The schematic diagram of the FCRD detection system active in the space domain is shown in Fig. 1. In FIG. 1, we can see that a Sagnac interference loop, with an asymmetrically integrated frequency shifter, constitutes a frequency-shifted interferometer.^{28}. An active fiber cavity is composed of two fiber couplers (C_{1} etc_{2}) with high split ratios, a sensor head (SH), a bidirectional erbium-doped fiber amplifier (Bi-EDFA), which is inserted into the frequency-shifted interferometer to form the active FCRD detection system in the space domain. The working principle of the active FCRD detection system in the space domain can be described as follows. The CW light generated by the tunable semiconductor laser (TSL) is coupled to the frequency-shifted interferometer via the circulator (Cir) and the C-fiber coupler_{0}. It produces two light beams circulating in the cavity of the fiber in counter-propagation directions. A Bi-EDFA, as shown in FIG. 1, is incorporated into the cavity of the fiber to amplify the intensity of the two light beams in the cavity. Therefore, it compensates for the inherent cavity loss, the light-sample interaction time is prolonged, and the two light beams decay more slowly. Small amounts of light beams escape from the fiber cavity and end up in the C-coupler_{0}, after having made each round trip in the cavity. The interference occurs between the two counter-propagating light beams exiting the fiber cavity after traveling the same number of round trips. If a frequency range of *Δf* is scanned from an initial frequency *F*_{0} over a period of *you*_{sw}the differential interference signal at the balanced detector (BD) will be a sinusoidal function of time *you*^{22.23}

$$begin{gathered} Delta I propto sumlimits_{m = 0}^{infty } {I_{m} cos left[ {2pi frac{{nleft( {mL + L_{0} } right)}}{c}f} right]} hfill {= }I_{m} cos left[ {2pi frac{{nleft( {mL + L_{0} } right)}}{c}(frac{Delta f}{{t_{sw} }}t + f_{0} )} right] hfill ;;; = i_{m} cos left[ {2pi cdot F_{m} cdot t + varphi_{0m} } right] hfill end{gathered}$$

(1)

where *not* is the refractive index of the single mode fiber, *L* is the length of the fiber cavity, *L*_{0} is a constant, *F* is the frequency offset generated by the frequency shifter, *vs* is the speed of propagation of light in vacuum, *φ*_{0m} is a fixed initial phase without consequence on our calculation, *F*_{m} =*not*(*mL*+*L*_{0}) *Δf* //( *side*_{sw}) (m = 0, 1, 2, …) is the frequency of oscillation which has a linear relationship with the number of round trips *m*^{22}. As*vs* is very large, the detection system can work under the low oscillation frequency *F*_{m}i.e. slow detection is achieved by using FSI and hence the cost is reduced *. I*_{m}is the intensity of the interference light after *m* round trips:

$$I_{m} = I_{0} cdot exp ( – malpha_{0} ) = I_{0} cdot exp ( – frac{l}{L}alpha_{0} )$ $

(2)

where *I*_{0} is the initial intensity, *I* = *mL* is the distance traveled by light in the fiber cavity, and (alpha_{0}) is the net loss in the cavity, which contains the total transmission loss of the fiber (alpha_{c}) and the gain *g* of Bi-EDFA in the fiber cavity, and can be given by:

$$alpha_{0} { = } , alpha_{c} – G{ = }alpha_{AR} { + }alpha_{ST} { + }alpha_{FS} { + }alpha_{IR } – G$$

(3)

where (alpha_{AR}) represents the fiber absorption loss, (alpha_{ST}) is the fiber diffusion loss, (alpha_{FS}) is the fiber fusion loss, and (alpha_{IR}) is the insertion loss of the components including the two fiber couplers, the bias controller and the sensor head (SH). (alpha_{c} { = }alpha_{AR} { + }alpha_{IR} { + }alpha_{ST} + alpha_{FS}) represents the inherent cavity loss, which is greater than the net cavity loss (alpha_{0}).

As can be seen from Eqs. (1)–(3), *I*_{m} is an exponential decay function of the propagation distance *I* while *F*_{m} is a linear function of*I* according to the report*I*= *mL.* Therefore, by linearly sweeping the frequency shift *F*and apply a Fast Fourier Transform (FFT) to Δ *I* we can obtain the Fourier spectrum of Δ *I* which display a series of exponentially decaying peaks located at*F*_{m}. After converting the frequency *F*_{m} at propagation distance *I*^{‘} using the formula *I*^{‘}=*I*+*L*_{0}=*heart rate*_{m}*/not*the Fourier spectrum of Δ*I*will become a ringing transient as the propagation distance^{22.23}, indicating that our proposed method belongs to a spatial domain FCRD technique performed using the FSI scheme. The distance required for the light intensity to decrease to 1/th of the initial light intensity can be defined as the ringing distance, which is analogous to the ringing time^{1}. When no external action is applied to the SH, the reduction distance (Lambda_{0}) can be written:

$$Lambda_{0} = frac{L}{{alpha_{0} }}$$

(4)

When an outside action*P*(magnetic field, pressure, deformation, etc.) is applied to the SH, additional loss (alpha_{s}) occurs, causing the ringing distance to change. In this case, the ringing distance (Lambda) can be rewritten as follows:

$$Lambda = frac{L}{{alpha_{c} + alpha_{s} – G}}{ = }frac{L}{{alpha_{c} + xi l_{s} P -G}}$$

(5)

where (alpha_{s} = xi l_{s} P), (xi) is the absorption coefficient induced by the external action^{26}and (l_{s}) is the length of the SH. By combining Eq. (4) with Eq. (5), we have:

$$(frac{1}{Lambda } – frac{1}{{Lambda_{0} }}){ = }frac{{alpha_{s} }}{L}{ = }frac {{xi l_{s} }}{L}P{ = }kP$$

(6)

Obviously, the external parameter is determined by measuring the ringing distances (Lambda) (with external action applied) and (Lambda_{0})(without the external action applied), and the reciprocal difference ((1/Lambda – 1/Lambda_{0})) of the descent distance changes linearly with the change of the external action, for an active FCRD sensor in the given spatial domain. Slope (k{ = }xi l_{s} /L) represents the sensitivity of the sensor to the detection activity, which can be adapted by adjusting the length of SH and the length of the cavity. Moreover, the detection limit is also an important indicator of system performance, which is defined as the minimum detectable external parameter.*P*_{min}. It can be obtained by taking the derivation of both sides of the equation. (6)

$$P_{min } = frac{1}{{kN_{e} L}} cdot frac{delta Lambda }{{overline{Lambda }}}$$

(seven)

where (N_{e} = Lambda_{0} /L) is the effective number of round trips traveled by the light in the ringing distance (Lambda_{0}), which essentially represents the multiple cycles of interaction in the HS. Therefore, compared to the detection limit of an intensity-based fiber optic sensor, our proposed sensor will improve the detection limit by a factor of*NOT*_{e}. (deltaLambda) and (overline{Lambda }) denote respectively the standard deviation and the mean value of the ringing distance in the condition of absence of external action applied to the SH. The report (delta Lambda /overline{Lambda }) can be defined as the stability of the proposed detection system, which is similar to the definition of stability in FCRD detection system in the time domain^{1}. Equation (7) indicates that the high detection limit can be expected with good stability, which is very important for the practical application. As the gain of Bi-EDFA can decrease the cavity loss and prolong the reduction distance, the space-domain active FCRD technique will have a larger effective circulation number and thus result in higher sensitivity and detection limit. . However, higher sensitivity usually results in smaller dynamic range, and vice versa.^{29}. That is, they must be compromised according to the requirements of practical applications.