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A low-noise transducer based on a fiber Fabry-Perot (FFP) cavity was used as a pickup for an acoustic guitar. A distributed feedback (DFB) laser was locked to a 25 MHz-wide resonance of the FFP cavity using the Pound-Drever-Hall method. The correction signal was used as the audio output and was preamplified and sampled at up to 96 kHz. The pickup system is largely immune against optical noise sources, exhibits a flat frequency response from the infrasound region to about 25 kHz, and has a distortion-free audio output range of about 50 dB.
©2011 Optical Society of America
1. Introduction
Vibration and strain sensors based on fiber Bragg gratings (FBGs) exploit that the FBG reflection spectrum is narrow and shifts as strain is applied to the optical waveguide [1]. Consequently, FBGs have found many applications in e.g., structural health monitoring [2–4]. FBGs can also act as chemically inert photoacoustic transducers and have demonstrated an audio frequency response that can exceed 20 MHz, while retaining fair detection sensitivity even in capillary flow systems [5, 6].
Similarly, in-fiber Fabry-Perot (FFP) cavities made from two identical FBGs can be used for vibration and strain measurements, since the resonances in the cavity spectrum depend on the cavity length and thereby, again, on the strain applied to the waveguide [1].
Both types of fiber strain sensors have been applied recently as transducers for musical instruments [7, 8]. By detection and audio sampling of the reflected light from either an FBG or a low finesse in-fiber Fabry-Perot cavity our group has realized compact and very lightweight transducers (“pickups”) for guitars, violins, harmonicas and other musical instruments, in which the sound is generated to a large extent by the vibration of a soundboard. The subjective quality of the recorded music was comparable to that of commercial piezoelectric transducers, and the frequency range was found to encompass the entire audible region of the audio spectrum i.e. from about 20 Hz to 20 kHz. Fiber-optic vibration sensors have the additional advantage that they are very lightweight and a large number of sensors can be affixed to an instrument without changing its acoustic properties. Also, we found that they can be easily attached and removed from vintage instruments such as violins and guitars. Commercial piezo-pickups need to be fixed with strong glue for the best acoustic transduction.
On the other hand it was apparent that audio recordings using fiber optic pickups – and, indeed, all strain and vibration measurements that are based on light intensity measurements – are susceptible to optical noise arising from a variety of hard-to-control sources, such as laser emission and detector response as well as absorption losses at mechanically disturbed fiber/fiber couplers. In addition, birefringence induced by movement of the waveguide combined with the polarization sensitivity of fiber circulators and couplers was the cause for audible distortions. The optical noise is further compounded by electronic noise from the amplification of the detector signal as well as the analog to digital conversion of the electronic signal.
A second and equally serious shortcoming of strain detection based on intensity measurements from mechanically-stressed FBG or FFP transducers lies in the limited dynamic range of the vibration amplitude measurement. The sensitivity of the vibration measurement is directly related to the slope of the reflection spectra of the FBG or FFP cavity at the, presumably constant, laser wavelength. The spectral wavelength region of constant slope typically spans only a few tens or hundreds of picometers, and any vibration that causes the spectrum to shift out of this region will produce a distorted acoustic response or even “clipping” of the signal.
Finally, thermal drifts in the fiber optic cable as well as low frequency fluctuations in the laser source may require occasional resetting of the laser wavelength to the mid-reflection point of an FBG or FFP cavity fringe.
In this report we show that all three problems can be overcome using an interrogation method that locks the frequency of the probe laser tightly to the audio-modulated reflection feature. The audio and strain information is then contained in the correction signal that is generated to keep the laser locked. Previously, the Pound-Drever-Hall (PDH) method of laser frequency locking [9] has already been demonstrated to show remarkably low noise when used in frequency measurements of strain and vibration [10–13]. Since the strain information is contained in the frequency response of the transducer and is largely independent of the reflected light intensity, a guitar pickup based on a PDH method is expected to be immune to most optical noise sources. Also, infrasound variations of the FFP cavity reflection spectrum due to, e.g., thermal expansion or material fatigue will be compensated by the feedback mechanism.
2. Experimental
The Pound-Drever-Hall (PDH) laser stabilization technique is a widely used method for locking a laser to an optical cavity [9] and has a critical role in e.g. interferometric gravity wave detection [14].
The acoustic transducer designed for this work consists of a fiber Fabry-Perot cavity containing two 23 dB FBGs positioned 2 centimeters apart. The transmission fringes of this cavity have a linewidth of ~25 MHz in the region of the FBGs’ maximum reflection (Fig. 1 ). The narrow linewidth fringes are essential in this work for locking the DFB laser and for detecting small strain modulations of a few με.
