1. Introduction

It is well known that photons have linear momentum [1] and spin angular momentum in the form of circular polarization [2]. Following Allen et al.’s discovery that light with a helical wavefront also possess orbital angular momentum (OAM) [3], a vast range of applications for OAM of light have been proposed. These range from high bandwidth data transfer [4,5] and quantum cryptography [6–8], to optical tweezers [9,10] and imaging [11] for biological applications. Light with OAM has been generated using pitchfork holograms [12,13], spiral phase plates [14,15], Dove prisms [16], cylindrical lens mode converters [3,17], liquid crystal $q$-plates [18], metasurfaces such as sub-wavelength gratings [19,20], plasmonic nano-antennas [21,22], optical phased arrays [23] and microrings [24]. Similarly, methods for measuring the OAM of light have seen rapid development, including techniques based on forked diffraction gratings [6], interferometry [25], apertures [26,27] and image reformatting [28]. OAM of light can also be measured using the torque that it exerts on incident objects. Mechanical detection of spin angular momentum was first demonstrated by Beth [2] and the direct mechanical measurement of the OAM of light has been achieved using torsional pendulums [29] and microscopic particles [30]. However, mechanical detection of the OAM of light using on-chip devices has not been achieved to date. Such devices would enable OAM to be used as a fingerprint in quantum non-demolition measurement of photons [31], as well as provide a platform for interfacing free-space OAM carrying optical fields to nanophotonic components.

The orbital angular momentum $\hbar l$ of a photon is determined by the OAM quantum number $l$ describing its helical wavefront and is in principle unbounded [3]. As a result, OAM can be much larger than spin angular momentum which is limited to $\pm \hbar$. Although from a practical point of view efficient generation and measurement of large OAM is challenging, light beams with up to $l=300$ have been demonstrated [32]. The torque per photon due to a change in OAM of light upon its interaction with an optical element is given by $\tau = \hbar \Delta l /\Delta t$, where $\Delta l$ is the change of the OAM of light and $\Delta t$ is the time duration of the photon pulse. For continuous optical excitation, we can express the torque in terms of the power of the incident light:

Here we propose and analyze an optomechanical photonic crystal cavity to detect the torque exerted by light on a photonic nanostructure, allowing measurement of the OAM of light, as well as non-absorbing optical field detection via light’s OAM degree of freedom. Optomechanical photonic crystal cavities localize light to sub-wavelength volumes where it interacts strongly with nanomechanical resonances of the device, resulting in coupling between nanomechanical motion and the optical cavity resonance frequency and linewidth. These devices can support optical modes with a desirable combination of high-quality factor ($Q_o$) and large optomechanical coupling to the mechanical resonances of the cavity. This provides sensitive transduction of mechanical motion via the changes that it imparts on the optical cavity mode’s response, enabling measurement of small forces acting on the device [33,34]. For example, optomechanical cavities have been used to realize accelerometers [35] and integrated atomic force detection systems [36]. Optomechanical torque sensing [37,38] has recently been studied within the context of torque magnetometry [39,40] and spin detection of photons [41]. However, OAM detection has not previously been explored using a nanophotonic cavity optomechanics platform.

2. Device design

2.1 General operating principle

Figure 1 shows a schematic of the OAM detection device studied here. As shown in Figs. 1(a) and 1(b), it consists of a central suspended pad connected to one half of a “slot-mode” photonic crystal nanobeam cavity. The other half of the cavity is attached to the surrounding chip. The cavity supports an “air-band” optical mode [42] whose field is concentrated in the gap between the nanobeams, giving it high sensitivity to any motion of the nanobeam that changes the slot gap width, and as a result, the wavelength of the cavity mode. In this dispersive optomechanical coupling process, motion of the nanobeam is transduced onto the amplitude of light coupled out of the cavity, as described quantitatively below and in the subsequent sections. To monitor this optomechanical transduction, light can be coupled into and out of the cavity using an optical fiber taper waveguide evanescently interacting with the cavity, as in Ref. [37]. Dissipative coupling [37], where mechanical motion modulates the optical loss rate of the cavity, can also be used to transduce nanobeam displacement to an optical signal. However, this effect is not considered in this paper.

