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Frequency demodulation of an inhomogeneous medium multi-longitudinal mode fiber laser and its sensing application

Jian Xu, Tigang Ning, Jingjing Zheng, Li Pei, Jing Li, Jianshuai Wang, and Bing Bai

Open Access Open Access

Abstract

Dispersion characteristic could be a significant factor, which impacts the beat frequency of Multi-longitudinal mode fiber laser (MMFL). In this paper, the mechanism of beat frequency generation in inhomogeneous medium Multi-longitudinal mode fiber laser is discussed. Compared with cavity length-dependent fiber laser sensing system, the proposed model uses a several-millimeter-Fiber-Bragg-Grating (FBG) as the sensing head, which features both high sensitivity and compact size. We designed an experiment to exhibit possible sensing application based on the proposed theory as well.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Sensor based on Multi-longitudinal mode fiber laser has been researched in the past several years due to its merits of high sensitivity, high resolution, and ease of deployment. Various sensors have been proposed for the measurement of multiple physical parameters [1–4]. In the above works, the sensing information is obtained by directly monitoring the wavelength of the laser beam, while the measurement sensitivity and accuracy are largely limited due to the low resolution of the optical spectrum analyzer (OSA), which is typically 10 pm to 20 pm. There are many researches based on beat frequency (BF) demodulation [5–11], which could improve the sensitivity and resolution of the sensing system. However, lots of previous work, employ a cavity length-dependent fiber laser to sample the sensing parameter. The disturbance resulted from the environment (i.e., temperature, strain, etc.) leads to the change in the physical length of the laser cavity, ultimately causing the frequency shift of BF. Cavity length-dependent fiber laser sensing system needs a quite long fiber cavity to achieve high sensitivity, and it takes a gigantic space to deploy the sensing head.

In our previous work [12], we found that the dispersion characteristic could be a significant factor, which impacts the beat frequency of Multi-longitudinal mode fiber laser (MMFL), and a simplified theory has been proposed and experimentally demonstrated. Fiber laser sensing system based on this theory, which only employs a several-millimeter–Fiber-Bragg-Grating (FBG) as the sensing head, could obtain a several dozen times higher sensitivity compared with a traditional cavity length-dependent fiber laser sensing system.

In this paper, the mechanism of BF generation in inhomogeneous medium MMFL has been discussed and a more accurate model has been proposed. We also designed an experiment to demonstrate the feasibility of fiber laser sensing system based on the proposed model.

2. Theory and model

2.1 Longitudinal mode distribution in medium-filled cavity

Considering plane wave operating within the laser cavity, which is shown in Fig. 1. Two ideal mirrors (thickness = 0) are placed at each end of the cavity. The cavity is essentially a Fabry-Perot(FP) resonator, and the stimulated emission can be amplified and output stably only when the wavelength of the stimulated laser beam meets the condition of constructive interference:

$$\Delta \varPhi = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot 2nL = q \cdot \mathrm{2\pi }$$
where ΔΦ is the phase change for a round-trip, λq is the wavelength of laser in vacuum, q is an integer number, L is the physical length of the cavity, n is the refractive index (RI) of the medium. nL is the optical path length separating the mirrors. The constructive interference occurs when an integral multiple of half-wavelengths fit into the cavity spacing of length L. λq’=λq/n, which is the wavelength of laser in a certain medium.

 figure: Fig. 1.

Fig. 1. Standing wave in homogeneous medium cavity.

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Equation (1) can be rewritten as:

$${\nu _q} = \frac{{q\textrm{c}}}{{2nL}}$$

νq is the frequency of the stimulated laser. This indicates that laser exists within the cavity in the form of standing wave, and each standing wave is known as a longitudinal mode. For a given homogeneous medium cavity, there are many longitudinal modes, and the frequency difference between two adjacent modes can be expressed as:

$$\Delta \nu = \frac{\textrm{c}}{{2nL}}$$

In homogeneous medium cavity, the frequency difference between two adjacent modes is equal. If there are two successive mediums with different RI in the cavity, which is shown in Fig. 2:

 figure: Fig. 2.

Fig. 2. Standing wave in inhomogeneous medium cavity.