Fig. 1 (a) Complete and (b) partial transmission spectrum of the Fabry-Perot fiber Bragg grating cavity. The cavity has a free spectral range of 36.2 pm and a finesse around 180.
This cavity was attached to the guitar body (Dagmar Guitar) close to the bridge, where a conventional piezoelectric pickup (K&K Sound, Pure Classic) was mounted and used for comparison. The piezo pickup is a passive device and designed for use with classic guitars. A DFB diode laser (Avanex A1905 LMI), with a linewidth of 5 MHz at 1549 nm, was locked to the peak of a cavity fringe using the PDH technique. As the guitar body vibrates, the length of the cavity is modulated at acoustic frequencies, causing the fringe wavelength to be modulated accordingly. A correction voltage must be applied to the diode laser for the laser to remain locked to the moving cavity peak. This correction signal contains frequency components corresponding to the vibrations of the guitar body, and hence, may be amplified and recorded, or it may be fed to a speaker to reproduce the sound generated by the guitar (Fig. 2 ).
Fig. 2 Diagram of experimental setup. DFB: distributed feedback; PD: Photodiode; BT: Bias Tee; LPF: 50 kHz low pass filter.
The PDH locking technique operates in the optical frequency domain and does not rely on direct intensity measurements, which are subject to fluctuations in laser power. The reflection coefficient of the FBG cavity displays a minimum at the cavity resonance frequency due to the destructive interference of light reflected from the first FBG and circulating light within the cavity which leaks through the first FBG. The phase of the reflection coefficient also undergoes a 180 degree phase shift on resonance. This abrupt change in the sign of the phase angle provides a means of determining whether the laser is tuned above or below the resonant frequency. The phase of the reflected beam may be interrogated by generating frequency sidebands in the incident field through frequency or phase modulation of the laser. On resonance the laser carrier resonates in the cavity while the sidebands are reflected from the first FBG. Interference of the sidebands with the cavity reflected field generates a beat pattern that contains information about their relative mismatch. From a Bessel expansion of the modulated laser field, it can be shown that the total reflected power displays beat signals at the modulation frequency, with weaker terms also at higher harmonics [15] (Fig. 3 ).
Fig. 3 Top curve (red) Spectrum of cavity fringe with sidebands at 270 MHz, taken at control point (b) in Fig. 2; bottom curve (black) Error signal sampled at point (a) in Fig. 2.
Since the laser sidebands exhibit a reciprocal 180° phase shift, the superposition of the beat signals change polarity about the cavity resonant condition, and hence can be used as an error signal for the laser wavelength relative to the cavity length. The error signal can be extracted from the reflected cavity field by mixing the detector signal with the modulation signal applied to the laser, followed by low-pass filtering. Adjustment of the signal’s phase can then be used to generate an error signal with a symmetrical response about the zero-crossing point. The error signal is fed to a proportional-integral servo amplifier whose output is added to the laser current (Fig. 2). Our laser was current modulated to generate sidebands typically at around 240 to 270 MHz (Fig. 3). The slope of the error signal near resonance is greatest for high modulation-frequency to cavity-linewidth ratio. Besides better noise performance, high modulation frequencies lead to a tighter and faster lock of the laser to the cavity and, for this reason, were employed in this work. When the system is locked, the error signal drops to zero and the reflected intensity is at its minimum. Any mechanical disturbance of the FBG cavity will appear as a modulation in the correction signal which is fed back to the laser driver. The servo amplifier was designed to have a bandwidth of at least 30 kHz, i.e. exceeding the 20 Hz - 20 kHz acoustic frequency range of interest. A portion of the feedback correction signal was applied to a × 20 gain acoustic amplifier (National LME49710) and then sampled by Adobe Audition 3.0 software for recording and analysis.
Birefringence was induced through the FBG-writing process using UV light and broadens the cavity fringes marginally as we determined by inserting a polarization controller into the fiber optic cable. This effect did not influence our measurements.
The PDH method as described above is appropriate for pure frequency or phase modulation of the laser. In our work, the laser was current modulated which leads to amplitude as well as frequency modulation of the light. It has been shown that the presence of an AM component only slightly modifies the situation from that described above [16]. The error signals are no longer exactly anti-symmetric in shape, and there is a DC offset when the laser carrier frequency is off-resonance. This has the consequence that the laser may not lock to the exact peak of the cavity fringe. However, this is unimportant in the present application, since all that is required is that the laser can track a given position on the cavity fringe.
