1. Introduction

Interferometric gravitational wave detectors, which have observed black hole coalescence since last year [1], are so sensitive that optical forces on the mechanically compliant mirror have to be taken into account [2]. In fact, studies of such optical forces began in the 1970s, and they have included trapping of dielectric particles by lasers [3], cooling of atoms with radiation [4], and the effect of optical force on macroscale mechanical masses [5]. Benefiting from advances in semiconductor fabrication technologies, a host of nanophotonic devices have been used in the past decade to study the interactions between optical forces and mechanical objects [6–9]. Optical forces are manifested as radiation pressure [10, 11] and gradient forces [12, 13]. Radiation pressure originates from momentum transfer when light reflects from object interfaces, such as a micro-toroid cavity [6, 11], while gradient force is generated by a strongly varying light electromagnetic field distribution where objects are polarized and the charged interfaces are stressed by the electromagnetic field, such as in various double-coupled waveguides [8] and cavities [7, 9, 13]. Various nano-optomechanical devices have been used to explore a number of optomechanical phenomena and concepts, such as optical force cooling and amplification [14, 15], optomechanically induced transparency [16], optomechanical crystals with both photonic and phononic bandgaps [17], kand adiabatic wavelength conversion [18]. In another aspect, the concepts of nanoelectromechanical systems (NEMS), downscaled from microelectromechanical systems (MEMS), can be put to use in the optomechanical field and thereby yield optomechanical devices with versatile mechanical designs [19, 20], electrical and optical hybrid drives [21, 22], and more degrees of freedom of motion [23].

An optical nano-cavity is a critical component for many kinds of emitters, such as integrated laser sources [24], cavity quantum electrodynamics (QED) [25], and Purcell enhancement of spontaneous emission [26]. Electro-optic and thermo-optic effects are common ways to tune or modulate the cavities’ optical modes, but it is challenging problem to apply them to emitting cavities, because the electro-optic effect redistributes the carriers so that the emission conditions are greatly changed, while temperature affects the spontaneous emission factor and wavelength as well [27]. Mechanical approaches can be used to tune or modulate the emitting cavities without disturbing the atomic energy levels, and there is a recent report on a nanoscale emitter modulated by a mechanical oscillator [28]. However, the standing nanowire used in that emitter cannot be coupled with other photonic devices, and its piezoelectric driver is not suitable for large-scale integrated electronic/photonic circuits. Optomechanical oscillators can overcome these problems and are compatible with industrial fabrication technology. For example, on the basis of the silicon on insulator (SOI) platform, double-coupled identical one-dimensional (1D) photonic crystal (PhC) cavities [29] and slotted PhC slab cavities [30] have been used to modulate on-chip optical signals. However, these optomechanical oscillators are pumped and probed using the same optical cavities; if they are used to modulate on-chip emissions, it becomes difficult to extract the emission signal from the pump light in one optical channel because the pumping power is several orders higher than the emission power. Thus, it is necessary to optically isolate the pump and emission cavities; until now, however, there has been no reported optomechanical oscillator having optically isolated pump and probe cavities. Here, we have designed and conducted an experimental verification of an lumped mechanical oscillator connecting two optically isolated cavity pairs (each pair is a set of double-coupled identical 1D PhC cavities), in which there is no disturbance between the two optical channels at all. It is foreseeable that it will be a suitable platform for optomechanical control of on-chip emitters.

2. Mechanical design and simulation

The fabricated optomechanical oscillator system is shown in Fig. 1(a), and its schematic diagram is depicted in Fig. 1(b). There are two pairs of side-coupled identical 1D PhC cavities (pair I for probing and II for pumping), and for each pair, one of the coupled cavities is fixed and connects the input and output waveguides, while the other cavity is supported by a movable shuttle. The unreleased regions, fixed by the silicon dioxide underneath, are anchors, and the shuttle is connected to the anchors through folded nanobeam springs. Thus, a typical lumped mechanical oscillator in microscale is formed, in which the rigid body consists of the shuttle and the movable cavities of the pairs of coupled cavities, and the four symmetrically arranged folded nanobeam springs constrain the lumped oscillator in one desired in-plane degree of freedom (DOF). The motion axis is shown in Fig. 1(b), where the motion direction is perpendicular to the 1D PhC cavities, so the gap in each pair of coupled cavities, which determines the optical coupling strength, can be changed through mechanical movement of the lumped oscillator. It can be seen that the two pairs of coupled cavities are optically isolated and possess their own independent input and output waveguides, but they share one mechanical oscillator and can be modulated by the same mechanical mode of the oscillator. As a functional device, one pair of coupled cavities [pair II in the lower part of Fig. 1(a)] is used to pump the mechanical oscillations, while the other pair [pair I in the upper part of Fig. 1(a)] is used as the target of the modulation by the mechanical oscillations. If we implement any wavelength-sensitive function in cavity pair I, we can modulate this function by optomechanical pumping of cavity pair II. For example, when narrow linewidth emitters are embedded in cavity pair I, we can modulate the light emission with this process. In this study, we didn’t use any emitters or functions, and thus, the target cavity worked as an optical probe for the mechanical oscillations.

