1. Introduction

The phase-shifting technique, which is effective for optical path difference (OPD) or displacement measurement, is widely used in three-dimensional optical information processing, such as surface roughness [1], surface profilometry [2,3], surface deformation [4], and optical fiber sensing applications, such as temperature [5], strain [6] and refractive index [5,7].

In phase-shifting technique, the phase difference between the object beam and the reference beam is varied by introducing the phase shift, the phase distribution of the interference signal is calculated by phase-shifting algorithms. A simple method to introduce the phase shift is to mount one of the reflectors of the interferometer on a piezoelectric transducer (PZT), and apply suitable voltages to drive the PZT [8]. The distortion of PZT cause phase shift errors due to a series of problems such as linear miscalibration, nonlinear sensitivity and spatial non-uniformity. The phase-shifted signals can also be obtained by narrowband lasers [9], optical fiber couplers [10], fiber Bragg gratings [11], or the frequency tunable laser [12]. However, in order to ensure the accuracy of the phase shift, these methods generally impose strict requirements on the stability of light source, the accuracy of wavelength control and the stability of the initial cavity length. Researchers such as Han et al. [13], Wang et al. [14] and Hao et al. [15] have proposed some phase recovery algorithms with unknown phase shift to eliminate these problems. However, these methods usually require considerable computation, and increase the compute load, which will influence the measurement speed.

In this paper, we propose an optical fiber F-P interferometer based on phase-shifting technique and birefringence crystals. The quadrature phase-shifted signals are obtained by using the four birefringence crystals with different thicknesses, and demodulated by using traditional four-step phase-shifting algorithm. The introduced phase shift is only related to the thickness of the birefringence crystal, which guarantees the stability of the phase shift. In addition, due to the lever amplification effect of the small refractive index difference on OPD, accurate phase shifts can be obtained with the lower crystal thickness fabrication accuracy. The proposed interferometer is experimentally verified to have high stability under the sinusoidal sonic signals of 21 kHz and 40 kHz, and can adapt to a larger measurement range.

2. Principle of the interferometer

The configuration for the optical fiber F-P interferometer based on phase-shifting technique and birefringence crystals is shown schematically in Fig. 1. The optical fiber F-P interferometer consists of a sensing interferometer and a demodulation interferometer. The system use broadband source having small coherence length and thus interference signal appearance is limited to areas near the zero OPD. Since the light pass sensing interferometer and demodulation interferometer in tandem, the total OPD of the system is the superimposing result of the OPD of sensing interferometer and the OPD of the demodulating interferometer. The two OPD should be matched to provide areas near zero OPD. That is to say, the demodulation interferometer is used to introduce an appropriate OPD to compensate that of the sensing interferometer. When the OPD in both of the interferometers matches, the interference signals appear.

 

Fig. 1 Schematic of the optical fiber F-P interferometer based on phase-shifting technique and birefringence crystals.

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In Fig. 1, the optical fiber F-P sensor forms the sensing interferometer. Light from a superluminescent light-emitting diode (SLED) broadband source is launched into the sensing interferometer through a 1 × 2 fiber coupler. The reflected optical signal from the sensing interferometer is split into four channels by a 1 × 4 fiber coupler, and each beam of the reflected optical signal projects into the demodulating interferometer. The demodulating interferometer consists of fiber collimators, polarizers, birefringence crystals with different thicknesses, and analyzers. The polarization directions of polarizer and analyzer have a 45° angle with the optical axis of the birefringence crystal. The four outputs from the demodulating interferometer are converted into electrical signal by four photoelectric detectors (PDs). The data acquisition (DAQ) acquires the four outputs of PDs, and transmits them to a computer for further processing.

Figure 2 shows the configuration of the optical fiber F-P sensor. The polyphenylene sulfide (PPS) polymer diaphragm is used as the sensing diaphragm. The F-P cavity is formed by the end face of fiber and the inner surface of PPS diaphragm. A ferrule and a D-type capillary are used to support and align the fiber. The F-P cavity length is adjusted by a displacement table. The gap between the D-type capillary and the ferrule forms a vent hole that balances the internal and external air pressure, and the vent hole allows the F-P sensor to accommodate different ambient pressure. The whole structure is fixed with epoxy glue.

 

Fig. 2 Configuration of the optical fiber F-P sensor.

