Optical Fiber Spool Acoustic Wave Resonances Employing a Mach-Zehnder Interferometer in Vacuum Chamber

The longitudinal, flexural, and torsional modes of acoustical resonances are presented for a long and small thickness cylindrical fiber spool. The fiber spool is built using a monomode optical fiber tightly wound in a helix form around a cylindrical Styropor. The acoustical resonance values are obtained using a Mach Zehnder interferometer with the fiber spool in the long (2 km) arm and a small linewidth laser. To reduce spurious signals, the interferometer was placed inside a vacuum chamber. The measured and theoretical results reasonably agree up to the first fifteen resonant longitudinal modes and for the first five flexural and torsional resonant modes, with a span from 10 Hz up to 200 Hz.

Also, laser noise reduction can be obtained using the filtering action of high-quality optical cavities [4], integrated phase noise filters [18], or actively stabilized optical delay lines [19]. Those applications require the evaluation of the laser parameters such as Lorentzian linewidth [20]- [21], and several techniques have been developed for linewidth measurements [22], comprising: delayed selfheterodyne [23]- [24]; self-homodyne detection with short delay Mach-Zehnder interferometer (MZI) [25]; Brillouin induced self-heterodyne [26]; strong coherent envelope [27]; and heterodyne offline digital signal processing [28]- [29]. The use of the self-homodyne fiber MZI has the advantage of using only one laser with no distinct local oscillator, and the authors employed this technique to analyze narrow linewidth lasers [30].
However, in the above applications concerning small linewidth lasers, the optical fiber spool (OFS) acoustical induced resonance effects may degrade the detected signals due to possible false positive signals or low signal to noise ratio (SNR). Indeed, OFS detectors based on fiber optic pressureinduced refractive index changes [31]- [32] are used for high sensitivity hydrophones [31]. In those applications, the fiber provides immunity to electromagnetic interference, including 3D shaped

II. THEORY OF ACOUSTIC FIBER REEL NATURAL RESONANCES
The free resonance of finite length and thin circular cylindrical shells has been studied through one and half century [34]. Exact solutions are complex [35], and W. Flügge introduced solutions for a few of the simplest types of shells [36]. The exact solution was available [37], but complex calculations are required [35]. Donnel [38] followed by Mushtari and Vlasov presented a simplified theory for quasi-shallow shells called Donnel-Mushtari-Vlasov (DMV) Theory [39], suitable for transversely loaded thin isotropic elastic shell of uniform thickness. An analytical and closed solution for open isotropic circular cylindrical shells has also been obtained [40], [41]. Furthermore, filled cylindrical shells have been analyzed [42]- [44]. We use here the DMV equations [35], [41], where the orthogonal displacements in the x direction (ux), y direction (uy), and z direction (uz) are shown in the thin shell of Fig. 1 representing an OFS.  The approximate solution with independent variables assumes harmonics displacements u, resulting in the following equation [35], where the longitudinal displacement ux in the direction x refers to the amplitude A, and so on: , h is the shell thickness, and R is the cylinder radius. Also, in the above equation, the angular frequency parameter Ω is defined as [35]: where ρ is the fiber shell density, ν is the Poisson ratio, and E is the Young's modulus. In addition, the parameter λ = π R/ L is defined, where the cylinder length L corresponds to the half wavelength in the longitudinal direction x . The solution of Eq. (1) requires that the matrix determinant must be zero. In this way, the free wavelength angular frequency ω can be calculated for a given λ. For an extremely long and small-thickness cylinder, the length L >> 0 , the half wavelength tends to infinity and λ tends to zero. So, Eq. (1) is reduced to: The solution of Eq.
The roots of the second order determinant of Eq. 3 give the flexural and torsional modes (FTM), also called radial and circumferential modes, of the OFS resonance. They are given by [35]: In this experiment, an external acoustic pressure signal is incident on the OFS to induce resonance Pa, and its density is ρFO ≈ 2.2 × 10 3 kg/m 3 . Also, the Poisson ratio is ν = 0.17.
Using the above equations and parameters, the OFS (with small thickness) free resonance frequencies (f = ω/2π) can be calculated. Using Eq. 4, the results for LMM can be calculated. The FTM can be calculated using (5). The mode n=0 is called breathe; n = 1, dipole; n = 2, quadrupole bell; n = 4, octupole. Complete results up to 200 Hz will be presented at Table I.
The thin crust Love-Timoshenko Theory [35] gives the following equation for the FTM resonances: The Styropor cylinder influence was not considered on the above analysis. The comparison of theoretical values with the experiments are examined in the following Sections.