 

Fig. 1 (a) Scanning electron microscope (SEM) image of the fabricated device. Insets: magnified SEM images of the pairs I (left) and II (right) of the coupled one-dimensional (1D) photonic crystal (PhC) cavities, respectively (b) Schematic of the device shown in (a), where there are two pairs of coupled 1DPhC cavities, and for each pair, one of the coupled cavities is supported by a lumped mechanical oscillator. The oscillator is indicated by the magenta dashed box in (b). (c) Finite element method (FEM) simulated motion of the lumped oscillator’s first order in-plane antiphase (I_AP,1) mechanical mode, where bendable folded nanobeams work as springs. (d) Frequency f, (e) effective mass m_eff, and (f) spring constant k versus beam length l and width w of the nanobeam springs. (g-i) I_AP,1 mode shapes of three spring dimensions with extreme w and l: (g) l = 6 μm and w = 470 nm, (h) l = 18 μm and w = 470 nm, and (i) l = 18 μm and w = 180 nm. Their corresponding mechanical parameters are marked in (d), (e), and (f).

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The devices were fabricated from silicon on insulator substrate with a 300-nm-thick device layer by electron beam lithography (EBL) and subsequent inductively coupled plasma reactive ion etching (ICP-RIE). The suspended structure was released from the silicon dioxide by using hydrogen fluoride (HF) vapor. Cavity pairs I and II were designed for different wavelength bands: ~1.2 and ~1.5 μm, so their dimensions were different. For the device shown in Fig. 1(a), 1D PhC cavities in pair I (left inset) had a beam width of 360 nm, an invariable PhC lattice constant of 240 nm, 40 holes in the Bragg mirror on each side of the cavity, hole diameters of 160 nm to 60 nm (tapered from cavity center to both sides), and a slot gap of 90 nm between the coupled cavities; the corresponding dimensions in pair II (right inset) were 660 nm, 277.5 nm, 39, 180 nm to 40 nm, and 90 nm.

Each nanobeam spring of the fabricated lumped oscillator had a beam length of l = 8.5 μm and a beam width of w = 270 nm. Using the finite element method (FEM) in simulations of the mechanical eigen-frequencies of the structure in Fig. 1(a), we found the first-order in-plane antiphase (I_AP,1) mechanical mode of the lumped oscillator As depicted in Fig. 1(c), the shuttle and two movable cavities oscillate with maximum displacement, and there is no observable deformation in either movable cavity, so the shuttle and movable cavities can together be regarded as a rigid body. The deformations only happen in folded nanobeams, and each two adjacent nanobeams bend in opposite directions to achieve the largest displacement of the movable cavities during mechanical oscillation; this is the antiphase mechanical mode (the in-phase mode will be described in section 2). Folded nanobeams are common spring designs in conventional MEMS [31]; here, we have downscaled them to make them applicable to nanophotonic devices. The largest displacement in the designed direction for changing the coupled cavities’ gaps provides both cavity pairs with the most sufficient optomechanical interactions, so the I_AP,1 mechanical mode is the most desired one for this optomechanical lumped oscillator. Compared with another nano-optomechanical device with large deformed areas [30], long and thin nanobeams are better for suppressing damping of the mechanical oscillations [32].

