF

We perform measurements of the different cavity mode tuning mechanisms using a two laser pump and probe scheme. In this scheme, a strong pump beam is coupled into either the fundamental bonded (TE_+,0) or anti-bonded (TE_-,0) cavity mode. In order to measure the self-mode tuning of a given mode the pump laser frequency is swept across the cavity mode resonance producing a bistability curve such as that shown in Fig. 3(b). A fit to the bistability curve yields the self-mode resonance tuning versus dropped power into the cavity for a fixed input power. The dropped cavity power is then converted into an internal photon number via the intrinsic Q-factor of the cavity mode [30] (measured to be Q_i≈6×10⁴ for both the fundamental bonded and anti-bonded modes using a calibrated fiber Mach-Zender interferometer). In order to obtain the cross-mode tuning between bonded and anti-bonded modes the pump beam wavelength is stepped across the resonance of one of the modes. At each step in the pump wavelength a weak probe beam is scanned across the other mode. Converting the dropped pump power into a stored photon number (again using the intrinsic cavity Q-factor), and fitting a Lorentzian line-shape to the probe scan yields the desired cross-mode tuning curve. In these measurements a pump power of ~300 µW and a probe power of ~100 nW were used to ensure only the pump beam produces a significant tuning of the modes. The measured self and cross tuning curves of the smallest slot-gap (40 nm) zipper cavity, with the strongest optomechanical coupling, are shown in Fig. 4(a).

These measurements yield four tuning slopes s_ij≡Δω_ij/N_i, where subscripts i, j∊{e,o} label the cavity mode (e for the even bonded mode, o for the odd anti-bonded mode), N_i is the stored cavity photon number of the pump mode, and Δω_{i j} is the induced frequency shift in the jth mode due to the ith pump mode (note for self-tuning there is only one mode and laser; the jth mode is the same as the ith pump mode). For the limited model in which we consider optomechanical, thermo-mechanical, and thermo-optic sources of mode tuning, the induced frequency shift in a given mode can be separated into two components, one which is mechanical in nature and depends upon the reaction of a given mode to the deflection of the nanobeams, the other which is independent of the type of mode (bonded or anti-bonded) and is only dependent upon the intensity of the internal cavity field (photon number):

where δα_i is the induced beam deflection (change in the half-slot-gap extent) due to the ith pump mode, g_j is the optomechanical coupling coefficient of the jth mode, and here the intensity-dependent coefficient, c_to, is associated with the thermo-optic tuning of the cavity (other tuning mechanisms, such as due to the Kerr effect, are also included in this term but are expected to be negligible in comparison to the thermo-optic effect). The beam deflection can be related to the optically-induced forces acting on the nanobeams, f_i=-h̄(g_i+g_tm)N_i, where the first term is the ith mode radiation pressure force as described above, and the second term is due to general thermo-mechanical effects (parametrized by the constant g_tm). Here we assume that the thermo-mechanical forces depend only upon the photon number, and not the type of mode, which is a good assumption due to the negligible difference in the optical absorption (or generated heat) expected for the two mode types. Putting this all together, the tuning slopes can be related to the four coefficients describing the opto-mechanical, thermo-mechanical, and thermo-optic per-photon forces/dispersion:

where k is the spring constant associated with differential in-plane motion of the zipper cavity nanobeams (which must be determined self-consistently based upon our choice of “amplitude”, α=(1/2)s).

Solving the set of coupled equations for the four cavity tuning coefficients yields (for α≡s/2, i.e., g_e>0, g_o<0),

where $\xi \equiv 1\sqrt{-\left({\Delta }_e+{\Delta }_o\right)}$, and we have defined relative tuning slopes Δ_e=s_ee-s_oe, Δ_o=s_oo-s_eo, and Δ_c=s_eo-s_oe. From the measured and fit tuning slopes shown in Fig. 4(a), we extract the following values for the optomechanical and thermal coefficients of the smallest slot-gap zipper cavity: $\sqrt{\bar{h}⁄k}{g}_e=1.44\pm 0.04$, $\sqrt{\bar{h}⁄k}{g}_o=-0.083\pm 0.021$, $\sqrt{\bar{h}⁄k}{g}_{\mathrm{tm}}-0.29\pm 0.01$, and $\sqrt{-c_{\mathrm{to}}}=0.68\pm 0.02$, all in units of $\sqrt{\mathrm{MHz}⁄\mathrm{photon}}$. The uncertainty in the measured coefficients is dominated by systematic error in the internal cavity photon number (~5% due to uncertainty in cavity Q and optical input power).

