In this section, a theoretical mode of DFB fiber laser were built on the effective cavity model, and then the three mirror cavity model of the self-mixing effect which has been extensively analyzed in the literatures [9] [10] [11] was use to approach and schematize.

DFB fiber laser provided lasing from the result of the signal generation by the gain medium and the feedback by the grating.

A single phase shift DFB fiber laser cavity and the three mirror cavity model are shown in Figure 1. The phase shift at z = z θ , this position determines the uni-directionality of the laser output [12]. P_pin and p_lout are the forward and

backward light power at z = 0, and p_rout and p_seed are the forward and backward light power at the fiber end facet z = z_L which reflected by the external reflector. In our simulation, pumped from left end, output is from right end, and the length of the Er-Yb co-doped fiber DFB laser is 50 mm, optimum coupling coefficient around 150 mm⁻¹ and the phase-shift located at z = 29 mm, L_efft is 6.5 mm. Also, output power depends on the grating strength κ ( z ) . Here grating is not apodized therefore coupling coefficient κ ( z ) is constant. Intensity builds around the place of phase shift in such a cavity, went into the gratings, intensity drops dramatically. So effectively signal is confined around phase-shift by two strong reflectors of length L₁ and L₂, thus the reflection coefficients of the grating approximated as [13]:

r i ≈ − tanh ( κ i L i ) , i = 1 , 2 , ⋯ (1)

For most practical applications, the gain is small compared to the coupling coefficient, Optimum position of phase-shift and optimum grating strength define the optimum reflectivities of these two grating segments on both sides of the phase shift.

The penetration depth D which is known as the effective distance that fields propagate into a grating, then the total length of the effective cavity could be calculated by added them up [13].

L e f f = D 1 + D 2 ≈ − ( r 1 2 κ 1 + r 2 2 κ 2 ) (2)

Here D₁ and D₂ are the penetration depths on the left and right of the phase shift into the grating.

In the F-P cavity, the signal light and the pump light transmitted in the fibers along the forward and backward, and reflected by the each end of the part of grating, thus analyzed the influence of pump power, internal loss and length of fiber grating in Er³⁺-Yb³⁺ co-doped DFB fiber lasers [14]. To analysis the effect of self-mixing, a seed light represented the feedback light from the external mirror [15] was set as shown in Figure 1.

Assuming that the external reflector was drove by a sinusoidal signal, so the length of external cavity changed periodically. The variable quantity of the external cavity can be expressed as follows:

Δ L e x t ( t ) = A cos ( ω 0 t + φ 0 ) (3)

ω 0 , φ 0 were the circular frequency and the initial phase of the driving signal.

In weak feedback regime, the effective reflection of the right end of the DFB FL could be expressed as:

r = r 2 + ( 1 − r 2 ) 2 r 1 exp ( − i ω τ ) (4)

ω is the frequency of the laser, c is the light velocity in vacuum, τ = 2L_ext/c is the external round-trip time. r 1 , r 2 are the reflectivity of the end of the grating and the surface of the feedback.

Thus, the seed light can be described as follows:

P s e e d = ( 1 − κ ) r 2 P s o u t (5)

κ is the coupling coefficient, P s o u t is the output power of the DFB fiber laser.

As our previous study [15], the laser output power is given by

p l a s e r = ε ( 1 − r 1 ) P r o u t exp ( − α s L + p s a b s / p s s + p p a b s / p s s ) (6)

ε is the total attenuation factor, α s is the small signal absortion coefficient, p s a b s , p p a b s are powers of the absorbed in one round trip, p s s is the power of the saturated absorption.

The output power of SMI signal can be obtained based on the above study was shown in Figure 2.

A self-mixing system using DFB fiber laser is showed in Figure 3. The pump light of 980 nm was coupled into the Er³⁺-Yb³⁺ codoped DFB fiber laser through a wavelength-division multiplexer (WDM 1) in the common port of the WDM.

A single longitudinal (SL) mode lasing (1550 nm) emitted from the DFB FL, coupled into another WDM 2. The signal arm focused onto a loud speaker through a collimated lens. The initial distance from the lens to target was 5.0 cm. On the surface of the speaker stuck a mirror, and a function generator (Tektronix AFG3102) was used to driver it. The monitoring arm was detected by a photo diode (PD) followed by a signal processing circuit, which includes transimpedance amplifier, filters and fur amplifier.

In the specific experiment of microvibration measurement based on the all fiber distributed feedback fiber laser self-mixing micro interference vibration sensor system, the feedback object is piezoelectric ceramic, the vibration range is from nanometer to centimeter, and its vibration amplitude is proportional to the driving voltage. Therefore, by changing the voltage of the driving signal provided by the signal generator, simple harmonic microvibration of nanometer can be achieved. However, due to the limitation of the range of the output signal of the signal generator (the minimum is 20 mV), the minimum amplitude of PZT that can be achieved in the experiment is 69 nm.

The spectral linewidth of laser was measured by a spectrograph (Yokogawa AQ6370C Optical Spectrum Analyzer) shown in Figure 4. In Figure 4, the 3 dB bandwidth of the laser is about 0.01 nm. But the real spectral bandwidth is much narrower than which is shown in Figure 4, because of the resolution ratio of the spectrograph is 0.02 nm, the measurement below it is not reliable.