The beam length l and width w of the nanobeam springs, shown in Fig. 1(c), are critical parameters for the lumped oscillator. Here, we investigated the oscillator’s frequency f, effective mass m_eff, and spring constant k as functions of l and w in FEM simulations [Figs. 1(d), 1(e) and 1(f)]. As shown in Fig. 1(d), f decreases with l and increases with w. The nanobeam dimensions of the fabricated device whose mechanical mode shape is shown in Fig. 1(c) are marked in Fig. 1(d) as (c). In addition, other extreme sizes of the nanobeams [(g) l = 6 μm and w = 470 nm, (h) l = 18 μm and w = 470 nm, and (i) l = 18 μm and w = 180 nm] are marked, whose mechanical mode shapes are depicted in Figs. 1(g), 1(h), and 1(i). The oscillator’s f can be increased by shortening and widening the spring beams, but for springs that are too stiff, the cavities cannot be approximated as a rigid body of a lumped oscillator because the springs and cavities have comparable stiffnesses in this case. For example, in Fig. 1(g), the mechanical mode of a lumped oscillator with short and wide spring beams has bent cavities. There are also limits on flexible nanobeam springs, because if the beams are too long and too thin, such as in Fig. 1(i), they cannot support the rigid body stably, and twist and sticking will easily happen after they are released from the silicon dioxide layer. The effective mass m_eff is calculated as $m_{eff}=\text{ρ}{{\displaystyle \int }}^{\text{}}{\left(\frac{\left|Q\right|}{\text{max}\left|Q\right|}\right)}^2\text{d}V$, where ρ = 2329 kg/m³ is the density of silicon, Q is the displacement field of the mechanical mode, and V is the volume of the lumped oscillator. As shown in Fig. 1(e), when the spring beams are narrow, m_eff slightly increases with l, but for wider beams such as w = 400 nm or more, m_eff drastically increases with l because the masses of the nanobeam spring can no longer be neglected in the mass-spring system, for example, in the oscillator in Fig. 1(h). The spring constant k in Fig. 1(f) is calculated as $k={\left(2\pi f\right)}^2{m}_{eff}$, whose tendency is the same as that of f: the longer and thinner the spring beams are, the smaller k becomes. There are upper and lower limits on k in the actual device: first, k cannot be too high to ensure that the movable cavities will have neglectable bendings; second, k cannot be too low in order to avoid twisting or sticking during the release process. Thus, we chose k of around 3 N/m (f = ~1.1 MHz), which satisfies both the upper and lower limits. In addition, low effective mass is preferred for this lumped oscillator (the effect of the effective mass on optomechanical interactions will be analyzed in the following section), so in Fig. 1(e), point (c) (l = 8.5 μm and w = 270 nm) in the left lower region along the contour of k = ~3 N/m [dashed line in Fig. 1(e)] was chosen for the fabricated device. As we previously asserted, the structure consisting of the shuttle and the two movable cavities is regarded as the rigid body in this lumped oscillator, so this structure should be much more rigid than the folded nanobeam springs. Simulated by the FEM, the lowest in-plane mode frequency of the shuttle and the movable cavities is 6.38 MHz, which is much higher than that (I_AP,1 mode, 1.15 MHz) of the entire lumped oscillator structure, so the previous assertion is reasonable.

3. Optomechanical interactions

Double-coupled 1D beam PhC cavities are well known and have been frequently employed in optomechanical studies [7,29], but this beam type of cavities are clamped at both ends and are not suitable for our purpose. The separation of the optical modes in the single pair of coupled cavities cannot be so large, and even if we employ two pairs of coupled beam cavities operating at different wavelengths, it is not straightforward to introduce an appropriate mechanical coupling between them. However, in this section, as it is a well-known optomechanical configuration with a very strong optomechanical interaction, a clamped configuration of coupled 1D PhC cavities will be used as a reference for analyzing the optomechanical interactions of the lumped-oscillator-supported configuration of coupled cavities. The mechanical mode of the clamped configuration is the in-plane bending deformations of the coupled 1D PhC cavities, but for the lumped-oscillator-supported configuration, the movable cavities move translationally in the plane during mechanical oscillation and their bending deformations are negligible [Fig. 1(c)]. Thus, these two configurations have different optomechanical interactions.