3. Results and discussion
Our analysis attempts to address the parameters that are most important for assessing musical instrument transducer quality, i.e. response time and frequency response curve, the signal-to-noise level, and dynamic range of the amplitude measurement.
An ideal transducer exhibits a very low noise floor, a sensor bandwidth from a few Hertz to 44 kHz (the rate at which Compact Disks are sampled) and a flat frequency response curve from about 50 Hz to 20 kHz, i.e. over the entire audible region. The experiments described below give these parameters but they are not able to represent the quality of the sound recording. We have deposited into the journal’s electronic databank six audio files that were recorded using the acoustic guitar, an electric solid-body guitar, a solid-body bass guitar as well as a cello [17].
The amplitude of the soundboard vibration could not be measured directly while the acoustic guitar was played, but it can be estimated from the measured modulation of the cavity fringe wavelength (Fig. 2). A modulation by about 5.7 pm (2.2 pm) was observed when the E2 (E4) string was played. Using a gauge factor of 0.78 this modulation amplitude corresponds to a strain of 4.7 με (1.8 με) [18]. Using a calibration obtained for a different guitar in an earlier publication, we can estimate the amplitude of vibration as approximately 90 μm (35 μm), which is consistent with earlier measurements (30-70 μm) for a slightly less “loud” instrument [7, 8].
The dynamic range of the transducer was determined by measuring the audio amplitude when the guitar was played or the body was tapped with a small hammer. The Signal-to-Noise ratio (SNR) was obtained by comparison of this amplitude with the noise level in the frequency regime of the fundamental string frequencies (80-400 Hz). It was found that the pickup worked largely distortion free in a SNR range of up to 50 dB as was evident from the small variation of the DC component of the photodetector output (indicated as (b) in Fig. 2). The guitar could be played at a SNR = 60 dB without unlocking the laser from the cavity resonance, but when the body was tapped with a hammer, the limited dynamic range of the servo caused the laser to fall out of lock with the cavity fringe at about SNR = 60 dB.
The noise level is determined largely by a 60 Hz hum from inadequate grounding as well as a Universal Serial Bus (USB) clock signal at 1 kHz that is associated with the acoustic amplifier. Importantly, the optical system itself did not give rise to a high noise level, i.e. the pickup is largely immune to changes in transmission and polarization in the optical cables, and is not influenced strongly by intensity fluctuations of the laser or detector noise. Also, the system remained frequency-locked for several hours and was immune to temperature changes.
The frequency response of the transducer was determined in two different ways. First, the Fourier transform of waveforms, which were obtained when tapping the guitar body and plucking a string, were analyzed for their highest observable frequency component. The waveforms were sampled at 96 kHz and recorded simultaneously using the built-in PZT pickup and the FFP cavity transducer. Only for the FFP cavity transducers we observe frequency components from 8 Hz to about 30 kHz, i.e. spanning the entire audible range of the acoustic spectrum (Fig. 4 ), whereas the range of the PZT transducer does not exhibit any response above about 25 kHz.
Fig. 4 Top: Waveform of an E4 string that was plucked about 5 cm from the bridge of the guitar. The black, top curve shows the response of the FFP transducer and the lower, red curve shows the signal from the PZT transducer, both sampled at 96 kHz. Bottom: Fourier transforms of the above waveforms.
For a more quantitative comparison a simple experiment was conducted. A speaker was placed 20 cm from the guitar and emitted a frequency ramp from 20 kHz to 0 kHz at a rate of 1.9 Hz/s. The response of the PZT and FFP cavity transducers was recorded and analyzed by calculating the Fourier transform of each 15 Hz interval. The resulting excitation-emission matrix (EEM) spectra are shown in Fig. 5 and display a pronounced emission response at the excitation frequency. Also, overtones of the excitation frequency are clearly visible, as well as a large number of resonances which are characteristic for the guitar and the placement of the sensor on the soundboard. At these resonances the acoustic excitation causes the guitar body to vibrate at a broad range of frequencies near the excitation frequency, leading to a broadening of the emission spectrum [18].
Fig. 5 Excitation-emission matrix spectrum showing the response of the guitar’s soundboard to sound emitted from a speaker at different excitation frequencies. The FFP transducer reproduces the fundamental frequencies from about 30 Hz to 20000 Hz, as well as its overtones. Horizontal lines at multiples of 1 kHz are due to a weak clock signal from the computer’s USB port. At frequencies above 13 kHz the FFP transducer response near the fundamental of the excitation frequency (black curve) shows a 5-10 dB stronger signal compared to the PZT pickup (red). The colour scale bar ranges from −220 to −180 dB. The curve in the right panel is the Fourier transform of the FFP cavity response at an excitation frequency 4 kHz.
