Cavity optomechanics with sub-wavelength grating mirrors

We design, fabricate and study a novel platform for cavity optomechanics: a silicon nitride membrane patterned as a sub-wavelength diffraction grating. Using the grating as one mirror of a Fabry–Perot cavity, we realize an optical finesse of F = 2830 ± 60, corresponding to a grating reflectivity of R = 0.998. The finesse we achieve appears to be within a factor of two of the limit set by material absorption. We study the finesse as a function of wavelength and optical spot size in order to elucidate various optical loss mechanisms. We find that the cavity exhibits birefringence, and establish that it, too, is a source of optical loss. We then characterize the mechanical motion. We observe hundreds of normal modes, and find the fluctuating amplitude of one of them to be very well described by a Boltzmann distribution. By injecting a red-detuned cooling laser, we optically cool all of the modes that we observe. The lowest effective temperature we achieve is Teff ≈ 1 K.


Introduction
Cavity optomechanics is at this point a mature research field, in which many different platforms have been demonstrated which couple the energy in a mechanical resonator to the energy stored in an optical cavity [1]. The ideal optomechanical system combines low optical and mechanical losses, low mass, and strong coupling between the optical and mechanical degrees of freedom. While it is difficult to realize all of these criteria simultaneously, a hybrid 'membrane-inthe-middle' Fabry-Perot cavity largely does so. In this platform, a thin silicon nitride (SiN) membrane, functioning as a high-Q mechanical resonator with low optical loss, is placed within a high-finesse optical cavity comprised of two conventional highly-reflective dielectric mirrors. When first demonstrated [2], this system achieved an unprecedented optical cooling efficiency despite the low reflectivity (R = 0.18) of the SiN membrane, a consequence of the very low losses that were realized. Substantially lower mechanical and optical losses were subsequently demonstrated in such a system by the use of stoichiometric silicon nitride, Si 3 N 4 [3].
From the outset it was recognized that certain experiments, such as the observation of mechanical quantum jumps, would be possible in such systems only with membranes of considerably higher reflectivity. As interest developed in experiments employing multiple membranes within a cavity [4][5][6], other potential benefits of stronger optomechanical coupling became evident. Multiple highly-reflective membranes can be used to increase the effective cavity length [6], for example, thus placing optomechanical systems with low mechanical frequencies and large optical linewidths in the resolved-sideband regime required for groundstate cooling.
Remarkably, there is a well-established method to increase the reflectivity of a dielectric membrane to nearly unity, while in fact reducing its mass, which is to remove material so as to pattern it as a diffraction grating with a sub-wavelength period. Such devices have been used in vertical cavity surface emitting lasers [7], where the inferred reflectivity was R = 0.995. They have recently been demonstrated in devices designed as optomechanical systems. A reflectivity of R = 0.95 was demonstrated in a mechanically compliant device with resonances in the MHz range [8], and in our early work we obtained [9] a reflectivity of R = 0.994 in a patterned SiN membrane, accompanied by a mechanical mode having a quality factor of Q = 7.5 × 10 5 .
In this work, we explore the use of a sub-wavelength grating patterned in SiN as an optomechanical element in considerably more detail. We start by establishing a region of parameter space amenable to fabricating a grating with near-unity reflectivity, and calculate the limiting reflectivity set by material absorption. We fabricate these devices and incorporate one of them as an end mirror in a standard two-mirror Fabry-Perot cavity. We measure a finesse as large as F = 2830 ± 60, and explore various optical loss mechanisms related to the geometry of the confined mode and the wavelength. We find a clear signature of birefringence in the cavity and discuss its consequences. We then illustrate the mechanical properties of these devices. The mechanical quality factor is not severely degraded by fabrication of the grating, and we realize optical cooling from room temperature to an effective temperature of T eff ≈ 1 K.
All of the results presented here are made with a conventional two-mirror cavity. The results we obtain, however, should be relevant to work in which these highly-reflective membranes are used in a 'membrane in the middle' cavity. At the same time, the two-mirror cavity is interesting in its own right; it is a far simpler system requiring less alignment, and a next-generation twomirror cavity incorporating a grating membrane should be capable of reaching the resolvedsideband regime required for optical cooling to the mechanical ground state [10,11].

Optical properties of sub-wavelength grating mirrors
The operating principle of sub-wavelength grating mirrors has been discussed thoroughly in the literature, and a comprehensive review has recently appeared [12]. Briefly, when a diffraction grating with period a is illuminated at normal incidence by means of a plane wave of wavelength λ, diffracted beams appear at angles θ m given by sinθ m = mλ/a, where the order number m is an integer. The distribution of power among the diffracted orders depends on the details of the grating, and it has long been known that by appropriately 'blazing' the grating, it is possible to optimize the amount of light scattered into a desired order. If the grating periodicity is smaller than the wavelength, a < λ, it is readily seen that the only diffracted orders that are allowed correspond to m = 0, either transmitted or reflected at normal incidence. By appropriate design of the grating, it is possible to force nearly all of the scattered light into the reflected beam. While superficially resembling blazing, the phenomenon is better described as destructive interference between the directly-transmitted beam and radiation modes that are coupled to leaky guided modes of the grating structure [13][14][15][16][17].