Optical transverse electric (TE) odd modes of double-coupled 1D PhC cavities have much weaker optomechanical interactions compared with their TE even modes, so here, we will ignore all the TE odd modes. The first four orders of optical TE even modes (TE_e,N, N = 1,2,3,4) of both cavity pairs I and II are investigated, and in Fig. 2, their electric field profiles overlap the displacement profiles of the mechanical modes of the two different optomechanical configurations [Figs. 2(a) and 2(b) respectively show the conventional clamped optomechanical configurations of cavity pairs I and II, while Figs. 2(c) and 2(d) show the proposed lumped-oscillator-supported optomechanical configurations of cavity pairs I and II]. Figure 2(a) is the separating phase of the first-order in-plane bending mechanical mode of the clamped cavity pair I, while Fig. 2(b) is the approaching phase of the same mechanical mode of the clamped cavity pair II. To show the cavity parts of the optomechanical oscillator system proposed in this paper, the shuttle and spring parts are omitted from Figs. 2(c) and 2(d). In Fig. 2(c), the lower coupled cavity is the movable one, and the green arrows indicate the movement direction in which the movable cavity approaches the fixed cavity, while in Fig. 2(d), the upper coupled cavity is movable, and the green arrows indicate the movement direction in which the movable cavity moves away from the fixed cavity. The volume of the movable part of the lumped oscillator [Fig. 1(c)] is much larger than those of the deformed parts of the clamped cavity pairs I and II [Figs. 2(a) and 2(b)], so m_eff (53.1 pg) is larger than those of the clamped cavity pairs (3.10 pg for cavity pair I and 6.97 pg for cavity pair II). The mechanical frequencies of the clamped configurations in Figs. 2(a) (7.91 MHz for cavity pair I) and 2(b) (11.6 MHz for cavity pair II) are higher than those of the lumped-oscillator-supported configurations in Figs. 2(c) and 2(d) (1.15 MHz for both cavity pairs). Apparently, the lumped oscillator is suitable for low-speed modulation only, while the mechanical frequencies of the clamped cavity pairs are still not fast enough for communication applications. Zero-point fluctuation x_zpf is a critical parameter in optomechanics in the quantum region [15]. Recently, quantum-mechanical studies of optomechanical devices have been done on the macroscale and at room temperature [32], and the optomechanical oscillator proposed in this study may also be able to be used in future quantum-mechanical studies. Thus, here, we calculate the zero-point fluctuations for the optomechanical systems in Fig. 2, as $x_{\text{zpf}}=\sqrt{\frac{\hslash }{2\times m_{\text{eff}}\times 2\pi f}}$, where ℏ = 1.05e-34 N∙m∙s is the reduced Plank constant. The clamped cavity pairs I and II [Figs. 2(a) and 2(b) respectively] have x_zpfs of 18.51 and 10.21 fm, respectively, and the lumped oscillator [see Figs. 2(c) and 2(d); Fig. 1(c) shows the complete view of the mechanical mode] has x_zpf = 11.74 fm, which is comparable to the values of the clamped cavity pairs. Moreover, for the lumped oscillator, there is much room to increase x_zpf. Here we show some example for such possible improvement. First, the mass of the shuttle in the device shown in Fig. 1(a) is triple that of two movable cavities, but in our newly improved design for a future device, mass of the shuttle is less than that of the movable cavities, so the total effective mass of this future lumped oscillator will be half that of the device demonstrated here. Second, the frequency of the lumped oscillator can be cut in half by redesigning the nanobeam springs. As a result, x_zpf of the oscillator can be made double the current value (11.74 × 2 = 23.48 fm), which is larger than that (18.51 fm) of the clamped cavity pair I. Ground state cooling of the optomechanical oscillators has been realized by combination of the conventional cryogenic pre-cooling and the laser cooling [14, 15]. In fact, our setup used in this paper can pre-cool the sample down to 4 K, and although the frequency of the oscillator is too low to satisfy the resolved-sideband limit [14, 15], the optomechanical ground-state cooling scheme proposed for the unresolved-sideband situation would be possible for our oscillator [33, 34]. Thus, this type of lumped oscillator is potentially good for studies and applications in the quantum regime, such as ground state cooling.

 

Fig. 2 Coupling of optical and mechanical modes of different optomechanical configurations: (a) clamped cavity pair I, (b) clamped cavity pair II, (c) lumped-oscillator-supported cavity pair I, and (d) lumped-oscillator-supported cavity pair II. Pair I of coupled cavities is designed to have wavelengths within a 1.2 μm band, while pair II is designed for wavelengths within a 1.5 μm band. For each configuration, the first four orders of the optical transverse electric even modes (TE_e,N, N = 1,2,3,4) of the coupled 1D PhC cavities (both pair I and II) are depicted. In order to illustrate the cavity parts, the shuttle and springs of the lumped oscillator, as shown in Fig. 1(c), are not shown in Figs. 2(c) and 2(d).

