A low-noise transimpedance amplifier for the detection of “ Violin-Mode ” resonances in advanced Laser Interferometer Gravitational wave Observatory suspensions

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I. INTRODUCTION
2][3][4][5][6] These shadow-sensors-one per suspension fibre-each comprised an optical emitter and detector, bracketing the illuminated fibre.The emitter provided a collimated beam of illumination from a Near InfraRed (NIR: λ = 890 nm) multi-LED source, 7 and this cast a vertical shadow of the illuminated fibre onto its facing, photodiode-based, detector.The detector was configured to monitor, with extremely high precision, any lateral displacement of the fibre's shadow.Ultimately, each detector was in the form of a differential "synthesized splitphotodiode"-this having a negligible dead-band between its pair of sensing photodiode elements. 8The purpose of the full shadow-sensing detection system was first to monitor any lateral "Violin-Mode" (VM) resonances that might be excited on these highly tensioned silica fibres, at frequencies in the range 500 Hz-5 kHz-which spanned the gravitational wave detection bandwidth; 9 and, as a secondary task, to record any "large" amplitude "pendulum-mode" motion of the test-mass and its suspension fibres at frequencies of ∼0.6 Hz-such that all of this unwanted oscillatory motion, which might mimic or obscure the detection of gravitational waves, then could be suppressed by active cold-damping. 10,11 he shadow-sensing system was required to have an overall fibre-(i.e., shadow-) displacement sensitivity of 100 picometres (rms)/ √ Hz at 500 Hz (the fundamental VM resonance frequency for the suspension fibres), across a ±0.1 mm range of fibre position.
Initially, a transimpedance (photocurrent-to-voltage) amplifier was researched for use as the low-noise "front-end" electronics to a single photodiode-based shadow-sensor, with the shadow of the illuminated fibre falling over one vertically orientated edge of the rectangular sensor.In this way, a lateral vibration, or simple displacement, of the silica fibre's shadow altered the photocurrent flowing through the photodiode (PD).
This single-PD-input amplifier, whose salient features are described in Sec.II, was developed subsequently into a differential amplifier, which was itself interfaced to the split-PD-based shadow-sensor, mentioned above.Here, the fibre's shadow fell over the central (common vertical) edge of two adjacent rectangular elements in the detector.An additional, beneficial, side-effect of this combination of differential detector and amplifier was that proper shadow-alignment with the detector could be carried out more straightforwardly-by virtue of the natural pendulum-mode motion of the monitored fibre and its suspended test-mass.This high-performance differential amplifier is described in Sec.III.
It turned out that vibrational motion of a silica fibre and its attendant shadow at a frequency of 500 Hz, and with an amplitude of 100 pm (rms), equated to an AC photocurrent modulation at this frequency of approximately 4 picoamps (rms), flowing in the single PD detector, and flowing in anti-phase in the split-photodiode detector's two elements.It was conceivable, though, that "very large" Violin-Mode vibrations with amplitudes of up to 1 μm (rms) might be excited in the suspension fibres, and such signals clearly would have to be accommodated within the amplification chain as wellwith neither distortion, nor clipping.Moreover, pendulummode motion of a suspended test-mass, at ∼0.6 Hz, could certainly attain excursions of ±100 μm about the monitored fibre's quiescent position.In addition, each photodiode element's standing DC photocurrent, arising from steady illumination by its NIR source, was found to be ∼50 μA.Therefore, ostensibly, the detection amplifier would have to be designed in such a way as to handle a very wide range indeed of photocurrents.
However, it was appreciated that this range could be reduced significantly, with other attendant benefits as described below, if the DC and 0.6 Hz photocurrents were handled separately from the VM signals.

