(Color online) A typical shadow sensor implementation using a split photodiode. The response is linear until a shadow edge crosses any of the eight corners of the two photocells. In the implementation shown, the linear range has been increased by rotating the photocells with respect to the fiber. The rotation also ensures that the shadow from thin fibers never falls entirely within the gap between the photocells.
(Color online) Schematic of the experiment. The two polarizers in combination with the polarizing beamsplitter allowed the intensity of the two beams to be varied independently. The wave plate, in combination with the beamsplitter acted as a rotator to prevent light reflected off the fiber from reentering the laser cavity. Lens 1 focused the beam at the edge of the fiber (seen here in cross section as a grey disk) and Lens 2 recollimated the beam. The signals from the two photodetectors were used to feed back to the PZT actuator which caused the lens to follow the fiber at low frequency. Our signal was the error signal of the compensation filter. (See Fig. 4.)
(Color online) (a) Diagram of the silica fiber suspension. The Viton pads provide some acoustic isolation and damping. The leaf spring (in this case a MiniMGAS filter14) provided vertical vibration isolation above . The fiber being sensed is shown with the laser beam focused on it. It was the thinnest of all the fibers in the suspension with a diameter of and length 15.7 cm. The other fibers and the intermediate masses were necessary for vibration isolation. Note that most of the suspension is monolithically constructed from fused silica to prevent rubbing friction. The fused silica was attached to the metal upper part of the suspension by fashioning the large aluminum bob as a collet into which the topmost fused silica rod piece is clamped. The weights provided 518 MPa of tension, equivalent to 54% of the breaking strength of the fiber under test. (b) The tungsten fiber suspension. The fiber being sensed was 27.4 cm long and had a diameter of . The fiber was clamped by the two dumbbells, the bases of which were split along a diameter and then bolted back together to form plate clamps. The upper two fibers (for vibration isolation) are steel fibers. By moving weights from the variable isolation mass to the tensioning mass, the tension on the fiber under test could be changed while keeping the entire suspended mass constant. This was necessary to keep the leaf spring properly loaded.
Simplified block diagram of the signal flow. The actuator moves the focusing lens laterally at low frequency (Fig. 2) so that the focused beam follows the fiber motion. This keeps the detector close to its linear operating point. The two poles of roll the transfer function off above about 1 Hz, and all the other transfer functions are simple gains in the active band of the servo. The input marked “laser noise” corresponds to the noise on the light in the signal beam where it hits the signal photodetector, and includes shot noise. The fact that the laser noise is input after the detector is an approximation. In reality, the amount of noise from the laser that is registered by the photodiode depends on the amount of light passed by the fiber, in other words it depends on the fiber position . However, since the fractional variation of the intensity of light passed by the fiber was normally very small, the dependence of the laser noise on the fiber motion could be ignored. Therefore, near the operating point, we can to a good approximation add the laser noise linearly. The reference signal contains the intensity fluctuations of the reference beam picked off at the beam splitter. The output signal (or just “the signal”) is identical to the error signal except for the gain of the ac-coupled amplifier .
(Color online) A typical amplitude spectral density of the signal for the tungsten fiber loaded with 26.2 g. The peaks marked with arrows are violin modes of the fiber. The white noise level (indicated by the blue horizontal line through the data) is consistent with predictions at , which is 30% above shot noise. This corresponds to a sensitivity floor of . The noise is expected to rise at frequencies below a few hundred hertz, due to laser noise. The broad peak between about 750 Hz and 1.5 kHz may be due to parasitic interference.(, 50 averages, Hanning window.)
(Color online) Histogram of the first 100 s of a typical set of raw data. The ordinate axis is logarithmic which makes a Gaussian distribution appear as an inverted parabola. In this view, any outliers become obvious. However, none are apparent.
(Color online) Fits to the resonance peaks of modes 3 through 14, assuming white background noise. The vertical axis represents the power spectral density and the horizontal axis gives the frequency [Hz]. In the fits, the resonant frequency was fixed while the peak height and loss angle were allowed to vary. If the peak is expressed as , where is the angular resonance frequency, and is the angular frequency, then the area under the peak is approximately . Some of the fits are slightly distorted by a nonwhite background, or by a nearby peak due to the near-degenerate orthogonal mode of oscillation. However, the fits are shown on a semilog scale, so the distortions evident at or near the level of the background noise are in fact very small.
(Color online) The rms amplitude of the fiber modes compared with the prediction from Eq. (10). The square root of the area under the mode peaks shown in Fig. 7 gives the rms displacement of the fiber at the point where it is interrogated by the focused beam. The area under each mode peak was estimated from the fits. In order to get an actual mode amplitude from the integrated peaks, we had to correct for the fact that the detector is more sensitive to some modes than others depending on the placement of the beam on the fiber relative to the modal nodes.
(Color online) Simulated data sampled at 10 kHz, consisting of a 1000 Hz sinusoidal signal (lighter, green, inner trace) and Gaussian noise. The sum of the two is shown by the darker, blue, outer trace.
(Color online) Quadratures of demodulation at 1 kHz. (Simulated data.)
(Color online) Energy innovation, . (Simulated data.)
(Color online) Histogram of energy innovation. (Simulated data.)
(Color online) Average energy innovation vs averaging time for two data sets corresponding to modes of a loaded fused silica fiber (031602A1) and a loaded tungsten fiber (030602). Also shown is data constructed at a point away from any modes of the tungsten fiber.
(Color online) The upper window shows the histogram of the energy innovation for the tungsten fiber mode at 2864.30 Hz. The fiber is loaded at 306 MPa or 94% of its breaking strength. The lower window shows the probability of getting at least one event having energy innovation equal to or greater than indicated on the abscissa, based on the assumption that all events are part of the main distribution. This probability is what we call the probability of an event being consistent with the main distribution. In this case, no events had less than 20% probability of being consistent with the main distribution.
(Color online) Histogram of the energy innovation for the silica fiber mode at 3271.75 Hz. The fiber is loaded at 518 MPa or 54% of its breaking strength. One event had less than 20% probability of being consistent with the main distribution, thus qualifying as an outlier. However, there was no corresponding event in the other mode being monitored at 4620.13 Hz. Thus, the outlier was most likely not due to a fiber event.
(Color online) Cumulative histograms for the energy (not energy innovation) of the lower mode of the tungsten fiber (at 94% of breaking load) and the fused silica fiber. The ordinate axis shows the rate at which events exceed the corresponding level on the abscissa. The scale is determined by the number of points in the data, which is determined by the length of the data set and by the spectral bandwidth over which the energy is estimated. Here, the bandwidth is the demodulation bandwidth, 1.18 Hz for the fused silica fiber and 10 Hz for the tungsten fiber.
Datasets acquired fall into four categories.
Salient features of the data sets. The description of the data groups are given in Table I. In the mode number column, a “-” indicates a frequency at which no mode is present. The average energy is calculated by integrating the appropriate region of the power spectrum and subtracting the background. The number of candidate events is the number of datapoints in the energy innovation time series of sufficiently high magnitude that each had less than a 20% chance of being consistent with the main distribution. The candidate events do not qualify as fiber events unless they are coincident in both modes examined and anticoincident with the noise. As the final column shows, this was never found to be the case.
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