Sub-wavelength grating mirrors have been demonstrated in a variety of material systems. The simplest platform, and probably the one most suitable for cavity optomechanics, is shown in figure 1(a). It consists of a dielectric slab with an index of refraction n that is perforated with lines or holes, giving an index contrast of n:1. One-dimensional gratings comprised of a periodic array of rectangular segments separated by air have been demonstrated using InP (n = 3.17) [18] and Al 0.6 Ga 0.4 As (n = 3.2) [7]. The sizes of these grating mirrors were 12 µm × 12 µm and 10 µm × 10 µm, respectively. A more sophisticated structure made of tailored ridges on a silicon (n ≈ 3.5) substrate allowed a one-dimensional grating with a spatial extent of many square millimeters to be demonstrated [19], owing to the rigidity of the substrate. Two-dimensional gratings consisting of a square array of circular holes have been demonstrated in silicon [20] and InP [8]. A high index contrast allows a broader reflection bandwidth [17], but gratings with a much lower index contrast (3.45:3.17) have been demonstrated [13].
In this work, we consider a one-dimensional grating comprised of a periodic array of dielectric 'fingers' with index of refraction n ≈ 2.2 appropriate to SiN, each finger having a trapezoidal shape with mean width w and thickness t, as illustrated in figure 1(a). The trapezoidal shape is motivated by study of devices we have fabricated, as will be discussed in more detail in section 3.1. The widths at the bottom and top of each finger are taken to be w bottom = w + 80 nm and w top = w − 80 nm, on the basis of measured samples. The periodicity a is chosen to be smaller than the wavelength of the light to be used, λ = 1560 nm. In between the fingers is vacuum, as are the half-spaces above and below the grating. For a given index of refraction n, the parameters a, w, and t can all be varied in order to tailor the reflectivity. Clearly the idealized structure cannot be completely realized, as the size of the structure will be finite, and fabrication defects will limit the periodicity of the structure. Nevertheless, this structure is amenable to calculation, and provides the starting point for our mirror design.

Infinite size grating; plane-wave illumination
Calculation of the diffraction pattern for the structure shown in figure 1(a) under plane-wave illumination can be accomplished by means of rigorous coupled-wave analysis (RCWA) [21]. RCWA is a frequency-domain technique in which the total electromagnetic field is expressed as a countable superposition of plane waves in the regions inside and outside a grating, related by boundary conditions. Outside the grating, the field consists of an incident plane wave and a finite number of forward-and backward-scattered diffracted waves. Within the grating, the field is described as an infinite number of forward-and backward-traveling plane waves with wave vectors given by Floquet theory. When the expressions for the plane waves are coupled by applying Maxwell's equations at the interfaces, the resulting set of equations is reduced to an eigenvalue problem for which user-friendly code has been written [22]. A color plot of the reflectivity of such a structure, as a function of period and finger width, is shown in figure 1(b). We have taken λ = 1560 nm, t = 470 nm, and n = 2.18 + 1.5 × 10 −4 i. The polarization of the incident beam is taken to be parallel to the grating fingers, and denoted as TE. The peak reflectivity is R ≈ 0.9988, and a relatively large region is found in the vicinity of w = 580 nm and a = 1420 nm where the reflectivity exceeds R = 0.998. For incident light polarized orthogonally to the grating fingers (TM), the reflectivity R < 0.1 over most of the domain shown in figure 1(b). By contrast, a simple unpatterned dielectric slab (w → a) with the same index of refraction and thickness exhibits a reflectivity of R ≈ 0.34.
Curves such as those shown in figure 1(b) can be used to try to find a region of parameter space exhibiting high reflectivity while having relatively low sensitivity to fabrication imperfections. For the case of grating fingers with a rectangular cross section, an analytic solution to the diffraction problem has been derived [17], facilitating study with dimensionless combinations of grating parameters. While we have concentrated on gratings designed for TE illumination, gratings with a high reflectivity for polarization perpendicular to the finger direction (TM) have been demonstrated as well [7,23]. Alternatively, a two-dimensional grating of holes can be used to construct a device whose reflectivity is independent of polarization [14]. Figure 2(a) illustrates calculations of reflection R, transmission T , and absorption A (R + T + A = 1) for two gratings we have fabricated with parameters in the high-reflectivity zone of figure 1(b), as a function of wavelength. The thickness is t = 470 nm, the period is a = 1425 nm, and the average finger widths are w = 620 nm (solid curves) and w = 660 nm (dashed curves). Ultimately, we are interested in using the gratings as end-mirrors in a Fabry-Perot cavity, so recalling that the finesse of a low-loss Fabry-Perot cavity can be expressed as where R 1 and R 2 are the reflectivities of the cavity mirrors, figure 2(a) shows (1 − R) rather than R. Clearly it is absorption that ultimately limits the reflectivity at all wavelengths. At λ = 1550 nm, the grating with w = 620 nm is far from absorption-limited, and T A. The grating with w = 620 nm exhibits a substantially higher reflection at λ = 1550 nm, but in this case T ≈ 2A. It is also interesting to note that the absorption of the gratings, A ≈ 1.2 × 10 −3 , is approximately twice that of an unpatterned dielectric slab with the same thickness, A slab = 5.3 × 10 −4 . In all cases the polarization of the incident light has been taken parallel to the grating fingers; the reflectivity for the orthogonal polarization varies monotonically from approximately R = 0.03 to 0.4 over the range of wavelengths shown.