II. THE PROTOTYPE SINGLE-INPUT VIOLIN-MODE AMPLIFIER
Initially, a single-PD-input/dual output transimpedance amplifier was researched, such that the AC modulated VM photocurrent and the standing DC photocurrent-plus any very low frequency modulation-would produce separate AC and "DC" output voltages, these outputs both being proportional to their respective photocurrents, although via different resistive transduction paths.
As regards the amplifier's primary AC output, it was a requirement of the detection system that it should be able to monitor additionally a number of harmonics of the fundamental VM resonance, since these Eigenmodes also could be excited on the silica fibres-perhaps up to the tenth harmonic.
Therefore, the shadow-sensor's prototype transimpedance amplifier needed the following characteristics: 1. High gain over a VM (AC) bandwidth covering at least a frequency range of 500 Hz-5 kHz: a transimpedance gain of 120 M would generate an acceptable 19.2 V signal (peak-peak) at the maximum anticipated level of VM AC photocurrent.2.An ultra-low-noise level at the VM (AC) output over the frequency range specified in requirement 1, above.However, "noise gain peaking" was anticipated within the above bandwidth, 12 due to the large PD plus cable capacitance, and, clearly, this effect would have to be mitigated in the amplifier's design, at the outset.3. Very low VM (AC) signal gain at ∼0.6 Hz, so that pendulum-mode motion of the test-mass/suspension would not interfere with VM detection at frequencies of 500 Hz, and above.4. A separate DC output for (ultimately, each of) the PD element(s), with a composite shadow-displacement range capability of ±100 μm, say, which would be sensitive also to the anticipated ∼0.6 Hz pendulum-mode motion of a test-mass/mirror and its suspension fibres.Here, a transimpedance gain of just 120 k would give a quiescent output of magnitude 6 V at the anticipated level of DC standing photocurrent; and then, a pendulum-mode modulation at ∼0.6 Hz, and with an amplitude of 4 μA, would superimpose an easily discernible ±0.48 V excursion about this quiescent value.5.In addition to the foregoing requirements, it was required that any overload of the VM (AC) amplifier should occur with essentially symmetrical clipping at the amplifier's output, such that a (cold-) damping system still could apply (e.g., electrostatic) damping forces with the correct phase.

A. The single-PD transimpedance amplifier design
A transimpedance amplifier with a single-PD input was designed to address the requirements listed 1-5 above, and its circuit diagram is shown in Figure 1(b), with a subsequent modification that was made to it shown in Figure 1(a).
Circuits similar to that shown in Figure 1(b)-if taken from the PD detector up to the node bearing the label V'have been described previously for the purposes of "rejecting ambient light," 13 or for the "control of ambient light." 14In this work, however, the op-amp based circuit shown in the figure does not discard the DC photocurrent, but rather uses it in order to create a separate "DC" output channel-to be employed, ultimately, (i) for aligning the shadow onto a dual-PD sensor, (ii) for calibrating the DC, and hence AC, responsivities to fibre-(shadow-) displacement, and (iii) for sensing the pendulum-mode motion mentioned above-also employed in (i).
In Figure 1, I Photo is the full (DC + AC) photocurrent generated in the PD by the illuminating source, with the fibre's shadow falling over one edge of the rectangular PD.In the following discussion this current has been decomposed explicitly into its steady-state and time-varying (VM) parts, such that I Photo = I DC + i.
In Figure 1(b) the "debout" (i.e., non-inverting) integrator feedback around IC3, involving IC2 and its two resistors R, and capacitors, C, forces the mean, steady-state, DC output of IC3 always to be close to zero volts.A zero-average input to the integrator can occur only when no DC photocurrent flows through the resistor, R f , and under these conditions the output of IC2, labelled V DC in the figure, must be frozen at whatever happened to be its existing-negative-value at the time the "zero volts average" condition was achieved at the output of IC3.Accordingly, the full DC photocurrent, I DC , must flow from the anode (A) of the PD, which is a virtual Earth point (at the inverting input to IC3), down through resistor R i , and into the output of IC2, which acts as a DC current-sink.Thus, V DC = −I DC × R i , as indicated in the figure: the DC transimpedance gain of the amplifier is therefore 120 k .
On the other hand, none of the AC photocurrent i can flow through resistor R i , because the output of the sluggish integrator IC2 cannot follow this HF signal so as to produce a suitable driving voltage for it across resistor R i -due the long RC time-constant of integration.However, the output of transimpedance op-amp IC3 is free to respond to photocurrent i, by taking up an AC voltage at its output given by V' Indeed, for frequencies f < 13.2 kHz, any AC photocurrent i must flow essentially through the transimpedance resistor R f , alone.Thus, the AC Violin-Mode transimpedance gain of the amplifier-to the node labelled V' -is 1.2 M .
In sum: the non-inverting integrator feedback forces the DC and "Violin-Mode" AC photocurrents to follow the different paths indicated in the figure .Furthermore, in Figure 1(b) the VM signal V' is AC coupled by the high-pass filter C 2 -R 2 , for frequencies above ∼160 Hz, to the input of a voltage amplifier having a gain of A = 101 (for this single-PD amplifier; and A = 100 for the differential amplifier), before being additionally band-limited in its response on the high-frequency side by the low-pass filter R 3 -C 3 , which has a single-pole roll-off above 12.5 kHz.
The output of this filter was the final "Violin-Mode" output voltage of the amplifier, V AC , as indicated in Figure 1.From V' onwards the mean value of this AC voltage was zero, so that saturation/clipping of large signals were symmetrical.

