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Abstract
A high-sensitivity low-frequency fiber optic hydrophone based on weak value amplification (WVA) is proposed and demonstrated. A polarization maintaining (PM) fiber with a length of 0.8 m wound around a polycarbonate (PC) tube is used as the sensing element. Theoretical analysis shows that the PM fiber in a WVA measurement scheme responds to underwater acoustic pressure with unprecedented sensitivity. The prototypical hydrophone based on such a scheme can sense underwater acoustic disturbance as weak as 1.3×10−6 Pa/Hz1/2 at 10 Hz, with a flat frequency response in the low-frequency band of 0.1–50 Hz. The experimental result agrees well with the theoretical prediction to within 0.5 dB.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, fiber optic hydrophone technology, with the salient characteristics of immunity to electromagnetic interference, passivity, inherent networkability, stability in harsh environments, high sensitivity, wide dynamic range and small footprint, has been providing solutions for applications in both the civilian and military fields [1–10]. Worldwide research and development effort of fiber optic hydrophone has waxed and waned but never really stopped since its advent in the late 1970’s at the U.S. Naval Research Laboratory [11]. The basic operational principle of the fiber optic hydrophone is that a sound source generated pressure wave signal in water acts on the hydrophone probe, which is transmitted to the fiber coupled to a mechanical structure to induce a deformation of the structure and the fiber along with it. This will in turn produce a change in one or more of the physical attributes of the light propagating in the fiber core, namely its phase, wavelength, polarization, or intensity [12–16]. Such a change is demodulated to obtain information about the sound source generating the wave disturbance, such as its intensity, frequency content, and bearing.
Currently fiber optic hydrophones roughly fall into two categories, those based on optical interferometry, which demodulate the phase change to reproduce the useful signal, and those relying on frequency shifts, including most fiber Bragg grating and fiber laser based schemes. Fiber optic interferometric hydrophones rely on demodulated optical phase signal [17–19]. Their sensitivity depends ultimately on how precise and how small such phase variation can be measured. Because of the weak coupling efficiency between the fiber and the sound pressure [20], the minimum detectable phase change is mainly improved by either increasing the fiber length or increasing the sensitivity of the mechanical structure. However, both measures run the risk of increased noise, complexity, and footprint of the sensor [21–25].
Most fiber optic hydrophones work in the frequency range of a few tens Hz to a couple of kHz, playing to the demands of military and civilian applications [26–28]. In recent years however, it is increasingly recognized that detection of low-frequency sound disturbance in the ocean has unique advantages: the disturbance itself transmits farther and suffers lower distance-related attenuation, and the disturbance originates from intrinsic structural flexural modes of a vessel. But designing low-frequency fiber optic hydrophones with good sensitivity posts some obvious challenges. As a hydrophone is essentially a mechanical oscillator whose excitation depends on its effective coupling with the surrounding fluid (an obvious mismatch in impedance to begin with), physics dictates heavier/bulkier and softer oscillator construct [29–31]. This not only runs counter to the basic premises of fiber optic sensors, but is technically difficult to achieve with the prevailing modalities of fiber optic hydrophones.
The emergence of the weak value amplification technique provides an alternative approach to increasing sensitivity of fiber optic hydrophone without a concurrent increase in its noise level, complexity and size [32]. In the context of foundational research in quantum measurement theory, Aharonov, Albert, and Vaidman expounded in 1988 that quantum mechanics offers a much greater variety of measurement scenarios [33]. In particular, in a weak measurement scenario, there exists an amplification mechanism, which can increase the measurement sensitivity by several orders of magnitude without a simultaneous increase of technical noise [34]. Any signal that introduces an optical phase change can be monitored with the method of weak measurement, which has been utilized in numerous fields in high precision detection of miniscule variations of physical quantities, such as the phase of an optical beam [35], spin Hall effect of light [36], the concentrations of chiral compounds in dilute solutions [37], and surface plasmon based refractive index variation [38].
Here we report on an implementation of a low-frequency fiber optic hydrophone based on weak value amplification (WVA) realized in a common-path polarization dependent fiber optic interferometer. The proposed hydrophone mainly comprises two integrated linear polarizers and a segment of polarization-maintaining (PM) fiber. The latter, of a short length of only 0.8 m is wound on a PC tube. This simple hydrophone was tested, making use of a standard hydrophone as the reference, in a water tank with low frequency vibration excitation, to determine its sensitivity as a function of excitation frequency.
