The optical set-up shown in fig. 3.3 is used to measure transduction. The laser power is heavily attenuated below 1 µW to avoid thermal drifts of the WGM mode and the trans- duction beam can be frequency tuned using the intracavity laser EOM. The transmission of the transduction beam is sent into a spectrum analyser which computes the FFT.
Figure 3.3: A simplified experimental set-up to characterise and optimise WGM trans- duction of the thermal motion of a microsphere-cantilever, defined as the signal to noise ratio (S/N) of the mechanical peak in the FFT of the transmission. An amplified voltage source is used to tune the laser frequency via the intracavity EOM, and the piezo-stack (PZT) at the clamped end of the cantilever is used to change the DC coupling distance. Offset thermal locking is possible because the input light is heavily attenuated at ≈ 1 µW.
Two variables will affect the S/N; the coupling distance, d, and the laser detuning with respect to the WGM. The dependence of the S/N on laser frequency is investigated by setting the microsphere-cantilever at an equilibrium DC coupling distance, d0 to the
tapered fibre, and measuring the transmission as the intracavity EOM tunes the laser frequency across the WGM. Next the laser frequency is thermally stabilised whilst d is varied with the piezo-stack (PZT).
3.3.1 Results
Following the experimental procedure detailed above, the S/N can be measured using a single trace FFT as shown in fig. 3.4. Here, the coupling distance d is decreased and the S/N of the FFT of the transduction signal is measured, normalised to the maximum peak amplitude3.
3
Figure 3.4: The FFT of the transduction beam at various coupling distances d shows a peak at the mechanical frequency of the c.o.m. of a microsphere-cantilever (this mode is cooled in chapter 4). The FFT is normalised to the maximum S/N.
The S/N relationship with d when the laser is thermally locked at a detuning of ∆ ≈ γi
2
is shown in fig. 3.5 c). The relationship between S/N and laser detuning is plotted in fig. 3.5 a) at a set coupling distance of d ≈ 180 nm. The graph of fig. 3.5 b) showing the
Figure 3.5: Optimising WGM transduction: a) The S/N as a function of coupling distance d (blue data), where the S/N is normalised to the peak maximum amplitude. The data is fitted with eq. 3.14. The power coupled to the WGM on-resonance is also shown in black data, where γe/γi = (4.4 ± 0.6)e−(11.6±1.2)d. b) The relationship between laser detuning
with respect to the WGM resonance and S/N is measured (red data) and fitted using eq. 3.14 (red line). The transmission, T , is simultaneously measured (black data) with a Lorentzian fit over the WGM profile, obtaining (30 ± 1) MHz FWHM.
S/N versus ∆0 at a fixed coupling distance d0 ≈ 180 nm, is fitted with eq. 3.14 such that
gom(d0) = γiGom, η, and γom(d0) = ηKeγi can be found. The same parameters can be
deduced using fig. 3.5 a) (S/N versus d), fitted with the d dependent version of eq. 3.14. However, the parameters are now gom(d) = Gomγie−ηd, and γom = Keγiηe−ηd, which can
can be found by analysing fig. 3.5 a) (black data) showing the normalised power coupled to the WGM on-resonance, Pc, as a function of d, fitted with eq. 1.33 (pg. 25) from chapter 1,
to obtain γe
γi. This is compared to the WGM lineshape at d0 = 180 nm (fig. 3.5 b) (black)),
whose FWHM is equal to γe+ γi at d0. A value of γi(d0) = (19.4 ± 1.9) MHz is obtained.
The optimum d is found to be 150 nm, which corresponds to the distance at which the maximum power is coupled into the WGM. Using a red-detuned transduction beam also increases the S/N. These results are in very good agreement with those obtained by the authors in [45]. The S/N dependence on the sign of the detuning indicates that both dispersive and dissipative changes in the coupling are important, unlike conventional optomechanical systems such as F-P cavities where dispersion is dominant. For those systems, β = 0 and F [T ] is symmetric around ∆0 = 0. Asymmetry arises when there is
dissipative coupling [45] which has also been found in a split-beam nanocavity [146]. The fitting of figs. 3.5 a) & b) to eq. 3.14 is achieved numerically, providing fitted values of γom,
η, and gom, shown in table 3.1.
Transduction Fitted γom(d=180 nm) Fitted η Fitted gom (d=180 nm)
relationship (MHz/nm) (m−1) (MHz/nm)
Laser detuning 1.61 ± 0.05 14 × 106∗ 8.25±0.9
Coupling distance 2.78 ± 1 (8.6 ± 0.5) × 106 5.49±1.5
Ref. [45] 0.4 12 to 14 6.7
Table 3.1: The numerically fitted values for the dispersive (gom) and dissipative (γom)
coupling parameters from fitting eq. 3.14 to figs. 3.5 a) & b). The laser detuning and the coupling distance is varied in a) and b) respectively. Also included is the parameter η which is the decay constant that determines the variation of gom, γom with d. The superscript
* indicates that this parameter is kept fixed during fitting. The bottom panel displays parameters measured in ref. [45].
The corresponding values of gom and γe obtained from fitting the coupling distance
dependence (fig. 3.5 a)), versus the laser detuning dependence (fig. 3.5 b)) agree relatively well with one another, although values obtained from the coupling distance graph have a larger error. This is related to the tendency of the taper to drift closer to the microsphere at d < 100 nm, possibly due to electrostatic forces. In comparison to values obtained by the authors of [45], gomagrees within error, but the value of γomobtained here is between 5 to
7 times larger. This is likely related to the inclusion of a scattering rate γsin the analysis of
[45], as well as geometric differences between the microsphere and tapered fibre. The ratio
γom
well with the ratio obtained using an alternative method in chapter 5 (section 5.4)4. By decreasing the coupling distance to maximise gom, γom, it is possible to obtain higher S/N
and coupling parameters comparable to those found for near-field evanescent coupling between a toroid WGM and nanostring i.e. γom=13 MHz/nm, gom = 290 MHz/nm is
measured in [84].
A clear message is that the transduction should be placed obtained near critical cou- pling and be red-detuned from the WGM to provide the highest S/N.
3.3.2 Calibration of Units
Although the Fourier transform provides information on the transduction S/N, further information such as mode temperature and the thermal fluctuations of the microsphere- cantilever can be extracted using the power spectral density (PSD), which describes the energy content per unit frequency. The units of the PSD are m2 Hz−1, and the square root of the background noise level in the PSD sets the transduction sensitivity.
The measured transmission of the WGM transduction beam is a voltage from the photodetector which must be converted to meters. This can be achieved by modulating the microsphere-cantilever at a frequency larger than its fundamental mechanical frequency (in fig. 3.6 a sinuisodial modulation of 3.3 kHz is used for a microsphere-cantilever with a 2.8 kHz c.o.m. frequency) so that the displacement of the cantilever follows that of the shaking frame, and not operated as an accelerometer. The peak amplitude of this shaking is defined by the magnitude of the voltage sent to the piezo-stack, which displaces d by 9.1 µm per 150 V.
Using the calibration curve in fig. 3.6, the specific scaling for this sample is (0.142 ± 0.023) µm V−1. This process is repeated over a range of driving frequencies and averaged each time the experiment is changed. The work conducted in this thesis has resolved displacements as small as 3 pm, verified by driving the piezo-stack with a small sinusoidal voltage.