 

Fig. 1. Overview of the device geometry and elements. (a) Top view of a slot-mode photonic crystal cavity geometry and simulation of its fundamental optical mode’s electric field ($x$ component) distribution. (b) Isometric view of the OAM detector. A helical light beam illuminates the square pad which is suspended by thin supports. Depending on the structure of additional material attached to the central square pad (blue), light exerts a torque on the pad due to its change in OAM during reflection, refraction, absorption, or transmission. The cavity from (a) is attached to the square pad by a hanger whose dimensions $w_h$ and $l_h$ are indicated in (c). The pad motion when excited by a source of torque is transferred to the nanobeam as shown by the simulated displacement profile shown in (c). For a given frequency of actuation, this motion is due to both displacement of the pad and excitation of the nanobeam mechanical modes. Dimensions not specified in the figure are varied in the studies described in the text.

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Optical actuation of the central pad by OAM shifts the nanobeam’s centre of mass position through both “twisting” of the central pad, and through excitation of the mechanical “bouncing” mode of the nanobeam. Examples of these types of displacement can be seen in the simulated displacement profile of the device when its central pad is driven by an external source of torque shown in Fig. 1(c). The OAM of light incident onto the central pad of the device can be converted to torque in several ways. If the torsional pad is coated with an absorptive layer and illuminated by helical light, the OAM of light changes from $l$ to $l^{\prime }=0$. On the other hand, if the torsional pad is patterned with a suitably engineered metasurface, a helical beam with OAM number $l$ can be changed during transmission or reflection to, in principle, arbitrary OAM number $l^{\prime }$ defined by the metasurface geometry.

2.2 Slot-mode nanobeam optomechanical cavity design

The device presented here is designed from a silicon nitride (SiN) membrane (370 nm thickness). Silicon nitride is chosen because of its low mechanical [43] and optical [44] loss, and its high tensile stress, which makes it suitable for large-scale suspended devices with mechanical quality factors exceeding $Q_m=10^8$ [35,45,46]. The square central pad is $25~\mu \textrm {m}\times 25~\mu \textrm {m}$ in size and is suspended by $2~\mu \textrm {m}$ wide tethers whose length is varied in the following studies to tune the device’s mechanical properties.

Each of the two nanobeams forming the photonic crystal cavity has width $w=1~\mu \textrm {m}$ and is perforated by holes to form the nanocavity. The nanobeams are separated by a 100 nm gap. This cavity shares some characteristics with photonic crystal zipper cavities [47]. However, unlike the zipper cavity, the slot-mode photonic crystal cavity supports optical modes concentrated in the gap between two nanobeams, as in two-dimensional slot mode photonic crystal cavities [42]. This is advantageous for maximizing the magnitude of the dispersive optomechanical coupling. The nanocavity holes have uniform radius of 206 nm and are periodically spaced with a lattice constant that tapers quadratically from 600 nm in the outer mirror section to 558 nm in the center of the cavity. The holes are offset toward the outer edge of each nanobeam by 72.5 nm from center. This feature is needed for the cavity to support slot-modes. Additional details of the device design are provided in [48].

Finite-difference time-domain simulations were used to calculate the field profile of the cavity’s fundamental slot-mode, as shown in Fig. 1(a). From these simulations this mode is predicted to have a wavelength $\lambda = 1428$ nm, optical quality factor $Q_o >10^6$, and mode volume $V \sim 0.19 \ (\lambda /n_{\textrm {SiN}})^3$ (defined by the peak field strength). Although in the following analysis of this system we will calculate the optomechanical coupling to the hybridized mechanical modes of the full device, the cavity’s baseline optomechanical performance is quantified by its optomechanical coupling coefficient of $g_{\textrm {OM}}/2\pi >64$ GHz/nm to the fundamental mechanical bouncing mode of the nanobeams. The effective mass of this mode is $m_{\textrm {eff}}=27$ pg, and is small compared to the slot-mode cavity in Ref. [42] due to its one-dimensional nature. For calculation of $V$, $g_{\textrm {OM}}$, and $m_{\textrm {eff}}$, we have used the definitions in Ref. [49] and COMSOL finite element simulations.