In the remainder of the paper we will first introduce the theory of WVA, especially as it pertains to the small phase change measurement in an optical fiber interferometer and the working principle and the construct of the hydrophone probe, followed by the presentation and discussion of the results of the fiber hydrophone experiment. The sensitivity and the frequency response, as well as the noise-equivalent pressure (the minimum detectable pressure) of the new fiber hydrophone are then experimentally established, demonstrating flat low-frequency response in the range of 0.1–50 Hz, with a noise-equivalent pressure of 1.3×10−6 Pa/Hz1/2 at 10 Hz, very close to the theoretical limit.
2. Methods
2.1 Theoretical background
It has been shown that the measurement of extremely small optical fiber phase can be realized by weak value amplification [32]. Consider a typical WVA experimental setup, where the initial polarization state of the photon, with the eigen-states represented by $|\textrm{H} $ and $|\textrm{V} $ for the two orthogonal polarizations (horizontal and vertical), is written as a superposition
This state (termed the pre-selection in the lingo of quantum weak measurement) then evolves into2.2 Design of fiber optic interferometer to realize weak value amplification
To realize a weak measurement system in the context of a fiber optic interferometric hydrophone, we design a simple all-fiber system in which the pre-selection and post-selection are accomplished by a pair of in-fiber integrated polarizers, whose optical axes are orthogonal to each other. In between these two polarization devices, the weak interaction occurs along a specially designed segment of PM fiber that admits and transmits simultaneously both an H and a V polarized light beam without any time-delay/phase difference between them in the absence of external perturbation. In this case the output light intensity would be null theoretically. External perturbation introduces phase shift between the two polarization, which is translated into drastic increase in the output light intensity through the weak value amplification mechanism.
Figure 1 depicts the optical-path diagram (Fig. 1(A)) and the evolution and manipulation of the photon polarization state (Fig. 1(B)) of the fiber optic hydrophone based on weak value amplification.
Fig. 1. Schematics of the optical path and the associated state of photon polarization evolution/manipulation. SFL: single frequency laser; PMF1 - PMF4: PM fiber; LP1, LP2: composite in-fiber linear polarizer device, with C1 - C4 representing collimators, P1 and P2 representing linear polarizers, which are integrated together to form LP1 and LP2; PD: Photodiode. (A) Optical path diagram, and (B) Evolution and manipulation of the state of photon polarization along the optical path depicted in (A). The two dashed-line boxes represent the two linear polarizer devices respectively; the green dot between PMF2 and PMF3 represents the spliced point, where PMF2 and PMF3 are spliced together, with their fast axes perpendicular to each other. The polarization orientations of light at several points along the optical path are given as: (i) between P1 and C2; (ii) in PMF2, where the two dark circular dots represent the panda eyes of a panda-type polarization maintaining fiber, whose fast axis is perpendicular to the line connecting the panda eyes; (iii) in PMF3; (iv) between C3 and P2 of the second linear polarizer device; (v) between P2 and C4.
In this design, we use two integrated in-fiber linear polarizer devices. The polarization-maintaining fiber linear polarizer is a relatively small device and consists of several components (two fiber collimators, and a C-band 45° polarizer of 1550 nm wavelength). The principle of the linear polarizer device is that the input light through a single fiber collimator and the 45° polarizer of 1550 nm wavelength, becomes a linear polarized light consistent with the direction of the polarizer's optical axis; finally, the linearly polarized light is coupled to polarization maintaining (PM) fiber of the output fiber collimator, and this output linear polarized light is at 45° angle to the fast axis of the PM fiber of the output fiber collimator.
The incident light from the SFL (single frequency laser, 5 mW output power) centered at 1550 nm with a bandwidth of 3 kHz, is pre-selected by the first in-fiber integrated linear polarizer device (extinction ratio of 10,000:1) with a pointing angle of 45° through the first PM fiber collimator. Then, the linearly polarized light beam enters the PM fiber at 45° angle to the PM fiber’s fast axis. The propagation speed of light with its electric field polarized along the fast axis and that along the slow axis are different in the fiber. If the PM fibers PMF2 and PMF3 are of equal length, and the rotation angle of the two fibers spliced together is 90°, the propagation time of light polarized in the vertical direction is the same as that of light polarized in the horizontal direction between the two linear polarizer devices.
