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Fiber-optic sensing is a field that is developing at a fast pace. Novel fiber-optic sensor designs and sensing principles constantly open doors for new opportunities. In this paper, we review a fiber-optic sensing technique developed in our research group called frequency-shifted interferometry (FSI). This technique uses a continuous-wave light source, an optical frequency shifter, and a slow detector. We discuss the operation principles of several FSI implementations and show their applications in fiber length and dispersion measurement, locating weak reflections along a fiber link, fiber-optic sensor multiplexing, and high-sensitivity cavity ring-down measurement. Detailed analysis of FSI system parameters is also presented.

As an offshoot of the fiber-optic communications industry, fiber-optic sensing has attracted considerable attention over the years [

Generally speaking, fiber-optic sensing research can be subsumed under two broad categories. One area of research is the design of new sensor types or structures for specific applications. The other area is the development of sensing systems that are intended to effectively detect sensor signals, analyze and interpret them. Frequency-shifted interferometry (FSI) is a fiber-optic sensing technique developed in our group [

Fiber-optic sensor multiplexing is an important application of FSI. Conventional fiber-optic sensor multiplexing techniques include spatial-division multiplexing (SDM) [

FSI provides an alternative way of carrying out cavity ring-down (CRD) measurement [

In this paper, we provide a review on the development of the FSI technique. We show various ways of implementing FSI and discuss their applications. We also analyze the parameters that affect the performance of FSI systems.

In frequency-shifted interferometry (FSI), a continuous lightwave at frequency _{0} and a frequency-shifted copy of it at _{0} + _{0} + _{0} to _{0} + _{i}

The earliest FSI developed is an asymmetric frequency-shifted Sagnac interferometer [_{in}_{0} exp(2_{0}) is launched into the interferometer from port 1, where _{0} is the field amplitude, _{0} is some initial phase. The two counter-propagating fields at port 1 are:
_{i}_{1} are the fiber section lengths as shown in the figure. Here
_{1} − _{0})/_{1} − _{0}). This relation can be used to measure fiber length and dispersion [

In [

Alternatively, an FSI can adopt a configuration akin to a Mach-Zehnder (MZ) interferometer as shown in _{1} and C_{2} are 50/50 fiber couplers, and R_{i}

Suppose CW input light at _{1}. It is divided equally into two parts, and the two lightwaves continue to propagate towards the reflectors. Each reflector R_{i}_{i}_{0} and _{1} + 2_{i}_{2}, where _{i}_{i}_{2} and the reflector. The differential interference signal from all the reflectors is a summation of sinusoids [_{i}_{i}_{i}_{i}_{i}_{1} + _{2} − _{0} so that (_{1} + 2_{i}_{2} − _{0}) ≈ 2_{i}_{i}_{i}_{i}_{i}

The remaining reflected components at _{1} output ports, which are effectively eliminated by the differential measurement. The intermixing of the fields at

A linear frequency-shifted Sagnac interferometer is a useful tool for multiplexing reflection-type fiber-optic sensors, since it can separate the sensor signals from the spatial domain and measure sensors' reflection spectra. As mentioned in the Introduction, FSI offers many advantages over conventional fiber-optic sensor multiplexing schemes.

Reflection-type sensors such as the ones based on fiber Bragg gratings (FBGs) are particularly suitable for FSI measurement [

Transmission-type sensors can also be interrogated by FSI systems if each sensor is used in conjunction with a reflector. For example, when the sensor array consists of gas cells with partial reflective mirrors, FSI is capable of identifying and quantifying chemical gases at different locations. In [

In most FSI fiber-optic sensor multiplexing demonstrations [

Based on sideband interference, an FSI system with only one interferometer arm can also be built [_{0} ^{i2πνt}. The modulator (an intensity modulator or phase modulator) in SA-FSI introduces sideband signals. When a phase modulator is employed, the output field of the modulator can be expressed as:
_{m}_{i}_{i}_{SA}_{DC}_{i}_{i}

In [_{3} phase modulator (PM) was used as the modulator, and a few weak FBGs were introduced as reflectors in the single fiber link. As a PM can be driven at a much higher RF frequency than a AOM, we were able to sweep the frequency shift

The principle of FSI can be applied to CRD measurement [

_{1} and C_{2} (e.g., 99.5/0.5 couplers). When CW light is sent into the interferometer from port 1 of C_{0}, lightwaves start to circulate in the RDC in opposite directions. A small fraction of the light exits the RDC each time when the light completes a round trip. If the cavity length _{2} + _{3} is longer than the coherence length of the light source, interference takes place at C_{0} between counter-propagating lightwaves that exit the RDC after the same number of round trips. As the differential interference signal contains a sinusoidal component for each round trip number _{S}_{1}+ _{2}+ _{4}− _{0}, _{m}_{m}_{s}_{m}_{m}_{2} with attenuation coefficient _{m}_{c}_{t}

By finding Λ, we can deduce the loss introduced by the sensing element in the RDC.

