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The output of the photodetector’s transimpedance amplifier is directed to one of two devices: an analog network analyzer (HP 4395A) and/or an FFT-based spectrum analyzer (Tektronix 5103A). The network analyzer (NA) is used for swept response (Section 4.3.1.1) and ringdown (Section 4.3.2) measurements. The spectrum analyzer (SA) is used for measuring thermal noise power spectra (Section 4.3.1.2). We will not review here the details of how network and spectrum analyzers work but rather provide an operational description.

4.5.4.1 Spectrum Analyzer Measurement

Operationally, the spectrum analyzer takes as its input a voltageV(t) and outputs a voltage power spectrum. The power spectrum is defined as the integral of the voltage power spectral density,

SV(Ω), over an effective bandwidthB (defined in non-angular frequency units):

hV2(Ω)iB≡

Z Ω+(2π)B/2

Ω−(2π)B/2

SV(Ω0)dΩ0/2π. (4.17)

The effective bandwidth of the spectrum analyzer can be defined relative to its output when supplied with a signal having a known, constant power spectral densitySV(Ω) =C0. In this special

case:

B=hV2(Ω)iB/SV(Ω). (4.18)

The SA is used to approximate power spectral densities by normalizing the measured power spectrum by its effective bandwidth. We denote this approximate power spectral density asS_VB(Ω):

SB_V(Ω)≡ hV2(Ω)iB/B≈SV(Ω). (4.19) This approximation is valid when the power spectral density of the underlying signal varies slowly on a scale set byB. If, on the other hand, the signalV(t) has all of its spectral content inside the

window Ω0±(2π)B/2, then the analyzer produces a value:

SBV(Ω0) =hV2i/B (4.20)

wherehV2_i=R∞

0 SV(Ω)dΩ/2π,is the rms

2 _{amplitude of the signal.}

We have verified that when operating the Tektronix 5103A in the “PSD units” mode, the ef- fective bandwidth is indeed given by Eq. 4.18. This was done by supplying the RSA input with calibrated white noise generated by a SRS DS345 function generator in “noise” mode. To obtain the effective bandwidth directly, we have the analyzer compute the power spectral density of a cali- brated sinusoidal input signalV(t) =V0sin(Ω0t), which gives the valueSVB(Ω0) =V02/(2B). A last

issue worth noting is that for noisy signals, a consistent outcome forSB

V(Ω0) is only obtained after

averaging numerous measurements; this averaging must take place in power units (a setting that must be specified — some analyzers by default average the logarithm of the power [56, 57]).

4.5.4.2 Network Analyzer Measurement

The HP 4395A network analyzer is based on a superheterodyne architecture. Internal to the network analyzer is a tunable local oscillator which produces a reference sinusoidal voltage (the “source signal”) at angular frequency Ωs, i.e.,Vs(t) =Vs0sin(Ωst). The source signalVs(t) is used to drive a piezo shaker coupled to the membrane. The signal produced by the etalon reflection photodetector’s transimpedance amplifier is V(t) = V0sin(Ωst+φ0). If the piezo is shaken near the resonance

frequency of the membrane (Ωs≈Ωm) and then abruptly shot off, the resulting “ringdown” signal is predicted to have the formV(t) =V0e−Ωmt/2Qmsin(Ωmt+φ0) according to Eq. 4.12. We denote the

slowly varying envelope at carrier frequency Ω byVΩ(t). For the ringdown,VΩm(t) =V0e

−Ωmt/2Qm_. The network analyzer is simultaneously used to measureVΩs(t), the slowly varying amplitude of

V(t) in the frame rotating at Ωs. It does this by mixing V(t) against a local oscillator VLO(t) =

V0

LOsin(ΩLOt) into the passband of a narrow bandpass (IF) filter centered at intermediate frequency Ωs−ΩLO, and then passing the output of this filter to an rms detector (digitally implemented by the HP4395A). IfV(t) =V0sin(Ωst+φ0), then the output of the rms detector is proportional to

VΩs(t) = V0. If V(t) = V0e

−Ωmt/2Qm_sin(Ω

mt+φ0), and if the bandwith of the IF filter (BIF) satisfies Ωm/Qm << 2πBIF << |Ωm−Ωs|, then the output of the rms detector for sufficiently short times (set by 2π/(Ωm−Ωs)) is proportional toVΩm(t)≈V0e

−Ωmt/2Qm_{. We use the network} analyzer in this single frequency (“zero span”) mode to perform ringdown measurements.

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spectrum analyzer noise floor photodetector noise floor laser intensity + shot noise thermal noise signal

predicted square membrane spectrum lorentzian fit) (

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Figure 4.4: Thermally induced vibrations of a membrane with dimensions {dm, wm} = {50 nm,500µm}. Lower plot: broadband thermal noise power spectrum obtained using an ef- fective bandwidth of B = 500 Hz >> Γm/2π. Units are in logarithmic power dissipated by the output of the photodetector transimpedance amplifier into 50 Ohm: 10Log[SB

V(Ω)/50Ohm/mW] dBm/Hz. Noise peaks are located at eigenfrequencies of a the square membrane resonator, with vertical lines representing the model Ωij = Ω11

p

(i2_+j2_{)/2. Diagonal modes are highlighted in}

pink. From the value of the fundamental frequency, Ω11/2π = 823.5 kHz, we infer a film tension

of T =935 MPa. Upper plot: a high-resolution measurement of the fundamental mode. For this measurement, an effective bandwidth ofB = 0.2 Hz is used and 3 scans are averaged. Becuase in this case B < Γm/2π, the shape of the power spectrum is proportional to the underlying thermal noise power spectral density, given by Eq. 4.10. A Lorentzian fit is shown, where the linewidth is constrained to the value obtained from a separate ringdown measurement, shown in Figure 4.6, for whichQm= Ωm/Γm= 1.4×106.