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Equation (1) can be rewritten in this situation:

$$\Delta \varPhi = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot (2{n_1}{L_1} + 2{n_2}{L_2}) = q \cdot \mathrm{2\pi }$$

In different medium, the wavelength of light changes while the frequency doesn’t, Thus, Eq. (3) can be expressed as:

$${\nu _q} = \frac{{q\textrm{c}}}{{2({n_1}{L_1} + {n_2}{L_2})}}$$

2.2 Longitudinal mode distribution in Fiber Bragg Grating Fabry-Perot (FBG-FP) cavity and the generation of beat frequency

We have built up an MMFL employing a chirped FBG (CFBG), a length of Er-doped fiber, and a uniform FBG (UFBG) as the cavity in our previous work, which is sketched in Fig. 3.

 figure: Fig. 3.

Fig. 3. FBG-FP cavity composed of CFBG and UFBG

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 figure: Fig. 4.

Fig. 4. The simplified FBG-FP cavity model.

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Clearly, both CFBG and UFBG are inhomogeneous medium, and the RI modulation is induced by the UV exposure. Here we assume for simplicity that in an infinitesimal length ΔL, the RI keeps a constant value n(L). The transmission time in an infinitesimal length ΔL, can be expressed as:

$$\textrm{d}\tau = \frac{{n(L)\textrm{d}L}}{\textrm{c}}$$

The total transmission time (single trip) in the whole FBG region:

$$\tau = \int {\textrm{d}\tau } = \frac{{\int {n(L)\textrm{d}L} }}{\textrm{c}}$$

The FBG-FP cavity could be simplified to an inhomogeneous medium cavity with two ideal mirrors placed at both end, see Fig. 4:

When a beam of laser operates within such a cavity, the phase changes of the standing wave in a round trip should meet the following condition:

$$\Delta \varPhi = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot 2[\int\limits_0^{{L_\textrm{C}}} {{n_\textrm{C}}(L)\textrm{d}L + } {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \int\limits_0^{{L_\textrm{U}}} {{n_\textrm{U}}(L)\textrm{d}L]} = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot 2(\textrm{c}{\tau _\textrm{C}} + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}) = q \cdot \mathrm{2\pi }$$
Where LC, LEr, LU are the physical length of CFBG, Er-doped fiber, UFBG, respectively. nC(L), nEr, nU(L) are the RI distribution of CFBG, Er-doped fiber, UFBG, respectively. Actually, the transmission time τ in CFBG or UFBG is wavelength-dependent, so the frequency of longitudinal mode ultimately can be rewritten as:
$${\nu _q} = \frac{{q\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}(\nu ) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}(\nu )]}}$$
Where τC and τU are light transmission time in CFBG and UFBG, respectively. The precise value of τ can be measured using an optical vector analyzer (OVA), for example, LUNA OVA 5000. The measurement result τtest, generally named “group delay” of an FBG, is the delay time when a beam of light operates a round-trip in FBG. Thus, τ=1/2τtest.

The frequency spacing between two adjacent longitudinal modes is no longer an equivalent value due to the various group delay of CFBG and UFBG, which is sketched in Fig. 5.

 figure: Fig. 5.

Fig. 5. Longitudinal mode distribution in FBG-FP cavity.

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The frequency spacing between two adjacent longitudinal modes in Fig. 5 can be expressed as:

$$\scalebox{0.92}{$\displaystyle\Delta {\nu _{i + n}} = \frac{{(i + n + 1)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + n + 1}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + n + 1}})]}} - \frac{{(i + n)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + n}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + n}})]}},(i,n = 1,2,3, \cdots )$}$$
Δνi + n refers to the frequency spacing between longitudinal mode Δνi + n and Δνi + n + 1. This indicates that the frequency spacing between two adjacent longitudinal modes is wavelength- dependent as well.