B. The transimpedance amplifier: Reducing input capacitance
The Hamamatsu S2551 silicon photodiode 15 was used in both the single and differential shadow-sensors, this having an effective detection area measuring 1.2 mm × 29.1 mm.Because of its large area (∼35 mm 2 ), this PD had a relatively large capacitance: 350 pF, at zero bias.However, as a consequence of an additional capacitance of 600 pF, or more, in parallel with the PD, due to a necessary coaxial cable run of ∼6 m between the photodiode(s) and the amplifier, a total input capacitance of C in ∼ 1000 pF was anticipated.There-fore, the necessarily small value of feedback capacitance C f (=10 pF) meant that this was, unfortunately, an ideal configuration for "noise gain peaking"-where the input voltage noise of op-amp IC3 is effectively multiplied-up by a factor of 1 + C in /C f (∼100, here). 12his undesirable effect was mitigated first by choosing a low-noise op-amp for IC3 (initially an AD743: 5.0 nV/ √ Hz, max., at 1 kHz; and later, because of its much smaller SMD footprint, an OPA627AU: 5.6 nV/ √ Hz, typ., at 1 kHz); and second by adding the circuitry shown in Figure 1(a), which followed a scheme described in Ref. 12.The prototype amplifier's original input circuit, shown in Figure 1(b), therefore was modified by the addition of the very low-noise JFET transistor, J 1 (BF862: ∼ 1 nV/ √ Hz), connected as a sourcefollower, along with its simple biasing circuitry.This transistor "bootstrapped" the signal from the photodiode's Anode (A)-such that any signal present at this "virtual-earth" point (arising from IC3's falling open-loop gain with rising frequency) was transferred to the photodiode's Cathode (K), so that very little signal voltage now appeared across C in .Consequently, a correspondingly reduced signal current now would be able to flow through the capacitance of the photodiode and its connecting cable-making them appear very much reduced in capacitance, and so greatly diminishing any noise gain peaking.The presence of transistor J 1 also provided a small but useful reverse bias to the photodiode (∼0.5 V), which slightly decreased its capacitance.
The addition of the transistor J 1 altered neither the DC nor the AC signal behaviour of the single-input amplifier, and so, from Figure 1(b), its "DC " output signal can be expressed straightforwardly in terms of the complex frequency s [s = jω, j ≡ √ (−1)], by a transimpedance relationship of the form, Clearly, for a truly DC component of photocurrent (s = 0) Eq. ( 1) reduces to the simple expression V DC = −I DC R i , as derived above.For the component values given in Figure 1(b), the expression in Eq. ( 1) is effectively that of a low-pass response, with a dominant pole at 161 Hz (and a second, HF, pole at 13.1 kHz): therefore, 0.6 Hz "pendulum-mode" signals were passed without attenuation to the amplifier's DC output.
In a similar fashion, the ratio of the amplifier's AC response to a VM signal photocurrent, to its DC response to a steady quiescent photocurrent (a convenient ratio) can be found to be . ( Here: In practice, the theoretical mid-band (∼1.48 kHz) AC/DC gain ratio given by Eq. ( 2) did not quite peak at the anticipated value of 101 × 10 = 1010 but was slightly reduced by the sharply circumscribed (four poles/two zeroes) pass-band of the amplifier, as discussed below.

C. Prototype amplifier: Practical NIR gain calibration
A calibration system was built to verify that Eq. (2) properly described both the AC and DC behaviours of this amplifier.This system involved irradiating the amplifier's single photodiode detector with a very low intensity beam derived from an OD50L NIR LED, using a pin-hole aperture, the beam consisting of a steady component of fixed intensity, plus a small sinusoidal intensity modulation on top of this at a known frequency, f. 17 The results of such measurements of the AC/DC gain ratio as a function of the frequency f agreed with the theoretical expression given by Eq. ( 2) at a ±1% level up to ∼100 kHz, i.e., well above the required VM bandwidth of 5 kHz.Consequently, as mentioned above, a bench measurement of the shadow-sensor's quasi-static responsivity to shadow-(i.e., suspension fibre-) displacement (hereinafter called the "DC responsivity"), allowed the corresponding AC responsivity to be inferred with a high degree of confidence from the measured AC/DC gain ratio at any given VM frequency, f.This was the technique used to calibrate the amplifier's AC responsivity as a function of f. 16,17 In practice the gain was found to be 1000 mid-band at 1.48 kHz and 990 at 1 kHz (and 976 ± 4 mid-band for the differential amplifier, described below, at 1.48 kHz).