The discussion has until now assumed an index of refraction with imaginary part Im(n) = 1.5 × 10 −4 , taken from a measurement using a low-stress SiN membrane at λ = 1064 nm [24]. The limiting finesse achievable, in which all cavity losses are set by grating absorption, is 2π/A = 5200. At λ = 1550 nm it is likely that the absorption is somewhat lower, as the photon energy is further below the SiN bandgap. Indeed, interferometric measurements [25] R)) (red), transmission T (blue), and absorption A (black) versus wavelength for two of the gratings we have fabricated. The grating thickness is 470 nm, the period is w = 1425 nm, and the solid (dashed) curves refer to gratings with a mean finger width of w = 620 nm (w = 660 nm). It is noteworthy that, while the reflection and transmission of the two gratings differ substantially from each other and vary strongly with wavelength, the absorption is nearly constant (1.2 × 10 −3 ) and the same in the two cases. The index of refraction is taken to have imaginary part Im(n) = 1.5 × 10 −4 . The absorption of an unpatterned dielectric slab with the same thickness is A = 5.3 × 10 −4 . (b) Reflectivity as a function of incident angle for the grating with w = 620 nm at λ = 1550 nm; the red (blue) curve corresponds to rotations about an axis parallel (perpendicular) to the grating fingers. The inset shows the reflectivity over the range of −0.09 to 0.09 rad.
of the Brownian motion of a silicon cantilever using light with a wavelength of λ = 1310 nm showed substantially more heating than measurements made with λ = 1550 nm, despite the fact that both wavelengths are below the silicon bandgap. An RCWA calculation made with Im(n) = 6 × 10 −6 , appropriate to stoichiometric SiN [3], yields a result similar in appearance to that of figure 1(b), but characterized by a peak reflectivity of R ≈ 0.999 95. The corresponding absorption-limited finesse is F = 125 000.

Confined optical mode
The RCWA calculation assumes plane-wave illumination, and it is of interest to consider how the result would be modified for a confined beam such as is encountered in a Fabry-Perot cavity. While more sophisticated treatments are possible, it is instructive to simply examine the angular dependence of the reflection in the plane-wave case, and compare it to a characteristic diffraction angle for a confined beam. Figure 2(b) shows the reflectivity as a function of incident angle for one of the gratings of figure 2(a) at a wavelength of λ = 1560 nm. The two curves correspond to rotations around an axis parallel (red) and perpendicular (blue) to the grating fingers. The reflectivity starts to plummet for incident angles larger than θ = 0.095 rad, but it is perhaps more illuminating to note (inset to figure 2(b)) that for rotation angles of θ = 0.05 rad about an axis perpendicular to the grating fingers, the quantity (1 − R) doubles from the value at θ = 0. The far-field diffraction (half-)angle of a Gaussian beam is θ = tan −1 (λ/(πω 0 )), where ω 0 is the spot size at the beam waist [26]. Taking θ = 0.05 rad, one finds a corresponding characteristic waist size of ω 0 ≈ 10 µm at λ = 1560 nm. Thus, a cavity whose finesse is limited by the reflectivity of the grating mirror could be expected to attain its maximum possible finesse only for waist sizes appreciably exceeding ω 0 ≈ 10 µm.

Device fabrication
The membrane used in this work has dimensions of 1.2 mm × 1.2 mm × 470 nm. It is patterned with 81 different gratings, with various periods and finger widths appropriate to the highreflectivity zone of figure 1. The fabrication procedure is as follows: silicon (100) wafers are cleaned by a standard procedure to remove organic and ionic contaminants, and immediately transferred to a low-pressure chemical vapor deposition (LPCVD) furnace. In LPCVD, SiN is deposited using a recipe that produces silicon-rich, low-stress SiN. The stress is measured to be 190 ± 35 MPa, using a commercial film stress measurement system. After deposition, both surfaces of the wafer are spin-coated with photoresist and membrane patterns are defined on the back surface by photolithography. The membrane patterns are then transferred to the SiN by CHF 3 plasma with a reactive ion etch (RIE). The photoresist on both surfaces is stripped off, and the silicon that has been exposed by the RIE process is etched in a KOH solution. The KOH etch stops when it reaches the SiN on the front side of the wafer, creating membranes. After the KOH etch, the wafer with the membranes is briefly dipped in HCl solution, rinsed with deionized water, dried, spin-coated with e-beam resist and diced. E-beam lithography is then used to define the grating structure in the resist and a second RIE step, with the same parameters as the first, is used to transfer the grating pattern to the underlying membrane. Finally, the e-beam resist is stripped away, and the sample is dried in a critical-point drier. Figure 3 shows test samples made to study various steps in the fabrication procedure.

Cavity
The prototypical platform for cavity optomechanics is shown in figure 4(a). It consists of a Fabry-Perot cavity comprised of a mechanically compliant, nominally flat mirror (right), separated by a distance L from a concave mirror (left) with radius of curvatureR. In our case, the mechanically compliant mirror is one of the 50 µm × 50 µm sub-wavelength gratings just described. For L <R, such a cavity supports stable optical modes whose characteristics are easily related to the cavity geometry as long as the transverse size of the mirrors is sufficiently large [26]. The mode waist is located on the flat mirror, with a spot size ω 0 that becomes very small near the stability boundaries of the cavity, L → 0 and L →R. We shall be concerned with cavity lengths L =R(1 − ), where 1, in order to have a spot size smaller than the grating mirror and a free spectral range ν = c/(2L) commensurate with the scanning range of our laser. For such a geometry, it may be shown that the waist size ω 0 and the spot size on the curved mirror ω 1 are given by If the spot sizes ω 0 and ω 1 are not substantially smaller than the size of the mirrors, then these equations are not valid and a numerical analysis is required [26,27].
For a cavity with a plano-concave geometry such as illustrated in figure 4(a), the cavity resonance frequencies are given by Here the integer N labels the longitudinal modes, and integers m 0 and n 0 label transverse modes. The modes with transverse mode indices such that m + n is even are located near the fundamental modes (m = n = 0), and spaced by δν = ν √ /π . The modes with m + n odd are located between the fundamental modes, and also spaced by δν. This mode structure is shown in figure 4(b).