Normally, each longitudinal mode oscillates independently of other modes. The combined total output of the laser is the composition of all the longitudinal modes, whose amplitude can be expressed as:

$$E(t) = \sum\limits_{q = 1}^N {{E_q}\cos (2\pi {\nu _q}t + {\varphi _q})}$$
Where Eq is the amplitude of the longitudinal mode q, φq is the phase of the longitudinal mode q. The intensity is given by the absolute square of total amplitude:
$$\begin{array}{l} I(t) = {|{E(t)} |^2} = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{E_i}{E_j}\cos (2\pi {\nu _i}t + {\varphi _i})\cos (2\pi {\nu _j}t + {\varphi _j})} } \\ = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{E_i}{E_j}\frac{1}{2}\{ \cos [2\pi ({\nu _i} + {\nu _j})t + ({\varphi _i} + {\varphi _j})] + \cos [2\pi ({\nu _i} - {\nu _j})t + ({\varphi _i} - {\varphi _j})]\} } } \end{array}$$
When using a photodetector (PD) to convert the laser beam into radiofrequency (RF) signal, only low-frequency component left due to the PD has a certain frequency upper limit. The RF signal therefore can be expressed as:
$$P(t) = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{E_i}{E_j}\frac{1}{2}\{ \cos [2\pi ({\nu _i} - {\nu _j})t + ({\varphi _i} - {\varphi _j})]\} } }$$

This is the beat frequency of two different longitudinal modes, νi-νj. A simplified situation that the generation of beat frequency between four longitudinal modes is sketched in Fig. 6.

 figure: Fig. 6.

Fig. 6. The generation of beat frequency.

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The order number of longitudinal mode beat frequency means that the mode numbers interval between the two longitudinal modes. For example, the first order longitudinal mode beat frequency is generated between νi and νi + 1, νi + 1 and νi + 2, νi + 2 and νi + 3. This can be extended to the situation where the number of modes is infinite. There are various signal peaks for the same order beat frequency, due to the frequency spacing between two longitudinal modes is nonuniform distribution, which is different from homogeneous medium fiber laser.

2.3 Wavelength induced longitudinal mode beat frequency shift

Rewriting Eq. (10), the longitudinal mode beat frequency between two casual longitudinal modes can be expressed as:

$${\nu _{i,j}} = \frac{{i\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} - \frac{{j\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _j}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _j})]}},(i,j = 1,2,3, \cdots )$$

If the frequency spacing between longitudinal mode i and j is close, and the group delay of FBGs at frequency νi and νj is approximately equal (τC(νi)≈τC(νj), τU(νi)≈τU(νj)), Eq. (14) degenerates to:

$${\nu _{i,j}} = \frac{{(i - j)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} = \frac{{(i - j)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _j}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _j})]}},(i,j = 1,2,3, \cdots )$$

This is very similar to the situation in homogeneous medium fiber laser. But the same order longitudinal mode beat frequency for different i and j is still varied, though the frequency difference between two longitudinal mode beat frequencies at same order is far less than those at different orders.

In such an inhomogeneous medium Multi-longitudinal mode fiber laser system, the dispersion characteristic is a significant factor, impacts the longitudinal mode beat frequency. If the bandwidth of UFBG is narrower than it of CFBG, the gain bandwidth is related to UFBG, the bandwidth of output laser spectrum is closely equal to the bandwidth of UFBG (it still determined by the bandwidth of UFBG and total losses of cavity, but the former is primary factor). The resulting laser modes in such situation is sketched in Fig. 7.

 figure: Fig. 7.

Fig. 7. The longitudinal modes with wavelength-dependent.

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UFBG is employed to the sensing head to convert the environment distribution into the wavelength shift of spectrum, and ultimately leads to the frequency change of longitudinal modes (Fig. 7, black line to red line). The frequency spacing of each longitudinal mode varies in such procedure due to the group delay of cavity is wavelength-dependent and the resulted longitudinal mode beat frequency shifts as well. If the total group delay (τC+τU) increases with the wavelength, the longitudinal mode frequency would conversely decrease, which is described in Eqs. (14) and (15).

2.4 Improving the sensitivity and stability

Taking the derivative of Eq. (15) on frequency ν, after some rearrangement we finally get:

$$\frac{{\textrm{d}{\nu _{i,j}}}}{{\textrm{d}\nu }} ={-} {\nu _{i,j}} \cdot \frac{{\textrm{c}[\frac{{\textrm{d}{\tau _\textrm{C}}({\nu _i})}}{{\textrm{d}\nu }} + \frac{{\textrm{d}{\tau _\textrm{U}}({\nu _i})}}{{\textrm{d}\nu }}]}}{{\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})}}$$

The slope rate of νi,j is mainly determined by the dispersion characteristic (group delay) of CFBG and UFBG. Generally speaking, dτC/dλ equals a constant value and dτU/dλ is approximately zero (not exceeding the 3-dB bandwidth) [12,13], the beat frequency sensitivity of such a laser sensing system, therefore, is depending on the dispersion rate of CFBG, which is strongly related to the fabrication. The sensitivity can be easily improved by using a CFBG with higher dispersion rate.