The resulting contributions from this model to the cavity mode tuning are shown graphically in Fig. 4(b). Qualitatively, the resulting contributions to the measured mode tuning are consistent with expectations. In particular, the strong radiation pressure force and dispersive nature of the even bonded mode results in the most significant mode tuning when light is coupled into this mode (i.e., the “ee” configuration). In contrast, the weak dispersive nature of the odd anti-bonded mode means that when pumping on this mode not only is the radiation pressure force small, but any thermo-mechanical force is also transduced more weakly for this mode (resulting in the very small tuning in the “oo” configuration). The sign of the thermo-optic coefficient, which should result in a red-detuning of the mode, is found to be in accordance with expectations. The more difficult response to untuit, due to the thermo-mechanical effect, is found to result in an effective force that pushes the nanobeams apart with a magnitude in between that of the radiation pressure force of the even and odd supermodes of the zipper cavity.

A quantitative comparison of the measured and simulated contributions to the mode tuning and relevant forces can also be made. For an effective mechanical spring constant of k=13.18 N/m (estimated from the mode-shape, motional mass m_x≈53 pg, and mode frequency Ω_M/2π=2.5 MHz for the fundamental in-plane mechanical mode of the zipper cavity structure [25]), the corresponding optomechanical coupling coefficients are g_e/2π=202.4±5.6 GHz/nm and g_o/2π=-11.63±2.9 GHz/nm. These should be compared with the theoretical values estimated from FEM simulation at the SEM-measured slot-gap for this device (s=40 nm), which are g_e/2π=208 GHz/nm and g_o/2π=-11.4 GHz/nm. A similar zipper cavity structure was also tested that did not have the thin beam extensions and with ten times the mechanical spring constant. The resulting cavity tuning rate in this structure is dominated by the thermo-optic effect, and is measured to be -0.69 MHz/photon, similar to the pump-probe measured value of c_to=-0.47±0.03 MHz/photon for the device studied in Fig. 4. From measurements of the cavity tuning rate versus substrate temperature of the stiff structure (-1.87 GHz/K), the estimated temperature rise rate in the cavity of Fig. 4 (using the pump-probe measured c_to) is β=2.5×10^-4 K/photon. For the largest pump powers used here this corresponds to a maximum cavity temperature rise of about 14 K. FEM simulations of the thermo-mechanical properties of the cavity structure show an increase of the inter-beam slot-gap per Kelvin of temperature rise in the central cavity region equal to δs_tm=0.021 nm/K. This yields a simulated change in beam displacement factor per cavity photon of δα_tm=(δs_tm/2)β≈2.6 fm/photon, or a thermo-mechanical force coefficient of g̃_tm=-(k/h̄)δα_tm≈-50 GHz/photon, in good correspondence with the pump-probe measured value of g_tm=-40.8±1.4 GHz/photon.

 

Fig. 4. (a) Measured tuning curves for the four pump-probe configurations. The solid (dashed) lines correspond to a linear fit to the self- (cross-) mode tuning data. (b) Optomechanical, thermo-mechanical, and thermo-optic tuning contributions as determined from a fit of the model coefficients.

Download Full Size | PDF

In conclusion, we have introduced a pump-probe method to completely characterize optomechanical, thermo-mechanical, and thermo-optic contributions to the static tuning of a zipper optomechanical cavity. Generally, this approach is useful for guided-wave devices in which near-field coupling between two (or more) elements is used to produce the optical gradient force, and for which the different supermodes of the composite structure introduce similar thermal and differing radiation pressure effects. We demonstrate the method experimentally using a strongly opto-mechanically coupled device, fabricated using a technique that allows for fine control over a nanoscale slot-gap which sets the magnitude of the optical gradient force. Using a device with a measured 40 nm slot-gap, we characterized all tuning mechanisms, and find in particular that the static tuning of the device is dominated by optical forces (g _OM≈200 GHz/nm equivalent to 0.13pN/photon). Beyond cavity optomechanics, the creation of ultra-small (tens of nanometer) slot gaps in photonic devices is interesting for cavity qauntum electrodynamics [25, 31], nonlinear optics [32], and sensing [33], due to the extremely large per-photon electric field strengths that build up in such nano-slots.