## Abstract

A new approach for self-mixing interference in fiber ring laser and its application to vibration measurement is presented. Fourier transform method is proposed to improve the measurement accuracy of such an active sensing, in which the basic frequency of self-mixing interference signal is used to determine the frequency of vibration, and the stop order of the harmonic components is proportional to the amplitude of vibration. We have demonstrated this system to measure the microscopic vibration of a remote target. Some errors due to this method are discussed. The maximum error of the amplitude is about λ/10 and the maximum error of the frequency is about 10%, showing a good agreement with the simulative results.

© 2009 OSA

## 1. Introduction

Self-mixing interference (SMI) in laser is available for the measurement of distance, displacement and velocity. Many researchers have analyzed the theory and observed the experimental phenomena of SMI in lasers, such as the gas laser, the laser diode and DFB laser [1–7]. So far, research on SMI in fiber laser is less reported. SMI in fiber laser has the same phase sensitivity with other kinds of lasers, shifting of one fringe corresponding to displacement of λ/2, moreover it is easy to realize multi-channel measurement based on its high output power [8]. Because of the wide gain bandwidth of erbium-doped fiber and enough power of fiber ring laser, many channels with different wavelength can coexist. Every channel can form a SMI system. Different parameter at different position can be monitored at the same time with simple system and less devices. Its potential application in flexible measurement and multiplexing make it significant to research in novel fiber active sensing network.

Modulated and non-modulated methods are two basic kinds of ways for displacement or vibration measurement by SMI. Modulated method, such as phase modulation and wavelength modulation, can exhibit a high accuracy to nanometer dimension, but at the expense of a complicated set-up [9,10]. Non-modulated method is usually used for the situations where micron dimension precision is adequate, such as fringe counting method with precision of λ/4 and peak-to-peak value method covering the 0-λ/4 amplitude range of vibration [11,12].

In this letter, SMI in fiber ring laser has been theoretically analyzed and the output power expression is deduced. The relationship between harmonic components and amplitude of vibration is discussed. The measurement error is considered and simulated. The experimental results show a good agreement with the simulative results.

## 2. Theory of SMI in fiber ring laser

The model of fiber ring laser with optical feedback is presented in Fig. 1 , where ${P}_{pump}$ denotes the power of pump, ${P}_{laser}$ denotes the power of the fiber ring laser and ${L}_{ext}$ is the distance between the circulator end and the target. According to the expression of ${P}_{laser}$ and the equivalent reflectivity, which is influenced by optical feedback from the target, of fiber circulator end [8], the approximate expression of ${P}_{laser}$ is given as

*ω*is the emission frequency of fiber ring laser,

*λ*is the wavelength of the laser,

*c*denotes the velocity of light in vacuum, ${P}_{0}$ is the power without optical feedback,

*m*is constant coefficient.

## 3. Theory of Fourier Transform method for vibration measurement

Fourier transform (FT) method can be adopted for vibration measurement when the target vibrates sinusoidally. Assumed that the frequency and amplitude of the vibration are ${f}_{0}$ and ${A}_{0}$ respectively, the distance between the circulator end and the target ${L}_{ext}$ can be expressed as

where ${L}_{0}$ denotes the initial distance between the circulator end and the target,*t*is the time, and

*φ*is the initial phase.

Combining Eq. (1) and Eq. (2), the output of the laser can be written as

where*f*is its argument of frequency, ${J}_{2n}(a)$ and ${J}_{(2n+1)}(a)$ are even and odd order Bessel function of the first kind with argument

*a*. The modulus of $F(f)$ is expressed as

By Eq. (6), the frequency spectrum (FP) of SMI signal is composed by ${f}_{0}$ and its harmonic components. The amplitude of the *N*th-order harmonic component is proportional to the value of *N*th-order Bessel function of the first kind. Figure 2(a)
shows the value of Bessel function changing with the order *n* and the argument *a*. For a certain *a*, the value of Bessel function is promptly decayed after a certain *n*. The relationship between the certain *n* and *a* is approximately linear, shown in Fig. 2(a). For a certain *a*, the maximum *n* satisfying ${J}_{n}(a)\ge \xi $ is named stop order. For $\xi =0.001$, the expression of the fitted linear line which is shown in Fig. 2(b) is as

According to Eq. (1),the intensity of fiber laser is a cosine variation with ${L}_{ext}$ at weak feedback, which is similar to DFB laser. The simulative output powers of fiber laser are shown in Fig. 3(a) . ${L}_{ext}$ is modulated by a sinousoidal signal, and the modulation amplitudes are 3μm, 2μm and 1μm respectively for $\lambda =1.55$μm and ${f}_{0}=4$Hz. The sampling frequency is 5000Hz and the number of sampling points is 2500. The inclination has not been found from the simulative signals, which is distinguished from DFB lasers [13]. Figure 3(b) shows the FP of SMI signals. We can find that the maximum order of harmonic components decreases with the decrease of ${A}_{0}$. The basic frequency is 3.7Hz, so the relative error of ${f}_{0}$ is 7.5%. The stop orders are 31, 23 and 14, and the calculated ${A}_{0}$ are 2.90μm, 2.00μm and 1.02μm respectively.