We have implemented such a cavity in a small vacuum chamber maintained at a pressure of 1.3 × 10 −5 Pa (1.3 × 10 −7 mbar) with an ion pump. All of the results to be discussed here employ a grating with period a = 1410 nm and finger width w = 620 nm. The cavity length L can be varied with a stick/slip actuator with a resolution of 30 nm. The curved mirror is characterized by a radius of curvatureR = 25 mm and diameter of φ = 7.75 mm. It has a dielectric coating with a reflectivity given by 1 − R = 1.6 × 10 −4 ± 2 × 10 −5 at λ = 1560 nm. In order to limit losses at the curved mirror, we envision a cavity in which the length L is sufficiently far from the stability boundary that the spot size of the confined mode on the curved mirror is no greater than approximately half the radius, or ω 1 = 1.5 mm. Using (2), this implies R > 790 nm, corresponding to a waist size of ω 0 > 8.3 µm.
Light is coupled into the cavity by means of the apparatus shown in figure 5. A tunable 'probe' laser is passed through a Faraday isolator and mode-matched into the cavity by means of two lenses. The (linear) polarization of the light entering the cavity can be rotated by means of a half-wave plate, in order to align the polarization relative to the grating orientation. An electrooptic modulator (EOM) can be used to phase-modulate the laser so as to generate sidebands to probe the cavity modes shown in figure 4(b), as discussed in section 4.2, or to derive a Pound-Drever-Hall (PDH) error signal. The beam reflected by the cavity is spatially separated from the ingoing beam by means of the Faraday isolator. Light transmitted by the cavity exits the vacuum chamber and can be monitored with both a photodiode and a camera.

Transmission, reflection, finesse
Figure 6(a) shows the cavity transmission and reflection curves in the vicinity of a resonance at our target wavelength of λ = 1560 nm. The curves were made by sweeping a tunable laser (linewidth < 100 kHz) by means of a piezoelectric actuator. The laser frequency is swept over a time of 1 ms, and the frequency axis is calibrated by adding sidebands (not shown) to the laser, at a frequency of 28 MHz. The transmission peaks of the carrier and sidebands are all fit to Lorentzians, a process that is repeated 20 times. The transmission peak has a full-width at halfmaximum of 3.03 ± 0.08 MHz, corresponding to a finesse of F = 1980 ± 50. The uncertainty is dominated by the standard deviation (type A evaluation) of the 20 measurements. From (1) we can then infer (1 − R 2 ) = 3.0 × 10 −3 ± 8 × 10 −5 . The primary transmission peak visible in figure 6(a) is that of a TEM 00 mode. In addition, a small, much broader peak is visible at a frequency 30 MHz higher. This peak is caused by excitation of a higher-order mode due to imperfect mode matching. In the optomechanical cooling to be described in section 5.2, light injected into the cavity is tuned to well within one A 'probe' laser is phase-modulated at 28 MHz by means of an EOM and sent into a Faraday isolator. The (linear) polarization of the emerging beam can be adjusted by means of a λ/2 plate, and the beam is mode-matched into the cavity with two lenses. The beam reflected by the cavity is spatially separated from the ingoing beam by means of the Faraday isolator, and the reflected signal is detected, high-pass filtered (HPF), and demodulated at 28 MHz to derive a PDH error signal. The error signal is used to maintain the probe laser resonant with the cavity at low frequencies ( f < 12 kHz), and to infer the membrane displacement at frequencies ( f > 12 kHz) outside the servo bandwidth. A second 'pump' laser, amplified with an erbium-doped fiber amplifier (EDFA), can be combined with the probe laser by means of a fiber beamsplitter (FBS) and simultaneously injected into the cavity. It is beat against the probe laser, and offset-locked to a frequency ν ≈ 6 GHz relative to the probe. When the pump is in use, a low-finesse filter cavity with a free spectral range of 12 GHz is used to minimize unwanted reflected pump light in the displacement detection signal.
cavity linewidth of the TEM 00 mode, and the radiation pressure associated with higher-order modes such as this one is negligible. Figure 6(b) shows the cavity finesse as a function of wavelength over a range of λ = 1500-1608 nm. At each wavelength, the error bars correspond to the standard deviation of 20 measurements. The highest finesse we measure is F = 2830 ± 60 at λ = 1509 nm, corresponding to (1 − R) = 2.06 × 10 −3 ± 5 × 10 −5 . Between λ = 1527 and 1560 nm the finesse plummets to a minimum value of F = 107 ± 8. We believe this is a consequence of resonant optical coupling to high order transverse modes with high losses, rather than an actual diminution of the grating reflectivity in this wavelength interval. This loss mechanism has been observed before [28], and is exacerbated by working near the critical point of a cavity with a small mirror. Not only does the mode density increase when the critical point is approached, as  6. (a) Transmission peak (red) and reflection dip (blue) at λ = 1560 nm for a cavity using a grating with period a = 1410 nm and mean finger width w = 620 nm as the output coupler. The main transmission peak is that of the TEM 00 mode; the small peak observed at a frequency 30 MHz higher reflects excitation of a higher-order mode due to imperfect mode matching. (b) Finesse versus wavelength for the same grating. While the reduction in finesse above λ = 1565 nm is presumably due to lower grating reflectivity, we believe that the drop between λ = 1527 and 1560 nm is a consequence of resonant optical coupling to high order transverse modes with high losses.
shown in (3), but the losses of the higher-order modes increase due to their greater spatial extent relative to the small mirror. It is notable that the scatter in the measured finesse for wavelengths in the vicinity of λ = 1500-1530 nm is far greater than the measurement uncertainty associated with each point. We believe that these (repeatable) fluctuations are also due to resonant optical coupling to high order transverse modes.