In a second, slower scan (rate: 0.14 Hz/s) at frequencies from 500 Hz - 0 Hz, resonances of the guitar strings are clearly visible (Fig. 6 ). At these resonant frequencies the guitar body itself vibrates less strongly at the excitation frequency and most of the excitation energy is transferred to the strings, leading to peaks along the EEM diagonal (Fig. 6, lower panel). Helmholtz resonances from the vibrating air mass in the guitar body can also be observed and were identified by sealing the sound holes. In this experiment the signal to noise ratio was frequency dependent and was found to be higher than 40 dB in the region of the fundamental string frequencies when the guitar is excited using the external speaker. The typical signal to noise ratio for a single plucked string is about 30-50 dB, depending on the force of the attack.
Fig. 6 Excitation-emission matrix spectrum similar to Fig. 5 but over a smaller range of excitation and emission frequencies. Some of the negative peaks along the diagonal line (lower graph) are attributed to string resonances at 84.2 Hz, 110 Hz, 146.8 Hz, 196 Hz, 247 Hz, and 329.6 Hz. Broad positive resonances at 180 Hz, 315 Hz and 440 Hz are possibly Helmholtz resonances, whereas other features can be assigned to resonances of the guitar’s sound plate itself. Horizontal lines at multiples of 60 Hz are due to noise from the electronic circuitry. The colour scale bar ranges from −130 to −70 dB. The curve in the right panel is the Fourier transform of the FFP cavity response at an excitation frequency 200 Hz.
Aside from the electronic noise sources at 60 Hz and 1000 Hz that were mentioned above, one may also be concerned about optical sources of noise. We believe that laser frequency noise does not strongly contribute to the white noise floor in Figs. 5 and 6. For example, the noise floor in the region of 80-400 Hz is about 60 dB lower than the amplitude of the resonant peaks (Figs. 4, 5 and 6), and since the peak values correspond to 1-2 με, the noise floor is on the order of 1 nε(Hz)1/2. This is a few orders of magnitude higher than the ultimate limit achievable by eliminating all sources of noise other than laser frequency noise. This ultimate limit can be estimated assuming a white noise frequency spectrum, with a plateau given by the laser linewidth (Δν = 5 MHz). The laser-frequency noise spectral density is then given by S = (Δν/π)1/2 = 1.3 kHz Hz-1/2. Multiplication with the fiber gauge factor of 0.78 yields the ultimate limit of 10 pε(Hz)1/2. Thus, our noise level is far above the expected laser-frequency contribution, which is the most important external source besides electronics and one can conclude that the observed white noise is largely due to electronic sources.
The lower part of Fig. 5 shows the response of both transducers at an emission frequency within a 15 Hz window of the excitation frequency, i.e. along the diagonal in the EEM spectrum. Both transducer signals decrease as the excitation frequency increases, which may be attributed to the slight drop-off in the volume of the speaker. More importantly, the fiber-optic transducer response rolls off less at frequencies above about 13 kHz compared to the PZT pickup, thereby leading to a flatter frequency response curve. Indeed, only with the FFP transducer we observed the first overtone of the excitation frequency to about 30 kHz and the fifth overtone to over 40 kHz [17]. While one would then expect that the fiber-optic pickup produces a higher pitched (“tinnier”) sound, the opposite is true. A direct comparison of songs recorded simultaneously with both pickups indicates that the fiber optic pickup also reproduces lower frequencies more accurately and, admittedly subjectively, produces a more natural sound.
The broader frequency response of the FFP cavity sensor compared to the PZT transducer is not unexpected since the FFP transducer considerably lighter (26 mg excluding the adhesive tape) than the PZT transducer (~150 mg excluding cables). Also the FFP cavity sensor responds not only to out of plane vibrations [8] but also to in plane vibrations along the fiber optic cable [12], whereas the PZT transducer is sensitive mostly to out-of-plane vibrations. The upper limit of the acoustic frequency range (~30 kHz) is presently given by the speed of the servo loop and not by the cavity finesse. Assuming an even faster response of the servo-loop the frequency range of the sensor itself may be further increased, if the fiber optic sensor was embedded in a small groove inside the top plate of the guitar and fixed under a layer of varnish. The fiber optic sensor would then be a part of the guitar body and its acoustic impedance will be optimally matched.