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In Figs. 2(a) and 2(b), the mode profiles extend along the cavity beams with the mode orders. However, in the clamped optomechanical configurations, the maximum displacements of the in-plane differential mechanical mode happen at the cavities’ centers and the displacements at both ends are zero, so the interactions of the higher order optical modes with the cavities’ deformations are not as significant as those of the first-order optical modes. The simulated curves in Figs. 3(a) and 3(b) verify this picture. For both TE_e,1 optical modes in cavity pairs I and II, the optomechanical coupling constants (g_OM = 2π∙dv/dx, where v is the optical frequency of the coupled cavities and x is the maximum relative displacement between the two coupled cavities) [7, 35] in the clamped configurations are slightly higher than those in the lumped-oscillator-supported configurations, because although the TE_e,1 optical modes concentrate in the central region of the cavity pairs, there are still tiny bending deformations in the clamped configurations. In the lumped-oscillator-supported configurations, the g_OMs of both cavity pairs increase with the optical mode order, because the entire slot gaps between the coupled cavities change the same distance and the higher order optical modes have a larger proportion of the mode volume inside the slot gaps relative to that in the entire region than in the case of the first-order optical modes. Thus, for higher order optical modes of both cavity pairs, the lumped-oscillator-supported configurations have significantly stronger optomechanical interaction strengths compared with the clamped configurations. In addition, compared with cavity pair II, cavity pair I has 2.6 times larger g_OM, because the beam width of cavity pair I is narrower; there is a stronger evanescent field in the slot gap between the coupled cavities. Another quantum parameter is the optomechanical coupling rate g₀, calculated as g₀ = g_OM∙x_zpf; it is the optical frequency shift induced by zero-point fluctuation of the optomechanical system [36]. g_OM is defined as 2π∙dv/dx, where x is the maximum relative displacement between two coupled cavities, and g₀ = g_OM∙x_zpf can be directly used for lumped-oscillator-supported coupled cavity pairs, while for the clamped configuration, the relative x_zpf between the coupled cavities is double their respective x_zpf because of the differential displacement; hence, the g₀s in the clamped coupled cavity pairs are calculated as g₀ = g_OM∙2∙x_zpf. As shown in Figs. 3(c) and 3(d), the relations between the g₀s of clamped and lumped-oscillator-supported cavity pairs are different from those of the g_OMs because of the different x_zpfs of the configurations. Although in Fig. 3(c) and 3(d), the g₀ of the lumped-oscillator-supported configurations is lower than that of the clamped configurations for both cavity pairs, as aforementioned, the x_zpf of lumped oscillator can be optimized to twofold the current value; hence, the g₀ of cavity pair II in the lumped-oscillator-supported configuration can be larger than that in the clamped configuration, while the g₀ of cavity pair I in lumped-oscillator-supported configuration can be quite close to the value in the clamped configuration. In conclusion, for optomechanical interactions characterized by g_OM, the lumped-oscillator-supported configuration has an advantage over the clamped one, and this advantage becomes more obvious as the order of the optical mode increases. For optomechanical interactions in the quantum regime characterized by g₀, the lumped-oscillator-supported configuration has comparable performance with the clamped one, so this optomechanical lumped oscillator can be potentially used in the quantum era.

 

Fig. 3 (a) Optomechanical coupling constant g_OM of the first four order optical transverse electric (TE) even modes of the clamped and lumped-oscillator-supported cavity pair I. (b) Optomechanical coupling constant g_OM of the first four order optical TE even modes of the clamped and lumped-oscillator-supported cavity pair II. (c) Optomechanical coupling rate g₀ of the first four orders of optical TE even modes of the clamped and lumped-oscillator-supported cavity pair I. (d) Optomechanical coupling rate g₀ of the first four optical orders of TE even modes of the clamped and lumped-oscillator-supported cavity pair II.

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Besides the I_AP,1 mechanical mode, there are other mechanical modes in this lumped oscillator. The mechanical and optomechanical properties of other mechanical modes are helpful for studying the measured RF mechanical spectrum of the lumped oscillator and understanding the versatile optomechanical interactions of the double-coupled 1D PhC cavities, such as relative out-of-plane motions and rotations. Figure 4 shows nine mechanical mode shapes we were interested in, and Table 1 lists their respective mechanical and optomechanical parameters. Figures 1(a) and 1(c) are the first-order out-of-plane antiphase (O_AP,1) and in-phase (O_IP,1) mechanical modes, respectively. In O_AP,1 mode, the two movable cavities of the respective cavity pairs I and II move in opposite directions, that is, there is a 180° phase difference between two cavity pairs, while in O_IP,1 mode, they always move in the same direction and there is no phase difference. The first-order torsional in-phase (T_IP,1) and antiphase (T_AP,1) mechanical modes are shown in Figs. 4(b) and 4(d), which rotate roughly around the shuttle’s motion axis. T_IP,1 and T_AP,1 modes respectively have 0 and 180° phase differences between the rotations of the two movable cavities. Figure 4(e) is the first-order in-plane in-phase (I_IP,1) mechanical mode. Figures 4(f) and 4(g) are second-order in-plane antiphase (I_AP,2) and in-phase (I_IP,2) mechanical modes, in which the displacements mainly happen in the cavity beams. Figures 4(h) and 4(i) are third-order in-plane antiphase (I_AP,3) and in-phase (I_IP,3) mechanical modes, in which the shuttle’s two wings supporting the movable cavities bend. In fact, for the higher order in-plane modes, this oscillator cannot be regarded as the lumped one because the shuttle and the movable cavities are not rigid body anymore. However, the oscillator proposed in this paper is named lumped oscillator for the most desired I_AP,1 mode, although the higher order modes of the oscillator are also mentioned in the simulation analyses and the following experimental results. As shown in Table 1, the g_OMs in the in-plane mechanical modes are much larger than those in the out-of-plane mechanical modes, while the g_OMs in the out-of-plane mechanical modes are roughly one-order larger than those in the torsional mechanical modes. Thus, the coupled 1D PhC cavities are more sensitive to the in-plane gap changes.