3. Optomechanical transduction of mechanical motion

To study the interaction between torsional excitation of the central pad and both the nanobeam’s center of mass position and its vibrational motion, we performed numerical simulations and developed a semi-analytic coupled harmonic oscillator model. Figure 2(a) shows a cartoon representation of this model, where the central pad and the nanobeam are represented as oscillators with masses and natural frequencies of $m_1$, $\omega _1$ and $m_2$, $\omega _2$, respectively. The coupling between the torsional pad and the nanobeam is represented by a spring with natural frequency of $g_m$. Variables $x_1$ and $x_2$ are the maximum displacement from equilibrium of the torsional pad and nanobeam in the $\hat {x}$ direction, respectively. Note that $x_1$ is related to radius $r$ and angle of rotation $\theta$ defined in Fig. 1(c) by $x_1=r\cos \theta \Delta \theta$. The equations of motion for this system are:

 

Fig. 2. (a) Schematic of the coupled oscillator model describing the interaction of the nanobeam bouncing mode and the central pad twisting mode. (b) Simulated displacement $x_2$ of the nanobeam as a function of the frequency of a torsional drive applied to the central pad edges, for device parameters $l_s=12\,\mu$m, $w_h=7\,\mu$m and $l_h=1\,\mu$m. Blue dots are simulated points, and the red dashed lines are fits from the coupled oscillator model.

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The validity of this model can be evaluated using finite element method software (COMSOL) to calculate the mechanical and optomechanical properties of our OAM detection device. Figure 2(b) shows the result of a simulation of the maximum displacement of the nanobeam when we drive the central pad by applying a torque at varying frequency $\omega _d$. This torque has been implemented in the simulation by applying tangential forces to the sides of the torsional pad. An initial stress of $S_0 = 1~\textrm {GPa}$ in the SiN layer was included in the simulations. Predictions from the coupled oscillator model described by Eq. (3) are shown by a red dotted line and fit the finite element simulation results well with $g_m$ as a fitting parameter. Note that for these simulations, $\gamma _{1,2}$ corresponding to mechanical quality factor $Q_m = 500$ are fixed in COMSOL. Although the $Q_m$ of these devices could be much higher, we chose $Q_m = 500$ for visualization purposes.

The peaks in Fig. 2(b) correspond to resonant excitation of two mechanical modes of the device: the twisting mode of the central pad, and the bouncing mode of the nanobeam, which for the central pad support length chosen for this simulation have widely separated frequencies of $\omega _1/2\pi = 4.81$ MHz and $\omega _2/2\pi = 5.96$ MHz, respectively. The twisting mode peak is stronger as it is driven directly, while the bouncing mode is driven indirectly through the nanobeam’s mechanical coupling to the central pad. However, Eq. (3) predicts that $x_2$ can be enhanced if the oscillators are tuned near resonance ($\omega _1=\omega _2$). This can be achieved in our design by tuning the central pad twisting mode frequency via changing the support length $l_s$ labeled in Fig. 1(c).

Displacement profiles of the mechanical modes obtained from COMSOL simulations are shown in Figs. 3(a) and (b) for different values of $l_s$. Figure 3(c) shows the simulated $l_s$ dependence of their resonance frequencies. The nanobeam bouncing mode frequency does not depend on $l_s$, while the frequency of the central pad twisting mode decreases with increasing $l_s$. For values of $l_s$ where the modes are not on resonance, their mode profiles are dominantly twisting– or bouncing–like, as shown in Fig. 3(a). Near $l_s = 10~\mu \textrm {m}$ they are on-resonance, and an avoided crossing is observed due to modal coupling. This coupling is evident in the modes’ mechanical displacement profiles when $l_s$ is tuned to the center of the anti-crossing: as shown in Fig. 3(b) the twisting and bouncing modes are hybridized into even and odd combinations. The degree of splitting between their frequencies is related to the mode coupling $g_m$.

 

Fig. 3. Displacement profiles of the hybridized central pad twisting mode and nanobeam bouncing mode for (a) $l_s = 20\,\mu \textrm {m}$ and (b) $l_s = 10\,\mu \textrm {m}$. These modes are highlighted in the simulated mode frequency vs. $l_s$ plot in (c) by color coded square points that match the frequency labels in (a) and (b). The mode profiles show that when $l_s$ is set so that the modes are not near resonance, as in (a), the twisting (left) and bouncing (right) modes are distinct. In contrast, when $l_s$ is set so that the modes are near resonance, as in (b), the modes become hybridized. Dependence of first (open circles) and second (filled circles) hybridized mode (d) optomechanical coupling coefficient and (e) optomechanical frequency shift per unit applied torque.