## 4. Error analysis

According to Eq. (10), if ${\varphi}_{0}=k\pi ,k=0,1,2,\cdots ,$ the amplitude of the basic frequency and odd harmonic components are 0. This must be avoided in experiment. According to Eq. (7), the relationship between $\delta {A}_{0}$ and $\delta n$can be expressed as

The error transferred to ${A}_{0}$ from *n* is very small. ${A}_{0}$changes $\lambda /14.82$ when *n* changes 1. The theoretical error of FT method is $\lambda /14.82$, less than $\lambda /4$ in fringe counting method.

For $\lambda =1.55$μm, ${f}_{0}=4$Hz and ${A}_{0}$ changing from 0.1μm to 3μm with the step of 0.1μm, the demodulated frequency is 3.7Hz. The maximum error of ${A}_{0}$ is 0.12μm which is about $\lambda /10$. The relative error is larger when the amplitude of vibration is less than 1μm, so the low limit of the amplitude is 1μm. For ${A}_{0}=2$μm and ${f}_{0}$ changing from 1Hz to 60Hz with the step of 5Hz, the demodulated amplitude is 2.00μm when ${f}_{0}\ge 5$ Hz. The demodulated amplitude is 1.67μm when ${f}_{0}=1$ Hz. Both the errors of ${A}_{0}$ and that of ${f}_{0}$ are larger for ${f}_{0}=1$Hz, because at a certain sampling frequency the lower ${f}_{0}$ is, the lower the accuracy of frequency is. The number of points of FT also impacts the resolution of frequency. By Eq. (8) the sampling frequency ${N}_{f}$ must satisfy the condition as following

The maximum measurement frequency depends on the sampling frequency and the amplitude of the vibration.

## 5. Experiment results

The experiment setup is shown in Fig. 4 . The pump light of 208mW was coupled into the ring through a wavelength-division multiplexer (WDM). One of the output ports of coupler was connected to a photodetector (PD) for observation and the other took the signal back into the ring. Port 2 of circulator A was connected to a fiber Bragg grating (FBG) with center wavelength of 1549.94nm and 3dB width of 0.012nm to stabilize the laser emission frequency. Because of the narrow line width of FBG, the mode hopping is neglected. The feedback light reflected by the target was connected into the system through port 2 of circulator B. The target was fixed on a piezoceramic transducer (PZT). The output power was about 10 dBm without feedback.

Figure 5(a) shows a experimental signal of SMI. A sinusoidal signal was launched to PZT with the frequency of 5Hz and driving voltage of 20V. The amplitude of PZT is proportional to the driving voltage and the amplitude of PZT at 20V corresponds to 1.32μm. Noise is inevitable in experimental system, so ξ is increased properly in order to increase the accuracy of the stop order. Figure 5(b) shows the frequency spectrum of the signal. The basic frequency is 4.9Hz and the demodulated amplitude is 1.30μm.

Changing the driving voltage of PZT when the frequency was 5Hz, the demodulated results are listed in Table 1
. The first row are the driving voltages increasing from 20 to 60V with step of 5V, the second row are the reference amplitudes, the third and last rows are the measurement results and errors. The maximum error is 0.14μm, less than *λ*/10.

According to Eq. (6), the initial phase of SMI signal at the basic frequency from its real and imaginary parts of the Fourier coefficient $\phi =\mathrm{arc}\mathrm{tan}[\mathrm{Re}F({f}_{0})/\mathrm{Im}F({f}_{0})]$. Figure 6 shows the reconstructive sinusoidal vibrations in experiment. The solid lines are reference vibrations of PZT and the broken curves are the reconstructive vibrations.

## 6. Conclusion

SMI in fiber ring laser is proposed to measure sinusoidal vibration and flexible vibration measurement is realized. FT method is used to increase the accuracy of the measurement. The precision of fringe counting method is λ/4 in theory and the precision of FT method is λ/14.82 without modulation. In experiment, the maximum error of the amplitude is about *λ*/10 and the maximum relative error of the frequency is about 10%. High SNR is obtained because of the high power of fiber laser. Fiber ring laser has higher output power than other DFB laser and its wavelength stability is similar to DFB laser. Moreover fiber ring laser has potential to support many channels with different wavelength because of its wide gain bandwidth and enough power. The main limitation is that the instantaneity of FT method is worse than fringe counting method. The system is sensitive to temperature because FBG which is used to stabilize the wavelength is sensitive to temperature. It is not obviously sensitive to surface or alignment.

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