 

Fig. 4 Various mechanical modes of the lumped oscillator: (a) the first order out-of-plane antiphase mode, (b) the first order torsional in-phase mode, (c) the first order out-of-plane in-phase mode, (d) the first order torsional antiphase mode, (e) the first order in-plane in-phase mode, (f) the second order in-plane antiphase mode, (g) the second order in-plane in-phase mode, (h) the third order in-plane antiphase mode, and (i) the third order in-plane in-phase mode. In (a-d), cavity pair I is on the right and cavity pair II is on the left; in (e-i), cavity pair I is on the top and cavity pair II is on the bottom.

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Table 1. Mechanical and optomechanical parameters of various mechanical modes

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4. Experimental results

First, we acquired the optical transmission spectra of cavity pairs I and II, respectively. Figure 5(a) is the setup for measuring the optical transmission spectrum of cavity pair II. A tunable laser source (TLS) launches light into the waveguide by end-fire coupling with the objective lens. After going through cavity pair II, light is collected by the objective lens at the end of the waveguide, and the light is then guided into the photodetector (PD). A PC is used to synchronize the TLS wavelength and PD signals. The measured optical spectrum is plotted in Fig. 5(b). The first three orders of optical TE modes are clear, but the fourth order can hardly be recognized. The cavities’ confinement gets weaker as the mode order increases, and quasi-continuum optical modes emerge in the cavity as the wavelength increases [37]. The optical spectrum of cavity pair I with a wavelength band around 1.2 μm was measured using another setup, shown in Fig. 5(c), where TLS and PD are replaced by a supercontinuum laser source (SLS) and a spectrometer. The SLS injects broad-band light into cavity pair I, and the grating spectrometer records the spectral information from the upward scattered light above the cavity. The measured optical spectrum of cavity pair I is plotted in Fig. 5(d), wherein the first four orders of the optical TE modes can be clearly seen.

 

Fig. 5 Optical characterization of the device. (a) Setup for measuring optical spectrum of cavity pair II. TLS, tunable laser source; PD, photodetector. (b) Measured optical spectrum of cavity pair II. (c) Setup for measuring optical spectrum of cavity pair I. SLS, supercontinuum laser source. (d) Measured optical spectrum of cavity pair I. TE_o,N and TE_e,N (N = 1,2,3,4) are the Nth order transverse electric odd and even modes, respectively.

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As our FEM analyses showed, some of the oscillator’s mechanical modes have quite different strengths of optomechanical interaction with cavity pairs I and II. Some mechanical modes have too small g_OMs to be visible to the corresponding cavity pairs. Here, to comprehensively measure the mechanical modes, we probed them using both cavity pairs. Two measurement schemes, whose setups are shown in Figs. 6(a) and 6(b), were used: in scheme A, mechanical oscillations are probed by cavity pair II; in scheme B, mechanical oscillations are probed by cavity pair I. In both schemes, the lumped oscillator is pumped by cavity pair II, as shown in Figs. 6(a) and 6(b). The CW light from the pumping TLS is amplified by an erbium doped fiber amplifier (EDFA) and modulated by an electro-optic modulator (EOM) before being launched into cavity pair II. The wavelength of the pump light [λ = 1550 nm, shown in Fig. 5(b)] is within the quasi-continuum mode band of cavity pair II, because the EDFA limits the wavelength in this narrow range. Because of the higher Q factors, the lower-order optical modes of cavity pair II should be used to pump the oscillator [21], but unfortunately, the wavelengths of these optical modes are out of the EDFA’s range. In scheme A, a fiber coupler (FC) is used to couple the pump and probe light together before the device, and a tunable filter (TF) is used to filter the pump light from the probe light before the photomultiplier (PMT), while in scheme B, the FC and TF are not needed because the probe and pump light are separated by the optically isolated cavity pair I and II. In both schemes, a vector network analyzer (VNA) acquires the RF signal of the mechanical oscillations from the PMT and stimulates the modulation of the EOM. The wavelengths of the probe light in schemes A and B were set at 1507.6 nm [TE_e,2 optical mode, Q = 3800, shown in Fig. 5(b)] and 1245.8 nm [TE_e,2 optical mode, Q = 2700, shown in Fig. 5(d)], respectively. The device was loaded into a vacuum chamber (lower than 10⁻⁶ Torr) during the measurement to avoid air damping.