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In the interferometer, the spectrum of SLED conforms to the Gaussian shape, it can be expressed as

In order to obtain the quadrature phase-shifted signals, the birefringence crystals have different thicknesses of $D$, $D+\pi /\left(2k_0\Delta n\right)$, $D+\pi /\left(k_0\Delta n\right)$, and $D+3\pi /\left(2k_0\Delta n\right)$, which represent the thicknesses of channel 1 to channel 4, respectively.

We can obtain the phase value $\Phi (L)=k_0(\Delta nD-2L)$ by using the traditional four-step phase-shifting equation [16]

In order to demonstrate the proposed interferometer, the optical fiber Fabry–Perot interferometer based on phase-shifting technique and birefringence crystals is set up. The broadband source is a SLED with a central wavelength of 750 nm and the FWHM is 20 nm. The transmission fiber type is a single mode optical fiber with a core/cladding diameter of 4/125 μm, and the demodulation interferometer and the sensing interferometer are connected via 3 dB couplers. The birefringence crystals made of MgF₂ are used. The four crystals are processed to the same thickness, and we rotate the crystals to obtain four different thicknesses seen by light beam. The thickness of crystal can be obtained by using the phase trace method [17]. The crystal thickness of the four channels after the rotation is 7.651 mm, 7.667 mm, 7.682 mm, and 7.698 mm respectively, and the refractive index difference between O-ray and E-ray of $\Delta n\approx \mathrm{0.012.}$ The length of the F-P cavity is 45.98 μm.

Figure 3 shows the phase error simulation results of the proposed interferometer when cavity length of sensing interferometer varies in the range $[-10\mu m,10\mu m]$ with actual thicknesses of the four birefringence crystals in the experiment being used. The phase error is in the range from −0.019 rad to 0.032 rad. The main source of error is because the actual thicknesses of the four birefringence crystals are approximately orthogonal since theoretical required thicknesses cannot be achieved. In addition, the envelope of low-coherence interference fringes also cause the phase error through dependence of interference intensity on cavity length. It can be seen that the proposed interferometer obtains high measurement accuracy.

 

Fig. 3 Phase error simulation results of the proposed interferometer when cavity length of sensing interferometer varies in the range $[-10\mu m,10\mu m]$ with actual thicknesses of the four birefringence crystals in the experiment being used.

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3. Experimental results and discussion

The experiment is carried out to verify the effectiveness of the proposed interferometer. We compare the performance of the proposed interferometer with the calibration microphone (B&K, 4191) in the same experimental environment. The experiment schematic is shown in Fig. 4(a).

 

Fig. 4 Schematic of the experimental setup. (a) The sensing interferometer is the optical fiber F-P sensor. (b) The sensing interferometer is composed of the fiber end face and the glass surface fixed on the nanopositioning stage. (c) The experimental photo. ① Birefringence-crystal-based Demodulator. ② F-P sensor. ③ Calibration microphone. ④ F-P interferometer controlled by nanopositioning stage.

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The sonic emission device is a PZT to achieve the conversion from electrical signal to mechanical vibration. The signal generator inject sinusoidal signal into PZT. The mechanical vibration of PZT energizes ambient air to emit sonic wave. The emission surface of PZT and the diaphragm surface of optical fiber F-P sensor are parallel to each other. As comparison, the calibration microphone is placed close to the optical fiber F-P sensor. The emission surface of PZT is also parallel to the receptive surface of calibration microphone.

We demonstrate the response of the proposed interferometer to sinusoidal sonic signals under frequency of 21 kHz. In this experiment, the signal generator drives PZT to transmit sinusoidal sonic signals under frequency of 21 kHz to the optical fiber F-P sensor and the calibration microphone. The cavity length of the optical fiber F-P sensor is modulated by the sonic signals, and the interference signals are appeared. We acquire the quadrature phase-shifted signals that are normalized and they are shown in Fig. 5(a). Then according to the quadrature phase-shifted signals, the phase value is calculated by four-step phase-shifting algorithm. The centerlines of the optical fiber F-P interferometer and the calibration microphone are obtained from the mean of the adjacent peaks and valleys of the corresponding results, respectively. The phase value and the centerline of the optical fiber F-P interferometer are shown in Fig. 5(b). The signal-to-noise ratio (SNR) of the phase value is 73 dB. The response signal and the centerline of the calibration microphone are shown in Fig. 5(c). The fast Fourier transform (FFT) of the phase value and the response signal of the calibration microphone are shown in Fig. 5(d). It can be seen that the signal waveforms from the proposed interferometer and the calibration microphone are very consistent to each other. They are all consistent with the actual signal. The normalized SD of the calibration microphone centerline is 1.97 times larger than the optical fiber F-P interferometer. The stability of the proposed interferometer is better than the calibration microphone.

