Recently, the frequency-shifted interferometer (FSI) has been studied and applied to fiber-optic sensor sensing, which attracts the interest of many researchers^[1–11]. FSI utilizes a frequency shifter [usually, an acoustic optic modulator (AOM)] asymmetrically inserted in a Sagnac loop, producing location-resolved information for multiple sensors without resorting to wavelength-division or time-division multiplexing. Since the Sagnac interferometer is self-referenced, no coherence requirement is needed for the light source. FSI also results in a high signal-to-noise ratio (SNR) and does not require a fast detector^[9]. However, the interference signal of FSI does not contain wavelength information of the light reflected from the sensor. If we need to obtain wavelength information, it is necessary to use either a tunable laser source or a tunable filter to scan the optical wavelength, which causes more complexity and adds to the cost.

In this Letter, we propose a novel modified FSI (MFSI) based on a compound interferometer structure to address the above problems. We modified the optical path structure, adding a Mach–Zehnder interferometer (MZI) in the conventional FSI, forming a compound interferometer. Two counter-propagating light waves go through different paths and encode wavelength information into the phase of the interference signal. We adopt the Hilbert transform (HT) to convert the phase into a wavelength^[12–14]. We calibrate the system with a laser source of known wavelength to establish the relationship between the initial phase of the interference signal and this known wavelength. Thereafter, the unknown wavelength can be calculated from the initial phase. Here, we demonstrate only the capability of extracting wavelength information, without sensors in the MFSI. Our method is tested with interference signals obtained from the MFSI system, the results show it is feasible, simple, and with low cost, and it is vital to extend the applications of FSI.

The schematic diagram of the MFSI based on the compound interferometer is depicted in Fig. 1.

A 50/50 fiber coupler ${\mathrm{C}}_1$, path A, and path B constitute a Sagnac loop, and an AOM in path B acts as a frequency shifter. An MZI is formed by a circulator (CIR-2) and a 50/50 fiber coupler ${\mathrm{C}}_2$ (included in path A), with slightly different path lengths $L_b$ and $L_c$. The phase difference (PD) between path $L_b$ and $L_c$ is a function of wavelength, and therefore the wavelength information can be extracted from this PD. A polarization controller (PC) is employed to improve the interference visibility.

After passing through CIR-1, the light signal is split equally at ${\mathrm{C}}_1$ and input in the clockwise (CW) and counterclockwise (CCW) directions. Then, upon returning to ${\mathrm{C}}_1$, each time light passes through the AOM, and its frequency is up-shifted by an amount equal to the frequency of the acoustic driving signal $f$. The intensity signal $\mathrm{\Delta }I$ at the balanced detector (BD) is^[8,15]where A is the transmission coefficient, ${\kappa }_A$ and ${\kappa }_B$ are the transmission coefficients of path A and path B, ${\gamma }_1$ and ${\gamma }_2$ are the coupling ratios of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$, $\lambda$ is the wavelength of the light source, $E_0$ is the input field amplitude, $n(\lambda )$ is the refractive index of the fiber which, considering dispersion, is a function of wavelength, $c$ is the speed of light in vacuum, $f$ is the amount of frequency shift caused by the AOM, $\mathrm{\Delta }L=L_b-L_c$ is the length difference of path A, $L=(L_{\mathit{a}}+L_c+L_d+L_e-L_f)$ is a constant, $L_a,L_b,L_c,L_d,L_e$, and $L_f$ are the fiber lengths in Fig. 1. In Eq. (1), the differential output $\mathrm{\Delta }I$ is a sine function of $f$ when we sweep the frequency of the frequency shifter linearly, $f$ changes with time, and $\mathrm{\Delta }I$ is a sine function changing with time $t$.

From Eq. (1), the initial phase (when $f=f_o$) is a function of wavelength. In signal processing, HT is used to construct an analytic signal, which can calculate envelope (instantaneous amplitude) and instantaneous phase. Therefore, we use HT to extract the wavelength from the initial phase.

Suppose $x(t)$ is the real signal corresponding to Eq. (1); using HT, it can be converted to an analytic sequence: $Z(\mathrm{t})=x(t)+j\overline{x}(t)$, where $\overline{x}(t)$ is the HT of $x(t)$. The instantaneous phase can be expressed as

Combined with Eq. (1), it can also be expressed as

As described in Eq. (4), the frequency $f$ is a function of time, and there is a unique single value of the instantaneous phase at any given time measured in radians. It would oscillate between $-\pi \mathrm{a}\mathrm{n}\mathrm{d}\pi$ because of the periodicity of the arctangent function. After unwrapping the instantaneous phase and fitting it to a line, it can be seen that the slope of the line is $2\pi n(\lambda )L/c$, and the intercept is $-2\pi n(\lambda )\mathrm{\Delta }L/\lambda$, which contains the wavelength of the light source. By calculating the instantaneous phase intercept of HT, we can get the initial phase including the wavelength. If there is another light source with a wavelength of ${\lambda }^{\prime }$, to calculate their initial PD, we can get the following expression as

Suppose ${\lambda }^{\prime }=\lambda +\mathrm{\Delta }\lambda$, using Taylor expansion, thenand $\mathrm{\Delta }\varphi (t)$ can also be expressed as

In Eq. (6), there is a corresponding relationship between the PD and the wavelength difference of two interference signals with different wavelengths, and it is related to $\mathrm{\Delta }L$. Keeping $\mathrm{\Delta }L$ very short and temperature stabilized, $\mathrm{\Delta }L$ can be considered as a constant. Hence, the relationship between PD and wavelength difference can also be considered stable. Therefore, we can take this relationship and the initial phase of a known wavelength as the initial parameters of the laser measurement system. By using the light source with known wavelength to calibrate the system, the initial phase of the known wavelength and the relationship between the wavelength difference and the PD can be obtained. Due to the periodicity of phase, the interference signals of two different wavelengths with a PD of $2\pi$ will coincide. So, we can use this feature to calibrate the system.