 

Fig. 6 (a, b) (a) Setup of measurement scheme A, where mechanical oscillations are probed by cavity pair II. EDFA, erbium doped fiber amplifier; EOM, electro-optic modulator; FC, fiber coupler; TF, tunable filter; PMT, photomultiplier tube; VNA, vector network analyzer. The TLS for pumping is always set at 1550 nm in the experiment since the EDFA only works in this band. (b) Setup of measurement scheme B for probing mechanical oscillations by cavity pair I, where the scattered light from cavity pair I is collected with the top objective lens. (c) Measured RF mechanical spectra of the lumped oscillator, where the red and blue curves are probed by cavity pairs II and I, respectively. (d, e) Detailed spectra of I_AP,1 mechanical mode probed by cavity pair II (d) and I (e). (f, g) FEM simulated motions of the first order in-plane mechanical modes of respective unmovable 1D PhC cavities in cavity pairs I (f) and II (g). The featured SEM images of unmovable cavities with their own suspended waveguides are shown for the sake of clarity of the simulated structures.

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The RF mechanical spectra of the device probed by cavity pairs I and II are shown in Fig. 6(c). To assign various mechanical modes to the peaks, we compared these peaks’ frequencies with FEM simulated frequencies of various mechanical modes. If there are several simulated mechanical modes with frequencies close to a measured peak, there should be only one simulated mode with a significantly larger g_OM than the others, and this is the mechanical mode corresponding to the measured peak. As we all know, the bigger g_OM is, the more sensitive the optical mode becomes to the mechanical mode, and thus the higher the peak is in the RF spectrum. If g_OM is too small, the mechanical mode will be invisible in the RF spectrum. The simulated and measured frequencies are compared in Table 2, and the visibilities are determined from the respective RF spectra in Fig. 6(c). The peaks labeled ① and ③ appear in both RF spectra and are the O_AP,1 and I_AP,1 mechanical modes, respectively. Theoretically, the O_AP,1 mechanical mode has large g_OM for both cavity pairs, and it is visible in both RF spectra in the experiments, but the O_IP,1 mechanical mode’s g_OM for cavity pair II is 3.8 times smaller than that for cavity pair I, so peak ② (O_IP,1 mechanical mode) appears only in the spectrum probed by cavity pair I. All the torsional mechanical modes shown in Figs. 4(b) and 4(d) are absent from both spectra because of their small g_OMs. The I_AP,1 mechanical mode possesses the maximum g_OM in both cavity pairs, so its oscillation amplitudes in both RF spectra are the largest. Figures 6(d) and 6(e) show detailed spectra of the I_AP,1 mechanical mode probed by cavity pairs II and I, respectively. They show an obvious deterioration in the mechanical Q factor (Q_m = 13,467 probed by cavity pair II and Q_m = 5,918 probed by cavity pair I), which is attributed to the optical force damping by cavity pair I [32]. Besides the lumped oscillator’s mechanical modes, peaks ④ and ⑤ are the mechanical modes of the unmovable cavities for cavity pairs I and II. Because the unmovable cavities are not absolute rigid bodies, bending is inevitable. Figures 6 (f) and 6(g) show the FEM simulated mechanical mode shapes of peaks ④ and ⑤; scanning electron microscope (SEM) images of the corresponding unmovable cavities are also shown to illustrate the waveguide configurations. Peaks ⑥ and ⑦ correspond the I_AP,2 and I_IP,2 mechanical modes shown in Figs. 4(f) and 4(g). For these two modes, their g_OMs in cavity pair I are much larger than those in cavity pair II, so they are visible only in the RF spectrum probed by cavity pair I. As was expected from the FEM simulations, the I_AP,1 mechanical mode with the maximum g_OM in both cavity pairs shows the largest oscillation amplitude in both RF spectra probed by the two cavity pairs, and it is the most appropriate mechanical mode of the lumped oscillator for optomechanical control of an integrated emitter, because the largest g_OM in cavity pair II can generate the largest optical force to pump the lumped oscillator and the largest g_OM in cavity pair I can yield the largest wavelength detuning of the optical mode during the mechanical oscillation.