III. THE DIFFERENTIAL VIOLIN-MODE AMPLIFIER
Following the successful performance of the prototype amplifier, connected to a single rectangular photodiode sensor, it was clear that a differential sensor would improve the signal-to-noise ratio by a factor of √ 2. Two closely adjacent Hamamatsu S2551 photodiode elements, orientated so that the fibre's shadow fell along their common vertical edge, and sensing differentially, could in principle double the size of the available AC photocurrent signal, whilst offering a degree of common-mode rejection, as well.

A. The differential amplifier: From concept to construction
The outline design of the Transimpedance Amplifiers block is shown in Figure 2. It is essentially a dual version of the prototype amplifier, but with an intermediate, differential, AC amplification stage.For the photodiode element PDa in Figure 2, I a represents the "DC " value of its photocurrent, inclusive of any pendulum-mode modulation, whilst i a represents any HF modulation of this photocurrent due to transverse VM oscillation of its monitored fibre (and shadow).Similar nomenclature has been used for element PDb.Note that the modulation photocurrents i a and i b flowed automatically in anti-phase for the "synthesized split-photodiode" detector: if the silica-fibre's shadow moved over the common edge of elements PDa and PDb, so as to move off PDa (by any FIG.2. Schematic diagram of the Violin-Mode shadow-sensor connected to its "Transimpedance Amplifiers" block (differential amplifier), with the "synthesized split-photodiode-" based sensor being shown in plan view.The mirrored prism split the collimated incident NIR illumination so that it, together with a variable fraction of the fibre's shadow, fell onto each photodiode sensor element-as if the separate photodiodes formed a single split-photodiode detector, with negligible dead-band between the two elements.Here, the full photocurrents flowing in photodiode elements PDa and PDb have been designated, respectively, i PDa = I a + i a and i PDb = I b + i b (refer to the text).
This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 130.159.82.88 On: Thu, 15 Jan 2015 15:11:31 amount) and onto PDb, then the illumination of PDa, and thereby its photocurrent, i PDa , increased; but as the shadow moved off element PDa it moved automatically onto element PDb, thereby decreasing i PDb by closely the same amount by which i PDa was increased. 16The amplifier's two "DC " voltage outputs, labelled VDC,a and VDC,b in the diagram, were derived from the DC photocurrents I a and I b , respectively, via a transimpedance gain of R = 120 k .Similarly, the intermediate AC signal voltages VDC,a and VDC,b were derived from i a and i b , respectively, but in this case via a ten-timeslarger transimpedance gain of R = 1.2 M .These two transimpedance signals were then differenced, and their resulting differential voltage was amplified further by a factor of ×100, so as to produce the final Violin-Mode output voltage VM AC.
In this way, and measured from an individual photodiode element to its own respective pair of outputs, the overall AC transimpedance gain was measured to be (976 ± 4) × the DC gain (mid-band), or approximately 1% lower than that of the single-input amplifier, as expected, given the slightly lower value of the AC gain stage in this amplifier.
The full circuit diagram of the differential amplifier is shown in Figure 3, and the realization of two such amplifiers on a pair of small Printed Circuit Boards, using surface-mount components, is shown in Figure 4.It is noteworthy that an earlier version of the differential amplifier, which was built using through-hole mounted components, gave an equally good noise performance in the pass-band, but did not conform quite so closely to the theoretical high frequency performance as did the surface-mount amplifiers-as exemplified by Figure 5. Four differential VM amplifiers and shadow-sensors were built, one per suspension fibre of a full test-mass mirror suspension.