The light propagates in the PM fiber PMF2 with its polarization along the fast axis will be going through the PM fiber PMF3 with polarization along the latter’s slow axis. When the two polarized light beams reach the linear polarizer in the second polarization device, the two polarized light will be combined to form another linearly polarized light. And the polarization direction of this linearly polarized light is almost completely perpendicular to the optical axis of the linear polarizer. Thus the output collimator of the second device outputs the weak-power light from the linear polarizer. Finally the weak-power light is received by the photodiode.
A part of the PM fiber between the two linear polarization devices is wound on a PC tube to form the hydrophone probe. This probe plays the role of weak coupling between the dynamic variable of the system, the polarization vector, with the momentum of the photon, in the weak measurement system, which is promoted by the underwater acoustic signal. Such an interaction is designed to introduce an optical phase difference between the H and V polarization, which quantitatively connects the fiber length change to the intensity of acoustic pressure change. Thus, the acoustic pressure could be detected through the power of the output light according to the principle of weak value amplification.
2.3 Mechanical construct of sensor head
Our hydrophone probe is mainly composed of a PM fiber winding between the two linear polarizer devices on the outer surface of an elastic PC tube. The shift of fiber phase caused by underwater acoustic pressure variation is mainly composed of two parts: one part of the phase shift is produced by the fiber length change caused by the radial change of the elastic cylinder, the other part of the phase shift is caused by the elasto-optical effect produced by the acoustic pressure. Thus the phase change of the fiber is directly related to the length change of the fiber coil, as a result of the change of the outer radius of the PC tube. Here we establish the mechanical model of the elastic cylindrical shell along with the fiber coil wound on its outer surface (Fig. 2) to analyze the phase sensitivity of the fiber hydrophone. The change of the radius can be obtained as [39–41]
Fig. 2. (i) Horizontal cross-section of the fiber hydrophone probe: a and b represent inner radius and outer radius of the PC tube, respectively; Ps represents the underwater acoustic pressure. (ii) Vertical cross-section of the fiber hydrophone probe: black dots represent PM fiber windings.
The phase shift of fiber hydrophone can be written as
Therefore, the phase shift caused by the fiber length variable is given as
where N is the number of windings in the coil of fiber. And the phase variation caused by the refractive index change of the fiber is given by where ${p_e}$ is the elasto-optical coefficient of the fiber.The total phase sensitivity S is
According to Eq. (7), when $r = b$, $\epsilon $ can be expressed as
3. Experiment
3.1 Underwater acoustic test setup
In order to generate low-frequency underwater acoustic disturbance to test the new hydrophone, a moving-water-column method is adapted. This setup is composed of a large vibration platform driven by an amplified voltage signal from a signal generator, a water tank placed on the vibration platform, and a standard hydrophone (Bruel and Kjar Model 8014, referred to as BK hydrophone henceforth), as shown in Fig. 3. When the signal generator produces a voltage signal of a certain frequency, the vibration platform moves up and down at this frequency, and in doing so, it lifts the water tank up and down at the same frequency, generating an acoustic pressure modulation at a given fixed depth in the water (referenced to the fixed laboratory floor).
Fig. 3. Underwater acoustic test setup. SFL: single frequency laser; PD: photodetector; PMF1, PMF2, PMF3, PMF4: polarization maintain fiber; LP1, LP2: linear polarizer device. Fiber hydrophone and B&K hydrophone were placed in the water at the same depth (fixed by thin, rigid wires not shown here) and are 20 mm from each other horizontally, in the water tank placed on a vibration platform.
The fiber optic hydrophone based on WVA measurement system is tested by placing it in the water tank at the same depth as the standard hydrophone, with the two separated horizontally by about 20 mm, which is much smaller than the shortest acoustic wavelength. When the underwater acoustic signal is generated, the resulting fiber phase shift and relative intensity of the output light are recorded. The piezoelectric hydrophone used as the standard hydrophone gives the local pressure reading on the fiber hydrophone. The test is performed in the frequency range of 0.1–200 Hz.
