the resonator or the substrate. In our case, a 1–10 MHz frequency phonon propagating in silicon with elastic wave speed ≈ 5000 m/s has a wavelength of 0.5 - 5 mm, which is of the same order as the square dimensions of the membrane (wm = 0.25–1 mm), the thickness of the chip (dchip = 0.2–0.5 mm), and the square dimensions of the chip (wchip = 5 mm–10 mm)! Recent work by the Cornell group with Ignacio Wilson-Rae [58] has taken some of these considerations into account.
They predict that for square and circular membrane geometries similar to ours, the quality factor limit due to acoustic coupling to the support should indeed scale as wm/dm. They compare their prediction to measurements made on dm= 112.5 nm, wm = 250-µm-square membranes developed in-house with T = 1200 MPa and using a somewhat different processing method than Norcada.
Their Q prediction is a factor 10–100 higher than measured. Accounting for this discrepancy by adding an ad-hoc internal loss term, they are able to predict to within factor of∼2 the differences in the quality factors of different higher harmonics for the same square membrane. It is interesting to note that the quality factors they observe agree to within a factor of∼2 with the value of Q≈106 that our results would phenomenologically predict for a membrane of their dimensions.
Chapter 5
Membrane-in-the-Middle Apparatus
In this chapter I present a top-down description of our first-generation “membrane-in-the-middle”
apparatus, including the cavity, nanopositioning system, vacuum chamber, vibration isolation, and optical layout. The description herein is intended to furnish the backdrop for the experiments described in later chapters. Most of the hardware in the system described here was developed in the autumn and winter of 2008. At the time, the “mechanics” project in the Kimble group was unfunded but heavily resourced, having recently decommissioned two laboratories (“lab 11” and
“lab 9”) dedicated to our Fabry-Perot-based CQED project. By dint of my familiarity with those projects, Cindy Regal, Scott Papp, and I were able to carve out an exclusive claim to a hodge-podge of valuable equipment. In particular, we decided to construct the new experiment around in-house diode and titanium-sapphire lasers operating at NIR (800 – 950 nm) wavelengths, along with a warehouse of optics at those wavelengths, including the supermirrors used for the optical cavity.
We also gathered vacuum components, a nanopositioning stage, vibration isolation material, servo electronics, RF synthesizers and spectrum analyzers, computers — you name it. Much of the lore and technology that has been inherited, particularly with regards to the development of high-finesse Fabry-Perot cavities (a craft with a long history in the Kimble group), will be lifted out of historical context here. I begin, rather, where the dust has settled, referring to former theses as needed.
5.1 Design Criteria
In the previous chapter it was shown that high-stress, stoichiometric Si3N4membranes could exhibit exceptionally high quality factors. They also have small mass (∼10 ng) and a geometry that en- ables them to be integrated into an an optical cavity [7], making them an attractive candidates for optomechanical studies [13]. Coupled with preliminary data (Section 6.4) and documented evidence [36, 13] that Si3N4 could exhibit low optical absorption in the NIR, and drawing from the seminal
Figure 5.1: Initial nanopositioning of the membrane between the cavity mirrors (the cameraman is holding his breath). In this chapter we explain the architecture and operation of the “membrane- in-the-middle” apparatus pictured here.
work of the Harris group [7], we set about constructing our own “membrane-in-the-middle” (MIM) cavity apparatus in October of 2008. Our design criteria deviated in some respects from the Harris group’s, and can be summarized as follows:
1. The cavity mirrors should support a finesse of F > 105, as we anticipated the extinction coefficient for our films to be<10−5 in the NIR. To provide some leeway, we also anticipated the need to “tune” the cavity finesse by operating on the side of the mirror coating curve.
2. The cavity length should be long enough to enable operation in the good cavity limit, but short enough to support a spot size smaller than the (6,6) mode of a 500-µm-square membrane (deemed a sweet spot based on our mechanical Qmeasurements at the time, see Chapter 4).
To obtainκ(HW HM)∼1 MHz, this compromise would require aL∼1 mm cavity with the smallest radius of curvature mirror substrates we had available at the time,Rc= 5 cm.
3. The cavity length must be tunable to enable referencing/locking to the laser. This would be necessary since, for a ∼ 1 mm cavity, the free spectral range, F SR ≡ c/2L ∼ 100 GHz, is much larger than the mode-hop free range of our diode and ti-sapph lasers.
4. The membrane should be separate from the cavity and attached to a nanopositioning stage with five alignment degrees of freedom. We had little idea how difficult alignment would be, so we required a positioning system with both fine and coarse tuning capability. Fine resolution along the cavity (z) axis should be sub-nm (<< λ). Fine resolution along perpendicular (x, y) axes should be sub-µm (much smaller than the nodal spacing for the 6,6 mode of a 500- µm-square membrane). Fine tip/tilt resolution should be << L/(Rc· F) ∼1 µrad. Coarse
resolution along the cavity axis should be longer than the cavity length, > 1 mm. Coarse resolution in thexandy direction should enable complete removal of the membrane from the cavity, and hence be larger than the 5 mm chip dimensions.
5. The membrane should be attached to a stiff, short-range piezo shaker for ringdown measure- ments and in order to provide a dither at the mechanical frequency for displacement calibration.
6. The membrane chip should be mounted in a fashion which does not appreciably reduce the quality factor of the membrane.
7. The MIM cavity should be supplied with dedicated vibration isolation. Isolation should ideally limit vibration of the membrane with respect to the cavity mirrors to ∼λ/F ∼100 pm at seismic frequencies (<100 Hz).
8. The MIM cavity should be enclosed inside a vacuum chamber. It must be large enough to accommodate the cavity, membrane nanopositioning system, and a vibration-isolation stack.
It must also be able to achieve a moderately high vacuum of∼10−6mbar in order to eliminate the possibility of air damping on mechanical quality of the membrane.
We now discuss how these challenges were met by the various components.