III. EXPERIMENTAL SETUP
The experimental setup used here is shown in Fig. 2. A narrow linewidth He-Ne laser operating at 632.8 nm continuous wave is lens-coupled to a bare monomode optical fiber. After the surface fiber modes were eliminated, the fiber is introduced in a vacuum chamber with adjustable inside air pressure. The transmitted light goes to the monomode fiber optical MZI made with a 50% fiber coupler, a short arm (1 m) and a long 2 km arm with the cylindrical OFS. Both arms are combined in a second coupler and a polarization controller is adjusted for maximum homodyne beat signal. The chamber inside pressure was kept at 1.2 Torr. The measured laser linewidth at this small pressure was 220 Hz [30]. Finally, the He-Ne homodyne mixed fiber signal is carried outside the vacuum glass chamber and detected by a silicon PIN photodiode. The He-Ne laser polarization controller was adjusted to avoid polarization flipping between the two laser eigenstates (parallel and perpendicular) [45]. IV. EXPERIMENTAL RESULTS After performing a characterization of the OFS shown in Fig. 1, a 75 Hz sinusoidal sound produced by a speaker was placed at a distance r equal to 1.2 m outside the glass chamber, as shown in Fig. 2.
Also, this chamber and the laser were mechanically isolated from the environment by double isolation optical tables. At 1.2 Torr, the environment noise was around 23 dB below the signal level [46]. The   Table I. The experimental results employed a Spectrum Analyzer based on Fast Fourier Transform (FFT).
The instrument error can be verified with the 60 Hz signal and its harmonics, as shown in Fig. 3 Fig. 2) is necessary to decrease the acoustical noises and to obtain the sharp resonances peaks shown in Fig. 3 [46]. The results without vacuum are inaccurate and it is almost impossible to distinguish the specific resonances [46]. Additionally, the PSD intensity level of the measured resonances are shown in the last column of  [30]. Indeed, the FFT frequency span shall be less than the laser linewidth of 220 Hz to achieve the best SNR using the MZI configuration of Fig. 2. In our case we used an FFT span of 200 Hz. The obtained laser white noise PSD flat level [46] of -82 dBVRMS /√Hz (shown in Fig. 3) is well above the noise level (without light) of -105 dBVRMS /√Hz (not shown here). This small noise level at 1.2 Torr is equivalent to 60 nW/√Hz and the laser PSD level is 12 W√Hz [46].
The results using the DMV Theory of Section II are presented once more in Table II for the FTM resonances. However, they are supplemented by numerical results based on the Love-Timoshenko Theory [35] and results [30] using commercial Finite Element Analysis packets: NASTRAN [42] and COMSOL [47].
The simplified DMV Theory results (presented in Table II   A possible further explanation for discrepancies between theory and experiment might be: fiber spool wound imperfections; Styropor influence [43,44,48]; environment noises and vibrations [4,49]; theoretical approximations; laser finite linewidth; MZI outside linear modulation range [50], linear dynamic range [51]; and fiber light-polarization Brillouin effects [52]. However, the frequencies measured here are much lower than the GHz Brillouin frequencies scattered in optical fibers. The technique presented here might be useful for resonances measurements at higher frequencies if a fiber spool (resonating at ultrasound frequencies) was inserted in one arm of the MZI shown in