However, the extremely high dispersion rate leads to the instability of longitudinal mode beat frequency due to the ununiform distribution of each longitudinal mode, see Fig. 8.

 figure: Fig. 8.

Fig. 8. The beat frequency “group”.

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Considering a simplified situation of only first-order longitudinal mode beat frequency exists. The generated longitudinal mode beat frequencies have multiple peaks due to the frequency spacing of longitudinal mode is not equal. The beat frequency of same order exists in a form of a “group”. The difference between maximum beat frequency and minimum beat frequency of same order can be weighted by a parameter νBW, which is the bandwidth of corresponding beat frequency.

If some longitudinal mode disappeared for some uncertain reason (i.e., hole burning effect), the corresponding beat frequency would disappear simultaneously (blued mode and beat in Fig. 8), which leads to the drift of beat frequency “group”. This greatly affects the accuracy and stability of such laser sensing system.

The bandwidth of longitudinal mode frequency, νBW, should be minimized to avoid the above problem. We can learn from Fig. 6 that there are fewer beat frequencies in a “group” with higher order number, thus the bandwidth is narrower, which is helpful to increase the stability. Besides, νBW can be expressed as:

$${\nu _{\textrm{BW}}} = {\nu _{i,j,\textrm{MAX}}} - {\nu _{i,i,\textrm{MIN}}} \propto \frac{{\textrm{d}{\nu _{i,j}}}}{{\textrm{d}\nu }},(i,j = 1,2,3, \cdots )$$

The sensitivity and stability, therefore, are contradictions in a certain degree.

2.5 Bi-longitudinal mode fiber laser

If carefully choosing the length of cavity and the bandwidth of UFBG, causing there are only two longitudinal modes in the laser, which leads to only one beat frequency, which is shown in Fig. 9.

 figure: Fig. 9.

Fig. 9. Beat frequency of Bi-longitudinal mode fiber laser.

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Only one longitudinal mode beat frequency is generated in this situation and it helps improve the stability even though a considerably high dispersion rate CFBG is used to obtain higher sensitivity. This Bi-longitudinal mode laser can be achieved when the following conditions are satisfied:

  • 1. The frequency spacing between two casually adjacent longitudinal modes is less than the bandwidth of gain curve (the bandwidth of stimulated laser, approximately equals to the bandwidth of UFBG):
    $$\Delta {\nu _{i + 1}} = \frac{{(i + 1)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + 1}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + 1}})]}} - \frac{{i\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} < \Delta {\nu _{\textrm{UFBG}}}$$
  • 2. The frequency spacing between three casually adjacent longitudinal modes is great than the bandwidth of gain curve:
    $$\Delta {\nu _{i + 2}} = \frac{{(i + 2)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + 2}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + 2}})]}} - \frac{{i\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} > \Delta {\nu _{\textrm{UFBG}}}$$

There are always two longitudinal modes in this situation when the above conditions are satisfied, leading to the unique longitudinal mode beat frequency to be detected, which is beneficial to improve the stability of the laser system.

3. Experiment

3.1 Configuration of the experiment

We have designed an experiment to prove the feasibility in sensing application of the proposed laser system. Figure 10 shows the structure of the experiment system. The fiber laser was mainly composed of a CFBG with high dispersion rate, a length of Er-doped fiber, and a UFBG. A 974 nm semiconductor laser was employed as pump, and the 974 nm pump light was launched into the resonant cavity of the laser through a wavelength division multiplexer (WDM) and an isolator. The stimulated laser (1550 nm) was split into two branches. One branch was received by an optical spectrum analyzer (OSA, YOKOGAWA AQ6370D), and the other was projected into a photodetector (PD, CONQUER KG-PD-20 G, Amplifier, ANRITSU G3H84), then monitored using an electrical spectrum analyzer (ESA, CEYEAR 4051E-S).

 figure: Fig. 10.