B. The differential amplifier: Signal and noise performance
A typical noise Power Spectral Density (PSD) of the differential amplifier, measured at its VM AC output, is shown in Figure 5.The −3 dB bandwidth was found from a fit of Eq. ( 2) to these noise data to lie between 226 Hz and 8.93 kHz.The theoretical pass-band is indicated by the labelled solid (blue) line in the figure and it is seen to be a very good fit to the measured noise PSD for frequencies above ∼50 Hz.Moreover, the peak, mid-band, noise PSD was found to lie very close indeed to the fundamental shot-noise limit, as indicated in the figure by the horizontal (red) line.Here, this limit was calculated from the two measured DC photocurrent values by assuming that the shot noise arising from the two photodiode elements was uncorrelated.The roll-off of the amplifier's gain towards both lower and higher frequencies was at a rate of −40 dB/decade.Given the measured mid-band noise PSD of −63.2 dBV(rms)/ √ Hz (≡ 692 μV(rms)/ √ Hz), the broad-band rms noise voltage was expected to be 68 mV (rms)-very close to the measured value, as seen in Figure 6.Note that the split supply rails for the amplifiers both benefited from "noise finessing" filters, in order to reduce voltage regulator noise. 18n actual VM signal resulting from an approximately 1 μm peak-peak acoustically induced fibre resonance, at a frequency of 1019.1 Hz, is also shown in Figure 6.Here, a single capture of the VM AC signal is seen to be approximately 10 V, peak-peak.Twenty three volts peak-peak (approx.

C. The differential amplifier: DC and AC responsivities
In practice, quasi-DC pendulum-mode signals were detected using the DC outputs of these amplifiers, and signals with amplitudes of up to ±140 μm, peak-peak, at 0.64 Hz, were detected. 9These same DC outputs were used also for  2), is indicated by the blue line.Here, the fibre's shadow fell over neither detector element, in order to avoid VM and other resonant peaks, and so the noise PSD is higher in this figure at 500 Hz, say, than when the shadow fell over the adjacent edge of both detector elements-as it must for proper VM detection.The reduction in noise PSD with the shadow present was approximately 0.8 dB, in fact.The noise reduction was due entirely to the lower levels of DC photocurrent, and so shot noise, in the two partially shaded PD elements. 7is article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:  displacement calibration, and in bench tests the DC responsivity of each photodiode detector + amplifier was measured by translating laterally a test fibre, mounted vertically on a motorised stage, through the NIR beam of each shadow sensor.In this way, the DC responsivity was found. 16Taken over the four detectors and their respective amplifiers, it was measured to be 10.44 kV m −1 of fibre displacement.From this value, the AC responsivity was deduced to be (9.43 ± 1.20) MV(rms) m −1 (rms), when taken over the four detectors/amplifiers, since the AC/DC gain ratio was measured to be 904 at this frequency. [9]

IV. CONCLUSIONS
The noise level measured for the differential amplifier was little different from the fundamental limit expected from (uncorrelated) shot noise in its two photodiode detector elements.Indeed, the "white noise" region of the differential amplifier's Power Spectral Density followed its theoretical passband very closely over the frequency range investigated, as seen in Figure 5, only deviating above this form for frequencies below ∼50 Hz, where 1/f noise began to dominate.Both versions of the amplifier exhibited close-to the theoretically expected signal gain, with the AC/DC gain ratio mid-band (1.48 kHz) being very close to the expected value of approximately 1000.Moreover, the signal gain at the "pendulum-mode" frequency of ∼0.6 Hz was, by a very useful factor of >70 000 (97 dB), smaller than that found mid-bandallowing Violin-Mode signals in the range 500 Hz-5 kHz to be detected, even in the presence of very much larger pendulum motion of the test mass and its supporting fibres.The differential amplifier's AC pass-band extended from 226 Hz to 8.93 kHz (−3 dB), and its very low noise level allowed it, in principle, to recover a 500 Hz sinusoidal Violin-Mode vibration of amplitude (69 ± 13) picometres (rms), in one second.Yet, this amplifier output could handle, without clipping or distortion, a similar signal of amplitude 2.4 μm (or ∼23 V), peak-peak.The subsidiary "DC " outputs had a designed (DC) responsivity to quasi-static shadow displacement that was lower than the mid-band AC responsivity by a factor of approximately 1000.These outputs therefore exhibited a much larger detection range for shadow displacement, with a bandwidth extending from true DC -160 Hz (−3 dB point).This allowed pendulum-mode signals at 0.64 Hz, and of amplitude exceeding 200 μm peak-peak, to be captured without attenuation.In summary, the differential Violin-Mode amplifier described above clearly met, or exceeded, all of its performance targets.
At the time of writing the Violin-Mode sensor system mentioned here has not been adopted for aLIGO, and, indeed, the need for VM damping has not yet been demonstrated.However, if it is found to be required, the current baseline solution is to use aLIGO's Arm Length Stabilisation system as a VM sensor. 19In fact, the issue of vacuum compatibility remains unresolved for the VM sensor mentioned here, because the Hamamatsu photodiodes used for the detector elements had been encapsulated, using an unknown epoxy.However, were it to become necessary, the issue of the epoxy for the photodiodes from this, or another, manufacturer probably could be resolved, and the LEDs and other components used are likely to prove vacuum compliant, or have This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 130.159.82.88 On: Thu, 15 Jan 2015 15:11:31 vacuum-compliant alternatives.Nevertheless, the VM amplifier described here may find other applications.