3.2 Hydrophone sensitivity prediction
In the experiment, the underwater acoustic pressure was measured by the standard hydrophone, and the corresponding phase variation of the fiber was calculated. At the same time, the variation of the output optical intensity of the fiber hydrophone, based on the WVA structure, can be used to deduce the phase shift variable. In this way, the phase change of the fiber calculated by using data from the standard hydrophone is compared with that of fiber hydrophone output. Therefore, the acoustic pressure sensitivity of the fiber hydrophone is very important for calculation of the phase change value in response to the underwater acoustic pressure signal.
The fiber hydrophone probe structural parameters are listed in Table 1.
Table 1. Fiber hydrophone structure parameters
According to Eq. (11), the acoustic pressure sensitivity S of the fiber hydrophone can be determined by the relation
And according to the principle of weak value amplification, the small shift of the fiber phase can be amplified and the $\kappa \; $value plays an important role for the demodulation of the phase shift, as can be seen from the relation given by Eq. (5),
With the fiber hydrophone placed in the water tank while the drive voltage to the vibration platform is turned off, the $\kappa \; $value can be deduced through the input light intensity of the pre-selection and that of post-selection. According to the relation given in Ref. [32], the PM fiber crosstalk plays an important role in the deduction of $\kappa \; $value:
where ${I_c} = {I_0}L\zeta \; $is the light intensity of the fiber crosstalk loss, $I_{out}^{\prime}$ is the light intensity of the post-selection without any acoustic disturbance in the water, ${I_0}\; $is the light intensity of the pre-selection, $L = \textrm{0}{.8\; }\textrm{m}\; $is the length of sensing fiber, $\zeta = {24\; \textrm{dB}/\textrm{m}}$ is the crosstalk constant of the PM fiber.Because the length of the front and back sections of the fiber is almost the same (to within 1.5 mm), with the difference being less than the beat-length (3.5 mm) of this particular type of polarization-maintaining fiber (YOFC model, PN: PM1550-125-18/250-Y), the overall phase difference between the fast-axis-polarized and the slow-axis-polarized components of light in the polarization-preserving fiber is less than π in the unperturbed state. In the perturbed state, the additional phase difference can be obtained from the experimentally measured light output intensities according to Eq. (14).
In this experiment, when the fiber hydrophone is placed in the water but without any acoustic pressure excitation, the relevant parameters measured and deduced are listed in Table 2
Table 2. Test results of this fiber hydrophone without excitation
In the experiment, the relational between the variation of light intensity $\varDelta I = {I_{out}} - I_{out}^{\prime}$ and the phase change of the sensing fiber at a single frequency is given as [32]
3.3 Hydrophone frequency response
The voltage signal of each fixed frequency is inputted to the vibration platform by a signal generator, and the test is performed in the frequency range of 0.1–200 Hz. The voltage output of the BK hydrophone and that of the photodetector are collected by an NI acquisition card, at the sampling rate of 50 kHz. The BK hydrophone is used as a standard for both calibration and subsequent measurement, and the photodetector receiving the light intensity is an AFR model (PN: GC-BPD-1117-200-A). Thus one can deduce the underwater acoustic pressure by the voltage signal of the standard hydrophone, and the light intensity signal by the voltage signal of the fiber hydrophone, corresponding to the same underwater acoustic pressure excitation.
Underwater acoustic pressure ${P_s}$ as measured by the standard hydrophone is given by
where ${A_s}\; $is the amplitude of the BK hydrophone output voltage at a single frequency, $\; {K_1} = 100$ an enlargement factor subsuming all the instrumentation gains, and $\eta = 42.3\; \mu \textrm{V/Pa}$ its pressure sensitivity.The variable output light intensity $\varDelta I$ of the fiber hydrophone is given as
where ${A_f}$ is the amplitude factor of the fiber hydrophone at a single frequency, and ${K_2} = 22.55 \times {10^3}\; mV/mW$ is the photoelectric coefficient of the PD. Obviously, As and Af are the amplitude values of the frequency response of the standard and fiber hydrophone, respectively. When a single frequency water pressure signal is generated in the water tank, the standard hydrophone and the optical fiber hydrophone both measure the underwater acoustic signal in the water tank, at the same time and at about the same location (within 20 mm), while the voltage signals of the two channels generated by the underwater pressure signal are collected in time domain separately. After the two channels of time-domain voltage signals are Fourier transformed, we obtain the two channels of frequency-domain signals, respectively. The amplitude values corresponding to a given output frequency of the signal generator are As and Af in the two frequency-domain signals.When the signal generator inputs a low-frequency voltage signal to the vibration platform, the fiber hydrophone in the tank outputs the corresponding low-frequency voltage signal. For the input low-frequency signal at 0.3, 0.2 and 0.1 Hz, we obtain the voltage signal from the fiber hydrophone and standard hydrophone as shown in Fig. 4.