Some reviews on fiber-based CRD techniques can be found in references [^{−4} was achieved for sodium chloride solutions, which is comparable to the result (3.2 × 10^{−5}) attained by a conventional fiber-based CRD system using a much longer taper [

In an FSI system, the spatial resolution is the minimum resolvable separation between two reflectors (minimum resolvable optical path length difference). In fiber-optic sensor multiplexing, this parameter dictates how closely sensors (such as FBGs) can be distributed along a fiber. In a linear frequency-shifted Sagnac interferometer or a SA-FSI system, the spatial resolution

As fiber is flexible and can be wound,

The spatial sensing range is the maximum distance between the furthest sensor and the system at which reliable measurement can be made (maximum measurable optical path length). Since _{i}_{i}_{i}_{i}_{i}_{max}_{step}_{step}

In practice, system loss must also be considered when one estimates the maximum sensing distance, if certain signal-to-noise ratio (SNR) is required.

As we have seen, both spatial resolution and spatial sensing range are determined by the sweep parameters of the optical frequency shifter. The following relation holds:

Depending on the requirements of a specific application, the spatial resolution and spatial sensing range can be optimized by choosing appropriate frequency shifter sweep range and sweep step, respectively. For example, if we have Δ_{step}

Up to now, the analysis of FSI has been under the assumption of monochromatic input light. To see the effects of dispersion in FSI, let us suppose that the input light is broadband and that it has a uniform spectrum centered at _{0} with a bandwidth of Δ_{i}_{i}_{i}_{i}_{i}_{0}) _{i}_{0}). Also note that the cosine term _{i}_{0}_{i}_{0}, and the dispersion effect due to bandwidth Δ_{i}

As a numerical example, let us consider the dispersion effect in a linear frequency-shifted Sagnac interferometer built with standard SMF-28 fibers. At _{0} = 1550 nm, the fiber group velocity dispersion parameter _{i}_{0} and at other wavelengths—the OPL difference is on the order of only 0.17 m, which is well below the width of the Fourier peak (∼10 m).

Three types of crosstalk may influence the performance of an FSI system, including spectral shadowing effect, discrete Fourier transform resolution, and unwanted reflections among reflectors.

When multiple sensors in a serial array have overlapping spectral features, spectral shadowing effects take place. To reach a specific sensor, sensing light needs to pass all upstream sensors. Therefore, the input light of the

As a form of crosstalk, spectral shadowing effects are not unique to FSI, but are common in conventional fiber-optic sensing systems [

In FSI, the system can resolve the reflection spectrum for every individual sensor. The spectral shadow experienced by the _{i}_{j}_{j}_{i}_{i}

As there is no spectral shadow for the first sensor, _{1} can be obtained directly from the reflection spectrum measured for the first sensor. The actual reflection spectra of subsequent sensors can be calculated sequentially by using

The interference signal Δ

If the sensing system contains multiple sensors in series, multiple reflections or inter-sensor interference may occur, which presents another potential source of crosstalk in FSI systems. Sensing light may make multiple trips between a pair or more sensors. If the two interfering sensing lightwaves follow the same routes and are bounced back and forth between sensors as shown in

Light reflected by two different sensors R_{i}_{j}_{i}A_{j}_{i}_{j}_{j}_{i}_{j}_{i}

Many interferometric fiber-optic sensing systems are susceptible to polarization effects [

Under our assumptions above, the Jones matrix for an optical component in an FSI system can be generally written as [^{iξ}^{iζ}^{2} +|^{2} = 1. The matrix can be characterized by

The Jones matrix of a cascaded system is the product of its constituents' Jones matrices [_{1}_{2}_{3}_{4}_{5}…_{N}_{i}

A FSI system is bidirectional, since light propagates through the same optical elements from both directions. Given the Jones matrix _{ij}^{i}^{+}^{j} U_{ji}

It can also be shown that the backward propagation Jones matrix for a cascaded system has the property [_{i}

For a frequency-shifted Sagnac interferometer, suppose the Jones matrices for clockwise propagation and counterclockwise propagation are _{c}_{a}