The cavity length and the reflectivity of the FBGs (23 dB) was chosen such that the width of the cavity fringes (25 MHz) is comparable to the laser linewidth (5 MHz). The modulation frequency (about 250 MHz) was then chosen to be larger than the cavity fringe width, so that the frequency sidebands fall outside the fringe spectrum. The cavity length has little influence on the strain responsivity, since cavities of different length will exhibit the same fringe frequency shift when probing the same strain (the sensitivity will increase only slightly with length due to the decrease in the cavity fringe width). The upper limit of the vibration amplitude is limited only by the tuning range of our laser (about 25 GHz) and corresponds to measurements of about 0.16 mε. This corresponds to vibration amplitudes (3 mm) that are much larger than typically achievable by a guitar body!
The sensor system is expected to work at a wide range of ambient temperatures. From the thermal expansion of the silica fiber (dL/LdT = 4.1×10−7 K−1) and its thermo-optic coefficient (dn/dT = 7.97×10−7 K−1) we determine a thermal cavity fringe frequency shift of 120 GHz/K or 9.2 pm/K. Considering the tuning range of the laser (25 GHz) we arrive at temperature tuning range of over 20K. This readily encompasses the relevant range for a guitar player of about 100C to 30°C. The centre of the temperature range may be readily changed in the range of about 80K by adjusting the temperature - and thereby the centre frequency - of the laser diode.
4. Summary and conclusion
In conclusion, we demonstrated the use of a Fabry-Perot (FFP) cavity as a pickup for an acoustic guitar. The FFP cavity was interrogated using the PDH-frequency locking method and was thereby rendered largely immune to noise arising from optical intensity fluctuations. We have made distortion-free audio recordings of musical pieces from infrasound (~8 Hz) to 30 kHz with a 50 dB dynamic range in acoustic power. The ultimate limit of the sensors sensitivity was not determined as sensitivity is not overly important in our particular application, but we have determined that the noise in our measurements arises from electronic sources and not optical sources, giving hope that future developments may be able to further increase the dynamic range of the measurements, if needed.
Ratiometric intensity measurements on a FFP cavity or a single FBG pickup present an alternative to the PDH frequency locking method. This method compensates for the intensity fluctuations arising from the lightsource. When using FBG reflectors in series they may even compensate intensity fluctuations due to bending losses and birefringence from the optical cable. However, ratiometric measurements exhibit the same limited dynamic range as we have observed in our earlier work [7, 8] and they also require periodic resetting to the mid-reflection point of the FFP cavity fringe or FBG attenuation spectrum due to low frequency thermal drifts. In addition the PDH method does not only increase the sensitivity, but also reduces noise due to laser intensity fluctuations.
Before the pickup can be adopted by guitar manufacturers and musicians the cost and complexity of the system needs to be reduced. We note that all components are off-the-shelf electronic and optical parts while the laser driver can be easily replaced by a home-built current source. The bulky and expensive function generator can be replaced by a low-cost voltage-controlled oscillator (VCO). The oscilloscope is for visualization of signals during testing which may be done by a computer with a DAQ interface, if needed. We estimate that a dedicated sensor system may be built from components costing $1000 or less. This is comparable to the cost of a studio-quality ribbon microphone.
As was mentioned in earlier work [7, 8], the pickup is lightweight and may be multiplexed into a sensor array. It is therefore conceivable that multiple FFP cavity sensors may be embedded in the guitar body and – depending on their position – provide different timbres. Some high-end musical instruments already incorporate two or even more piezo pickups, giving a more balanced sound. Of course, one then requires a separate laser, laser driver, detector and locking electronics for each channel, whereas the power supplies, high-frequency generator and preamplifer may be shared.
Finally, we want to emphasize that audio recordings are extremely revealing with regards to the performance of any audio-range vibration sensor, since very slight distortions are easily audible even by untrained ears. A sensor that performs to high-fidelity audio standards should be more than adequate for almost all mechanical sensing applications.
Acknowledgments
HPL, JM, CN, DG and JB thank the Canadian Institute for Photonic Innovations and PARTEQ Innovations for funding this work. JM and JB also thank Queen’s University and the Natural Sciences and Engineering Research Council (NSERC) of Canada for financial support. SA thanks CNR-INO for supporting his visit at Queen’s University. XG acknowledges support through an NSERC Discovery Grant. We also thank Steve Adolph (PARTEQ) for his advice on the electronic design, Peter Swanson (Dagmar Guitars) for building the guitar, Paul Langlois (guitar) and Wolfgang Rau (cello) for recording some of the music files as well as Lucas Agosta, and Ron Elbaz for assistance with the experiments.
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