Fig. 4. Time-domain measurement at low frequencies of response of the fiber hydrophone and standard hydrophone. The blue trace represents the output photoelectric signal value of the optical fiber hydrophone based on the WVA scheme, and the red trace is the output electric signal value of the standard hydrophone. In a one-minute time window, the low-frequency signal of 0.1, 0.2 and 0.3 Hz were applied to the vibration generator.
As can be seen from Fig. 4, the standard hydrophone has only apparent noise output and no obvious time-domain response for frequencies at and below 0.3 Hz.
To shed light on the resonant excitation and the general frequency response of the fiber-wrapped compliant cylinder in water, the eigenfrequencies of the PC tube submerged in water is determined using finite element method [42]. The simulation results of the lowest-mode configuration (corresponding to the lowest-frequency eigenmode with f0 = 47.76 Hz) is depicted in Fig. 5. Note that in this calculation the fiber coil (of 5.5 turns) wound on the cylindrical shell was neglected.
Fig. 5. Lowest-frequency (47.76 Hz) mode excitation of the PC tube in water. The color bar represents normalized displacement amplitude.
It can be seen from Fig. 5 that the vibration mode of the first order corresponds to an axial-symmetric expansion/contraction, telescopically outward from the axis of the hollow cylinder to the outside along the radial direction, which is consistent with our understanding of the mechanical behavior of the fiber hydrophone probe.
The mechanical system depicted in Fig. 5 can be thought of as providing a strain-gain mechanism to the fiber coil wound around it [43,44], with a magnification of the radial direction amplitude that is related to the eigenfrequency of the PC tube. The gain factor$\; \; \beta $ can be obtained as
According to Eq. (13), the acoustic phase sensitivity of the fiber hydrophone is $- 173.03\;\textrm{dB}\;({{\textrm{Re}\; \textrm{rad}}/\mathrm{\mu}\textrm{Pa}} )$ from the theoretical calculation of fiber hydrophone probe. The particular values of the resonant frequency and the damping coefficient associated with our compliant cylindrical shell probe dictate that the acoustic phase sensitivity of the fiber hydrophone should be decreasing beyond about 50 Hz according to Eq. (19). The total sensitivity Sa of the fiber hydrophone at each frequency can be expressed as
The results of the acoustic pressure sensitivity measurement of the fiber hydrophone as a function of frequency in the band of 0.1 - 200 Hz is given in Fig. 6, represented by the blue data points (with error bars), along with the corresponding theoretical relation represented by the orange-colored curve.
Fig. 6. Acoustic pressure sensitivity as a function of frequency. Orange-colored curve is the theoretical sensitivity Sa when Q = 0.86; Blue data points (with error bars) are sensitivity values at various frequencies measured in the experiment; Acoustic frequency range is 0.1 Hz - 200 Hz.
It can be seen from Fig. 6 that the sensitivity to underwater acoustic pressure calculated theoretically is in excellent agreement with that obtained experimentally when the damping coefficient Q is 0.86. The difference between the theoretical and experimental values is less than $0.5\; dB({Re\; rad/\mu Pa} )$ across the entire frequency band of 0.2 - 50 Hz.
The sole exception occurs at 0.1 Hz, where abnormally large difference between the theoretical and experimental values exists. This is due to the fact that the standard hydrophone (BK hydrophone) gives uncertain readings at this low frequency: The nominal amplitude value (from Fourier transform) of the BK hydrophone is 0.07 mV when the acoustic frequency is 0.1 Hz, but the noise value measured by the BK hydrophone is 5 mV as can be seen from Fig. 4. This noise value is much greater than the nominal amplitude value, resulting in a large uncertainty in the measured result.