Spatial mode size
We now address how the finesse depends on the cavity length, and correspondingly the geometry of the confined optical mode. The subject of diffraction losses in cavities with very small mirrors, as encountered in optomechanics, has recently been addressed [27]. In the present work, an additional loss mechanism arises for a grating of small spatial extent, which is the angular dependence of the reflectivity of the grating. As we will show, the optimum finesse we are able to achieve seems to be limited by both diffraction losses and the angular response of the grating. In order to accurately measure the cavity length L, we first use an EOM to generate sidebands on the laser frequency ν l with frequencies ν l ± ν rf , where ν rf 3 GHz is generated by a synthesizer. The free spectral range of our cavity is ν = c/(2L) ≈ 6 GHz. We sweep the laser frequency ν l and adjust ν rf so that the sidebands probe adjacent fundamental resonances simultaneously, at which point ν rf = ν/2 and L = c/(4 ν rf ). The mode ν N ,1,0 is not located exactly midway between modes ν N ,0,0 and ν N +1,0,0 , and thus is not resonant with the carrier. A plot of the finesse as a function of cavity length at λ = 1560 nm is shown in figure 7(a). The dominant contribution to the error bars is the uncertainty (type B evaluation) in adjusting ν rf to the exact centers of the cavity resonances. The finesse increases and reaches a maximum with increasing cavity length, and then diminishes as the cavity is brought to the stability boundary. Qualitatively, we can understand this, as the size of the TEM 00 mode waist diminishes when L →R, reducing the losses due to the limited spatial extent of the grating mirror. If the waist size is made too small, however, the grating reflectivity drops as discussed in section 2.2; in addition, power may be lost at the curved mirror if the spot size ω 1 is no longer small compared to its diameter. In order to be more quantitative, we use (2) relating the spot sizes on the curved mirror and the grating to the distanceR − L between the cavity length L and the stability boundarỹ R. Unfortunately, the radius of curvatureR is not known well enough in advance to allow a useful inference ofR − L from the length measurements shown in figure 7(a). In order to determineR − L, we employ a spectroscopic measurement of the higher-order transverse modes of the cavity. We phase-modulate the laser at ν rf and sweep the carrier ν l about the mode ν N ,1,0 (figure 4(b)), and adjust ν rf so that the lower sideband is resonant with the fundamental resonance ν N ,0,0 at the same time that ν l is resonant with the mode ν N ,1,0 . At this point, ν rf = ν/2 − ν √ /(2π ), and can be readily inferred since ν is known. It is worth noting that (2) relating the spot size ω 0 to is strictly valid only in the limit that the mirror size is very large compared to the spot size, while our grating mirror is only d = 50 µm on a side. Figure 7(b) shows the finesse as a function of the spot size on the grating, where the spot size has been determined by use of (2). The error bars reflect only measurement uncertainties; the actual waist size becomes less and less certain as it increases towards the size of the grating mirror, since (2) becomes less and less valid. Subject to this caution, the finesse peaks for an inferred waist size of ω 0 ≈ 16 µm, at which point the spot size on the curved mirror is ω 1 = 770 µm, well below the mirror radius. It is apparent that the peak finesse is limited not by the curved mirror, but rather by a compromise between losses beyond the edge of the grating and the loss of grating reflectivity with angle associated with spatial components comprising beams with a waist below ω 0 = 16 µm. The inset to figure 7(b) shows a circle with a radius equal to ω 0 ≈ 16 µm superposed upon the grating.
Indeed, we saw in section 2.2 that a plane wave incident on a grating such as ours at an angle θ corresponding to a waist size of ω 0 ≈ 10 µm would experience a value of (1 − R) twice that at normal incidence. The fact that the measured finesse drops to approximately half its peak value when the waist is reduced to a size of ω 0 ≈ 10 µm is highly suggestive that the dependence of the grating reflectivity on angle is the reason. One can estimate the loss in finesse as the waist size increases by means of a crude model, in which the intensity distribution at the waist is taken to be Gaussian with spot size ω 0 , the flat mirror is taken to be a square of side d with perfect reflectivity R = 1, and the space outside the mirror is taken to have reflectivity R = 0.34, corresponding to unpatterned SiN. The curved mirror is also taken to have infinite spatial extent and perfect reflectivity, so that all optical losses occur where the Gaussian beam spills over the edge of the small square mirror. The optical losses are then given by κ = (1 − R )(1 − erf 2 (d/( √ 2 ω 0 ))), and the corresponding finesse F = 2π/κ is shown in the dotted line in figure 7(b). It is notable that the observed finesse exceeds the predictions of this crude model for all waist sizes above ω 0 ≈ 17 µm, consistent with the fact that the actual spatial modes of a cavity with small mirrors deviate from the modes of infinite-size mirrors by adopting a shape that gives a lower round-trip diffraction loss [27]. The fact that the finesse reaches a sharp peak, rather than a plateau, suggests that it is currently limited by both of the mechanisms discussed here, and is strong motivation to pursue a grating with a larger area.

Polarization
Since the one-dimensional grating we employ is highly polarization dependent, as discussed in section 2.1, one might assume that the cavity would function as a linear polarizer. This is in fact not the case, as shown in figure 8, relating the polarization at the output of the cavity to that at the input. In figure 8(a), the injected light is polarized vertically, parallel to the grating fingers; the exact polarization angle is found by maximizing the peak transmitted power. This is the configuration used for all of the results described to this point. The red curve is a polar plot of the incident power transmitted through a polarizer oriented along an axis described by polar angle θ relative to the horizontal. The blue curve is a similar plot of the power transmitted on resonance (normalized to its maximum power as a function of θ ), and it is clear not only that the polarization is elliptical, but that the principal axis of the ellipse has been rotated by 90 • relative to the incident polarization.