For an arbitrary input Jones vector ∣_{x} s_{y}^{T}, the output polarization of clockwise propagation is ∣_{c}_{c}_{a}_{a}_{c}_{a}_{c}_{a}^{iζ}^{−}^{iζ}

The system for a linear frequency-shifted Sagnac interferometer is more complicated. As shown in _{i}_{ci}_{ai}_{A}_{B}_{arrayi}_{arrayi}_{arrayi}_{arrayi}_{12} ∈

To optimize the interference fringe visibility for an arbitrary input polarization state ∣_{ci}_{ci}_{ai}_{ai}_{ci}_{ai}_{ci}_{12} ∈ _{ci}_{ci}_{ai}_{0}, _{0}, _{A}_{A}_{A}_{B}_{B,}_{B}

It was found that if we can control _{A}_{B}_{A}_{A}_{A}_{B}_{B,}_{B}

This suggests that in a practical FSI sensor interrogation system, we may keep the setup to the left of coupler C_{2} in the control center, and we can optimize the interference fringe visibility for all the sensors, although we do not have access to the sensor array.

In an FSI-CRD system, the polarization state evolves as light makes multiple passes through the ring-down cavity. We can denote the Jones matrices of the system components as shown in _{cm}_{am}_{RDCm}_{L}U_{S}^{m}U_{L}

We wish to optimize the interference fringe visibility for every _{A}_{B}_{A}_{B}_{S}_{A}_{B}_{L}_{S}_{A}_{B}_{B}U_{L}U_{A}_{S}_{L}U_{S}_{cm}_{am}_{cm}_{am}

In summary, fiber-optic sensing has become an important frontier of the sensing industry. Frequency-shifted interferometry is a versatile addition to the tool box. With the help of a CW light source, a frequency shifter, and a slow detector, FSI is able to undertake very different tasks. An FSI system can be used to measure fiber length and dispersion [

The authors declare no conflict of interest.

A frequency-shifted Sagnac interferometer formed by connecting the output of a 50/50 fiber directional coupler with an optical frequency shifter asymmetrically.

Experimental setup of a frequency-shifted Sagnac interferometer for fiber length and dispersion measurement [_{1}: 50/50 fiber direction coupler; AOM: acousto-optic modulator.

A linear frequency-shifted Sagnac interferometer with multiple reflectors. C_{1} and C_{2}: 50/50 fiber directional couplers, R_{i}

Typical setup of a linear frequency-shifted Sagnac interferometer for fiber-optic sensor multiplexing. LS: light source; CIR: circulator; BD: balanced detector; AOM: acousto-optic modulator; PC: polarization controller; C_{1} and C_{2}: 50/50 fiber directional couplers; S_{i}

Overlapping FBG spectra measured by an FSI system that uses a light source consisting of a broadband ASE source and a tunable filter.

A single-arm frequency-shifted interferometer. R_{i}

An FSI-CRD system with a fiber loop cavity. C_{0}: 50/50 fiber directional coupler; C_{1} and C_{2}: highly unbalanced fiber directional couplers.

A typical FSI-CRD sensing system [_{0}: 50/50 fiber directional coupler; C_{1} and C_{2}: highly unbalanced fiber directional couplers.

Comparison between the normalized Fourier peaks with and without dispersion effects. The thick black dashed curve is the Fourier peak contributed by light at _{0} without dispersion, while the red curve is the Fourier peak computed from ^{20}.

Spectral shadowing effect. The (_{a}_{i}_{i}_{−1}.

The effects of windowing in DFT. Given the same interference signal Δ

The effects of unwanted reflections. (

Input and output polarization of a frequency-shifted Sagnac interferometer. The coordinate system for the input polarization state ∣_{c}_{a}

Input and output polarization of a linear frequency-shifted Sagnac interferometer. The coordinate system for the input polarization state ∣_{ci}_{ai}_{i}

Input and output polarization of an FSI-CRD system. The coordinate system for the input polarization state ∣_{cm}_{am}

Contributions of interference caused by undesirable reflections for a linear frequency-shifted Sagnac interferometer system [

path A → R_{j}_{i} |
2_{i}_{1}+_{2}−_{0})_{i}_{j}_{0}/ |

path A → R_{i}_{j} |
2_{j}_{1}+_{2}−_{0})_{j}_{i}_{0}/ |

path A → R_{j}_{i} |
2_{i}_{j}_{0}/ |

path B → R_{j}_{i} |
2_{i}_{j}_{i}_{j}_{0}/ |