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After calibration, the initial parameters can be treated as constants. Then, we can measure the unknown wavelength through its initial phase and initial parameters.

The experimental setup is depicted in Fig. 1. The output power of tunable semiconductor laser (TSL) (Santec TSL-510) is set to 2 mW. The AOM (Brimrose AMM-100-20-25-1550-2FP) is swept from 90 to 110 MHz at steps of 0.02 MHz with a step time interval of 1 ms. The output of the BD (New Focus Model 2117) was sent to a data acquisition board (NI USB-6361) with a sampling rate of 100 kHz and synchronized with the AOM scanning cycle. The gain of the BD is set to ${10}^3$, with a corresponding bandwidth of 100 kHz. The optical backscatter reflectometer (OBR) (LUNA OBR4600) was used to make an MZI of path A, and it measured $L_b$ and $L_c$. The $\mathrm{\Delta }L$ is calculated as $-1.6\times {10}^{-4}\mathrm{m}$, which means $L_b$ is less than $L_c$.

A LabVIEW program was developed and used to control the AOM frequency sweep and to acquire and process the data by the computer.

The grating sensors in our laboratory are most concentrated on the wavelength range starting from 1548 nm, in order to facilitate the research of MFSI in the future; the free spectral range (FSR) of the measurement system starts from the wavelength of 1548 nm.

At the beginning of calibration, FSR was larger than 6 nm. However, $\mathrm{\Delta }L$ is affected by fiber attitude and ambient temperature, resulting in FSR changing unsteadily. In order to obtain a stable FSR, we adjusted the fiber attitude in a relatively stable environment at room temperature and observed the period through the TSL. During adjusting, there may be some stretching of $L_b$ or $L_c$, causing FSR to change to around 6 nm. Considering the situation of the experimental equipment and the convenience of calculation, we choose to adjust the FSR to 6 nm and keep it stable. In Fig. 2(a), we show the interference signal obtained for the AOM sweep range of 90–110 MHz for two different wavelengths. (Note, data in the small region near the end of the range is removed due to instability in AOM sweeping.) We set 1548 nm as the initial wavelength for calibration. The calibration results are shown in Fig. 2(b).

In Fig. 2(a), as can be seen, the interference signals at the 1548 nm and 1554 nm wavelengths have almost overlapped; according to the periodic characteristic of phase $2\pi$ and the method of system calibration, when the wavelengths of 1548 nm and 1554 nm coincide for the first time, it clearly illustrates that the wavelength difference of 6 nm corresponds to the PD of $2\pi$. According to the results and Eq. (5), if the refractive index does not change with the wavelength, the value is assumed to be 1.468. $\mathrm{\Delta }L$ can be calculated as $2.731\times {10}^{-4}\mathrm{m}$. Figure 2(b) shows their phase intercept.

After obtaining the corresponding relationship between the PD and wavelength difference, we saved the interference data to test our method on the actual signal processing and selected the interference data with TSL wavelength sweeping from 1548 nm to 1554 nm.

Figure 3(a) shows the 1548 nm to 1553 nm interference signal changes. Figure 3(b) shows their phase changes at the initial phase. Combined with Fig. 2, the initial parameter of the wavelength is set as 1548 nm, and the PD and the measured wavelength (MWL) are shown in Table 1.

It can be seen from Table 1 that when the wavelength of TSL increases, the initial phase value gradually decreases. The phase of 1554 nm is similar to the phase of 1548 nm. It can be inferred that the FSR of our measurement system is 6 nm. The little bit of inconsistency between 1548 nm and 1554 nm may be caused by the system noise and other factors affecting $\mathrm{\Delta }L$. Moreover, there is a mutation in the phase of 1553 nm; by subtracting $2\pi$ from the abrupt change of phase, the continuous change of phase can be ensured, which means that, for the unknown wavelength measurement, the wavelength measurement range is only the FSR. In addition, from the theoretical analysis and experiment test results, it is also known that the calculation of the phase demodulation wavelength using the arctangent function will produce periodic multi-values, that is, phase ambiguity^[16]. Although we use the unwrapped function of MATLAB to solve the phase, the intercept is still limited between $-\pi \mathrm{a}\mathrm{n}\mathrm{d}\pi$, which caused one initial phase to correspond to multiple different wavelengths (e.g.,  1548 nm or 1554 nm), thus limiting the measurement range of our system to the FSR of 6 nm. To solve the phase ambiguity problem, we can extend the phase ambiguity boundary by shortening $\mathrm{\Delta }L$. Also, to improve the accuracy, we can maintain the $\mathrm{\Delta }L$ stability.

We presented and demonstrated a novel MFSI based on the compound interferometer to obtain the wavelength of the light source. The wavelength information is converted into the interference phase by embedding an MZI in a conventional FSI. The laser wavelength measurement experiment is used to verify the compound interferometer. The absolute measurement of laser wavelength can be realized by using the known wavelength and the PD of the detection interference signal. The experiment results show that our MFSI system does not need a reference laser for the scanning wavelength, which can lower the cost and is of great significance for demodulation FBGs based on FSI.