From Fig. 4 one can see that the fiber hydrophone produces obvious time-domain signals in the frequency range of 0.1 Hz - 0.3 Hz, but the standard hydrophone has no obvious time-domain signal. However, the phase sensitivity of the fiber hydrophone can still be clearly measured at 0.2 Hz and 0.3 Hz as can be seen from Fig. 6. The reason for this is that the standard hydrophone can give an uncertain amplitude value at 0.1 Hz through a frequency-domain analysis, while it is capable of giving definitive amplitude values in frequency-domain measurement when the frequency is greater than 0.2 Hz. Obviously these signals of the standard hydrophone output are masked by noise in the time-domain measurement at 0.2 Hz and 0.3 Hz also. However, from Fig. 4, we expect that the fiber hydrophone based on WVA should behave normally at 0.1 Hz or even lower frequencies. This point needs further corroboration when low-frequency hydrophone calibration apparatus becomes available in our laboratory shortly.
The new fiber optic hydrophone, with a resonant frequency of about 50 Hz, has a frequency response that is flat from 50 Hz all the way down to at least 0.1 Hz. Above 50 Hz, its response falls off rapidly and at 200 Hz it cuts off.
3.4 Hydrophone noise-equivalent pressure
The hydrophone noise-equivalent pressure $\sigma$in this experiment is evaluated by the noise-equivalent phase Ns of the fiber hydrophone and expressed as
where Sa is the sensitivity of the hydrophone given in Eq. (20). The noise-equivalent phase level of the fiber hydrophone system is measured in frequency domain, and defined with the mean power density of the hydrophone output signal in the detectable frequency band but without an input signal, which is shown in Fig. 7.The experimental results in Fig. 7 show that the noise-equivalent phase has values below −120 dB re rad/Hz1/2 (at 50 Hz). And the average system noise level is -117.04 dB re rad/Hz1/2 from 0.1 Hz to 50 Hz. For example, at 10 Hz, the hydrophone noise-equivalent pressure $\sigma$ is calculated using Eq. (12) as
Information available in the public domain on state-of-the-art interferometric fiber optic hydrophones based on fiber-wrapped compliant cylindrical tubes [45–48] reveals that such sensors generally use a few tens meters of fiber, resulting in minimum detectable pressures (noise-equivalent pressure) of 3 to 5×10−5 Pa/Hz1/2, at frequencies ranging from a few tens to a few thousands of Hz (although values of the noise-equivalent pressure at 1 kHz were usually quoted). On the other hand, the fiber hydrophone based on WVA described here uses only 0.8 m of fiber, achieving a noise-equivalent pressure of 1.3×10−6 Pa/Hz1/2 at 10 Hz. With more than an order-of-magnitude increase in the minimum detectable pressure while without concomitant increase in fiber length and sensor footprint, the fiber hydrophone based on WVA demonstrates enormous potential for low-frequency weak underwater acoustic signal detection. That this type of application of WVA is advantageous is because of the technical noise mitigation, in addition to the signal amplification power of WVA [34]. Others have pushed experimental verification of these aspects of the weak measurement theory [49,50]. The ability of WVA in allowing table-top experiments to reach even the shot-noise limit [51,52] can be considered as indirect support of the results presented here.
4. Conclusion
We have presented a novel scheme of combining quantum weak value amplification and fiber optic polarization interference to construct a unique fiber optic hydrophone, using less than a meter of PM fiber wound on a PC tube as the sensing part. The new hydrophone demonstrated high underwater acoustic pressure sensitivity at unprecedented low-frequencies, and an order-of-magnitude increase in the minimum detectable pressure change. Another unique feature is that the proposed hydrophone does not require a phase demodulation process, as the WVA scheme gives the phase variation in terms of the output light intensity directly. As such, the new fiber optic hydrophone design, though still preliminary and prototypical, foreshadows a brand new way of designing high-performance hydrophones and other fiber optic sensors based on interferometry.
Funding
Leading Talents Program of Guangdong Province Program (00201507); State Oceanic Administration of China (201405026-01).
Disclosures
The authors declare no conflicts of interest.
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