 

Fig. 5 Experimental results of sinusoidal sonic signals under frequency of 21 kHz. (a) The quadrature phase-shifted signals that are normalized. (b) The phase value calculated by four-step phase-shifting algorithm and the centerline of the optical fiber F-P interferometer. (c) The response signal and the centerline of the calibration microphone. (d) FFT of the phase value and the response signal of the calibration microphone.

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In addition, we set the signal generator drives PZT to transmit sinusoidal sonic signals under frequency of 40 kHz to the optical fiber F-P sensor and the calibration microphone. The quadrature phase-shifted signals that are normalized are shown in Fig. 6(a). The phase value and the centerline of the optical fiber F-P interferometer are shown in Fig. 6(b). The SNR of the phase value is 75 dB. The response signal and the centerline of the calibration microphone are shown in Fig. 6(c). The FFT of the phase value and the response signal of the calibration microphone are shown in Fig. 6(d).

 

Fig. 6 Experimental results of sinusoidal sonic signals under frequency of 40 kHz. (a) The quadrature phase-shifted signals that are normalized. (b) The phase value calculated by four-step phase-shifting algorithm and the centerline of the optical fiber F-P interferometer. (c) The response signal and the centerline of the calibration microphone. (d) FFT of the phase value and the response signal of the calibration microphone.

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We can see the signal waveforms from the proposed interferometer and the calibration microphone are still consistent to each other at this frequency. They are still consistent with the actual signal. The normalized SD of the calibration microphone centerline is 2.63 times larger than the optical fiber F-P interferometer. The stability of the proposed interferometer is still better than the calibration microphone. It can be seen that the proposed interferometer can still achieve high stability at this frequency.

Figure 7 shows the experimental results under another sensing interferometer. The sensing interferometer under this experimental condition is composed of the fiber end face and the glass surface fixed on the nanopositioning stage (PI, P-752). The schematic of sensing interferometer is shown in Fig. 4(b). Figure 4(c) shows the experimental photo. The nanopositioning stage shift waveform is set to sinusoidal wave under frequency of 10 Hz. The displacement of the nanopositioning stage is set to 2 μm. The quadrature phase-shifted signals that are normalized, which are shown in Fig. 7(a) and in the range of $[100ms,250ms]$ are shown in Fig. 7(b). The phase value calculated by four-step phase-shifting algorithm is shown in Fig. 7(c). The signal intensity of the sensing interferometer composed with the optical fiber F-P sensor is larger than the fiber end face and the glass surface fixed on the nanopositioning stage, which cause the SNR of phase value in Fig. 7 is worse than Fig. 5 or Fig. 6. The SNR of the phase value is 62 dB. The FFT of the phase value is shown in Fig. 7(d).

 

Fig. 7 Experimental results of sinusoidal displacement under frequency of 10 Hz in the nanopositioning stage. (a) The quadrature phase-shifted signals that are normalized. (b) The quadrature phase-shifted signals that are normalized in the range of $[100ms,250ms]$. (c) The phase value calculated by four-step phase-shifting algorithm. (d) FFT of the phase value.

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It can be seen that the proposed interferometer can still recover the actual signal when the cavity length variation results in the corresponding phase value outside the range of $[-\pi ,\pi ]$. The proposed interferometer is effective in avoiding phase ambiguity and can adapt to a larger measurement range.

4. Conclusion

In this paper, the optical fiber F-P interferometer based on phase-shifting technique and birefringence crystals is proposed and demonstrated. We use the four birefringence crystals of different thicknesses to obtain the quadrature phase-shifted signals. The interferometer can accurately recover the sinusoidal sonic signals of 21 kHz and 40 kHz. In addition, the normalized SD of the calibration microphone centerline is 1.97 and 2.63 times larger than the optical fiber F-P interferometer under the sinusoidal sonic signals of 21 kHz and 40 kHz. The proposed interferometer has high stability. The interferometer is not limited in the range of $[-\pi ,\pi ]$, and can adapt to a larger measurement range. The high stability and high speed characteristics make the proposed interferometer suitable for harsh application such as aero-engine acoustic property measurement, space acoustic measurement in low pressure gas medium and so on.