We interpret this as a consequence of birefringence within the cavity, so that the cavity eigenstates have a small TM component. Therefore, after a large number of round trips inside the cavity, light injected with an initial TE polarization acquires a weak perpendicular polarization component. Because of the large difference in grating reflectivity for the two polarizations, the TM component is much more favorably transmitted. The form of the polar plot in figure 8(a) can be used to infer that the output polarization is an ellipse with its major axis horizontal and eccentricity ε ≈ 0.75. The circulating field striking the output coupler then must have an elliptical polarization with a ratio of horizontal to vertical electric field amplitudes of 1/ε √ T TE /T TM , where T TE (T TM ) is the (power) transmission of the grating for TE (TM) polarized light. It can then be shown that the power transmitted by the output coupler is where θ = tan −1 ε = 0.64, so that the output coupler transmits a factor of 2.78 more light than it would if there were no birefringence. Thus, the birefringence provides a substantial loss mechanism limiting the finesse. Taking the absorption given by RCWA (A = 1.2 × 10 −3 ), the transmission T TE of the grating would have to be T TE = 6.5 × 10 −4 in order to provide the finesse F = 1980 that we observe; in the absence of birefringence, the corresponding finesse would be F = 3100.
It is not at the moment clear which element is principally responsible for the birefringence. Since the stress in the grating fingers is larger along the length than across the width, as discussed in section 5.1, it is likely that the grating is birefringent. At the same time, stressinduced birefringence is a well-known phenomenon in high reflectivity dielectric mirrors unless special care is taken in mounting them [29], and in fact we have observed birefringence in symmetric cavities made employing dielectric mirrors identical to the curved one used here. An appealing possibility is to apply stress to the dielectric mirror in such a way as to create birefringence with the fast axis aligned parallel to the slow axis of any birefringence present in the grating. If the magnitude of the birefringence in the dielectric were made equal to that in the grating, the two would compensate each other, and the associated loss would be eliminated. Figure 8(b) shows a similar polar plot for the case that the polarization of the light injected into the cavity is perpendicular to the grating fingers. In this case the blue curve representing the transmitted power has the same functional dependence on θ as does the injected light, showing that the transmitted light has the same linear polarization as that which is injected. The normalization has been adjusted by a factor of two for clarity. Because of the very low reflectivity of the grating in this direction, light injected with TM polarization makes very few round trips of the cavity and thus does not significantly probe the cavity birefringence, resulting in a polarization state unchanged by the cavity.

Passive mechanical characterization
To probe the mechanical motion of the membrane, we implement the PDH setup shown in figure 5 with a modulation frequency ν rf = 28 MHz. Feedback is used to maintain the laser, with a power of approximately 150 µW, resonant with the cavity at frequencies below 12 kHz. The PDH error signal is proportional to the membrane displacement at higher frequencies, where feedback is ineffective.
In order to calibrate the PDH error signal, we sweep the laser frequency ν over a cavity resonance and digitize the PDH signal as a function of time. The sidebands of the error signal, separated from each other by 2ν rf = 56 MHz, allow us to calibrate the temporal axis in terms of optical frequency. The central feature of the PDH signal then has a known slope dV /dν. In practice, this calibration is compromised by low-frequency fluctuations in the cavity length as the laser is swept. For this reason, we sweep the laser frequency rapidly (but still slowly compared to the cavity decay rate), and average over 20 sweeps. The typical uncertainty in dV /dν associated with the standard deviation of the 20 measurements (type A evaluation) is 10%. Small changes in the cavity length dL are then related to small changes in the resonance frequency dν by dν/ν = −dL/L, from which we obtain the relation between signal voltage and cavity length for fixed laser frequency dV /dL = −(ν/L) dV /dν. L and ν are both known with sufficient accuracy that the uncertainty in dV /dL is dominated by the 10% uncertainty in dV /dν.
The red curve in figure 9(a) shows the mechanical spectrum made with a weak resonant probe as observed on a spectrum analyzer (resolution bandwidth 30 Hz) in the frequency range from 120 kHz to 1.85 MHz. Hundreds of mechanical resonances are present, and although the mode density increases with frequency, the resonances are well-resolved at even the highest frequencies shown.
To frame our discussion of the dynamics, we express the membrane displacement z(x, y, t) as a sum of products of normal modes g mn (x, y) with time-dependent factors q mn (t) z(x, y, t) = m,n q mn (t)g mn (x, y).
For a uniform square membrane, the mode functions g mn are given by g mn (x, y) = sin mπ x d sin nπ y d , where m and n are positive integers, and d is the length of a side of the membrane. In the absence of applied forces, the amplitudes q mn (t) satisfy the differential equation where F th (t) is the thermal Langevin force, whose power spectral density is given by The ω mn are the mechanical eigenfrequencies, and m eff is the membrane effective mass. For a uniform square membrane of mass m, the eigenfrequencies are related to each other by ω 2 mn = ω 2 11 (m 2 + n 2 )/2, and the effective mass m eff = m/4 [30]. intr mn is the intrinsic damping of the mode labeled by (m, n), related to the mechanical quality factor by intr mn = ω mn /Q mn . In thermal equilibrium at temperature T , the amplitude q mn (t) of each mode is given by equipartition, and the measured displacement z(x, y, t) is proportional to the mode function g mn (x, y) evaluated at the position of the laser spot.
Returning to figure 9(a), the fundamental mode has a frequency of ν 11 = 128 kHz, and the next two modes are at frequencies ν 12 = 193 and ν 21 = 209 kHz. The ratios of the mode frequencies, ν 12 /ν 11 = 1.51 and ν 21 /ν 11 = 1.63, are split about the ratio √ 5/2 ≈ 1.58 expected for a uniform square membrane. This is a consequence of the asymmetric stress induced in the membrane by patterning it with gratings; similarly, the mode eigenfunctions g mn (x, y) and effective mass m eff cannot be expected to have exactly the same forms as those for the uniform square case. By fitting the measured frequencies of the ten lowest-frequency modes to the result for a square membrane with different stresses in thex andŷ directions, we infer a stress of 158 ± 2 MPa along the grating fingers and 120 ± 2 MPa in the perpendicular direction.
