5.3 Hardware: Vibration Isolation, Vacuum System, and Optical Layout
5.3.3 Optical Layout: Locking and Probing the Cavity at Variable Detuning
5.3.3.3 Stabilizing the MIM System
For the standard Fabry-Perot cavity, changing the absolute frequency of the input laser ω0 (used to generate the science and locking beams) by a multiple of the cavity FSR does not affect the optomechanical coupling strength of the end-mirror as long as ∆ω0/2π << F SR. Rather (because the optomechanical interaction is a Raman process), the dynamics are only sensitive to the relative detuning ∆ between science beam and the cavity resonance frequency, ωc/2π=m×F SR. For a fixed input power, the task of stabilizing the system consists of just adjusting the cavity length or the laser frequency in order to maintain a stable detuning. The MIM system, however, has an extra degree of freedom that must be accounted for, since the optomechanical coupling of the membrane
g varies as a function of its position zm with respect to one of the end-mirrors (see Figure 3.1).
Because of this extra degree of freedom, we ideally must stabilize all three variables {∆, L, zm} to maintain a constant value for g. Equivalently, we must stabilize both ∆ and the phase of the intracavity standing wave at the position of the membrane: θ= 2ω0zm/c=ω0/F SR×zm/L. For a short cavity, it turns out that we can meet these two requirements using a simplified scheme.
First, we lock the cavity length to the laser frequency, ω0. This is the preferred method for a short cavity, since the cavity FSR = 202 GHz is much larger than the ∼ 10 GHz mode-hop free range of the lasers used in the experiment. In the experiments described in Chapters 6-9, we use the error signal derived from the transmitted locking field to apply feedback to the piezo beneath one of the end mirrors. The feedback signal is obtained by passing the raw error signal (mixer IF port) through an inverted pre-amplifier (Stanford Research Systems SR560), a simple op-amp integrator, and a single-pole 3 kHz low-pass filter (to reduce the servo gain at the first piezo resonance,≈10 kHz). The unity gain of the feedback loop is∼100 Hz. Stabilizing the laser-cavity detuning of the cavity to within 10% of the linewidth (FWHM) requires that the effective length noise of the cavity be reduced toδL = 10%×λ/2F ≈5 pm×(104/F), which is within an order of magnitude of the room-temperature Brownian displacement (rms) for the fundamental mode of the 50 nm x 500µm x 500µm membrane used in the initial cooling experiment (Chapter 9). This represents an interesting challenge that must be addressed in the future if a higher finesse cavity is used.
The second task is to stabilize θ. Here, we can make use of the fact that because the cavity is short, drift in laser frequency δω0 (typically GHz) is passively much smaller than the cavity FSR. By locking the cavity resonance to the laser frequency, the change in phase at position zm then becomes δθ = δω0/F SR×zm/L+ω0/F SR×δzm/2L ≈ 4πδzm/λ. For a thin membrane (|rm|2 1), the fractional change in the optomechanical coupling gm ≈ 2|rm|g0sin(θ) is given byδgm/gm≈cot(θ)δθ, so that at the position of maximal (gm= 2|rm|g0) and minimal (gm = 0) coupling the first-order sensitivity isδgm/gm= 0 and 4πδzm/λ, respectively. For many applications, particularly where we operate at gmax, we have found it sufficient to allow the absolute frequency of the laser to drift while the cavity is locked, and to make measurements on a timescale for which δzm/λ 1 (∼ 10 minutes to an hour when the membrane is at a point of maximal coupling, depending on temperature stability in the lab). This is the approach that has been taken thus far.
We have also set up a HeNe fringe at the exit mirror substrate and the membrane chip in order to calibrate the Picomotors (see Figure 6.4) and to monitor drift of the membrane position. We hoped to use this fringe to perform a slow feedback loop to stabilize the membrane position to within
∼10 nm. However, currently this application will have to wait until the damaged piezo lead on our membrane chip holder is replaced.