Fig. 10. The structure of the sensing system.

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The characteristic of CFBG has been reported in our previous paper [12]. The length of Er-doped fiber is ∼1.85 m, and the total length of cavity is ∼2 m, including a few single mode fiber which is connected to FBG. The central wavelength and 3-dB bandwidth of CFBG are ∼1549.9 nm and ∼0.5 nm, respectively. The dispersion rate of CFBG is approximately 1800ps/nm, with nonlinearity and ripple. The central wavelength and 3-dB bandwidth of UFBG are ∼1549.6 nm and ∼0.08 nm, respectively. The spectra and group delay of two FBGs were tested and sketched in Fig. 11. In fact, the precise value of dispersion rate is not required for this laser sensing system in engineering applications, which is also the advantage of this system.

 figure: Fig. 11.

Fig. 11. Spectra and group delay: (a) CBFG, (b) UFBG.

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The UFBG was bonded on a Terfenol-D rod via UV glue as a transducer (Fig. 12), which converts the change of external magnetic field into the deformation of UBFG, ultimately leading to the wavelength shift of UFBG. This finally causes the shift of longitudinal mode beat frequencies of the laser according to the above theory.

 figure: Fig. 12.

Fig. 12. The photograph of transducer.

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A solenoid coil system was employed to generate the controllable magnetic field, and the intensity of the generated magnetic field can be controlled by regulating the current flowing through the coil. The sensing head transducer was put in the center of the coil. The intensity of generated magnetic field has been set to 0 mT to 30 mT with a step of 5 mT in the experiment, which is essentially controlled by adjusting the current.

3.2 Experimental result

The spectrum of output laser beam is shown in Fig. 13 (a). The original spectra of UFBG and laser are quite similar in shape, and the shift between two spectra is resulted from the stretch of UFBG when bonding it on Terfenol-D rod.

 figure: Fig. 13.

Fig. 13. (a) The spectrum of stimulated laser, (b) The Frequency spectrum.

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Figure 13(b) shows the frequency spectrum of longitudinal mode beat frequency ranging 0 Hz to 2 GHz. It shows that the intensity and the signal-to-noise ratio (SNR) of the longitudinal mode beat frequency decrease with the frequency, which leads to the fact that the higher longitudinal mode beat frequency is not qualified for the measurement.

The spectra of stimulated laser were recorded as well when the external magnetic field increasing, which is shown in Fig. 14 (a). The wavelength shift of laser, is caused by the deformation of Terfenol-D rod, which is essentially due to the increase of the magnetic field. Figure 14 (b) shows the wavelength response to the magnetic field, which performs satisfying linearity. The slope of the response curve is approximately 9.57pm/mT, which is also the optical sensitivity of the proposed sensing system.

 figure: Fig. 14.

Fig. 14. (a) The spectra of laser with different magnetic field (b) the wavelength response.

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A longitudinal mode beat frequency of ∼250 MHz was chosen to be measured as a tread off, which has both enough SNR and sensitivity. The corresponding frequency spectrum was recorded at least three times when the intensify of magnetic field varies every 2.5 mT. The recorded frequency spectra were plotted in Fig. 15(a) (The spectrum with median central frequency was selected from the recorded data).

 figure: Fig. 15.

Fig. 15. (a) The frequency with different magnetic field, (b) The frequency response to the magnetic field.

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Figure 15 (b) shows the frequency response to the magnetic field of the laser sensing system. It was plotted with error bar, which is calculated by the multiple data recorded at the same intensify of magnetic field. Cyan-filled zone refers to the measured lower limit frequency and green-filled zone refers to the measured upper limit frequency, respectively.

3.3 Discussions about the result

To validate the proposed theory, we have developed a brief MATLAB program as we show in Code 1 (Ref. [14]) to calculate the expected frequency of LMBF using the tested group delay data of FBGs. Figure 16 shows the comparison between calculated LMBF and tested data. When the magnetic field is weak, the shift of calculated LMBF is tiny, which shows good coincidence with experiment. The prediction by the theory behaves more errors when the magnetic field exceeding 20 mT. However, the theory performs satisfying feasibility overall, and we believe there is still some unknow factors impacts the generation of LMBFs in fiber laser, and the essential mechanism of the LMBF in inhomogeneous medium deserves further research.

 figure: Fig. 16.