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The efficacy with which these modes can be used to detect OAM strongly depends on their optomechanical coupling coefficient $g_{\textrm {OM}}$. We calculate $g_{\textrm {OM}}$ using the perturbation theory discussed in Ref. [47,51] input with the simulated displacement profiles of the mechanical modes for the full device and the field profile of the optical cavity’s fundamental slot-mode. The resulting $g_{\textrm {OM}}$ of each mechanical mode for varying $l_s$ is shown in Fig. 3(d). We see that when the modes are not resonant, the nanobeam bouncing mode has nearly constant $g_{\textrm {OM}} \sim 32\,\textrm {GHz/nm}$, while the central pad’s twisting mode’s $g_{\textrm {OM}}$ is much lower. When the modes are tuned on-resonance, they have equal $g_{\textrm {OM}}$, as expected given their hybridized nature.

We further characterize the optomechanical properties of the device by calculating the optomechanical shift of the cavity mode frequency, $\Delta \omega _{\textrm {OM}}$, per $\textrm {fN}\cdot \textrm {m}$ of torque applied to the central pad at the calculated mechanical mode frequencies shown in Fig. 3(c). As shown in Fig. 3(e), when $l_s$ is set to tune the mechanical modes are on–resonance, $\Delta \omega _{\textrm {OM}}$ peaks due to excitation of the nanobeam bouncing mode by the twisting motion. The enhancement to $\Delta \omega _{\textrm {OM}}$ when the central pad twisting and the nanobeam bouncing modes are resonant is related to the resonator coupling $g_m$, as shown in Eq. (3). We study the dependence of this coupling on the geometry of the hanger connecting the resonators in Fig. 4(a), which shows $g_m$ for varying $w_h$, as calculated by fitting the anti-crossing of $\omega _m(l_s)$ for each $w_h$ to the coupled mode model with $g_m$ as a fitting parameter. We find that the mechanical coupling decreases with increasing $w_h$. Intuitively, this is due to the fact that as the width of hanger increases, the hanger becomes more centered on the pad ($\theta =90^{\circ }$) and the average horizontal center of mass motion of the nanobeam becomes negligible. For small values of hanger width, the see-saw mode of the nanobeam, whose frequency and mode profile are shown in Fig. 4(b), becomes close to resonance with the twisting and bouncing mode frequencies. Its coupling rate to the central pad’s twisting mode is much higher than that of nanobeam bouncing mode, reducing the amount of energy coupled to the nanobeam bouncing mode. Therefore, we chose $w_h=7\,\mu$m for the operation of our OAM sensor in the remainder of the analysis.

 

Fig. 4. (a) Coupling rate $g_{m}$ between the central pad twisting mode and the nanobeam beam bouncing mode for varying hanger width. This rate is extracted from COMSOL simulations of the hybridized mechanical mode frequencies. (b) Dependence of the mechanical frequencies of the nanobeam bouncing (blue) and see-saw (red) resonances on $w_h$.

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4. Torque sensitivity

To predict the sensitivity of this device to OAM generated torque, we need to consider sources of noise, including thermal Brownian motion ($\tau _{\textrm {th}}$), photon shot noise ($\tau _{\textrm {SN}}$), detector noise ($\tau _{\textrm {DN}}$) and back-action noise ($\tau _{\textrm {BA}}$). These noise-equivalent torques combine to determine the minimum detectable torque of the device: $\tau _{\textrm {min}}=\sqrt {(\tau _{\textrm {th}})^{2}+(\tau _{\textrm {SN}})^{2}+(\tau _{\textrm {DN}})^{2}+(\tau _{\textrm {BA}})^{2}}$ [37]. Applied torque in our system is related to $x_{\textrm {max}}$ by

Using Eq. (4) and the fluctuation dissipation theorem, the thermal noise equivalent torque is [52–54]:

The importance of optomechanical device parameters $Q_o$ and $g_{\textrm {OM}}$ for reaching a regime of OAM detection limited by thermal noise is revealed by expressions for the torque equivalent photon shot noise and detector noise. The shot noise optical power spectral density is given by $S_{\textrm {P}}^{\textrm {SN}}=2 \hbar \omega _{0} P_{\textrm {det}}/\eta _{qe}$ [55], and the corresponding torque equivalent noise is

