4.7 Mechanical vibration detection
4.7.1 Standing wave sensing system
We use the same experiment system to measure the mechanical resonance of the nanofiber.
The experiment setup is shown in Fig.4.7. The position of the dichroic mirror is set to have the gold nanosphere in the sensitive region. As we introduced before, our sensing system can record the displacement of the nanofiber with frequency higher than kHz.
Although we have the information of the fiber displacement, there is other informa-tion hidden in the signal’s frequency content, for example, the mechanical resonance of the nanofiber triggered by the vibration in the environment. In order to extract fre-quency information from information of fiber displacement, the Fourier Transform (FT) and Wavelet Transform (WT) are used for transforming mathematically our view of the signal from time-based to frequency-based. By FT, a raw time domain signal is broken down into constituent sinusoids of different frequencies. Consequently the frequency-amplitude representation obtained by FT presents a frequency component for each fre-quency that exists in the signal. The serious drawback of FT is that the information in time domain is lost in transforming to the information in frequency. So FT is appropriate for the situation when the frequency does not change a lot over time. However, for the spectrum characteristics of a time varying signal, for example, if we want to distinguish the resonance spectrum of the nanofiber when it’s pushed by a radiation force, FT will not be a perfect choice. The frequency generated at certain time will be easily covered by the background noise in larger time scale during the whole data acquisition.
In Fig.4.9, we show the Fourier transform of the signal acquired within 10 seconds with integration time τint =100 µs when there is only the background noise. The resonance frequency shown in this figure comes from the background mechanical vibration in the lab.
Therefore, it would be better to analyze the time localized signal with spectrum that also vary with time. We introduce another mathematics approach that can transform
4.7. MECHANICAL VIBRATION DETECTION 83
Figure 4.9: Fourier transform of the signal acquired within 10 seconds with integration time τint =100 µm. Inset: the Fourier transform in log scale.
signal from time-based to frequency-based while keep the information in time domain, which is the Wavelet Transform (WT).
A wavelet is a mathematical function that is used to divide a given function or continuous-time signal into different scale components [79]. Wavelet transform provides a time-frequency window that can be modulated. The width of the window changes with the frequency. When the frequency increases, the width of the time window becomes nar-rower to improve the resolution. The average value of the amplitude of the wavelet in the entire time range is 0, and it has limited duration and sudden frequency and amplitude.
We use the continuous wavelet analysis to deal with the 1-D signals we recorded.
We show in Fig.4.10 the WT of the signal acquired within 10 seconds with integration timeτint =100 µm, which is using the same set of data as the Fourier transform in Fig.4.9.
With wavelet transform approach, we manage to extract the spectrum of the signal over time. The maximum distinguishable frequency we have is 4 kHz, which is limited by the time resolution of photon number detection system. In principal it can be further improved by using a Time-Correlated Single Photon Counting (TCSPC) system with picosecond resolution.
The resonance frequency of the string produce wave is decided by the length and tension of the string. Similar effect as stringed instrument, the increase of tension on optical nanofiber increase its resonance frequency. To find the resonance frequency of the nanofiber, we increase continuously the tension applied on the nanofiber at 1 second intervals to trace the frequency shift. The tension was added to the optical nanofiber by the bending piezo on one side of the fiber holder, introduced in previous sections. Based on Mersenne’s laws, the relation between the resonance frequency of the string can be given by the tension T, the linear density of the stringµ and the length of the string L:
f = 1 2L
s T
µ. (4.11)
84 NANOFIBER SENSOR
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Figure 4.10: Wavelet transform of the signal acquired within 10 seconds with integration time τint =100 µs.
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Figure 4.11: Wavelet transform of the signal acquired within 10 seconds with integration time τint =100µs. Each 1 second apart, we increase the tension applied on the nanofiber.
4.7. MECHANICAL VIBRATION DETECTION 85 The WT of the signal is acquired within 10 seconds with integration time τint =100 µs, as shown in Fig.4.11. The added tension causes the resonance to increase because it takes a greater wave speed to move the wire. Therefore, we expect to see the shift of frequency towards higher frequency, and this can give us the information of fiber resonance frequency as well. However, in Fig.4.11, the frequency excited by added tension seems to be stable. The obtained frequency when we add tension is 400 Hz, 800 Hz, 1.2 kHz, 1.6 kHz, 2 kHz, 2.4 kHz, appears to be a harmonic spectrum. Since the distinguishable frequency of our system is limited, and the resonance frequency of a nanofiber can be higher, the resonance frequency might be out of our detection range.
We didn’t find the resonance frequency of the nanofiber. It might be due to the limit of the detection range or the sensitivity of this system is not enough for the radiation force we applied. Based on the suggestion by Arno Rauschenbeutel, homodyne detection system is a better choice for detecting the frequency of the fiber deflection. It measures the phase changing due to the birefringence caused by the bending of optical nanofiber, which gives higher sensitivity than the displacement system. Therefore, in the next section, we introduce our attempt to add a homodyne detection system on the original displacement system to have better resolution in vibration sensing.