Table 2. Comparison of simulated and measured mechanical modes

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As described in section 2, this optomechanical system can be applied to optical modulation, and cavity pair I can be set as the target of the modulation; hence, it is necessary to observe dynamic detuning of the optical modes of cavity pair I modulated by the pumped lumped oscillator. The measurement setup depicted in Fig. 7(a) is slightly different from that in Fig. 6(b); the PMT is replaced with a spectrometer. Figures 7(b), 7(c) and 7(d) are spectra of the first three orders of optical modes of cavity pair I with and without the lumped oscillator’s modulation. The pumping laser wavelength was still set at 1550 nm and the modulation frequency was set at 1.184 MHz, which is exactly at the oscillator’s I_AP,1 mechanical mode. As the pumping power increases (EDFA was varied over that range of 100 mA to 500 mA), the amplitude of the mechanical oscillation increases and the optical modes dynamically detune in a wide range. The spectrometer recorded the time integrated optical power spectrum. The peaks of all three order optical TE even modes were suppressed and broadened during the mechanical oscillation, and these effects were induced by the dynamical detuning. Compared with the results of the optical TE even modes, the spectra of the optical TE odd modes change only a little, which is consistent with the fact that their optomechanical interactions are much weaker. In addition, to evaluate the optical isolation between these two cavity pairs, we removed the SLS of the cavity pair I and kept the input power of the cavity pair II maximum (EDFA @ 500mA), but no signal can be detected by the PD from the top of the cavity pair I, which means the crosstalk between these two cavity pairs is negligible. The dynamic detuning results of this experiment provide a solid foundation for future optomechanical control of integrated emitters. For example, if an emitter instead of an external laser source is embedded in cavity pair I and the emission wavelength is aligned with the cavities’ TE even mode, the emission power will be modulated during dynamic detuning of the cavities’ optical mode (implemented with a pumped lumped oscillator) because of the Purcell effect [26].

 

Fig. 7 (a) Setup to observe dynamic detuning of the optical spectrum of optical cavity pair I modulated by the pumped lumped oscillator. (b-d) Spectra of the first (b), second (c) and third (d) order optical modes of cavity pair I modulated by the lumped oscillator with different pumping powers. In each figure, the left peak is the TE odd mode and the right one is the TE even mode.

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5. Conclusion

In conclusion, we designed and experimentally verified an optomechanical lumped oscillator that connects two optically isolated pairs of coupled 1D PhC cavities. The two pairs of coupled cavities are designed for wavelength bands of 1.5 um and 1.2 um, respectively, and one is used to pump the mechanical oscillations of the oscillator by optical force, while the other is used as the modulation target of the mechanical oscillations. FEM simulations were used to investigate the effects of the spring dimensions on the mechanical properties of the lumped oscillator, and based on the results, appropriate spring dimensions were designed. The optomechanical coupling constants, g_OMs, of the lumped-oscillator-supported configurations of cavity pairs I and II were calculated and were found to be higher than those of conventional side-clamped configurations of cavity pairs, especially for higher order optical modes. Although this optomechanical lumped oscillator is large, the calculated g₀s of the lumped-oscillator-supported cavity pairs I and II, which characterize their quantum optomechanical performance, can be optimized to be as good as those of clamped cavity pairs. Not only the most desired first-order in-plane antiphase mechanical mode was investigated; other mechanical modes of the oscillator were investigated as well, including various mechanical and optomechanical parameters that provide versatile DOFs for optomechanical applications. The oscillator’s RF spectra were experimentally measured using both cavity pairs as probes, and the results agreed with the simulated mechanical modes. Finally, the dynamic detuning of optical spectrum of cavity pair I was demonstrated with a pumped lumped oscillator. In conclusion, such a lumped oscillator that uses two optically isolated pairs of coupled cavities for pumping and being modulated has never been reported before, and it is suitable for optomechanical control of integrated lasers, cavity QED, and spontaneous emission. Furthermore, this kind of optomechanical oscillator may open a new door on the study of interactions between photons, phonons, and excitons.