Combining these noise sources, we can predict the minimum detectable torque of the device. This is shown in Fig. 5(a), where we have assumed that the measurement and source of torque is at frequency $\omega _m$ resonant with the lower frequency branch of the hybridized device modes from Fig. 3(c). From this, we see that $\tau _{\textrm {min}}$ minimizes near $l_s \sim 10\,\mu \textrm {m}$ where the central pad twisting and nanobeam bouncing modes are on resonance with another. This behavior is a result of the different $l_s$ dependence of $r_{\textrm {eff}}$ for nanobeam bouncing versus central pad twisting modes, as shown in Fig. 5(b) together with the behavior of $m_{\textrm {eff}}$ in these two regimes. The corresponding effective moment of inertia is shown in Fig. 5(c). These calculation were performed assuming $Q_{o}=10^{6}$, $Q_{m}=10^{6}$, cryogenic temperature (4 K) operation, $P_{\textrm {det}}=0.1 \,\mu \textrm {W}$, $\Delta l=1$ and hanger geometry $w_{h}=7\,\mu$m and $l_{h}=1\,\mu$m. From this we see that $\tau _{\textrm {min}}\approx 3.22\times 10^{-21}\,$N$\cdot$m/$\sqrt {\textrm {Hz}}$ is expected to be achievable. This sensitivity is similar to that of Refs. [37,38] despite the fact that its effective mass is orders of magnitude higher due to the pad and subwavelength grating.

 

Fig. 5. (a) The minimum detectable torque (left axis) and minimum detectable optical incident power (right axis) as a function of support length, with contribution from different noise sources shown. (b) Effective mass (left) and effective lever arm (right) as a function of support length. (c) Contributing factors to the thermal noise as a function of length of supports.

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5. Transmissive OAM detection

We next focus on the possibility of detecting light using the OAM degree of freedom by converting it to torque when the input optical field is transmitted through the central pad. This would allow detection of light without destroying its linear momentum and intensity, which could be used for a non-demolition photon detection scheme. To achieve this, we designed a sub-wavelength grating (SWG) shown in Fig. 6, whose on-chip chiral pattern converts OAM to mechanical torque. This SWG is a high refractive index contrast pattern [56] composed of amorphous silicon (a-Si) pillar structures with refractive index n$=3.62$ [20] patterned on the SiN central pad. High refractive index contrast gratings can operate off-resonance (reflective) [57,58] and on-resonance (transmissive) [20], and in both cases have demonstrated ultra-broadband operation and high capability of modulating wavefront phase. In our design, the transmissive SWG has a $20\,\mu$m diameter and consists of $450\,\textrm {nm}$ thick a-Si pillars. The pillars are arranged in a hexagonal lattice with lattice constant $\Lambda =360\, \textrm {nm}$. Varying the duty cycle as a function of angle, by changing pillars’ diameter from $110$ nm to $210$ nm in increments of $10$ nm, results in modulation of the local effective refractive index and corresponding spatially varying phase shift of light transmitted through the SWG that depends on the azimuthal angle. This allows the incident beam OAM state to be changed by $\Delta l$ determined by the SWG design. The physical principle of this SWG is similar to that discussed in [20,59].

 

Fig. 6. (a) Transmissive OAM detection system. a-Si (blue) pillars have been patterned on top of the SiN (green) optomechanical device in order to form an SWG. (b) Top view picture of the SWG designed for $\Delta l=1$, showing the azimuthally varying pillar diameters.

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Figure 7(a) shows the simulated optical phase and intensity profiles of a Gaussian beam with $10\,\mu \textrm {m}$ beam waist and $1\,\textrm {nm}$ linewidth after it is transmitted through two different SWGs: one designed for $\Delta l=1$ and one for $\Delta l=10$. These results were obtained using Lumerical finite-difference time-domain (FDTD) software. Comparing these results to the analytically calculated phase and intensity profiles of Laguerre-Gaussian (LG) beams with $l=1$ and $l=10$, also shown in Fig. 7(a), we see that the SWGs convert the input Gaussian beam ($l=0$) to a LG beam with $l=1$ or $l=10$, depending on the grating design. For large $l$ values, conversion efficiency will be impacted by the small size of the SWG relative to the large extent of the incident field in the transverse plane. One could design a larger SWG to circumvent this issue at the expense of increasing the effective mass and consequently lowering the device sensitivity.