We now focus attention on the mode labeled by (m, n) = (1, 2), which we have determined from measurements of mechanical ringdown [9] to have the highest quality factor, Q intr 12 = 7.5 × 10 5 , of any mode that we have measured. For a lightly-damped oscillator ( mn ω mn ), the q mn (t) can be expressed as where A mn (t) is a slowly-varying amplitude and φ mn (t) is a slowly-varying phase. In order to more fully characterize the mechanical excitation, the PDH signal is demodulated at frequency ν 12 by means of a lockin amplifier, with a bandwidth of 50 Hz. The lockin amplifier gives the instantaneous amplitude A 12 (t) g 12 (x 0 , y 0 ), where (x 0 , y 0 ) is the location of the laser spot. Figure 10(a) shows A 12 (t) sampled every 2 ms, where g 12 (x 0 , y 0 ) is taken to be that appropriate to a uniform membrane. It fluctuates over a range from nearly zero to 9 pm in a time of 40 s, with a mean amplitude of A(t) = 3.2 pm and a mean-square amplitude A 2 (t) = 12.5 pm 2 .
In the case of thermal equilibrium, Boltzmann statistics predict a distribution of the form and a relation between the first and second moments of A 2 / A 2 = 4/π ≈ 1.27. The mean and second moment of the data give A 2 / A 2 = 1.26. A histogram of A 12 (t) is shown in figure 10(b), in which the error bars reflect the uncertainty in the PDH calibration. Also shown is a fit to f 12 (A), in which the temperature is the only free parameter. The agreement is very good, and yields a temperature of 342 ± 4 K, where the uncertainty in the fit parameter does not include a contribution from the error bars. Indeed, fitting to the lower or upper extremities of the error bars yields a fit that is much less satisfactory, suggesting that our estimate of the uncertainty in the PDH calibration is on the high side. The fact that the extracted temperature is greater than the ambient temperature is not completely understood. It is important to note that equations concerning thermal equilibrium involve properties of the membrane, in addition to the measured oscillation amplitude. Thus, the effective mass in (8) is taken to be one-fourth of the physical mass of the patterned membrane, m eff = 5.36 × 10 −10 kg, and the eigenmodes g(x, y) relating the measured displacement z(x, y, t) to the mode antinode amplitude A(t) are taken to be those of an unpatterned membrane. Both of these choices are approximations, as we have not attempted an exact numerical solution, and may contribute to the discrepancy from room temperature. It may also be the case that excess technical noise is present atop the fundamental thermal noise, or that a very small blue detuning of the probe laser contributes a weak optical antidamping force. It is notable that if we infer the temperature only from the mean-square amplitude, rather than the full statistical distribution, we find T = 354 ± 71 K, where the (much larger) uncertainty arises from the PDH displacement calibration.

Optical cooling
The effect of radiation pressure is to add a generalized force F mn (t) [30] to the Langevin force in (6). In the case that the applied force is the radiation pressure associated with a Gaussian optical mode with spot size ω 0 centered at position (x 0 , y 0 ) on the membrane, the generalized force F mn (t) is given by where F RP (t) is the total applied radiation pressure force. In our case, ω 0 d, and to very good approximation F mn (t) ≈ F RP (t)φ mn (x 0 , y 0 ) for the low-order mechanical modes. For a high-Q mechanical oscillator, the radiation pressure is correlated with the mechanical motion, with the well-known consequence that the damping rate in (6) is modified. The new damping rate is given by intr → intr + RP , where RP is proportional to the cooling laser power, and may be positive or negative depending on the sign of the detuning [1]. It can then be shown from (6) that the mean-square displacement is equivalent to that given by equipartition but at an effective temperature T eff = T ( intr /( intr + RP )).
To implement optical cooling, the cooling ('pump') laser shown in figure 5 is combined with the probe laser by means of a fiber beamsplitter and simultaneously injected into the cavity. The probe laser power is set to approximately 200 µW for these experiments. In order that the reflection of the pump not perturb the displacement measurement, it is tuned near to an adjacent longitudinal mode, separated by 6 GHz, and a low-finesse filter cavity is used to separate the pump and probe beams. As shown in figure 5, the pump is beat against the probe laser, and offset-locked to a frequency ν ≈ 6 GHz relative to the probe. The exact offset frequency can be set by means of a synthesizer to allow precise control of the detuning of the cooling laser. The filter cavity is comprised of two flat glass plates, each coated with a reflectivity of R ≈ 0.7, and has a free spectral range of 12 GHz. The length of the cavity is set by means of a piezoelectric actuator to be resonant with pump light, so that the cavity reflection is minimized for the pump and maximized for the probe. No active stabilization of this cavity is required.