Fig. 16. The comparison between calculated LMBF and tested data.

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The response curve shows linearity only ranging 12.5 mT to 22.5 mT. This is partly because the unique characteristic of Terfenol-D, which is not linear proportional to the magnetic field. To be exactly, the deformation of Terfenol-D rod would be extremely small when the intensity of magnetic field not exceeding a certain value, which is the threshold of it. Thus, the shift of frequency is not distinct when the magnetic field is too weak.

Besides, a noticeable drift at each point occurs on the response curve. Figure 15 (b) shows the error bars as well, which indicates that, the frequency would drift though there is no change in magnetic field. This may cause by the competition between longitudinal modes. The frequency spacing between longitudinal modes is nonuniform, leading to the longitudinal mode beat frequency “group”, which has been illustrated in section 2.4. Indeed, it’s a defect of the sensing system which obstructs its application in reality and engineering. However, this work mainly aims to prove the feasibility of the proposed theory in sensing application, rather than makes it in practicality.

Figure 17 is a simple illustration of the response stability. The data was tested when the magnetic field was set to 8 mT, and the spectrum was recorded every 30s, sustained for 13 minutes. The central frequencies drift approximately 0.3 MHz, ∼1.5 mT when converted to magnetic field according to the sensitivity of 204KHz/mT. This is a considerable measurement error.

 figure: Fig. 17.

Fig. 17. (a) The spectra when the magnetic field is 8 mT, (b) The central frequency versus time.

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4. Conclusion

In this work, the generate mechanism of longitudinal mode beat frequency in inhomogeneous medium Multi-longitudinal mode fiber laser is discussed and experimentally illustrated. Compared with previous fiber laser sensing system, our work features higher sensitivity and more compact size, which is detailed in Table 1.

Tables Icon

Table 1. The comparison between this work and previous technology

View Table

The conventional fiber laser system converts the parameter under test into the physical length change of Fiber laser cavity, which ultimately leads to the shift of laser wavelength or beat frequency shift. Massive lengths of fiber are required is such fiber laser sensing system to achieve high sensitivity. Our work performs better sensitivity and more compact size due to the different sensing mechanism.

The stability of this work is reasonable low. The sensitivity and stability are a pair of irreconcilable contradictions, which has been discussed in section 2.4 and 2.5. We believe that the proposed sensing system would be extensive used in the future due to its high sensitivity and compact size.

Funding

State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (RCS2019ZZ007); National Natural Science Foundation of China (62005012).

Acknowledgements

This work is jointly supported by the National Natural Science Foundation of China (62005012), he State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (Contract No.RCS2019ZZ007).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. M. A. Gonzalez-Reyna, E. Alvarado-Mendez, J. M. Estudillo-Ayala, E. Vargas-Rodriguez, M. E. Sosa-Morales, J. M. Sierra-Hernandez, D. Jauregui-Vazquez, and R. Rojas-Laguna, “Laser Temperature Sensor Based on a Fiber Bragg Grating,” IEEE Photonics Technol. Lett. 27(11), 1141–1144 (2015). [CrossRef]  

2. X. Bai, J. Yuan, J. Gu, S. Wang, Y. Zhao, S. Pu, and X. Zeng, “Magnetic Field Sensor Using Fiber Ring Cavity Laser Based on Magnetic Fluid,” IEEE Photonics Technol. Lett. 28(2), 115–118 (2016). [CrossRef]  

3. C. Sun, M. Wang, J. Liu, S. Ye, L. Liang, and S. Jian, “Fiber Ring Cavity Laser Based on Modal Interference for Curvature Sensing,” IEEE Photonics Technol. Lett. 28(8), 923–926 (2016). [CrossRef]  

4. Z.-B. Liu, Z. Tan, B. Yin, Y. Bai, and S. Jian, “Refractive index sensing characterization of a singlemode–claddingless–singlemode fiber structure based fiber ring cavity laser,” Opt. Express 22(5), 5037–5042 (2014). [CrossRef]  