Chapter 6
Linear Optical Properties of the MIM System: Measurements
In this chapter I discuss how steady-state optical properties of our MIM system have been char- acterized in the lab, drawing heavily from the 1D formalism provided in the Chapter 3. With the exception of the first section (in which real index of the membrane is determined), all of the mea- surements described below were carried out in the autumn of 2008 through the summer of 2009, in preparation for the optomechanical cooling experiment described in Chapter 9. The ordering of the measurements is altered to follow the structure of chapter 3. In particular, I describe (1) the real index of refraction and thickness of the film, (2) the transmission and absorption losses of the cavity end-mirrors (via the finesse of the bare Fabry-Perot resonator), (3) optomechanical coupling of a membrane placed within the Fabry-Perot, and (4) reflectivity and absorption of the membrane as inferred from the linewidth and transmission of the MIM cavity.
6.1 Membrane Reflectivity and Thickness
Si3N4 membranes supplied to us by Norcada (see Chapter 4) are specified according to square dimensions (widthwm= 0.25, 0.5, and 1 mm) and thickness (dm= 30, 50, and 100 nm). For some time we have relied on these factory specifications, along with standard ellipsometric measurements of the real index of LPCVD Si3N4, to infer their mass and reflectivity [34]. The latter is important for estimating the optomechanical coupling. The former is important for understanding the mechanical properties of the system (e.g., the spectrum of vibrational frequencies). Whereas SEM images have confirmed thatwmis consistent with the factory specification at the∼1% level, a simple reflectivity measurement suggests that the actual thickness of the films can differ from the factory specified value by as much as 30%. This realization, coupled with substantial efforts to process our own films in the clean-room facility at the Caltech KNI and in Oskar Painter’s laboratory (efforts led by Painter graduate student Richard Norte and our post-doc Kang-Kuen Ni), have reinforced the need
Figure 6.1: Reflectivity of Si3N4membranes from Norcada for three nominal thicknesses (gray = 30 nm, orange = 50 nm, blue = 100 nm). The maximum value, |rm|max≈0.593 is consistent with a refractive index of|nm|= 1.98. Assuming a constant index for measurement wavelengthsλ={672 nm, 813 nm, 935 nm, 1064 nm}gives a result consistent with dm ={26 nm, 37 nm, 91 nm}. The curves at left are a qualitative fit to data for differentdm using|rm| vs. λfrom Eq. 3.4 with fixed
|nm|. The curves at right are a qualitative fit to data for differentdm using|rm|vs. dm from Eq.
3.4 with constant|nm|and the four differentλ.
to characterize their optical properties from scratch.
Fortunately, there are variety of simple ways to measure the thickness and refractive index of a thin film. If the index of refraction,nmof the film is known precisely at a specific wavelength, then a direct measurement of reflectivity at this wavelength gives the film thickness, and vice-versa. If neither is known exactly, then one can vary both, and use the fact that at a given wavelength the maximum reflectivity is independent of film thickness for negligible loss. For a low-loss (Im[nm]<<
1) film embedded in air/vacuum, this value is |rm|max = |n|nm|2−1
m|2+1 (see Section 3.1.1). This is the basic principle behind ellipsometry.
We have conducted our own thickness/index measurements by directly measuring power trans- mission Tm through the film at normal incidence. The measurement is made by focusing a laser beam through a membrane and onto a photodetector (Thorlabs PDA55). The membrane chip is secured to a flipper mirror mount so that measurements of input and transmitted power can be iter- ated. The laser source is switched between 670 nm (Oz Optics box diode), 813 nm (Schwarz-Optics Ti-Sapph), 935 nm (Toptica diode), and 1064 nm (Innolight YAG). Each measurement is repeated for three membranes with different nominal thicknesses: 30, 50, and 100 nm.
Results for this sequence of measurements are shown in Figure 6.1. Solid circles correspond to measured reflectivity,|rm|=√
1−Tm. Lines correspond to a model which assumes a constant real index from 670 nm –1064 nm (2% variation is expected from standard ellipsometric data [34]). From the reflectivity data for the thickest film, we obtain a maximum value of|rm|max≈0.593 near 750 nm. This is consistent a real index of 1.98. We fix this value and the thickness in the formula for|rm| (Eq. 3.4) to obtain the qualitative fits shown. The model is consistent with membrane thicknesses of 26 nm, 37 nm, and 91 nm, respectively. An independent measurement of the “50 nm” membrane
using the Filmetrix machine at the Caltech KNI obtains a thickness value of 38 nm. The Filmetrix measurement appears to be repeatable to within±3 nm for several membranes drawn from the same batch sent from Norcada.