The blue curve in figure 9(a) shows the mechanical spectrum in the presence of the cooling laser, detuned to the red of a cavity mode by 800 kHz, or approximately one-fourth of the cavity linewidth ν/F. This is near the optimal detuning (1/(2 √ 5)) ν/F for cooling in the limit of low mechanical frequency (far from sideband-resolved). The mean-square amplitude of most of the modes drops by over an order of magnitude. The modes are not cooled equally, a consequence of their different couplings g mn (x 0 , y 0 ) to the radiation pressure, different frequencies, and different mechanical quality factors. The black trace in figure 9(b) shows the displacement signal resulting when both the pump and probe lasers are present, but off resonance. The background level rises, a consequence of amplitude noise from the cooling laser (not completely removed by the filter cavity) entering the detection system. An opticallyinduced frequency shift is clearly seen in figure 9(b) as well. Figure 11(a) shows mechanical spectra made with a spectrum analyzer (resolution bandwidth 1 Hz) for a series of cooling laser powers. As the cooling laser power is increased, the mechanical spectra diminish in amplitude, broaden, and experience a frequency shift. The effective temperatures, inferred by integrating the power spectral densities in figure 11(a) to obtain q 2 12 and then using (10), are shown as a function of cooling laser power in figure 11(b). For small cooling laser powers, the width of the measured mechanical spectrum is limited by the resolution bandwidth of the spectrum analyzer, and for this reason points corresponding to effective temperatures above 3 K are not shown. For higher laser powers, the effective temperature drops inversely proportional to power, as expected, until it flattens out at T eff = 1 ± 0.2 K. Here the uncertainty is primarily due to the determination of the slope of the PDH signal relating signal voltage to mechanical displacement. Alternatively, the mean-square displacement of the antinode is reduced by a factor of 313 by optical cooling, which depends on neither the membrane effective mass m eff nor the PDH calibration. It is not clear at the moment why the cooling saturates. One possibility is local heating caused by material absorption as the circulating optical power increases; it is also possible that pump laser noise is not adequately removed by the low-finesse filter cavity as the power is increased. The issue is currently under investigation.

Conclusion and outlook
We have demonstrated sub-wavelength gratings in SiN to be a viable platform for optomechanics. Using a grating etched from a SiN membrane as the end mirror of a Fabry-Perot cavity, we demonstrated a cavity finesse of F = 1980 at the target wavelength of λ = 1560 nm, and a finesse of F = 2830 at λ = 1509 nm. The latter figure corresponds to a grating reflectivity of R = 1 − 2.06 × 10 −3 . RCWA calculations, assuming an absorption characterized by Im(n) = 1.5 × 10 −4 , imply R < 1 − 1.2 × 10 −3 , which suggests that we are not more than a factor of two away from the limitation imposed by material absorption. We have identified a number of mechanisms, in addition to absorption and scattering from fabrication defects, contributing to the optical losses. The reduction in grating reflectivity as the spot size is reduced would be immediately improved by use of a larger grating, as would the optical coupling to lossy highorder optical modes which can accompany working near the critical point of the Fabry-Perot cavity. Cavity birefringence is another mechanism providing substantial loss, apparently limiting the finesse at λ = 1560 nm from a potential F ≈ 3000 to 2000. This loss mechanism may be overcome if we are able to compensate birefringence in the membrane by appropriate stress of the dielectric mirror.
We are currently working to overcome these loss mechanisms. While the yield of our 50 µm × 50 µm grating structures is nearly 100%, we have already fabricated gratings with a 75 µm × 75 µm size with yield better than 50%. There are other approaches to obtaining a larger grating as well, including employing a structure with a substrate, making a two-dimensional grating, and the use of a dry technique to remove the e-beam resist in the final step. The twodimensional grating is of particular interest since it can be made polarization-insensitive and should not exhibit birefringence. Finally, we are investigating the use of stoichiometric SiN, which would give an absorption-limited finesse more than an order of magnitude higher.
We have shown that the mechanical motion of the membrane is well-described by normal modes whose slowly-varying amplitudes are characterized by a Boltzmann distribution, and we are able to optically cool one of the modes, at a frequency of 193 kHz, to T eff ≈ 1 K from room temperature. The current membrane is not particularly well adapted to cooling, as it was made very large in order to accommodate 81 gratings. We are currently fabricating smaller membranes with fewer gratings. A membrane 120 µm on a side would have a mass one hundred times smaller than the one described here, and the fundamental frequency of 1.3 MHz would bring it nearly into the resolved-sideband regime even with the present value of the finesse.
In addition to these straightforward extensions to the present work, a number of other interesting possibilities arise. Sub-wavelength grating mirrors with focusing ability have been demonstrated [31], allowing the possibility of an all-grating cavity with no dielectric mirror, in which the mechanical motion of both gratings could be coupled by radiation pressure. It may be possible to make a grating giving high TE and TM reflectivities for different wavelengths, allowing a richer variety of 'pump/probe' experiments. It was recently suggested in a theoretical paper [32] that by designing a photonic crystal mirror with a rapidly varying reflectivity as a function of wavelength, it should be possible to engineer optomechanical damping and antidamping without the need for an optical cavity. Indeed, sub-wavelength gratings such as those studied here can be designed to have resonant features with a reflectivity that varies very rapidly from zero to unity as a function of wavelength [12], which may allow such phenomena to be observed. An interesting question that arises is whether the motion of individual fingers in the grating can be observed as well. In the present sample, the frequency of the fundamental finger mode is ν 1 ≈ 1.8 MHz, at which point the density of membrane modes ν mn is very high (mean spacing between modes ≈ 3 kHz). Thus, even if the motion of a finger were transduced by the cavity, the response would not be distinguishable from that of a membrane mode. The situation is vastly more favorable for a smaller membrane. In a square membrane with a = 90 µm and finger length l = 50 µm, for example, the fundamental mode of a finger would fall midway between the (m, n) = (1, 1) and (m, n) = (1, 2) modes of the membrane, and thus would be easily identified.
Finally, there arises the interest in using these devices in a 'membrane in the middle' configuration. In such an application the grating size and cavity length should be less critical; the cavity length could be set far from the critical point so that the waist would be much larger, but any part of the spatial mode 'spilling over' the grating would still be confined to the cavity, and thus would not constitute a loss mechanism. Perhaps the most tantalizing possibility [6] is that in which the optomechanical coupling achievable by placing a small number of such gratings in a cavity would enable strong coupling between a single phonon and a single photon.