5. X. Tong, Y. Shen, X. Mao, C. Yu, and Y. Guo, “Fiber-optic temperature sensor based on beat frequency and neural network algorithm,” Opt. Fiber Technol. 68, 102783 (2022). [CrossRef]  

6. Z. Yin, L. Gao, S. Liu, L. Zhang, F. Wu, L. Chen, and X. Chen, “Fiber Ring Laser Sensor for Temperature Measurement,” J. Lightwave Technol. 28(3), 3403–3408 (2010). [CrossRef]  

7. L. Huang, L. Qian, L. Chen, L. Gao, and X. Chen, “Multilongitudinal mode fiber laser sensor for temperature measurement,” in 2012 Asia Communications and Photonics Conference (ACP), (2012), 1–3.

8. H. Zhang, J. Luo, B. Liu, S. Wang, C. Jia, and X. Ma, “Polarimetric multilongitudinal-mode distributed Bragg reflector fiber laser sensor for strain measurement,” Microw. Opt. Technol. Lett. 51(11), 2559–2563 (2009). [CrossRef]  

9. L. Gao, L. Chen, L. Huang, and X. Chen, “Multimode fiber laser for simultaneous measurement of strain and temperature based on beat frequency demodulation,” Opt. Express 20(20), 22517–22522 (2012). [CrossRef]  

10. L. Gao, L. Chen, L. Huang, S. Liu, Z. Yin, and X. Chen, “Simultaneous Measurement of Strain and Load Using a Fiber Laser Sensor,” IEEE Sens. J. 12(5), 1513–1517 (2012). [CrossRef]  

11. S. Liu, Z. Yin, L. Zhang, L. Gao, X. Chen, and J. Cheng, “Multilongitudinal mode fiber laser for strain measurement,” Opt. Lett. 35(6), 835–837 (2010). [CrossRef]  

12. J. Xu, T. Liao, J. Zhang, L. Liu, W. Zhang, J. Zheng, L. Pei, J. Li, J. Wang, and T. Ning, “Multi-longitudinal mode fiber laser sensing system based on resonant cavity dispersion-frequency mapping,” Opt. Fiber Technol. 73, 103045 (2022). [CrossRef]  

13. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]  

14. J. Xu, “LMBF calculator,” figshare, (2022), https://figshare.com/articles/software/code_zip/21617271/2

Supplementary Material (1)

NameDescription
Code 1       LMBF calculator

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (17)
Fig. 1.
Fig. 1. Standing wave in homogeneous medium cavity.

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Fig. 2.
Fig. 2. Standing wave in inhomogeneous medium cavity.

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Fig. 3.
Fig. 3. FBG-FP cavity composed of CFBG and UFBG

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Fig. 4.
Fig. 4. The simplified FBG-FP cavity model.

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Fig. 5.
Fig. 5. Longitudinal mode distribution in FBG-FP cavity.

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Fig. 6.
Fig. 6. The generation of beat frequency.

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Fig. 7.
Fig. 7. The longitudinal modes with wavelength-dependent.

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Fig. 8.
Fig. 8. The beat frequency “group”.

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Fig. 9.
Fig. 9. Beat frequency of Bi-longitudinal mode fiber laser.

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Fig. 10.
Fig. 10. The structure of the sensing system.

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Fig. 11.
Fig. 11. Spectra and group delay: (a) CBFG, (b) UFBG.

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Fig. 12.
Fig. 12. The photograph of transducer.

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Fig. 13.
Fig. 13. (a) The spectrum of stimulated laser, (b) The Frequency spectrum.

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Fig. 14.
Fig. 14. (a) The spectra of laser with different magnetic field (b) the wavelength response.

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Fig. 15.
Fig. 15. (a) The frequency with different magnetic field, (b) The frequency response to the magnetic field.

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Fig. 16.
Fig. 16. The comparison between calculated LMBF and tested data.

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Fig. 17.
Fig. 17. (a) The spectra when the magnetic field is 8 mT, (b) The central frequency versus time.