Then the delayed self-homodyne measurement technique was used to investigate the spectral linewidth. The laser beam is split into two beams, one of them delayed by 20 km single mode fiber, that can generated the beat signal by interferometer. The result is shown in Figure 5. In Figure 5, the result was measured by a spectrum analyzer (Tektronix RSA-3408B Real-Time Spectrum Analyzer) using Exponential curve fitting method (for the output power in dBm as a unit) could get that the 3dB linewidth of the optimized system is about 0.5 kHz which is much narrow than the conditional DBR FL. Because longitudinal mode spacing is inversely proportional to the laser cavity length, that by decreasing cavity

length of longitudinal mode spacing is greater than the bandwidth of the frequency selective device, make only a longitudinal mode bandwidth conditions, stable single longitudinal mode output can be obtained, inhibit mode jump.

A highly sensitive all fiber distributed feedback fiber laser is used in the microvibration measurement system to improve the system sensitivity. The driving signal of piezoelectric ceramics in the experiment is shown in Figure 6(a) and Figure 6(b). The lower channel has three color sinusoidal curves: red dotted

line, blue dotted line and black solid line. The red dotted line, blue dotted line and black solid line correspond to different amplitudes respectively: 97 nm, 169 nm and 2200 nm. In Figure 6(a), the three curves of the upper channel correspond to the waveform of self-mixing signal generated under the driving limit of each amplitude simulated at 380 Hz. The black solid line corresponds to a typical self-mixing interference signal waveform whose amplitude exceeds a complete wavelength. Accordingly, the red dotted line and blue dotted line are the waveform of sinusoidal self-mixing interference signal when the amplitude is lower than 200 nm. The three curves of the upper channel in Figure 6(b) correspond to the experimental results corresponding to the simulation results displayed in the oscilloscope.

When the vibration amplitude of piezoelectric ceramics is lower than 200 nm, the waveform of self-mixing interference signal appears like a sinusoidal curve, and the peak value of the waveform is proportional to the actual amplitude of piezoelectric ceramics. Piezoelectric ceramics are directly driven by a function generator, which can provide a minimum vibration signal amplitude of 20 mW. The amplitude of piezoelectric ceramics is changed by changing the driving voltage, and the vibration amplitude is kept below the range of 200 nm, so as to realize the measurement of microvibration with different amplitudes.

The comparison diagram of experiment and simulation results is shown in Figure 7. The black curve corresponding to the left vertical axis represents the actual amplitude of PZT in the range below 200 nm. The blue dot corresponding to the right vertical axis represents the experimental measurement value calculated from the peak to peak value of the mixed signal power displayed by the oscilloscope at the same amplitude as the black dot. The experimental results of the measurements shown in Figure 7 are consistent with the trend of the simulation

results. Obviously, using our system, the amplitude range of measurement can be significantly expanded than that of similar systems reported previously (previously close to 400 nm, now expanded to 70 nm).

In order to better compare the actual amplitude of the experiment and PZT in Figure 7, an error chart was made for comparing the experimental data of self-mixing microvibration measurement with the actual amplitude of PZT, as shown in Figure 8. The maximum measurement error is 9.17 nm.

The error can be attributed to the following influencing factors: the self-mixing interference signal graph displayed in the oscilloscope is the electrical signal graph through the photodetector and signal processing circuit, which may cause signal distortion due to amplification and filtering, leading to errors; Different heat dissipation state and uneven temperature distribution will affect the shift of grating center wavelength in all fiber distributed feedback fiber laser, resulting in calculation error.

The traditional research of self-mixing interference vibration measurement technology is mainly carried out under the condition of space light feedback, and the feedback light is easy to be affected by the surrounding environment; At the same time, due to the phenomenon that a large amount of energy is dissipated and absorbed in the space propagation of light, and it is difficult to arrange an effective transmission optical path in the actual environment, it is difficult to achieve remote measurement. However, all optical fiber transmission can avoid the above problems. Therefore, a 20 km long single-mode optical fiber was introduced into the third port of the wavelength division multiplexer and the collimating lens to achieve remote microvibration measurement.

A radio-frequency (RF) spectrum analyzer (Tektronix RSA-3408B real-time spectrum analyzer) was used to get the spectrums of the self-mixing signal which showed in Figure 9.

When the external single-mode fiber is 20 km long, the output power of all fiber distributed feedback fiber laser decreases to one fifth of the initial value, so it is difficult to observe obvious self-mixing interference images on the oscilloscope. However, the laser light intensity can also be measured by the photoelectric detector, and the spectrograph displays the corresponding spectrum as shown in Figure 7. Due to the high zero noise power of the instrument, it is unable to display the low frequency noise part. The self-mixing signal diagram in the figure shows that the driving frequency is 100 Hz, the peak driving voltage is 6 V, and the corresponding piezoelectric ceramic amplitude is 288 nm. This is due to the additional loss and dispersion effect in the external optical fiber, which limits the measurement range of remote microvibration sensing.