 

Fig. 7. (a) FDTD simulated transmitted optical field intensity and phase profiles of SWGs of type $\Delta l=1$ and $\Delta l=10$ for a Gaussian incident field. Also shown are the ideal optical field intensity and phase profiles of LG modes with $l = 1$ and $l = 10$. (b) Fidelity of conversion of a Gaussian beam transmitted through the $\Delta l=1$ SWG to the $l = 1$ LG mode as a function of the incident light wavelength.

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Deviations between the ideal profiles and the simulated transmitted profiles are also visible in Fig. 7(a), and indicate that the SWG conversion efficiency is not ideal. This efficiency can be benchmarked by considering the fidelity ($F$) of the output compared to a perfect LG beam, and SWG transmission ($T_{\textrm {SWG}}$), as defined in [60]. Finite-difference time-domain simulations predict $F = 0.90$ and $T_{\textrm {SWG}}$ = 0.92 for the $\Delta l=1$ SWG input with a Gaussian field at a wavelength of 840 nm, which scatters light into the approximate $l = 1$ field labeled SWG ($l=1$) in Fig. 7. This yields an OAM conversion efficiency $F \times T_{\textrm {SWG}}= 0.83$. Using this conversion efficiency, together with Eq. (1) and assuming an input beam modulated at $\omega _m$, we predict the minimum detectable optical power scattered from $l =0$ to $l = 1$ to be $8.7\,\mu \textrm {W}/\sqrt {\textrm {Hz}}$, as shown as a function of $l_s$ in Fig. 5(a). The sensitivity of the device linearly scales with the change in OAM and can reach less than a $1\,\mu \textrm {W}/\sqrt {\textrm {Hz}}$ using the SWG designed for $\Delta l=10$. This assumes the same operating conditions and device parameters as the torque sensitivity analysis above, and that the optical field is modulated with unity contrast at the frequency of the lower branch of the device’s hybridized mechanical resonances.

6. Pulsed operation

Next, we study the sensitivity of our device when it is driven with optical pulses instead of a harmonically modulated continuous wave. The SWG used above has a wide optical bandwidth of operation, as shown by the wavelength dependence of its fidelity shown in Fig. 7(b), allowing short optical pulses to drive the device. The optical power of a pulse train whose pulse width $\Delta t$ is much smaller than the repetition time $1/f_r$ can be expressed as $P(t)=\sum _{m=-\infty }^{+\infty }n\hbar \omega _{c}f_{r} e^{i2 \pi m f_{r} t}$ where $n$ and $\omega _{c}$ are photon number per pulse and the carrier frequency of the incident light, respectively. This shows that in order to resonantly drive the mechanical device, the pulse repetition rate should be set to mechanical frequency ($2\pi f_r=\omega _m$). In this case, the optical drive power of the pulse train is $P(\omega _m)=n\hbar \omega _{c}f_r$. Using $P(\omega _m)$ and Eq. (1), the exerted torque by the pulse train can be calculated. Figure 8(a) shows the predicted minimum detectable number of photons $n_{min}$ per pulse as a function of $l_s$, indicating that pulses with as few as $3.9\times 10^3$ photons can be detected. In plotting Fig. 8(a) we assume the ideal but achievable conditions of $Q_o=10^6$, $Q_m=10^8$ [45,46,61,62], $\Delta l=10$, $n_{\textrm {cav}}=10^{-3}$ and that the device is cooled to $T=10\,\textrm {mK}$ using a dilution refrigerator [63]. We also assume detector noise $P_{\textrm {DN}}=3.8\times 10^{-17}$W$/\sqrt {\textrm {Hz}}$, which is reachable by using a superconducting nanowire single-photon detector, for example Single Quantum Eos detector [64]. However, we note that the maximum count rate observable using these detectors is 80 MHz. Assuming a pulse length $\Delta t < f_r \sim 5\,\textrm {MHz}$, the maximum photon flux will exceed the detector maximum count rate. Given the assumed detector noise is three orders of magnitude smaller than the thermal noise in the proposed experiment, a strategy to circumvent this limitation will be to identify a detector that trades-off technical noise for improved maximum count rate.

 

Fig. 8. (a) Minimum detectable photon number per pulse with repetition rate on mechanical resonance, as a function of support length. Device parameters and operating conditions: $Q_o=10^6$, $Q_m=10^8$, $\Delta l$=10 and $T=10\,$mK. (b) Minimum detectable photon number per pulse as a function of readout interactivity photon number for support length $l_s=10\,\mu$m.