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Tables (1)

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Table 1. The comparison between this work and previous technology

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Equations (19)

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$$\Delta \varPhi = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot 2nL = q \cdot \mathrm{2\pi }$$
$${\nu _q} = \frac{{q\textrm{c}}}{{2nL}}$$
$$\Delta \nu = \frac{\textrm{c}}{{2nL}}$$
$$\Delta \varPhi = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot (2{n_1}{L_1} + 2{n_2}{L_2}) = q \cdot \mathrm{2\pi }$$
$${\nu _q} = \frac{{q\textrm{c}}}{{2({n_1}{L_1} + {n_2}{L_2})}}$$
$$\textrm{d}\tau = \frac{{n(L)\textrm{d}L}}{\textrm{c}}$$
$$\tau = \int {\textrm{d}\tau } = \frac{{\int {n(L)\textrm{d}L} }}{\textrm{c}}$$
$$\Delta \varPhi = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot 2[\int\limits_0^{{L_\textrm{C}}} {{n_\textrm{C}}(L)\textrm{d}L + } {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \int\limits_0^{{L_\textrm{U}}} {{n_\textrm{U}}(L)\textrm{d}L]} = \frac{{\mathrm{2\pi }}}{{{\lambda _q}}} \cdot 2(\textrm{c}{\tau _\textrm{C}} + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}) = q \cdot \mathrm{2\pi }$$
$${\nu _q} = \frac{{q\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}(\nu ) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}(\nu )]}}$$
$$\scalebox{0.92}{$\displaystyle\Delta {\nu _{i + n}} = \frac{{(i + n + 1)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + n + 1}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + n + 1}})]}} - \frac{{(i + n)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + n}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + n}})]}},(i,n = 1,2,3, \cdots )$}$$
$$E(t) = \sum\limits_{q = 1}^N {{E_q}\cos (2\pi {\nu _q}t + {\varphi _q})}$$
$$\begin{array}{l} I(t) = {|{E(t)} |^2} = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{E_i}{E_j}\cos (2\pi {\nu _i}t + {\varphi _i})\cos (2\pi {\nu _j}t + {\varphi _j})} } \\ = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{E_i}{E_j}\frac{1}{2}\{ \cos [2\pi ({\nu _i} + {\nu _j})t + ({\varphi _i} + {\varphi _j})] + \cos [2\pi ({\nu _i} - {\nu _j})t + ({\varphi _i} - {\varphi _j})]\} } } \end{array}$$
$$P(t) = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{E_i}{E_j}\frac{1}{2}\{ \cos [2\pi ({\nu _i} - {\nu _j})t + ({\varphi _i} - {\varphi _j})]\} } }$$
$${\nu _{i,j}} = \frac{{i\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} - \frac{{j\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _j}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _j})]}},(i,j = 1,2,3, \cdots )$$
$${\nu _{i,j}} = \frac{{(i - j)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} = \frac{{(i - j)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _j}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _j})]}},(i,j = 1,2,3, \cdots )$$
$$\frac{{\textrm{d}{\nu _{i,j}}}}{{\textrm{d}\nu }} ={-} {\nu _{i,j}} \cdot \frac{{\textrm{c}[\frac{{\textrm{d}{\tau _\textrm{C}}({\nu _i})}}{{\textrm{d}\nu }} + \frac{{\textrm{d}{\tau _\textrm{U}}({\nu _i})}}{{\textrm{d}\nu }}]}}{{\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})}}$$
$${\nu _{\textrm{BW}}} = {\nu _{i,j,\textrm{MAX}}} - {\nu _{i,i,\textrm{MIN}}} \propto \frac{{\textrm{d}{\nu _{i,j}}}}{{\textrm{d}\nu }},(i,j = 1,2,3, \cdots )$$
$$\Delta {\nu _{i + 1}} = \frac{{(i + 1)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + 1}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + 1}})]}} - \frac{{i\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} < \Delta {\nu _{\textrm{UFBG}}}$$
$$\Delta {\nu _{i + 2}} = \frac{{(i + 2)\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _{i + 2}}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _{i + 2}})]}} - \frac{{i\textrm{c}}}{{2[\textrm{c}{\tau _\textrm{C}}({\nu _i}) + {n_{\textrm{Er}}}{L_{\textrm{Er}}} + \textrm{c}{\tau _\textrm{U}}({\nu _i})]}} > \Delta {\nu _{\textrm{UFBG}}}$$
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