114705- 3 N
FIG.1.Schematic of the single-input amplifier.(a) A subsequent modification made to the amplifier shown in (b).This involved "bootstrapping" any signal present at the (large area) photodiode's Anode (A) to its Cathode (K), via the source-follower J 1 .This prevented any signal current from flowing through the capacitance of the photodiode and its connecting cable: refer to the text.Here, a small reverse bias V KK ∼ 0.5 V appeared across the photodiode.(b) "Debout," or non-inverting, integrator feedback via IC2 (together with the two resistors labelled R and the two capacitors labelled C), forced the mean output voltage V of IC3 always to be close to zero volts, causing the DC and "Violin-Mode" AC components of the photocurrent, I DC and i, respectively, to follow the different paths indicated in the figure (refer to the text).The voltage gain A = 101.
120 k , C = 100 nF, R = 100 k , C 2 = 100 nF, R 2 = 10 k , C 3 = 4.7 nF, and R 3 = 2.7 k .Equation (2) effectively expresses the relative sensitivity of the amplifier's AC and DC outputs to changes in photocurrent due to the same displacement of a fibre's shadow across (the edge of) the PD detector: in the AC case a change occurring at a frequency s (= jω) for a sinusoidal modulation at angular frequency ω; and in the DC case a change occurring from one steady DC value to another, as the fibre's shadow made a quasi-static displacement.Practically, this relationship was used to calibrate the sensitivity to VM displacement of the fibre, at any given frequency f (=ω /2π ), relative to the-straightforwardly measurablerate of change of DC output voltage V DC with silica fibre (shadow) position.Refer to Ref. 16.

FIG. 3 .
FIG. 3. Circuit diagram of the Violin-Mode differential shadow-sensing transimpedance amplifier sensor (with photodiode sensing elements PDa and PDb).Three complementary outputs were made available for driving long cables, if necessary (refer to the text).This also allowed positive versions of the two DC outputs, VDC, a and VDC, b, to be used.The INA103 instrumentation amplifier is a 1 nV/ √ Hz device.Four such amplifiers were constructed-one per silica fibre.Two of these amplifiers are shown, as constructed, in Figure 4.For coaxial output cables > 2.5 m in length, 600 resistors were added directly in series with the six outputs, to prevent oscillation.This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 130.159.82.88 On: Thu, 15 Jan 2015 15:11:31

FIG. 5 .
FIG. 5. Noise Power Spectral Density of the differential amplifier as a function of frequency, measured at the VM AC output of the amplifier.The fundamental mid-band shot noise limit is marked by the horizontal line in the figure, at −63.2 dBVrms/ √ Hz.The theoretical amplifier Passband, as described by Eq. (2), is indicated by the blue line.Here, the fibre's shadow fell over neither detector element, in order to avoid VM and other resonant peaks, and so the noise PSD is higher in this figure at 500 Hz, say, than when the shadow fell over the adjacent edge of both detector elements-as it must for proper VM detection.The reduction in noise PSD with the shadow present was approximately 0.8 dB, in fact.The noise reduction was due entirely to the lower levels of DC photocurrent, and so shot noise, in the two partially shaded PD elements.7 130.159.82.88 On: Thu, 15 Jan 2015 15:11:31

FIG. 6 .
FIG. 6. (Left) The differential amplifier's VM AC output, showing an acoustic excitation of the fundamental Violin-Mode of a 95 mm long silica fibre sample, 0.4 mm in diameter, using a distant loudspeaker.A single capture of the resulting Violin-Mode waveform is shown, at approximately 10 V, peak-peak, and so the amplitude of the test fibre's transverse Violin-Mode motion was approximately 1 μm, peak-peak.The resonant frequency was found to be 1.0191 kHz.(Right) Photo of the Tektronix TDS-3052B oscilloscope screen, showing the amplifier's typical broadband output noise level to be ∼68 mV (rms) at the amplifier's VM AC output-in agreement with theory for this wide frequency band.The horizontal bars have been set approximately at the ±2σ levels.