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As can be seen in Fig. 8(a), the measurement sensitivity is maximized when $l_s$ is chosen to tune the central pad twisting mode and the nanobeam bouncing mode onto resonance. To study the impact of backaction from the readout laser, which is significant for the idealized device parameters assumed here, in Fig. 8(b) we show $n_{\textrm {min}}$ as a function of intracavity photon numbers $n_{\textrm {cav}}$ for $l_s=10\,\mu$m (when torsional pad and nanobeam bouncing modes are in resonance). This shows that $n_{\textrm {min}}=3.9\times 10^3$ is the optimal operating condition for these device parameters. While more sensitive detection of the OAM, e.g. at the single photon level, has been demonstrated [16,65], this is the first analysis of the potential performance of an integrated on-chip optomechanical approach.

In practice, a technical challenge will be maintaining the device at 10 mK when the readout and OAM fields are incident upon it. While this problem has been studied in silicon photonic crystal devices [66], which despite being nominally transparent at the readout wavelength suffer from two-photon and surface-state related absorption, the performance limits of silicon nitride nanophotonic devices at dilution fridge temperatures are less well characterized. Also, note that the low optimal intracavity photon number of $10^{-3}$ will help reduce heating effects. The effect of heating from the OAM field will be reduced by the fact that this field is not enhanced by a cavity. For the $n_{\textrm {min}}$ predicted above and a repetition rate of 5 MHz set by the mechanical frequency, the average optical power incident on the membrane is approximately 3.3 nW. If the SiN optical absorption at the 850 nm operating wavelength is comparable to that observed in Ref. [44]’s cavity ($Q_o > 10^6$), this would correspond to approximately 3.3 fW of absorbed power that needs to be extracted by the cryogenic cooling system to maintain the desired operating temperature.

Finally, we emphasize that since this scheme transforms the field profile of the optical input, fully harnessing its non-destructive detection capabilities will require components for manipulating non-trivial optical modes. Alternatively, it may be possible to implement cascaded SWGs that reverse the field transformation of the initial OAM detector.

7. Refractive and absorptive detection

The device studied here also allows OAM detection in refractive or absorptive schemes without the need of patterning an SWG on the central pad. The OAM change, and resulting torque, when a photon is absorbed by the central pad is given by the total OAM of the input field, and is independent of the central pad geometry. However, this scheme is destructive as the photons are destroyed upon detection. Implementation of absorptive detection with the device studied here would require operation at shorter wavelengths below the transparency window of SiN, utilization of multiphoton or impurity related absorption processes, or modification of the central pad material to enhance its optical absorption. Circularly polarized light with spin angular momentum also can be detected using this scheme and the sensitivity analysis is equivalent to when the OAM change is $\Delta l=1$. Note that in all of the above studies, linear momentum transfer can in principle excite drum-like modes of the pad. However, since the drum-like mode of the central pad is a few MHz separated from its torsional mode, the OAM and linear momentum transfer can be studied independently.

In the case of a refractive detection scheme, which can be realized with a planar surface, the change in OAM is given by $\Delta l = 0.5(\cos \theta _i/\cos \theta _r+\cos \theta _r/\cos \theta _i)l$ where $\theta _i$ and $\theta _r$ are incident and refracted angles [67]. In this scheme, the magnitude of an incident field’s OAM and linear momentum are conserved while their directions are changed.

8. Conclusion

In conclusion, we have designed an optomechanical system that enables non-destructive measurement of light via the torque induced by its OAM on the device. The device has a torque sensitivity of $\tau _{\textrm {min}}=3.22\times 10^{-21}\,$N$\cdot$m/$\sqrt {\textrm {Hz}}$, allowing OAM detection of optical fields with a sensitivity of $P_{\textrm {min}}=8.7\,\mu \textrm {W}/\sqrt {\textrm {Hz}}$, assuming $Q_{o}=10^{6}$, $Q_{m}=10^{6}$ and cryogenic temperature $T=4\,$K. Considering the existing state-of-the-art performance of similar SiN devices, detection of $3.9\times 10^3$ photons in a single pulse is achievable. This number could be further reduced by using SWGs with higher order OAM conversion ($\Delta l >10$) and designing a photonic crystal nanobeam cavity with a lower mechanical frequency.